id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
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|---|---|---|---|---|---|
1108.4723
|
arxiv-papers
| 2011-08-23T23:55:25 |
2024-09-04T02:49:21.738451
|
{
"license": "Public Domain",
"authors": "Jie Ren, Kai-Kit Wong and Jianjun Hou",
"submitter": "Kai-Kit Wong K Wong",
"url": "https://arxiv.org/abs/1108.4723"
}
|
|
1108.4770
|
1
A Classification Framework for
Web Browser Cross-Context Communication
IVAN ZUZAK
School of Electrical Engineering and Computing, University of Zagreb
Unska 3, 10000 Zagreb, Croatia
izuzak@gmail.com, ivan.zuzak@fer.hr
MARKO IVANKOVIC
Google Inc.
Brandschenkestrasse 110, CH-8002 Zurich, Switzerland
ivankovic.42@gmail.com, markoi@google.com
IVAN BUDISELIC
School of Electrical Engineering and Computing, University of Zagreb
Unska 3, 10000 Zagreb, Croatia
ibudiselic@gmail.com, ivan.budiselic@fer.hr
Demand for more advanced Web applications is the driving force behind Web
browser evolution. Recent requirements for Rich Internet Applications, such as
mashing-up data and background processing, are emphasizing the need for
building and executing Web applications as a coordination of browser execution
contexts. Since development of such Web applications depends on cross-context
communication, many browser primitives and client-side frameworks have been
developed to support this communication. In this paper we present a
systematization of cross-context communication systems for Web browsers. Based
on an analysis of previous research, requirements for modern Web applications
and existing systems, we extract a framework for classifying cross-context
communication systems. Using the framework, we evaluate the current ecosystem
of cross-context communication and outline directions for future Web research
and engineering.
Keywords: Web browsers, Web applications, browser execution contexts, cross-
context communication, mashups, systematization, classification, evaluation
## 1 Introduction
The evolution of the Web may be seen in the evolution of Web applications
provided to satisfy user demand [1]. Accordingly, Web browsers are rapidly
evolving to support execution of such applications. Where early Web
applications were simple interlinked documents, recent Web applications, known
as Rich Internet Applications (RIA) [2], show an increase in functionality,
user-friendliness and responsiveness, and therefore in complexity. One recent
trend is seen in mashups [3, 4], Web application portletization [5], personal
learning environments [6] and complex widget-based applications, such as
Geppeto [7], in which the client-side of the Web application is designed and
executed as a composition of semi-isolated Web browser contexts, such as
frames. This trend is emphasized even more with the recent introduction of Web
workers [8] through which browsers provide Web applications with GUI-less
background processing contexts similar to threads in operating systems. In
essence, Web browsers are evolving into environments for execution of Web
applications [9], similar to operating systems that execute multi-process and
multi-threaded applications.
Development of modern Web applications therefore depends on Web browsers
supporting interaction between contexts, similar again to supporting inter-
process communication in operating systems. However, enabling cross-context
communication has historically been a difficult task due to the Same-origin
security policy (SOP) [10] implemented in browsers. SOP almost completely
restricts Web applications executing in a browser from communicating with
entities on different trust domains, also called origins. As a consequence of
implementing SOP, for a long time Web browsers lacked native primitives both
for same-origin and cross-origin cross-context communication. Consequently,
Web applications developers used and often misused insecure browser primitives
intended for other purposes, like browser cookies and window location fields,
to enable cross-context data transfer [11, 12, 13].
However, driven by industry demand for modern Web applications and browser
compatibility, new primitives for cross-context and cross-origin communication
are being standardized and implemented as a part of the HTML5 group of
standards [14]. At the same time, many client-side frameworks are being built
on top of both unstandardized and standardized primitives, offering support
for legacy browsers, cross-browser support and many other features like
security and high-level programming models. Today, Web researchers and
engineers face a complex ecosystem of cross-context communication systems in
which it is often difficult not only to discern each system’s capabilities and
benefits over other systems, but also to be aware about the issues affecting
the operation and usage of such systems. Therefore, the field of Web
engineering [15], as “the application of systematic, disciplined and
quantifiable approaches to development, operation, and maintenance of Web-
based applications”, should provide better support for understanding and
managing cross-context communication in Web applications.
In this paper we present a systematization of the Web browser cross-context
communication ecosystem. Our systematization provides both a broader and a
deeper view of cross-context communication through the following
contributions. First, we analyze previous research results related to this
field. Second, we define a multi-dimension framework for classification of
cross-context communication systems. Fourth, we consistently apply the defined
framework to existing cross-context communication systems. Although some
browser primitives and systems analyzed in our work have also been analyzed in
previous research, these previous analyses were not systematic and were mainly
focused on security aspects. Moreover, our framework includes criteria that
reflect cross-context communication requirements of modern and next-generation
Web applications, such as Web worker support, reliability, discovery and high-
level communication models. Lastly, we analyze the evaluation results and give
directions for future cross-context communication research and engineering
practice.
The remainder of the paper is organized as follows. In Section 2, we introduce
basic concepts of cross-context communication in Web browsers. In Section 3 we
give an overview of existing research related to cross-context communication.
Section 4 presents our classification framework and evaluation of existing
cross-context communication systems. Section 5 discusses the presented
framework and evaluation results, also proposing directions for future work.
Section 6 concludes the paper.
## 2 Web Browser Contexts And Cross-Context Communication
In the context of the architecture of the WorldWideWeb [16], browsers are
user-agents which fetch and execute server resource representations, i.e. HTML
documents and applications. Browsers manage the execution of each Web
application using semi-isolated environments called browser execution
contexts, sometimes also called script contexts [17, 18, 19]. Web applications
may be built from many parts, each part executing in its own context. For
example, a mashup Web application [4] may contain a widget for displaying
locations on a map together with a widget for displaying Wikipedia information
on specific locations, each in its own context.
Each browser execution context contains an event loop which coordinates
events, user interaction, rendering and networking of the part of the Web
application executing within that context [14]. Most importantly, event loops
coordinate the execution of JavaScript scripts of the Web application. Since
JavaScript is a single-threaded language with no concurrency primitives, an
event loop also executes in a single thread of execution. However, since the
execution of an event loop in one context is independent of the execution of
other contexts, event-loops of different contexts may execute concurrently.
Furthermore, each context has an associated origin derived from the URI from
which the Web application part executing in that context was retrieved. The
origin [22] is a tuple consisting of the normalized scheme, host and port
parts of an URI, for example (“http”, “www.example.com”, “80”). Notably, the
origin does not include the path, query and fragment parts of an URI. The
origin is an important property of browser execution contexts as it was and
still is the basis for designing browser security policies, as explained later
in this section.
Two types of browser execution contexts exist: window contexts and worker
contexts. A window context is an environment in which Web applications are
presented to the user through the use of a graphical user interface (GUI)
[14]. Window contexts consist of a browsing context that displays the user
interface and an event loop that interprets JavaScript scripts and manages GUI
interaction and other events. Examples of window contexts are browser windows
and tabs, iframe objects and frames in a frameset [14]. In contrast, worker
contexts, introduced only recently with the Web Workers specification [8],
consist only of an event loop [8] and may be thus considered as GUI-less
window contexts. Furthermore, worker contexts must be associated with at least
one parent window or worker context. According to the number of parent
contexts they may be associated with, worker contexts are further divided into
two classes: dedicated worker contexts and shared worker contexts. While
dedicated worker contexts are associated with only a single parent context,
shared worker contexts may be associated with multiple parent contexts.
Window contexts are created either directly by the browser as a result of the
user requesting the execution of a Web application resource located at a
specific URI or programmatically from already executing window contexts. In
contrast, worker contexts may be created only programmatically from already
existing contexts. Furthermore, browser execution contexts may create and nest
other contexts which enable the parent and child context to maintain a
programmatic and sometimes visual relationship. Window contexts may nest both
other windows contexts, specifically frames and iframes, and worker contexts.
Nesting of window contexts enables composition of GUIs of multiple Web
application parts on a single screen, for example a portal page containing
many widgets. In contrast, worker contexts may nest only other worker
contexts.
In essence, a Web browser is a platform which executes Web applications, where
each Web application is a set of browser execution contexts hierarchically
organized into a tree starting from a browser window or tab context. Figure 2
shows a simplified view of a Web browser which executes two Web applications.
Web application A consists of three window context; the top-level window
($c_{1}$) and two nested iframes ($c_{3}$ and $c_{4}$). Similarly, Web
application B consists of one window context and two hierarchically nested
worker contexts; the top-level window ($c_{2}$) and two dedicated Web workers
($c_{5}$ and $c_{6}$).
Fig. 1. Simplified view of a Web browser executing two Web applications.
Fig. 1. Simplified view of a Web browser executing two Web applications.
The correct operation of multi-context Web applications depends on the
interaction between contexts. For example, widgets may need to exchange data
for display on a GUI, as shown for example in Figure 2 for communication
between context $c_{3}$ and $c_{4}$, or a window context may need to pass data
to a worker context for background processing, as shown in Figure 2 for
communication between context $c_{2}$ and $c_{5}$. Therefore, systems are
needed that enable cross-context communication as a process of transferring
data across browser context boundaries. We define cross-context communication
as any kind of data transfer between any two context executing in any two
browsers, which may be initiated programmatically. For example, we do not
consider user-driven copy-pasting or drag-and-dropping data from one context
to another [23] to be a cross-context communication process. Cross-context
communication may be thus seen as a generalization and extension of inter-
window communication and inter-iframe communication, terms which have usually
been used for denoting communication between window contexts [20, 21].
Historically, the main problem with development of cross-context browser
communication systems was related to security aspects of communication.
Specifically, in order to prevent many types of attacks on their users, Web
browsers implement a security policy called the Same-origin policy (SOP). The
SOP policy [22] restricts scripts executing in browser contexts to
communication only with contexts with the same origin and with server
resources with the same origin. In other words, cross-origin communication, as
shown in Figure 2 for communication between contexts $c_{1}$ (origin $o_{1}$)
and $c_{3}$ (origin $o_{3}$), was denied by the browser, while same-origin
communication, as shown on Figure 2 for communication between contexts $c_{3}$
(origin $o_{3}$) and $c_{4}$ (origin $o_{3}$) was allowed. Since most early
Web applications were either static or executed within a single context, they
did not require cross-origin cross-context communication capability and cross-
context communication primitives were not being provided by browsers or
developed as external systems.
However, as Web applications evolved and their requirements for cross-context
communication increased, developers started misusing other browser primitives
to achieve even rudimentary cross-origin cross-context communication. For
example, browsers disregard the same-origin policy for certain cases [10, 11,
18, 19] such as redirecting a window context to a new URI and accessing the
list of directly nested iframes of a window context [14]. Browser cookies also
have a different security policy which grants access based on the resource
origin but excluding the scheme and port parts and including the path part
[24, 25], which isn’t compatible with SOP. Only recently with the development
of the HTML5 specification [26, 14] have Web browsers started implementing
native and secure cross-origin cross-context communication primitives.
Consequently, security attributes of cross-context communication systems
varied from browser to browser and were the subject of extensive research,
while other attributes of communication were mostly unresearched. Still,
modern Web applications require communication features which have not been
systematically explored and analyzed, a problem we address in this paper.
## 3 Related Work
In this section we give an overview of research that contributed to the
analysis and systematization of cross-context communication systems. Most
previous research activities were focused on evaluating and comparing security
properties of cross-context communication systems based on browser primitives
not intended for cross-context communication. Other research was focused on
designing new browser primitives and client-side libraries that overcome
specific security deficiencies.
One of the earliest critiques of the lack of secure browser primitives for
cross-context communication was given by the proposal of the $<$module$>$ HTML
tag and API [26, 27]. Although it was never standardized or implemented, the
proposal inspired research of the later standardized HTML5 postMessage API
[14]. A list of several similar early research proposals is given in [28]. In
[26] two techniques for communication between window contexts are analyzed
with respect to confidentiality and authenticity; fragment identifier
messaging (FIM) and higher-level protocols based on FIM, and the postMessage
API.
In [29] the authors analyze whether existing and proposed browser mashup
communication primitives enable communication between two principals, browser
contexts for example, without ceding complete control to each other.
Vulnerabilities of primitives are illustrated through several proof-of-concept
attacks and recommendations for prevention are given. Furthermore, an
evaluation of design choices for access control aspects of communication
primitives is given; for example, using values versus objects for
communication. In [30] the authors demonstrate that the existing browser
security model was not designed to support multi-context Web applications, and
that as a consequence these are typically implemented insecurely. The paper
also emphasizes the need for higher-level communication abstractions and
presents a secure component model based on a publish-subscribe communication
abstraction. In [31] a critique of cross-context communication based on
browser cookies is given with respect to security. The paper also recognizes
the security disadvantages of using server-side proxies for cross-context
communication. Furthermore, a proposal for a secure publish-subscribe
communication system is given. In [11] the authors analyze incoherencies in
browser access control policies. A special part of this analysis are browser
resource types which may be shared among principals and their interaction,
which is a subset of cross-context communication systems.
In [20] the authors analyze inter-widget communication, a specific application
of cross-context communication, with the purpose of maximizing usability of
widget-based personal learning environments. The analysis is based on a
framework for categorization of inter-widget communication systems, an
approach similar to the one we present later in this paper. For example, the
authors distinguish between same-browser and cross-browser communication,
inter-widget and intra-widget communication, several types of event
distribution, such as broadcast and direct subscription, and several types of
security and semantic interoperability. However, the framework is strongly
focused on end-user usability of inter-widget communication and does not take
into account many technical dimensions of cross-context communication, such as
discovery, cross-origin support and reliability.
The most recent and broad analysis of browser primitives for Web application
interactivity is given in [32]. The authors present results of a usage
analysis of several new browser primitives implemented in browsers as a part
of the HTML5 group of standards. The results show that the postMessage API,
Web Storage API [33] and the Web SQL Database APIs [34] are being used
insecurely. Moreover, the authors give insights into why these primitives can
potentially be hard to use safely and propose the economy of liabilities
principle in designing security primitives - a primitive must minimize the
liability that the user undertakes to ensure application security. The authors
also propose several enhancements to the postMessage API to shift the burden
of verifying and ensuring security properties from the developer to the
browser.
Another relevant field of research is secure architectures for modern Web
browsers. In this research field, the browser and management of Web
applications are observed at a lower level with regard to operating system
integration and inter-process communication. Still, this work is the
foundation for implementing communication abstractions at the Web application
level. The security architecture of the open-source Chromium browser is
described in [35]. In Chromium, the traditional monolithic architecture of
browsers is replaced with an architecture based on two modules in separate
protection domains: a browser kernel, interacting with the operating system,
and a rendering engine, executing with restricted privileges in a sandbox. The
authors describe the security advantages of the architecture and describe how
other architectures make it difficult to implement cross-context communication
primitives. The architecture of Gazelle, a secure Web browser constructed as a
multi-principal operating system, is presented in [25]. Gazelle’s security
model protects principals, contexts from different origins, by separating
their resources into hardware-isolated protection domains. The authors
describe benefits of aligning the browser architecture with the SOP policy and
analyze cross-principal interaction possibilities.
Lastly, operating systems (OS) [36] have been extensively researched through
the last decades providing valuable experience for designing browsers as
multi-context execution environments. Specifically, multi-process applications
and inter-process communication mechanisms designed for OSes provide a
starting point for cross-context browser communication research. However,
these mechanisms were seldom taken into account in previous cross-context
communication research.
In conclusion, cross-context communication is still an unresearched field and
trailing behind industry requirements. First, since SOP has been a major issue
for the last several years, research has been focused mainly on security
aspects of cross-context communication and disregarded other aspects. Second,
since until recently only window contexts were in use, no prior research
includes worker contexts in their analyses. Third, many existing systems for
cross-context data exchange have similarly received little attention. Fourth,
there is no systematic approach aimed at analyzing cross-context communication
or a broad systematization of existing systems. Fifth, existing operating
systems IPC mechanism research has not been integrated into cross-context
communication research. Our systematization presented in the next section
addresses some of the stated challenges.
## 4 Systematization Of Cross-Context Browser Communication Systems
This section presents our systematization of cross-context browser
communication systems. The purpose of the systematization is three fold.
First, we provide a framework for classification of cross-context
communication systems. The developed framework is a multidimensional space in
which each dimension represents one system characteristic and dimension values
correspond to alternatives for that characteristic. A specific system design
corresponds to a point in the design space [37]. Second, we clarify the
current state of cross-context communication systems by applying the presented
framework to existing systems. Third, we propose future research and
engineering directions based on this evaluation of existing systems and future
Web application requirements.
Although the framework and evaluation of systems are presented separately,
their research and definition was interwoven. First we gathered existing
systems and evaluated those using dimensions from existing body of research
combined with existing engineering concepts from IPC mechanisms from operating
systems. Afterwards, we defined additional dimensions in order to enable
clearer understanding and comparison of systems. We iterated this process
until the set of dimensions covered most design choices of existing systems
and possible requirements of future systems.
### 4.1 Classification Framework
The classification framework consists of a set of dimensions relevant for
research and evaluation of cross-context communication systems. Each dimension
is described separately with rationale explaining its importance as an
explicit system characteristic and possible alternatives.
Type of system – We differentiate four types of systems with regard to span of
the system’s implementation, as shown on Figure 4.1. The most basic systems
are _browser primitives_ , i.e. mechanisms provided by the browser itself.
_Client-side frameworks_ are systems that build their logic on top of browser
primitives and don’t require any components outside the browser. These two
types of systems are pure client-side systems as they do not require any
server-side components. However, the following two types of systems require
external components. A system that additionally requires calls to a server
component but only to coordinate communication is called a _server-mediated
coordination framework_ , while a system that routes messages through a server
component outside the browser is called a _server-mediated communication
framework_.
Fig. 2. Types of cross-context communication systems.
Fig. 2. Types of cross-context communication systems.
The significance of this dimension is twofold. First, as the span of the
system’s implementation increases, the system’s run-time performance is
expected to decrease due to increased implementation complexity and network
traffic. Furthermore, server-mediated coordination and especially
communication frameworks may suffer from scalability issues as the number of
communicating contexts increases. Second, the increase of the implementation
span is commonly correlated with the increase in system functionality; browser
primitives offer basic communication mechanisms while third-party frameworks
build on top of them and offer ease of use and other features. For example,
because server-mediated communication frameworks route messages outside the
browser, these types of systems may also be used for communication between
contexts executing in different browsers.
Window context support and worker context support – These two dimensions
reflects the system’s support for communication with window contexts and
worker contexts, respectively.
Cross-origin support – This dimension denotes the system’s degree of support
for communication between contexts with different origins. Other than systems
that support only _same-origin_ cross-context communication and systems that
support _cross-origin_ cross-context communication, a third type of system
exists with regard to this dimension. These systems support communication
between contexts on the _same domain and path_. As more and more Web
applications are built from context of different origins, this dimension
determines the gradual increase in applicability of systems, from those
supporting only same-origin communication, to that supporting full cross-
origin communication.
Cross-application support – This dimension reflects the system’s support for
communication between contexts of different Web applications. As we explain in
Section 2, a Web application is defined with the top-level window context of a
Web browser, such as a browser window or tab, together with all other window
and worker contexts nested within that top-level context. Therefore, cross-
application cross-context communication systems support communication between
contexts nested within different top-level contexts.
This dimension is orthogonal to the Cross-origin support dimension because
contexts in different Web applications may have equal or different origins.
Furthermore, this dimension is also orthogonal to the Type of system
dimension. Although server-mediated communication frameworks usually do
support communication between contexts of different browsers, this behavior is
not implied. Similarly, systems with cross-application support need not
achieve this support using server-mediated communication.
The significance of this dimension is in its relation not only to cross-
browser communication, but also to communication between Web applications
executing in the same browser. Specifically, Web applications are increasingly
built as to provide local APIs for their remote services to other applications
executing in the browser [38]. For example, consuming URL shortening services
provided by the URL shortening Web application which the user uses.
Communication model – One of the more important dimensions is the
communication programming model [36] which the observed system exposes to Web
application developers. _Message-oriented_ communication is based on contexts
sending and receiving structured messages. In _shared memory_ systems,
contexts communicate indirectly by reading from and writing to a shared data
space. In _remote procedure call_ systems, communication is based on contexts
invoking procedures on remote context and receiving responses of invocations,
as if the procedures were implemented locally. In _publish-subscribe_ systems,
contexts communicate indirectly by publishing messages to channels, named
virtual entities, which route messages to all subscribers subscribed to those
channels.
The significance of this dimension is in its relation to ease of achieving
application goals. Application goals of different multi-context Web
applications are significantly easier to implement using a specific
communication model versus using other models. Lower-level programming models,
like message-oriented and shared memory, commonly require a larger code
overhead to implement application goals than higher-level ones, such as remote
procedure call and publish-subscribe.
Naming – Contexts that want to communicate must have a way to refer to each
other, a fundamental function of naming. This dimension denotes the type of
entity and its semantics used to refer to communicating entities when using
the observed system. In other words, programmers using a cross-context
communication system use these entities to refer to browser execution
contexts.
In most cross-context communication systems, contexts are referred to
directly, using _context object references_ which are JavaScript references to
browser contexts. These references may be obtained in several ways; for
example worker context references are obtained when creating Web workers while
window context references are usually obtained by special browser APIs for
traversing context hierarchies. Some systems also support forms of indirect
communication where naming entities are not context object references. Some
systems support referring to window contexts using the _document URI_ of the
document executing in that context. Other systems use _custom string names_ ,
for example channel names in publish-subscribe systems, or a combination of
context object references and custom string names, such as destination
procedures in remote procedure call systems.
Discovery – This dimension denotes whether or not the observed system supports
discovery of communicating entities, implemented through discovery of entities
used for naming. In widget-based applications consisting of multiple widget
contexts, widgets are often unaware of each other i.e. whether a specific
widget has even been loaded and if so, how to obtain a reference to it. For
example, the iGoogle portal generates random strings for widget iframes names
which complicates obtaining a reference to the wanted context manually.
Distribution scheme – This dimension denotes the kinds of distribution schemes
supported by the observed system. The _unicast_ scheme defines communication
towards a single context, the _multicast_ scheme defines communication towards
a defined set of contexts, and the _broadcast_ scheme defines communication
towards all contexts in the application, except the sender.
Maximum message length – This dimension denotes the limit in size of data sent
and received using the observed system. For message-oriented, publish-
subscribe and remote procedure call systems this denotes the maximum message
size while for shared memory systems this denotes the maximum size of a single
shared memory location. The system is _unrestricted_ with respect to this
dimension if it does not limit the size of data, while otherwise it is
_restricted to a specific size_ (in kilobytes or megabytes), for example,
restricted to 5kB. However, browsers may additionally restrict message lengths
of otherwise unrestricted systems for security or reliability reasons; for
example, to prevent memory depletion.
Transport system – Systems other than browser primitives are implemented using
existing cross-context communication systems for data transfer. This dimension
denotes the names of other systems used to implement the observed system. As
complex frameworks commonly inherit properties of underlying systems, such as
performance and browser support, knowledge of the underlying systems is an
important instrument for determining the suitability of a particular
framework.
Reliability – Since browser contexts run in parallel and may be created
dynamically by scripts executing in other contexts, the destination context
may not be ready for receiving messages at the moment in which the source
context is sending them. For example, in an aggregator Web application with
several widgets one widget may want to send a message to another widget which
has either not yet been created or not fully loaded. Therefore, some messages
may be unknowingly lost.
Therefore, communicating contexts need a mechanism to guarantee a certain
degree of reliable and fault-tolerant communication. This dimension denotes if
the observed system has such mechanisms or is otherwise considered
_unreliable_. Some systems implement a _retry mechanism_ by which the sender
retries the communication if no confirmation of success is received. Other
systems use a _queuing mechanism_ that delays communication until the receiver
is available.
Communication confidentiality – This dimension denotes whether or not
communication performed using the observed system is confidential. In other
words, communication is _confidential_ if no other contexts except the
intended receivers may read the communicated data. Otherwise, communication
confidentiality is _unsupported_.
Communication integrity – This dimension denotes the degree to which the
observed system restricts unauthorized modification of communicated data. If
the communicated data may be modified without authorization and if no
mechanisms are provided to receivers to check if the data was modified,
communication integrity is said to be _unsupported_ by the system. However, if
mechanisms for verifying data integrity are provided by the system,
communication integrity is said to be _verifiable_. Lastly, if communicated
data may not be modified without authorization, communication integrity is
said to be _supported_.
Authentication of sender and receiver – Authentication is the act of verifying
a claim of identity, which may be performed implicitly by the system or
explicitly by the sender or receiver. These two dimensions denote whether or
not the observed system supports that contexts sending or receiving data may
not falsify their identity. In other words, if a context sending data may
falsify its identity, established through naming, then communication does not
support sender authentication. Similarly, if a context receiving data may
falsify its identity, established through naming, then communication does not
support receiver authentication.
Authorization of sender and receiver – These two dimensions denote whether or
not the observed system supports that contexts sending data specify intended
receiver contexts or that contexts receiving data may specify from which
contexts data is to be accepted. The specification of these properties is
often expressed using authorization policies, which are often based on the
concept of origins or even finer-grained with respect to context URIs. In Web
applications using cross-context communication systems without support for
authorization, contexts receiving data must implement application-level logic
to support authorization policies, if such support is even possible to
implement. However, if supported, authorization policies usually either
specify a _single_ authorized context or specify _access control whitelists_
for authorizing multiple contexts.
Generality of applicability – This dimension denotes the degree of
applicability of the observed system when developing Web applications. Most
existing systems have been developed as _generic_ frameworks and may be used
for implementing cross-context communication in any Web application. However,
some systems are _application specific_ and may be used only in a subset of
applications. For example, some systems may be used only for communication
between contexts that host Google Gadgets. Although limited in use as is,
these systems are still considered in this paper since they represent a
substantial part of the cross-context communication ecosystem.
Browser support – This dimension denotes the names and versions of major Web
browsers which support the observed system. Internet Explorer, Firefox,
Chrome, Safari and Opera are considered major Web browsers.
### 4.2 Evaluation of Existing Systems
In this section we present an evaluation of existing cross-context browser
communication systems according to the framework established in the previous
section. However, as the number of existing systems and the number of
dimensions are both large, a full evaluation would require more space than
permitted. Therefore, we restrict our evaluation to subsets of existing
systems and framework dimensions.
First, the evaluation does not include server-mediated communication
frameworks. The number of these systems is potentially very large since any
kind of system for transferring data on the Web is applicable. We address this
issue further in the next section. For similar reasons, the evaluation does
not include various browser plugins and extensions, like Flash [39] and
Silverlight [40]. Furthermore, the evaluation does not include both numerous
patents published in this area [12, 41, 42, 43] and unimplemented research
projects.
Second, systems are not evaluated according to several security-related
dimensions, namely communication confidentiality, communication integrity, and
sender and receiver authentication, which have been addressed in previous
research. Furthermore, the browser support dimension was also left out due to
lack of time for thorough testing of all evaluated systems in all the major
browsers on all major operating systems.
Tables 4.2, 4.2 and 4.2 presents the summary of the evaluation. In the
following paragraphs, we give short notes on the evaluated cross-context
communication systems and give references for more detail on each system.
Table 1. Evaluation of existing cross-context communication systems according
to the Type of system, Window context support, Worker context support, Cross-
origin support and Cross-application support dimensions. Legend: Type of
system: browser = browser primitive, client side = client side framework,
server coord = server-mediated coordination framework, server comm = server-
mediated communication framework. Window context support, Worker context
support, Cross-application support: + = supported, - = unsupported.
Table 1. Evaluation of existing cross-context communication systems according
to the Type of system, Window context support, Worker context support, Cross-
origin support and Cross-application support dimensions. Legend: Type of
system: browser = browser primitive, client side = client side framework,
server coord = server-mediated coordination framework, server comm = server-
mediated communication framework. Window context support, Worker context
support, Cross-application support: + = supported, - = unsupported.
Cross-context communication system Type of system Window context support
Worker context support Cross-origin support Cross-application support Direct
access browser + - same-origin - FIM browser + - cross-origin - Window name
browser + - cross-origin - Cookies browser + - same domain+path + CrossFrame
server coord + - cross-origin - Complex window name server coord + - cross-
origin - RMR client side + - cross-origin - NIX client side + - cross-origin -
Frame Element browser + - cross-origin - postMessage browser + - cross-origin
- Channel messaging browser + + cross-origin - XSS interface client side,
server coord + - cross-origin - Google Closure client side, server coord + -
cross-origin - jQuery postMessage client side + - cross-origin - OMOS client
side, server coord + - cross-origin - Shindig rpc client side, server coord +
- cross-origin - easyXDM client side, server coord + - cross-origin - Window
post-Messge plugin client side + - cross-origin - jsChannel client side + -
cross-origin - Web intents server coord + - cross-origin + sMash server coord
+ - cross-origin - Shindig pubsub client side, server coord + - cross-origin -
OpenAjax Hub client side, server coord + - cross-origin - open-app client
side, server coord + - cross-origin - pmrpc client side + + cross-origin -
LocalStorage browser + - same-origin + WebDatabase browser + + same-origin +
IdexedDB browser + + same-origin + CrossDomain Storage server coord + - cross-
origin - WebWorker postMessage browser - + same-origin - jQuery hive client
side - + same-origin -
Table 2. Evaluation of existing cross-context communication systems according
to the Communication model, Naming, Discovery, Distribution scheme and Maximum
message length dimensions. Legend: Communication model: msg = message-
oriented, sh-mem = shared memory, rpc = remote procedure call, pubsub =
publish-subscribe. Naming: obj ref = context object references, doc uri =
document URI, custom str = custom string names. Discovery: + = supported, - =
unsupported. Distribution scheme: 1:1 = unicast, 1:N = multicast, 1:all =
broadcast. Maximum message length: max = unrestricted, X KB/MB = restricted to
X kilobytes/megabytes.
Table 2. Evaluation of existing cross-context communication systems according
to the Communication model, Naming, Discovery, Distribution scheme and Maximum
message length dimensions. Legend: Communication model: msg = message-
oriented, sh-mem = shared memory, rpc = remote procedure call, pubsub =
publish-subscribe. Naming: obj ref = context object references, doc uri =
document URI, custom str = custom string names. Discovery: + = supported, - =
unsupported. Distribution scheme: 1:1 = unicast, 1:N = multicast, 1:all =
broadcast. Maximum message length: max = unrestricted, X KB/MB = restricted to
X kilobytes/megabytes.
Cross-context communication system Communication model Naming Discovery
Distribution scheme Maximum message length Direct access sh-mem, rpc ctx obj -
1:1 max FIM sh-mem ctx obj - 1:1 2KB, max Window name sh-mem ctx obj - 1:1
2MB, max Cookies sh-mem string - 1:1, 1:N 4KB, max CrossFrame msg ctx obj -
1:1 2KB, max Complex window name msg ctx obj - 1:1 max RMR msg ctx obj - 1:1
max NIX msg ctx obj - 1:1 max Frame Element msg ctx obj - 1:1 max postMessage
msg ctx obj - 1:1 max Channel messaging msg ctx obj - 1:1 max XSS interface
msg doc URI, string - 1:1 max Google Closure msg ctx obj, string - 1:1 max
jQuery postMessage msg ctx obj - 1:1 max OMOS rpc string - 1:1 max Shindig rpc
msg, rpc string - 1:1 max easyXDM msg, rpc, pubsub doc URI, string - 1:1, 1:N,
1:all max Window post-Messge plugin rpc ctx obj, string - 1:1 max jsChannel
msg, rpc ctx obj, string - 1:1 max Web intents msg, rpc string + 1:1 max sMash
pubsub string - 1:1, 1:N, 1:all max Shindig pubsub pubsub string - 1:1, 1:N,
1:all max OpenAjax Hub pubsub string - 1:1, 1:N, 1:all max open-app pubsub
string - 1:1, 1:N, 1:all max pmrpc msg, rpc, pubsub ctx obj, string + 1:1,
1:N, 1:all max LocalStorage sh-mem string - 1:1, 1:N 5MB, max WebDatabase sh-
mem string - 1:1, 1:N 5MB, max IdexedDB sh-mem string - 1:1, 1:N max
CrossDomain Storage sh-mem string - 1:1, 1:N 5MB, max WebWorker postMessage
msg ctx obj - 1:1 max jQuery hive msg ctx obj, string - 1:1 max
Table 3. Evaluation of existing cross-context communication systems according
to the Transport system, Reliability, Authorization of sender, Authorization
of receiver and Generality of applicability dimensions. Legend: Reliability: -
= unreliable. Authorization of sender, Authorization of receiver: - =
unsupported, single = single authorized context, acl = access control
whitelist. Generality of applicability: generic = generic system, specific =
application specific system.
Table 3. Evaluation of existing cross-context communication systems according
to the Transport system, Reliability, Authorization of sender, Authorization
of receiver and Generality of applicability dimensions. Legend: Reliability: -
= unreliable. Authorization of sender, Authorization of receiver: - =
unsupported, single = single authorized context, acl = access control
whitelist. Generality of applicability: generic = generic system, specific =
application specific system.
Cross-context communication system Transport system Reliability Author. of
sender Author. of receiver Generality of app. Direct access - - - - generic
FIM - - - - generic Window name - - - - generic Cookies - - - - generic
CrossFrame FIM, direct access - - - generic Complex window name Window name -
- - generic RMR FIM - - - generic NIX opener property - - - generic Frame
Element Direct access - - - generic postMessage - - - single generic Channel
messaging - - - single generic XSS interface CrossFrame, postMessage - single
- generic Google Closure CrossFrame, post-Message, Frame Element, NIX, RMR
queueing - - generic jQuery postMessage FIM, postMessage - single - generic
OMOS CrossFrame, postMessage - acl - generic Shindig rpc CrossFrame, post-
Message, Frame Element, NIX, RMR queueing - - specific easyXDM Complex window
name, FIM, NIX, post-Message retries acl - generic sMash CrossFrame - acl acl
generic jsChannel postMessage queueing single single generic Window post-
Messge plugin FIM, postMessage - - - generic Web intents postMessage,
WebWorker postMessage, LocalStorage - - - generic Shindig pubsub Shindig rpc
queueing - - specific OpenAjax Hub NIX, CrossFrame, postMessage - - - generic
open-app Shindig pubsub queueing - - specific pmrpc postMessage retries acl
acl generic LocalStorage - - - - generic WebDatabase - - - - generic IdexedDB
- - - - generic CrossDomain Storage postMessage, LocalStorage - acl - generic
WebWorker postMessage - - - - generic jQuery hive WebWorker postMessage - - -
generic
The simplest system for communication between same-origin window contexts is
direct access [14] which enables the sender to access the memory space of the
receiver, including variables and functions, as if it was local to the sender.
In order to achieve cross-origin communication, this primitive system was
later replaced with manipulations of browser mechanisms which ignore the
cross-origin constraint. The fragment identifier messaging (FIM) system [13]
uses the location property of window context objects which contains the URI of
the document loaded in the context. The location property enables any sender
to write but not to read the fragment part of the receiver’s URI. Only the
receiver context can read the URI fragment data making the fragment identifier
a simple form of shared-memory. However, the message size of this system is
limited by browser restrictions on URI length. A system similar to FIM was
developed using the window.name property of window objects [44]. Furthermore,
browser cookies, intended for session storage, were also used for implementing
shared-memory communication, however with a same-domain communication
restriction [14, 24].
Due to low reliability, cross-browser support and sometimes inappropriate
programming models of these systems, new message-oriented frameworks were
developed on top. The CrossFrame framework [45] is an extension of FIM while
the complex window name framework [46] is an extension of the window.name
method. However, in order to enable a message-oriented model, the frameworks
use a server component to initiate communication. These systems were
accompanied by other frameworks based on browser-specific features enabling
message-oriented cross-origin communication, namely the RMR system [45] on
WebKit based browsers (Safari, Chrome), the NIX system [45] on Internet
Explorer browsers and the FrameElement system [45] on Gecko based browsers
(Firefox).
The standardization of cross-context communication was addressed by the HTML5
specification. The specification defines two APIs for secure, reliable and
message-oriented cross-origin communication: the postMessage API [14] and the
Channel messaging API [14]. As most new versions of popular Web browsers
implement these two APIs, they have become the de facto standard for message-
oriented cross-context communication. However, most Web applications need to
support older browsers that do not implement the new HTML5 APIs. This
motivated the development of many frameworks, like XSSinterface [50], the
Google Closure library [49] and jQuery postMessage [51], that implemented the
postMessage or similar message-oriented interface using either browser APIs
or, as a fallback, previously described message-oriented frameworks.
Furthermore, some Web applications require a different communication model to
achieve application goals. OMOS [52], the Shindig RPC feature [45], easyXDM
[46], the window postMessage plugin [53], jsChannel[54] and Web intents [38]
are all frameworks that extend the described message-oriented systems to
provide a remote procedure call model. Similarly, SMash [30], the Shindig
pubsub feature [45], OpenAjax Hub [55], open-app [56] and pmrpc [57, 58]
frameworks provide a publish-subscribe communication model. Many of these
frameworks are also based on the postMessage API and fall back to other
systems for older browsers. Among these systems, easyXDM, jsChannel, Web
intents and pmrpc are the most recently developed and have several advantages
over other systems, such as providing reliability and discovery features.
Along with HTML5, other specifications also introduced new APIs for storing
data which enable shared-memory cross-context communication in a standardized
way. The LocalStorage API [33] provides a simple structure of key-value pairs,
the Web SQL database API [34] provides an offline SQL database, while the
Indexed Database API [47] provides a compromise between the simplicity and
speed provided by the previous two specifications. All of the mentioned
specifications support communication only between contexts on the same origin
and are not yet implemented by all major browsers. An extension of the Web
Storage specification to enable cross-origin communication is implemented in
the Cross Domain Storage framework [48] inspired by the XAuth protocol
implementation [59]. The framework is based on a combination of the Web
Storage same-origin shared-memory and cross-origin postMessage API.
Web Workers were introduced in recent years and therefore a small number of
systems support communication with worker contexts. The Web Worker
specification [8] defines an API, almost exact to the postMessage API, for
message-oriented communication between window contexts and directly nested
worker contexts, and between two directly nested worker contexts. The jQuery
Hive plugin [60] reduces the code overhead of communicating with worker
contexts and is still based on a message-oriented communication model. Lastly,
the previously mentioned Web SQL database APIs and Indexed Database API
specifications also support worker contexts in addition to being accessible
from window context.
## 5 Discussion
In this section we discuss relevant aspects of the proposed framework and
performed evaluation. Based on the discussions, we give an aggregated view of
the evaluated cross-context communication ecosystem and propose several
beneficial directions for future work.
First, both the framework and evaluation show that there are many relevant
dimensions to cross-context communication. However, we do not claim that the
extracted dimensions are completely orthogonal and therefore dependencies
between dimensions may exist. For example, a publish-subscribe communication
model implies support for the multicast distribution scheme and indirect
naming. Although system evaluation and trend analysis would be more succinct
and clearer with orthogonal dimensions, we do not believe this is a
significant drawback.
However, we do propose that some dimensions be researched in more depth. The
best example is the definition of server-mediated communication frameworks of
the type of system dimension. Server-mediated communication frameworks, as
currently defined, include any system that enables data exchange outside the
browser. The importance of these types of systems is in their support for
communication between contexts executing in remote browsers. As hinted by the
currently developing peer-to-peer API in the HTML Device specification [61],
this type of communication is becoming more important. However, the difficulty
in systematizing of this subset of cross-context communication systems is in
that it includes not only systems developed specifically for cross-context
communication but also any kind of service outside the browser which can be
used for data transfer. This, for example, includes APIs for accessing the
file system, cloud services for publish-subscribe messaging, such as PubNub
and PusherApp, and even e-mail and social-networking services like Twitter.
Therefore, our future work includes a deeper analysis of these types of
systems to provide a finer and more useful granularity of values for the type
of system dimension.
The proposed framework does not yet address the possible requirements of a
cross-context communication system with regard to creating contexts with which
it supports communication. For example, the easyXDM framework may be used to
communicate with a context only if the framework was used to create the
context. This feature enables easier setup of multi-context Web applications
but also limits the usage of the system in Web applications not under the
control of the developer, for example iGoogle.
Furthermore, the framework does not address performance-relevant aspects of
cross-context communication systems, such as the size of libraries which need
to be downloaded by the Web application using the system and communication
latency. As these aspects are becoming more important for modern Web
applications and especially Web applications optimized for mobile devices,
research in this field should take them into consideration.
Some dimensions common in operating systems have been left out of the
framework. For example, inter-process communication systems are differentiated
based on their synchronicity, denoting whether send and receive primitives
block execution until the other party responds (synchronous) or do not block
execution (asynchronous). Although synchronous cross-context communication may
be implemented in Web browsers, this is exceedingly impractical due to the
asynchronous event-based execution model of browsers. All evaluated systems
are based on asynchronous communication and therefore the dimension has been
left out of the framework.
Furthermore, as noted in the previous section, a complete systematization of
the ecosystem should include an evaluation of all existing cross-context
communication systems across all dimensions. Therefore, future work in this
area should include an evaluation of the left out security and browser support
dimensions, as well as evaluating representative server-mediated communication
frameworks.
The evaluation of existing systems gives the following insights and possible
areas for future work. As shown on Figure 5 a), a substantial number of
systems use server components for initiation of communication. Second, as
shown on Figure 5 b), only a small number of systems supports worker contexts
and even a smaller number of systems unify both window and worker context
communication.
Fig. 3. Aggregated evaluation results for Type of system, Window context
support and Worker context support dimensions.
Fig. 3. Aggregated evaluation results for Type of system, Window context
support and Worker context support dimensions.
Third, as shown on Figure 5 a), only one third of evaluated systems support
high-level communication models like remote procedure call and publish-
subscribe. These communication models are often preferable over message-
oriented and shared memory models since they require a smaller code overhead
for achieving application goals. Fourth, as shown on Figure 5 b), a small
number of systems unify and expose more than one communication model. As a
result, several cross-context communication systems must often be used to
achieve the required cross-context interaction, thus increasing application-
level complexity.
Fifth, as shown on Figure 5 a), although security features of cross-
communication systems have been the most researched, the authorization aspect
of security is still significantly underdeveloped. As concluded in [32], more
expressive mechanisms for authorization, such as the whitelist access control
model, should be integral parts of these systems, for both senders and
receivers. Lastly, as shown on Figure 5 b), context discovery is addressed by
only two of the evaluated systems, while only seven of the evaluated systems
support some form of reliable communication.
Fig. 4. Aggregated evaluation results for Communication model dimension.
Fig. 4. Aggregated evaluation results for Communication model dimension.
Fig. 5. Aggregated evaluation results for Authorization of sender,
Authorization of receiver, Discovery and Reliability dimensions.
Fig. 5. Aggregated evaluation results for Authorization of sender,
Authorization of receiver, Discovery and Reliability dimensions.
## 6 Conclusion
Web browsers are evolving at a rapid pace to support execution of modern Rich
Internet Applications (RIAs). In many ways, modern Web browsers are offering
the same execution services as operating systems: application execution,
reliability, security, resource management and others. For example, the
recently announced ChromeOS is an operating system almost completely based on
the Chrome Web browser.
Like multi-process desktop applications executing on operating systems, modern
Web applications are built from many browser execution contexts. Examples of
multi-context Web applications include widget-based applications (personal
learning environments) [7], background processing [8] and even platforms for
secure authentication [59]. Therefore, a fundamental requirement for Web
browsers is adequate support for cross-context communication, a field which
has historically been in disorder due to browser evolution.
Our systematization of cross-context communication is based on a
classification framework that establishes relevant properties of cross-context
communication systems. We show the usefulness of such a framework through a
broad analysis of many existing cross-context communication systems. The
usefulness of the analysis is twofold. First, it enables a thoughtful
understanding of each system and comparison to other systems. Second, the
analysis offers insights into many areas for both future research and
development of new cross-context communication systems in order to support
many new requirements of RIAs. As research results concerning widget portals
[62, 63, 6] and several cross-context communication systems [32] suggest, we
believe that future cross-context communication systems should be guided by
the principle of economy of liabilities [32]. In other words, cross-context
communication systems should hide the complexity of cross-context
communication by providing high-level functionalities, such as multiple
communication models, context discovery, unified window context and worker
context communication and ease of specifying authorization policies.
Furthermore, as high performance is an important requirement of Web
applications, further research should be focused in this direction.
Consequently, we have started with the development of a framework for testing
run-time performance of cross-context communication systems. Therefore, the
next revision of our framework will at least include a dimension which
reflects the size of the cross-context communication system’s libraries
required in the browser and the empirically measured latency when transferring
data between two contexts.
Acknowledgements
The authors acknowledge the support of the Ministry of Science, Education, and
Sports of the Republic of Croatia through the _Computing Environments for
Ubiquitous Distributed Systems_ (036-0362980-1921) research project.
Furthermore, the authors thank Sinisa Srbljic, Dejan Skvorc, Miroslav Popovic,
Klemo Vladimir, Marin Silic, Goran Delac, Jakov Krolo and Zvonimir Pavlic from
School of Electrical Engineering and Computing, University of Zagreb. Lastly,
the authors thank the following people for their support in research of cross-
context communication in Web browsers: Oywind Sean Kinsey, developer of the
easyXDM cross-context communication framework; Tobias Nelkner and Philipp
Rustemeier, members of the MATURE project; Scott Wilson, contributor to the
Apache Wookie project; Bodo von der Heiden, member of the Responsive Open
Learning Environments (ROLE) project; and Fridolin Wild from the Knowledge
Media Institute, Open University, UK.
References
## References
* [1] Y. Ding, L. Xu and D. Embley (2009), A Model of World Wide Web Evolution, Proceedings of the WebSci’09: Society On-Line, Athens, Greece, March 2009, http://www.websci09.org/proceedings/ (in press)
* [2] P. Fraternali, G. Rossi and F. Sánchez-Figueroa (2010), Rich Internet Applications, IEEE Internet Computing, Vol.14, No.3, May-June 2010, pp. 9-12
* [3] J. Yu, B. Benatallah, F. Casati and F. Daniel (2008), Understanding Mashup Development, IEEE Internet Computing, Vol.12, No.5, Sept.-Oct. 2008, pp. 44-52
* [4] S. Aghaee and C. Pautasso (), Mashup Development with HTML5, Proceedings of the 3rd and 4th International Workshop on Web APIs and Services Mashups, Ayia Napa, Cyprus, December 2010
* [5] O. Diaz, A. Irastorza, J. Sanchez Cuadrado and L. M. Alonso (2008), From page-centric to portlet-centric Web development: Easing the transition using MDD, Information and Software Technology Journal, Vol. 50, No. 12, November 2008, pp. 1210-1231
* [6] D. Renzel, C. Hobelt, D. Dahrendorf, M. Friedrich, F. Modritscher, K. Verbert, S. Govaerts, M. Palmer and E. Bogdanov (2010), Collaborative Development of a PLE for Language Learning, International Journal of Emerging Technologies in Learning, Vol. 5, No. 1, Jan. 2010, pp. 31-40
* [7] S. Srbljic, D. Skvorc and D. Skrobo (2009), Widget-Oriented Consumer Programming, AUTOMATIKA: Journal for Control, Measurement, Electronics, Computing and Communications, Vol. 50, No. 3-4, Dec. 2009, pp. 252-264
* [8] I. Hickson (Editor), Web Workers. W3C draft (accessed on Sept. 8, 2010), http://dev.w3.org/html5/workers/
* [9] A. Wright (2009), Ready for a Web OS?, In Communications of the ACM, Vol. 52, No. 12, Dec. 2009, pp. 16-17
* [10] M. Zawelski, Browser security handbook, (accessed on Sept. 8, 2010), http://code.google.com/p/browsersec/wiki/Main
* [11] K. Singh, A. Moshchuk, H. J. Wang and W. Lee (2010), On the Incoherencies in Web Browser Access Control Policies, IEEE Symposium on Security and Privacy, May 2010, Oakland, California, USA, pp. 463-478
* [12] H. Schneider (2008), Communication between browser windows, United States patent, patent number 7426699 (issued on Sept. 16, 2008)
* [13] C. Jackson and H. J. Wang (2007), Subspace: Secure CrossDomain Communication for Web Mashups, Proceedings of the 16th international conference on World Wide Web, Banff, Alberta, Canada, 2007, pp. 611-620
* [14] I. Hickson (Editor), HTML5: A vocabulary and associated APIs for HTML and XHTML, W3C draft (accessed on Sept. 8, 2010), http://www.w3.org/TR/html5/
* [15] Y. Deshpande, S. Murugesan, A. Ginige, S. Hansen, D. Schwabe, M. Gaedke and B. White (2002), Web Engineering, Journal of Web Engineering, Vol. 1, No. 1, Oct. 2002, pp. 3-17
* [16] I. Jacobs, N. Walsh (2004), Architecture of the World Wide Web, Volume One, W3C Recommendation, (accessed on Sept. 8, 2010), http://www.w3.org/TR/webarch/
* [17] D. Akhawe, A. Barth, P. E. Lam, J. Mitchell and D. Song (2010), Towards a Formal Foundation of Web Security, Proceedings of the 23rd IEEE Computer Security Foundations Symposium (2010), pp. 290-304
* [18] C. Reis (2009), Web Browsers as Operating Systems: Supporting Robust and Secure Web Programs, Doctoral Thesis (June, 2009), University of Washington
* [19] C. Reis, S. D. Gribble and H. M. Levy (2007), Architectural principles for safe web programs, Proceedings of the Sixth Workshop on Hot Topics in Networks, Atlanta, Georgia, November 2007
* [20] E. Isaksson and M. Palmer (2010), Usability and Inter-widget Communication in PLEs, Proceedings of the 3rd Workshop on Mashup Personal Learning Environments, Barcelona, Spain, September 2010
* [21] B. Hoisl, H. Drachsler and C. Waglechner (2010), User-tailored Inter-Widget Communication - Extending the Shared Data Interface for the Apache Wookie Engine, Proceedings of the 13th International Conference on Interactive Computer aided Learning, September 2010, Hasselt, Belgium
* [22] A. Barth, C. Jackson and I. Hickson, The Web Origin Concept, IETF Internet-Draft (accessed on Sept. 8, 2010), http://tools.ietf.org/id/draft-abarth-origin
* [23] S. Sire, A. Vagner, M. Paquier and J. Bogaerts (2009), A Messaging API for Inter-Widgets Communication, Proceedings of the 18th International World Wide Web Conference, Madrid, Spain, April 2009, pp. 1115-1116
* [24] A. Barth, HTTP State Management Mechanism, IETF Internet-Draft (accessed on Sept. 8, 2010), http://tools.ietf.org/id/draft-ietf-httpstate-cookie
* [25] H. J. Wang, C. Grier, A. Moshchuk, S. T. King, P. Choudhury and H. Venter (2009), The Multi-Principal OS Construction of the Gazelle Web Browser, Microsoft Research Technical Report (2009), http://research.microsoft.com/pubs/79655/gazelle.pdf
* [26] A. Barth, C. Jackson and J. C. Mitchell (2009), Securing Frame Communication in Browsers, Communications of the ACM, Vol. 52, No. 6, June 2009, pp. 83-91
* [27] D. Crockford, The Module Tag: A Proposed Solution to the Mashup Security Problem (accessed on Sept. 8, 2010), http://www.json.org/module.html
* [28] J. Howell, C. Jackson, H. J. Wang and X. Fan (2007), MashupOS: operating system abstractions for client mashups, Proceedings of the 11th USENIX workshop on Hot topics in operating systems, San Diego, California, USA, May 2007, pp. 1-7
* [29] A. Barth, C. Jackson and W. Li (2009), Attacks on JavaScript Mashup Communication, Proceedings of the Web 2.0 Security and Privacy Workshop (2009)
* [30] F. De Keukelaere, S. Bhola, M. Steiner, S. Chari and S. Yoshihama (2008), SMash: Secure Component Model for Cross-Domain Mashups on Unmodified Browsers, Proceedings of the 17th International World Wide Web Conference, Beijing, China, April 2008
* [31] S. Bhola, S. Chari and M. Steiner (2007), Security for web 2.0 application scenarios: Exposures, issues and challenges, Proceedings of the IEEE Web 2.0 Security and Privacy Workshop, 2007
* [32] S. Hanna, R. Shin, D. Akhawe, P. Saxena, A. Boehm and D. Song (2010), The Emperor’s New API: On the (In)Secure Usage of New Client Side Primitives, Proceedings of the IEEE Web 2.0 Security and Privacy Workshop, May 2010
* [33] I. Hickson (Editor), Web Storage, W3C draft (accessed on Sept. 8, 2010), http://dev.w3.org/html5/webstorage/
* [34] I. Hickson (Editor), Web SQL Database, W3C draft (accessed on Sept. 8, 2010), http://dev.w3.org/html5/webdatabase/
* [35] A. Barth, C. Jackson, C. Reis and the Google Chrome Team (2008), The Security Architecture of the Chromium Browser, Stanford University, Technical report, 2008, http://seclab.stanford.edu/websec/chromium/
* [36] A. Silberschatz, P. B. Galvin and G. Gagne (2008), Operating System Concepts; 8th edition, Wiley, 2008, ISBN: 978-0470128725
* [37] M. Shaw and D. Garlan (1996), Software Architecture: Perspectives on an Emerging Discipline, Prentice Hall, April 1996, ISBN 978-0131829572
* [38] P. Kinlan (2010), Web intents, (accessed on Dec. 8, 2010), http://webintents.appspot.com/
* [39] E. Mullins, Local Connection Actionscript – Communicate between seperate Flash files, (accessed on Sept. 8, 2010), http://blog.circlecube.com/2008/03/tutorial/local-connection-actionscript-communicate-between-seperate-flash-files-tutorial/
* [40] A. Ghoda (2010), Introducing Silverlight 4, Apress, July 2010, ISBN:9781430229919
* [41] G. Clayton, V. Chen, A. Athsani and R. Martinez (2009), System and method of inter-widget communication, United States patent, patent number US 2009/0070409 A1 (issued on March 12, 2009)
* [42] G. Corvera, G. Moore, D. Thorpe and K. Kato (2009), Inter-frame messaging between different domains, United States patent, patent number 20090328063 (issued on December 31, 2009)
* [43] B. C. Appleton, S. Meschkat, T. Tran, A. Sah, Z. Wang, A. P. Schuck and J. R. Macgill (2010), System using router in a web browser for inter-domain communication, United States patent, patent number 7809785 (issued on October 5 2010)
* [44] K. Zyp (2008), window.name transport, (accessed on Sept. 8, 2010), http://www.sitepen.com/blog/2008/07/22/ windowname-transport/
* [45] Apache Shindig project, (accessed on Sept. 8, 2010), http://shindig.apache.org/
* [46] S. Kinsey, easyXDM framework, (accessed on Sept. 8, 2010), http://easyxdm.net/
* [47] N. Mehta, J. Sicking, E. Graff and A. Popescu (Editors), Web Database API, W3C draft (accessed on Sept. 8, 2010), http://www.w3.org/TR/IndexedDB/
* [48] N. C. Zakas (2010), Learning from XAuth: Cross-domain localStorage, (accessed on Sept. 8, 2010), http://www.nczonline.net/blog/2010/09/07/ learning-from-xauth-cross-domain-localstorage/
* [49] Google Closure Library, (accessed on Sept. 8, 2010), http://code.google.com/p/closure-library/
* [50] XSSinterface JavaScript library, (accessed on Sept. 8, 2010), http://code.google.com/p/xssinterface/
* [51] jQuery postMessage plugin, (accessed on Sept. 8, 2010), http://github.com/cowboy/jquery-postmessage
* [52] S. Zarandioon, Y. Danfeng and V. Ganapathy (2008), OMOS: A Framework for Secure Communication in Mashup Applications, Proceedings of the Annual Computer Security Applications Conference, Anaheim, California, USA, December 2008, pp. 355-364
* [53] Window postMessage plugin, (accessed on Sept. 8, 2010), http://postmessage.freebaseapps.com/
* [54] L. Hilaiel (2010), jsChannel, (accessed on Dec. 8, 2010), https://github.com/mozilla/jschannel
* [55] OpenAjax Hub 2.0, (accessed on Sept. 8, 2010), http://www.openajax.org/member/ wiki/OpenAjax_Hub_2.0_Specification
* [56] Open Application JavaScript Library, (accessed on Sept. 8, 2010), http://code.google.com/p/open-app/
* [57] I. Zuzak, M. Ivankovic and I. Budiselic Cross-context Web browser communication withunified communication models and context types, Accepted for publication: 34th international convention on information and communication technology, electronics and microelectronics, Opatija, Croatia, May 2011
* [58] I. Zuzak and M. Ivankovic, Pmrpc library, (accessed on Sept. 8, 2010), http://code.google.com/p/pmrpc/
* [59] Extended authentication (XAuth), (accessed on Sept. 8, 2010), http://xauth.org/info/
* [60] jQuery.Hive plugin, (accessed on Sept. 8, 2010), http://github.com/rwldrn/jquery-hive
* [61] I. Hickson (Editor), HTML Device, An addition to HTML – Peer-to-peer connections, W3C Editor’s draft (accessed on Sept. 8, 2010), http://dev.w3.org/html5/html-device/#peer-to-peer-connections
* [62] S. Sire and A. Vagner (2008), Increasing Widgets Interoperability at the Portal Level, Proceedings of the First International Workshop on Mashup Personal Learning Environments, Maastricht, The Netherlands, Sept. 2008, pp. 33-36
* [63] Sire, S., Bogdanov, E., Palmér, M., and Gillet, D. 2009. Towards Collaborative Portable Web Spaces. In Proceedings of the Second Workshop on Mash-Up Personal Learning Environments (2009).
|
arxiv-papers
| 2011-08-24T07:35:21 |
2024-09-04T02:49:21.743533
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ivan Zuzak (1), Marko Ivankovic (2), Ivan Budiselic (1) ((1) School of\n Electrical Engineering and Computing, University of Zagreb, Croatia, (2)\n Google Inc., Zurich, Switzerland)",
"submitter": "Ivan Zuzak",
"url": "https://arxiv.org/abs/1108.4770"
}
|
1108.4885
|
Blow-ups of locally conformally
Kähler manifolds
Liviu Ornea111Partially supported by CNCS UEFISCDI, project number PN-II-ID-
PCE-2011-3-0118., Misha Verbitsky222Partially supported by RFBR grant
10-01-93113-NCNIL-a, RFBR grant 09-01-00242- a, AG Laboratory SU-HSE, RF
government grant, ag. 11.G34.31.0023, and Science Foundation of the SU-HSE
award No. 10-09-0015., and Victor Vuletescu11footnotemark: 1
Keywords: Locally conformally Kähler manifold, locally trivial bundle, blow-
up.
2000 Mathematics Subject Classification: 53C55.
Abstract
A locally conformally Kähler (LCK) manifold is a manifold which is covered by
a Kähler manifold, with the deck transform group acting by homotheties. We
show that the blow-up of a compact LCK manifold along a complex submanifold
admits an LCK structure if and only if this submanifold is globally
conformally Kähler. We also prove that a twistor space (of a compact
4-manifold, a quaternion-Kähler manifold or a Riemannian manifold) cannot
admit an LCK metric, unless it is Kähler.
###### Contents
1. 1 Introduction
1. 1.1 Bimeromorphic maps and locally conformally Kähler structures
2. 1.2 Positive currents on LCK manifolds
3. 1.3 Fujiki class C and LCK geometry
2. 2 Blow-ups and blow-downs of LCK manifolds
3. 3 The proofs
## 1 Introduction
### 1.1 Bimeromorphic maps and locally conformally Kähler structures
A locally conformally Kähler (LCK) manifold is a complex manifold $M$,
$\dim_{\mathbb{C}}M>1$, admitting a Kähler covering
$(\tilde{M},\tilde{\omega})$, with the deck transform group acting on
$(\tilde{M},\tilde{\omega})$ by holomorphic homotheties. Unless otherwise
stated, we shall consider only compact LCK manifolds.
In the present paper we are interested in the birational (or, more precisely,
bimeromorphic) geometry of LCK manifolds.
An obvious question arises immediately.
Question 1.1: Let $X\subset M$ be a complex subvariety of an LCK manifold, and
$M_{1}{\>\longrightarrow\>}M$ a blowup of $M$ in $X$. Would $M_{1}$ also admit
an LCK structure?
When $X$ is a point, the question is answered in affirmative by Tricerri [Tr]
and Vuletescu [Vu]. When $\dim X>0$, the answer is not immediate. To state it
properly, we recall the notion of a weight bundle of an LCK manifold. Let
$(\tilde{M},\tilde{\omega})$ be the Kähler covering of an LCK manifold $M$,
and $\pi_{1}(M){\>\longrightarrow\>}\operatorname{Map}(\tilde{M},\tilde{M})$
the deck transform map. Since
$\rho^{*}(\gamma)\tilde{\omega}=\operatorname{\text{\sf
const}}\cdot\tilde{\omega}$, this constant defines a character
$\pi_{1}(M)\stackrel{{\scriptstyle\chi}}{{{\>\longrightarrow\>}}}{\mathbb{R}}^{>0}$,
with
$\chi(\gamma):=\displaystyle\frac{\rho^{*}(\gamma)\tilde{\omega}}{\tilde{\omega}}$.
Definition 1.2: Let $L$ be the 1-dimensional local system on $M$ with
monodromy defined by the character $\chi$. We think of $L$ as of a real bundle
with a flat connection. This bundle is called the weight bundle of $M$.
One may think of the Kähler form $\tilde{\omega}$ as of an $L$-valued
differential form on $M$. This form is closed, positive, and of type (1,1).
Therefore, for any smooth complex subvariety $Z\subset M$ such that
$L{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ is a trivial local system, $Z$
is Kähler.
The following two theorems describe how the LCK property behaves under blow-
ups.
Theorem 1.3: Let $Z\subset M$ be a compact complex submanifold of an LCK
manifold, and $M_{1}$ the blow-up of $M$ with center in $Z$. If the
restriction $L{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ of the weight
bundle is trivial as a local system then $M_{1}$ admits an LCK metric.
Proof: See 2.
A similar question about blow-downs is also answered.
Theorem 1.4: Let $D\subset M_{1}$ be an exceptional divisor on an LCK
manifold, an $M$ the complex variety obtained as a contraction of $D$. Then
the restriction $L{\left|{}_{{\phantom{|}\\!\\!}_{D}}\right.}$ of the weight
bundle to $D$ is trivial.
Proof: 2.
This result is quite unexpected, and leads to the following theorem about a
special class of LCK manifold called Vaisman manifolds (Section 2).
Claim 1.5: Let $M$ be a Vaisman manifold. Then any bimeromorphic contraction
$M{\>\longrightarrow\>}M^{\prime}$ is trivial. Moreover, for any positive-
dimensional submanifold $Z\subset M$, its blow-up $M_{1}$ does not admit an
LCK structure.
Proof: 2
### 1.2 Positive currents on LCK manifolds
The proofs of 1.1 and 1.1 are purely topological. However, they were
originally obtained using a less elementary argument involving positive
currents.
We state this argument here, omitting minor details of the proof, because we
think that this line of thought could be fruitful in other contexts too; for
more information and missing details, the reader is referred to [D1], [DP] and
[D2].
A current is a form taking values in distributions. The space of
$(p,q)$-currents on $M$ is denoted by $D^{p,q}(M)$. A strongly positive
current111In the present paper, we shall often omit “strongly”, because we are
only interested in strong positivity. is a linear combination
$\sum_{I}\alpha_{I}(z\wedge\overline{z})_{I}$
where $\alpha_{I}$ are positive, measurable functions, and the sum is taken
over all multi-indices $I$. An integration current of a closed complex
subvariety is a strongly positive current.
It is easy to define the de Rham differential on currents, and check that its
cohomology coincide with the de Rham cohomology of the manifold.
Currents are naturally dual to differential forms with compact support. This
allows one to define an integration (pushforward) map of currents, dual to the
pullback of differential forms. This map is denoted by $\pi_{*}$, where
$\pi:\;M{\>\longrightarrow\>}N$ is a proper morphism of smooth manifolds.
Now, let $\pi:\;M{\>\longrightarrow\>}N$ be a blow-up of a subvariety
$Z\subset N$ of codimension $k$, and $\omega$ a Kähler form on $M$. Then
$(\pi_{*}\omega)^{k}$ has a singular part which is proportional to the
integration current of $Z$.
This follows from the Siu’s decomposition of positive currents ([D1]).
Demailly’s results on intersection theory of positive currents ([D2]) are used
to multiply the currents, and the rest follows because the Lelong numbers of
$\pi_{*}\omega$ along $Z$ are non-zero.
Applying this argument to a birational contraction
$M\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}M^{\prime}$ of an
LCK manifold $M$, and denoting by
$\tilde{M}\stackrel{{\scriptstyle\tilde{\varphi}}}{{{\>\longrightarrow\>}}}\tilde{M}^{\prime}$
the corresponding map of coverings, we obtain a closed, positive current
$\xi:=\tilde{\varphi}_{*}\tilde{\omega}$ on $\tilde{M}^{\prime}$, with the
deck transform map $\rho$ acting on $\xi$ by homotheties. Then $\rho$ would
also act by homotheties on the current $\xi^{k}$, $k=\dim Z$, where $Z$ is the
exceptional set of $\tilde{\varphi}$.
Applying the above result to decompose $\xi^{k}$ onto its absolutely
continuous and singular part, we obtain that the current of integration $[Z]$
of $Z$ is mapped to $\operatorname{\text{\sf const}}[Z]$ by the deck transform
action. Since the current of integration of $Z$ is mapped by the deck
transform to the current of integration of $\tilde{\varphi}(Z)=Z$, the
constant $\operatorname{\text{\sf const}}$ is trivial; this implies that
$\pi(Z)\subset M^{\prime}$ is Kähler, with the Kähler metric obtained in the
usual way from $\tilde{\omega}$.
### 1.3 Fujiki class C and LCK geometry
A compact complex variety $X$ is said to belong to Fujiki class C if $X$ is
bimeromorphic to a Kähler manifold. The Fujiki class C manifolds are closed
under many natural operations, such as taking a subvariety, or the moduli of
subvarieties, and play important role in Kähler geometry.
This notion has a straightforward LCK analogue.
Definition 1.6: Let $M$ be a compact complex variety. It is called a locally
conformally class C variety if it is bimeromorphic to an LCK manifold.
The importance of the Fujiki class C notion was emphasized by a more recent
work of Demailly and Păun [DP], who characterized class C manifolds in terms
of positive currents. Recall that a Kähler current is a positive, closed
(1,1)-current $\varphi$ on a complex manifold $M$ which satisfies
$\varphi\geqslant\omega$ for some Hermitian form $\omega$ on $M$.
Demailly and Păun have proven that a compact complex manifold $M$ belongs to
class C if and only if it admits a positive Kähler current.
For an LCK manifold, an analogue of a Kähler current is provided by the
following notion (motivated by 2).
Definition 1.7: Let $M$ be a compact complex manifold, $\theta$ a closed real
1-form on $M$, $\Xi$ a positive, real (1,1)-current satisfying
$d\Xi=\theta\wedge\Xi$ and $\Xi\geqslant\omega$ for some Hermitian form
$\omega$ on $M$. Then $\Xi$ is called an LCK current.
It would be interesting to know if an LCK-analogue of the Demailly-Păun
theorem is true.
Question 1.8: Let $M$ be a complex compact manifold. Determine whether the
following conditions are equivalent.
(i)
$M$ belongs to locally conformally class C.
(ii)
$M$ admits an LCK current.
## 2 Blow-ups and blow-downs of LCK manifolds
We start by repeating (in a more technical fashion) the definition of an LCK
manifold given in the introduction. Please see [DO] for more details and
several other versions of the same definition, all of them equivalent.
Definition 2.1: A locally conformally Kähler (LCK) manifold is a complex
manifold $X$ covered by a system of open subsets $U_{\alpha}$ endowed with
local Kähler metrics $g_{\alpha}$, conformal on overlaps $U_{\alpha}\cap
U_{\beta}$: $g_{\alpha}=c_{\alpha\beta}g_{\beta}$.
Note that, in complex dimension at least $2$, as we always assume,
$c_{\alpha\beta}$ are positive constants. Moreover, they obviously satisfy the
cocycle condition. Interpreted in cohomology, the cocycle
$\\{c_{\alpha\beta}\\}$ determines a closed one-form $\theta$, called the Lee
form. Hence, locally $\theta=df_{\alpha}$. It is easily seen that
$e^{-f_{\alpha}}g_{\alpha}=e^{-f_{\beta}}g_{\beta}$ on $U_{\alpha}\cap
U_{\beta}$, and thus determine a global metric $g$ which is conformal on each
$U_{\alpha}$ with a Kähler metric. One obtains the following equivalent:
Definition 2.2: A Hermitian manifold $M$ is LCK if its fundamental two-form
$\omega$ satisfies:
$d\omega=\theta\wedge\omega,\quad\quad d\theta=0.$ (2.1)
for a closed one-form $\theta$.
If $\theta$ is exact then $M$ is called globally conformally Kähler (GCK).
As we work with compact manifolds and, in general, the topology of compact
Kähler manifolds is very different from the one of compact LCK manifolds, we
always assume $\theta\neq 0$ on $X$.
Let
$\Gamma{\>\longrightarrow\>}\tilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$
be the universal cover of $M$ with deck group $\Gamma$. As $\pi^{*}\theta$ is
exact on $\tilde{M}$, $\pi^{*}\omega$ is globally conformal with a Kähler
metric $\tilde{\omega}$. Moreover, $\Gamma$ acts by holomorphic homotheties
with respect to $\tilde{\omega}$. This defines a character
$\chi:\Gamma{\>\longrightarrow\>}{\mathbb{R}}^{>0},\quad\gamma^{*}\tilde{\omega}=\chi(\gamma)\tilde{\omega}.$
(2.2)
It can be shown that this property is indeed an equivalent definition of LCK
manifolds, see [OV2].
Clearly, a LCK manifold $M$ is globally conformally Kähler if and only if
$\Gamma$ acts trivially on $\tilde{\omega}$ (i.e.
$\operatorname{im}\chi=\\{1\\}$).
A particular class of LCK manifolds are the Vaisman manifolds. They are LCK
manifolds with the Lee form parallel with respect to the Levi-Civita
connection of the LCK metric. The compact ones are mapping tori over the
circle with Sasakian fibre, see [OV1]. The typical example is the Hopf
manifold, diffeomorphic to $S^{1}\times S^{2n-1}$.
On a Vaisman manifold, the vector field
$\theta^{\sharp}-\sqrt{-1}J\theta^{\sharp}$ generates a one-dimensional
holomorphic, Riemannian, totally geodesic foliation. If this is regular and if
$M$ is compact, then the leaf space $B$ is a Kähler manifold.
Example 2.3: On a Hopf manifold ${\mathbb{C}}^{n}\setminus\\{0\\}/\langle
z_{i}\mapsto 2z_{i}\rangle$, the LCK metric $\displaystyle\frac{\sum
dz_{i}\otimes dz_{i}}{|\sum z_{i}\overline{z}_{i}|^{2}}$ is Vaisman and
regular; the leaf space is ${\mathbb{C}}P^{n-1}$.
We refer to [DO] or to the more recent [OV2] for more details about LCK
geometry.
It is known, [Tr, Vu], that the blow-up at points preserve the LCK class. The
present paper is devoted to the blow-up of LCK manifolds along subvarieties.
In this case, the situation is a bit more complicated and a discussion should
be made according to the dimension of the submanifold.
Definition 2.4: Let $Y\stackrel{{\scriptstyle j}}{{\hookrightarrow}}M$ be a
complex subvariety. We say that $Y$ is of induced globally conformally Kähler
type (IGCK) if the cohomology class $j^{*}[\theta]$ vanishes, where $\theta$
denotes the cohomology class of the Lee form on $M$.
Remark 2.5: Notice that a IGCK-submanifold of an LCK manifold is always
Kähler.
Remark 2.6: By a theorem of Vaisman ([Va2]), any LCK metric on a compact
complex manifold $Y$ of Kähler type is globally conformally Kähler if
$\dim_{\mathbb{C}}Y>1$. Therefore, the IGCK condition above for smooth $Y$
with $\dim_{\mathbb{C}}Y>1$ is equivalent to $Y$ being Kähler.
Remark 2.7: Notice that there may exist curves on LCK manifolds which are not
IGCK, despite being obvioulsy of Kähler type. For instance, if $M$ is a
regular Vaisman manifold, and if $Y$ is a fiber of its elliptic fibration,
then $Y$ is not IGCK, as any compact complex subvariety of a compact Vaisman
manifold has an induced Vaisman structure (see e.g. [Ve1, Proposition 6.5]).
The main goal of the present paper is to prove the following two theorems:
Theorem 2.8: Let $M$ be an LCK manifold, $Y\subset M$ be a smooth complex
IGCK subvariety, and let $\tilde{M}$ be the blow-up of $M$ centered in $Y$.
Then $\tilde{M}$ is LCK.
Proof: See the argument after 3.
Theorem 2.9: Let $M$ be a complex variety, and
$\tilde{M}{\>\longrightarrow\>}M$ the blow-up of a compact subvariety
$Y\subset M$. Assume that $\tilde{M}$ is smooth and admits an LCK metric. Then
the blow-up divisor $\tilde{Y}\subset\tilde{M}$ is a IGCK subvariety.
Proof: See 3.
Remark 2.10: In the situation described in 2, the variety $\tilde{Y}$ is of
Kähler type, because it is IGCK. When $Y$ is smooth, $Y$ is Kähler, as shown
by Blanchard ([Bl2, Théorème II.6]). Together with 2, this implies the
following corollary.
Corollary 2.11: Let $M$ be an LCK manifold, and $Y\subset M$ a smooth compact
subvariety, such that the blow-up of $M$ in $Y$ admits an LCK metric. If
$\dim_{\mathbb{C}}(Y)>1$ then $Y$ is a IGCK subvariety.
Remark 2.12: Note that, from [Ve1, Proposition 6.5], a compact complex
submanifold $Y$ of a compact Vaisman manifold is itself Vaisman, and $\theta$
represents a non-trivial class in the cohomology of $Y$, so there are no IGCK
submanifolds of proper dimension $\dim_{\mathbb{C}}(Y)>0.$ This implies the
following corollary.
Corollary 2.13: The blow-up of a compact Vaisman manifold along a compact
complex submanifold $Y$ of dimension at least $1$ cannot have an LCK metric.
The proofs of these two theorems and of the corollary will be given in Section
3. As a by-product of our proof, we obtain the following:
Corollary 2.14: If $M$ is a twistor space, and if $M$ admits a LCK metric,
then this metric is actually GCK.
Proof: See 3.
Here, by a “twistor space” we understand any of the following constructions of
a complex manifold: the twistor spaces of half-conformally flat 4-dimensional
Riemannian manifolds, twistor spaces of quaternionic-Kähler manifolds, and
Riemannian twistor spaces of conformally flat manifolds.
Remark 2.15: $(i)$ A similar, weaker result is proven in [KK]. Namely, the
twistor space of half-conformally flat 4-dimensional Riemannian manifolds with
large fundamental group cannot admit LCK metrics with automorphic potential on
the covering. The proof uses different techniques from ours, and which cannot
be generalized neither to higher dimensions nor to quaternionic Kähler
manifolds.
$(ii)$ It was known from [Ga, Mu] that the natural metrics (with respect to
the twistor submersion) cannot be LCK. Our result refers to any metric on the
twistor space, not necessarily related to the twistor submersion. On the other
hand, as shown by Hitchin, the twistor space of a compact 4-dimensional
manifold is not of Kähler type, unless it is biholomorphic to
${\mathbb{C}}P^{3}$ or to the flag variety $F_{2}$ [Hi].
Remark 2.16: So far, we were unable to deal with the reverse statement of 2,
namely, to determine whether a smooth bimeromorphic contraction of an LCK
manifold is always LCK. In the particular case when an exceptional divisor is
contracted to a point, this has been proven to be true by Tricerri, [Tr]; we
conjecture that in the general case this is false, but we are not able to find
any example.
For GCK (that is, Kähler) manifolds, the answer is well known: blow-downs of
Kähler manifolds can be non-Kähler, as one can see from any example of a
Moishezon manifold.
Remark 2.17: We summarize the case of blow-up of curves on LCK manifolds.
Since rational curves are simply-connected, they are IGCK submanifolds, so
blowing-up a rational curve on a LCK manifold always yields a manifold of LCK
type. The case of the elliptic curves was partially tackled in 2. If $Y$ is a
curve of arbitrary genus contained in an exceptional divisor of a blow-up,
then it is also automatically a IGCK subvariety since the exceptional divisor
is so; hence again, blowing it up yields a manifold of LCK type.
To our present knowledge, the only examples of curves $Y$ on LCK manifolds $M$
with genus $g(Y)\geqslant 2$ are curves belonging to some exceptional
divisors. It would be interesting to prove that this is the case in general,
or to build out a counter-example.
## 3 The proofs
Lemma 3.1: Let $M$ be an LCK manifold, $B$ a path connected topological space
and let $\pi:M{\>\longrightarrow\>}B$ be a continuous map. Assume that either
(i)
$B$ is an irreducible complex variety, and $\pi$ is proper and holomorphic.
(ii)
$\pi$ is a locally trivial fibration with fibers which are complex
subvarieties of $M$.
Suppose also that the map
$\pi^{*}:H^{1}(B){\>\longrightarrow\>}H^{1}(M)$
is an isomorphism, and the generic fibers of $\pi$ are positive-dimensional.
Then the LCK structure on $M$ is actually GCK.
Proof: Denote by $\theta$ the Lee form of $M$, and let $\tilde{M}$ be the
minimal GCK covering of $X$, that is, the minimal covering
$\tilde{M}{\>\longrightarrow\>}M$ such that the pullback of $\theta$ is exact.
Since $H^{1}(B)\cong H^{1}(M)$, there exists a covering
$\tilde{B}{\>\longrightarrow\>}B$ such that the following diagram is
commutative, and the fibers of $\tilde{\pi}$ are compact:
$\begin{CD}\tilde{M}@>{}>{}>M\\\ @V{\tilde{\pi}}V{}V@V{}V{\pi}V\\\
\tilde{B}@>{}>{}>B\end{CD}$
Let $\tilde{B}_{0}\subset\tilde{B}$ be the set of regular values of
$\tilde{\pi}$, and let $F_{b}:=\tilde{\pi}^{-1}(b)$ be the regular fibers of
$\tilde{\pi}$, $\dim_{\mathbb{C}}F_{b}=k$. Since $B_{0}$ is connected, all
$F_{b}$ represent the same homology class in $H_{2k}(\tilde{M})$.
Denote the Kähler form of $\tilde{M}$ by $\tilde{\omega}$, conformally
equivalent to the pullback of the Hermitian form on $X$.
Since all $F_{b}$ represent the same homology class, the Riemannian volume
$\operatorname{Vol}_{\tilde{\omega}}(F_{b}):=\int_{F_{b}}\tilde{\omega}^{k}$
is independent from $b\in B_{0}$. This gives (recall the definition of the
character $\chi$ in (2.2))
$\operatorname{Vol}_{\tilde{\omega}}(F_{b})=\int_{F_{b}}\tilde{\omega}^{k}=\int_{F_{\gamma^{-1}(b)}}\gamma^{*}\tilde{\omega}^{k}=\int_{F_{\gamma^{-1}(b)}}\chi(\gamma)^{k}\tilde{\omega}^{k}=\chi(\gamma)^{k}\operatorname{Vol}_{\tilde{\omega}}(F_{b}),$
hence the constant $\chi_{\gamma}$ is equal to 1 for all $\gamma\in\Gamma$.
Therefore, $\tilde{\omega}$ is $\Gamma$-invariant, and $M$ is globally
conformally Kähler.
The above lemma immediately implies 2.
Corollary 3.2: Let $Z$ be the twistor space of $M$, understood in the sense
of 2. Assume that $Z$ admits an LCK metric. Then this metric is globally
conformally Kähler.
Proof: There is a locally trivial fibration $Z{\>\longrightarrow\>}M$, with
complex analytic fibers which are compact symmetric Kähler spaces, hence 3 can
be applied.
Remark 3.3: In the same way one deals with the blow-ups: the generic fibers
over an exceptional set of a blow-up map are positive-dimensional. Therefore,
3 implies 2.
We can now give The proof of 2:
If $\dim_{\mathbb{C}}(Y)>1$ the result follows from 2 and 2. In the case
$\dim_{\mathbb{C}}(Y)=1$ we cannot use this argument directly - see 2 \- so in
this case we argue as follows.
Assume $\tilde{M}$ has an LCK metric $\tilde{\omega}$ with Lee form
$\tilde{\eta}$. By 2, the restriction $\tilde{\eta}_{|Z}$ to the exceptional
divisor $Z$ is exact. Hence, after possibly making a conformal change of the
LCK metric, we can assume $\tilde{\eta}_{|V}=0$ where $V$ is a neighbourhhood
of $Z$. In particular, $\tilde{\eta}$ will be the pull-back of a one-form
$\eta$ on $M$. On the other hand, $\tilde{\omega}$ gives rise to a current on
$\tilde{M}$ (see also §1.2) and its push-forward defines an LCK positive
$(1,1)$ current $\Xi$ on $M$ with associate Lee form $\eta$. Clearly
$\eta_{|Y}=0$.
Possibly conformally changing now $\Xi$, we can assume that $\eta$ is the
unique harmonic form (with respect to the Vaisman metric of $M$) in its
cohomology class. Possibly $\eta_{|Y}$ is no longer zero, but remains exact.
We now show that $\eta$ is basic with respect to the canonical foliation
$\mathcal{F}$ generated on $M$ by $\theta^{\sharp}-\sqrt{-1}J\theta^{\sharp}$.
Indeed, from [Va2], we know that any harmonic form on a compact Vaisman
manifold decomposes as a sum $\alpha+\theta\wedge\beta$ where $\alpha$ and
$\beta$ are basic and transversally (with respect to $\mathcal{F}$) harmonic
forms. In particular, as a transversally harmonic function is constant, we
have
$\eta=\alpha+c\cdot\theta,$ (3.1)
where $c\in\mathbb{R}$ and $\alpha$ is basic, transversally harmonic (see [To]
for the theory of basic Laplacian and basic cohomology etc.).
Let now $S^{1}$ denote the unique homology class in $H_{1}(M)$ (call it the
fundamental circle of $\theta$) such that $\int_{S^{1}}\theta=1$ and
$\int_{S^{1}}\alpha=0$ for every basic cohomology class $\alpha$.
As any complex submanifold of a compact Vaisman manifold is tangent to the Lee
field and hence Vaisman itself, $Y$ is Vaisman with Lee form $\theta_{|Y}$.
Hence we deduce that the fundamental circle of $\theta$ is the image of the
fundamental circle of $\theta_{|Y}$ under the natural map $H_{1}(Y)\rightarrow
H_{1}(M)$.
We now integrate 3.1 on any $\gamma\in H_{1}(Y)$ and take into account that
$\eta_{|Y}$ is exact to get $c=0$. Hence, $\eta$ basic. It can then be treated
as a harmonic one-form on a Kähler manifold (or use the existence of a
transversal $dd^{c}$-lemma). This implies $d^{c}\eta=0$.
But then one obtains a contradiction, as follows. Letting $J$ to be the almost
complex structure of $M$, we see on one hand we have
$\int_{M}d(\Xi^{n-1})\wedge J(\theta)=\int_{M}(n-1)\Xi^{n-1}\wedge\theta\wedge
J(\theta)>0$
since $\Xi$ is positive. On the other hand, since $d(J(\theta))=0,$ it follows
that $d(\Xi^{n-1})\wedge J(\theta)$ is exact so $\int_{M}d(\Xi^{n-1})\wedge
J(\theta)=0,$ a contradiction.
The following result is certainly well-known, but since we were not able to
find out an exact reference we include a proof here.
Lemma 3.4: Assume $(U,g)$ is a Kähler complex manifold, $Y\subset U$ a
compact submanifold and let $c:\tilde{U}{\>\longrightarrow\>}U$ be the blow-up
of $U$ along $Y$. Then, for any open neighbourhood $V\supset Y$, there is a
Kähler metric $\tilde{g}$ on $\tilde{U}$ such that
$\tilde{g}_{|\tilde{U}\setminus c^{-1}(V)}=c^{*}(g_{|U\setminus V})$
Proof. (due to M. Păun; see also [Vu]).
1\. There is a (non-singular) metric on ${\mathcal{O}}_{\tilde{U}}(-D)$ (where
$D$ is the exceptional divisor of the blow-up) such that:
1.A. Its curvature is zero outside $c^{-1}(V)$, and
1.B. Its curvature is strictly positive at every point of $D$ and in any
direction tangent to $D$.
Indeed, if such a metric is found, everything follows, as the curvature of
this metric plus a sufficiently large multiple of $c^{*}(g)$ will be positive
definite on $\tilde{U}.$
2\. To finish the proof, we notice that the existence of a metric $h$ with
property 1.B is clear, due to the restriction of
${\mathcal{O}}_{\tilde{U}}(-D)$ to $D$.
Now let $\alpha$ be its curvature; then
$\alpha-i\partial\overline{\partial}\tau=-[D]$ for some function $\tau$, with
at most logarithmic poles along $D$, bounded from above, and non-singular on
$\tilde{U}\setminus D.$ Consider the function $\tau_{0}:=\max(\tau,-C)$ where
$C$ is some positive constant, big enough such that on $\tilde{U}\setminus
c^{-1}(V)$ we have $\tau>-C.$ Clearly, on a (possibly smaller) neighbourhood
of $D$ we will have $\tau_{0}=-C,$ such that the new metric $e^{-\tau_{0}}h$
on ${\mathcal{O}}_{\tilde{U}}(-D)$ also satisfies 1.A.
Now we can prove 2. Let $c:\tilde{M}{\>\longrightarrow\>}M$ be the blow up of
$M$ along the submanifold $Y$. Let $g$ be a LCK metric on $M$ and let $\theta$
be its Lee form. Since $Y$ is IGCK we see $\theta_{|Y}$ is exact. Let $U$ be a
neighbourhood of $Y$ such that the inclusion $Y\hookrightarrow U$ induces an
isomorphism of the first cohomology. Then $\theta_{|U}$ is also exact, so,
after possibly conformally rescaling $g$, we may assume $\theta_{|U}=0$ and
hence $g_{|U}$ is Kähler. In particular, $\mathrm{supp}(\theta)\cap
U=\emptyset.$ Now choose a smaller neighbourhood $V$ of $Y$ and apply 3. We
get a Kähler metric $\tilde{g}$ on $\tilde{U}$ which equals $c^{*}(g)$ outside
$c^{-1}(V)$, so it glues to $c^{*}(g)$ giving a LCK metric on $\tilde{M}$.
Acknowledgments. We are indebted to M. Aprodu, J.-P. Demailly, D. Popovici and
M. Toma for useful discussions. We are grateful to Mihai Păun for explaining
us the proof of 3.
## References
* [B] F.A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.
* [Bl1] A. Blanchard, Espaces fibrés kähleriens compacts, C.R. Acad. Sci. Paris 328 (1954), 2281–2283.
* [Bl2] A. Blanchard, Sur les variétés analitiques complexes, Ann. Sci. E.N.S. 73 (1956), 157–202.
* [D1] J.-P. Demailly, Analytic methods in algebraic geometry, Lecture Notes, École d’été de Mathématiques de Grenoble “Géométrie des variétés projectives complexes : programme du modèle minimal” (June-July 2007).
* [D2] J.-P. Demailly, Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1 (1992) 361-409.
* [DP] J.-P. Demailly, M. Păun, Numerical characterization of the Kähler cone of a compact Kähler manifold, math.AG/0105176 also in Annals of Mathematics, 159 (2004), 1247-1274.
* [DO] S. Dragomir, L. Ornea, Locally conformal Kähler geometry, Progress in Math. 155, Birkhäuser, Boston, Basel, 1998.
* [Ga] P. Gauduchon, Structures de Weyl et théorèmes d’annulation sur une variété conforme autoduale, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 4, 563–629.
* [Hi] N. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981) 133–150.
* [KK] G. Kokarev, D. Kotschick, _Fibrations and fundamental groups of Kähler-Weyl manifolds_ , Proc. Amer. Math. Soc. 138 (2010), no. 3, 997–1010.
* [Mu] O. Muşkarov, Almost Hermitian structures on twistor spaces and their types, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), no. 2, 285–297.
* [OV1] L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds, Math. Res. Lett., 10 (2003), 799–805.
* [OV2] L. Ornea, M. Verbitsky, A report on locally conformally Kähler manifolds, Contemporary Mathematics 542, 135-150, 2011.
* [To] P. Tondeur, Foliations on Riemannian manifolds, Universitext, Springer– Verlag 1988.
* [Tr] F. Tricerri, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politec. Torino 40 (1982), 81–92.
* [Va1] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (1980), 533–542.
* [Va2] I. Vaisman, A geometric condition for an l.c.K. manifold to be Kähler, Geom. Dedicata 10 (1981), 129–134.
* [Va3] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231–255.
* [Ve1] M. Verbitsky, Vanishing theorems for locally conformal hyperkähler manifolds, Proc. of Steklov Institute, 246 (2004),54–79.
* [Vu] V. Vuletescu, Blowing-up points on locally conformally Kähler manifolds, Bull. Math. Soc. Sci. Math. Roumanie 52(100) (2009), 387–390.
Liviu Ornea
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania. _and_
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Street 010702-Bucharest, Romania
Liviu.Ornea@imar.ro, lornea@gta.math.unibuc.ro
Misha Verbitsky
Laboratory of Algebraic Geometry, Faculty of Mathematics, NRU HSE, 7 Vavilova
Str. Moscow, Russia
verbit@maths.gla.ac.uk, verbit@mccme.ru
Victor Vuletescu
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania.
vuli@gta.math.unibuc.ro
|
arxiv-papers
| 2011-08-24T17:06:41 |
2024-09-04T02:49:21.753742
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liviu Ornea, Misha Verbitsky, Victor Vuletescu",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1108.4885"
}
|
1108.4922
|
# New Results on the $\pi^{+}\pi^{-}$ Electroproduction Cross Sections off
Protons
G.V. Fedotov R.W. Gothe
###### Abstract
In these proceedings we present preliminary $\pi^{+}\pi^{-}$ electroproduction
cross sections off protons in the kinematical area of 1.4 GeV $<W<$ 1.8 GeV
and 0.4 GeV2 $<Q^{2}<1.1$ GeV2. Our results extend the kinematical coverage
for this exclusive channel with respect to previous measurements. Furthermore,
the $\pi^{+}\pi^{-}$ electroproduction cross sections were obtained for
$Q^{2}$-bins of much smaller size. The future analysis of this data within the
framework of the JLAB-MSU reaction model (JM) will considerably improve our
knowledge on the $Q^{2}$ evolution of the transition $\gamma_{v}NN^{*}$
electrocouplings, in particular for the resonances with masses above 1.6 GeV.
###### Keywords:
Nucleon resonances, electromagnetic form factors, meson electroproduction
###### :
13.40.Gp; 14.20.Gk; 13.60.Le; 11.55.Fv
## 1 Introduction
Measurements of the $\pi^{+}\pi^{-}$ electroproduction cross sections
represent an important part of the efforts in the N∗ program with CLAS Bu11 ;
Ra11 . This program is focused on the evaluation of most excited proton
electrocouplings at photon virtualities (Q2) up to 5.0 GeV2,which will allow
us to pin down the active degrees of freedom in the N∗ structure at different
distance scales, and to access the non-perturbative strong interaction
mechanisms that are responsible for the formation of the ground and excited
nucleon states CL12 ; CR11 .
Single and charged double pion exclusive channels are two major contributors
to the meson electroproduction in the N∗ excitation region with different non-
resonant mechanisms. A successful description of all observables in these
exclusive channels with consistent N∗ electrocouplings offers evidence for the
reliable evaluation of these fundamental quantities. Moreover, the
$\pi^{+}\pi^{-}$ exclusive channel is very promising for the studies of N∗s
with masses above 1.6 GeV Bu11 ; Mo11 .
Our measurements of $\pi^{+}\pi^{-}$ electroproduction cross sections,
described in these proceedings, continue the previous studies of this
exclusive channel with the CLAS detector Fedotov08 ; Ripani . Our preliminary
data provid a complementary kinematical coverage: 1.4 GeV $<W<$ 1.8 GeV and
0.4 GeV2 $<Q^{2}<1.1$ GeV2 in comparison with the previously available
measurements, and offer more than a factor six smaller binning over $Q^{2}$.
This kinematical region is suitable to access the electrocouplings and the
$\pi\Delta$, $\rho p$ hadronic decay widths for high lying nucleon resonances
as $S_{31}(1620)$, $S_{11}(1650)$, $F_{15}(1685)$, $D_{33}(1700)$, and
$P_{13}(1720)$ that have substantial decay probabilities to the $N\pi\pi$
final states Mo11 .
## 2 Data analysis
Our analysis is focused on the evaluation of the fully integrated and nine one
fold differential $\pi^{+}\pi^{-}$ electroproduction cross sections off the
proton as defined in Fedotov08 . This information will allow us to determine
$N^{*}$ electrocouplings within the framework of the reaction model JM Victor
; Mo11 . The analysis is based on the experimental data taken with the CLAS
detector during the 2003 e1e run period with the electron beam energy 2.039
GeV. The procedure for the extraction of the above mentioned cross sections
incorporates: a) the selection of $ep\pi^{+}\pi^{-}$ exclusive events, and b)
the evaluation of detector efficiencies in the five dimensional reaction phase
space through Monte-Carlo simulations. One fold differential cross sections
were obtained by integrating the five fold differential $\pi^{+}\pi^{-}$
electroproduction cross section over all variables except of one of interest,
as described in Fedotov08 .
Figure 1: $Q^{2}$ versus $W$ distribution for the selected $ep\rightarrow
e^{\prime}p^{\prime}\pi^{+}\pi^{-}$ exclusive events. Cross sections were
determined inside the area shown by white borders. Hatched areas correspond to
the kinematical regions, where the data on $\pi^{+}\pi^{-}p$ electroproduction
cross sections are available from previous CLAS measurements Fedotov08 ;
Ripani .
The scattered electron was identified by the coincidence of the
electromagnetic calorimeter, Cherenkov counter, drift chambers, and time-of-
flight detector signals. Positive hadrons (protons and $\pi^{+}$) were
identified as particles leaving hits in the drift chambers and time-of-flight
scintillators. Further particle identification procedures were similar to
those described in Fedotov08 . The reaction events were selected in four
different topologies. In the first one all final hadrons were detected, while
in the remaning one of the final hadrons was reconstructed employing energy
and momentum conservation. Maximal statistics is achieved in the missing
$\pi^{-}$ topology. The distribution of the selected events over $W$ and
$Q^{2}$ is shown in Fig. 1. Differential and fully integrated cross sections
were obtained in the kinematical region depicted in Fig. 1 by the white
polygon. A considerable extension of the covered kinematical region is
achieved in comparison with the previous experiments Fedotov08 ; Ripani
(hatched areas in Fig. 1). The $\pi^{-}$ missing mass squared distribution of
the selected events is shown in Fig. 2 and peaks at the $\pi^{-}$ mass
squared.
Figure 2: Distribution of the events over the $\pi^{-}$ missing mass squared
that were selected after particle identification and kinematical cuts.
## 3 Preliminary cross sections
Figure 3: Fully integrated $\pi^{+}\pi^{-}$ electroproduction cross sections
obtained in our analysis (open circles) in comparison with the previously
available data Fedotov08 (filled diamonds).
Preliminary results on the fully integrated $\pi^{+}\pi^{-}$ electroproduction
cross sections obtained in our analysis are shown in Fig. 3 in comparison,
with the previously available data Fedotov08 . The cross sections obtained in
the current analysis are in a good agreement with previously available results
in the overlaping area of $Q^{2}$ and $W$. The $Q^{2}$-evolution of the
$\pi^{+}\pi^{-}$ electroproduction cross section determined in the entire
range of photon virtualities covered by our measurement is depicted in Fig. 4.
The $W$-dependencies of the fully integrated cross sections show resonant
structures in the second and third resonance regions for all $Q^{2}$-bins.
Therefore, our preliminary data demonstrate a good prospect for the extraction
of resonance electrocouplings and $\pi\Delta$, $\rho p$ hadronic decay widths
employing the JM reaction model Mo11 ; Victor .
Figure 4: $Q^{2}$-evolution of the fully integrated $\pi^{+}\pi^{-}$
electroproduction cross sections obtainbed in our analysis (all open symbols
and black bullets). Previously available data Ripani at $Q^{2}=$ 0.65 GeV2
are shown as black stars.
## 4 Conclusions
In our analysis of the CLAS e1e data obtained $\pi^{+}\pi^{-}p$
electroproduction cross sections extend considerably the kinematical range
covered in previous measurements of this exclusive channel. Preliminary
results on nine differential and integrated $\pi^{+}\pi^{-}$p
electroproduction cross sections were obtained in the kinematic region of 1.4
GeV $<W<$ 1.8 GeV and 0.4 GeV2 $<Q^{2}<1.1$ GeV2, and they are in a good
agreement with previously available results. The cross sections show clear
indication of resonance contributions to the second and third resonance
regions, offering encouraging prospects for the extension of our knowledge on
$N^{*}$ electrocouplings, in particular for the resonances with masses above
1.6 GeV.
## References
* (1) V. D. Burkert et al., J. Phys: Conf. Ser. $\bf{299}$, 012008 (2011).
* (2) R.W.Gothe, these proceedings.
* (3) I. G. Aznauryan, et al., arXiv:0907.1901 [nucl-th].
* (4) C.D.Roberts, these proceedings.
* (5) V.I.Mokeev, these proceedings.
* (6) G.V. Fedotov et al. Phys.Rev. C79:015204, 2009.
* (7) V. Mokeev, et al. Phys.Rev. C80:045212, 2009.
* (8) Nucl.Phys. A663:675-678, 2000.
|
arxiv-papers
| 2011-08-24T19:57:28 |
2024-09-04T02:49:21.759282
|
{
"license": "Public Domain",
"authors": "G.V. Fedotov, R.W. Gothe",
"submitter": "Gleb Fedotov V",
"url": "https://arxiv.org/abs/1108.4922"
}
|
1108.4936
|
# The Tension on dsDNA Bound to ssDNA-RecA Filaments May Play an Important
Role in Driving Efficient and Accurate Homology Recognition and Strand
Exchange
Julea Vlassakis Efraim Feinstein Darren Yang Antoine Tilloy Dominic
Weiller Julian Kates-Harbeck Vincent Coljee Mara Prentiss Harvard
University, Department of Physics, Cambridge, MA, 02138
prentiss@fas.harvard.edu
###### Abstract
It is well known that during homology recognition and strand exchange the
double stranded DNA (dsDNA) in DNA/RecA filaments is highly extended, but the
functional role of the extension has been unclear. We present an analytical
model that calculates the distribution of tension in the extended dsDNA during
strand exchange. The model suggests that the binding of additional dsDNA base
pairs to the DNA/RecA filament alters the tension in dsDNA that was already
bound to the filament, resulting in a non-linear increase in the mechanical
energy as a function of the number of bound base pairs. This collective
mechanical response may promote homology stringency and underlie unexplained
experimental results.
###### pacs:
87.15.ad, 87.15.La
## I Introduction
Sexual reproduction and DNA damage repair often include homologous
recombination facilitated by RecA family proteins Roca and Cox (1990);
Kowalczykowski and Eggleston (1994). In homologous recombination, a single
stranded DNA molecule (ssDNA) locates and pairs with a sequence matched
double-stranded DNA molecule (dsDNA). In the first step of the process, the
incoming ssDNA binds to site I in RecA monomers, resulting in a helical ssDNA-
RecA filament with 3 base pairs/monomer and $\sim 6$ monomers/helical turn
Chen et al. (2008). This helical ssDNA-RecA filament then searches dsDNA
molecules for homologous sequences by rapidly binding and unbinding dsDNA to
site II in RecA Chen et al. (2008). Thus, the sequence of the ssDNA in the
searching filament is fixed, and the system then searches through the
available dsDNA to find a sequence match for that ssDNA. The binding of dsDNA
to site II is very unstable, so if the dsDNA is not homologous to the ssDNA
bound to site I, the dsDNA rapidly unbinds from the ssDNA-RecA filament. If
the dsDNA is homologous, strand exchange should occur, probably via base-
flipping that transfers the Watson-Crick pairing of the complementary strand
from the outgoing DNA strand bound to site II to the incoming DNA strand bound
to site I Folta-Stogniew et al. (2004). This strand exchange reduces the
unbinding rate for the dsDNA Xiao et al. (2006). RecA is an ATPase, but in
vitro homology recognition and strand exchange can occur without ATP
hydrolysis Mazin and Kowalczykowski (1996); Menetski et al. (1990); Rosselli
and Stasiak (1990). Thus, each step in the homology search/strand exchange
process is fully reversible.
During the homology search and strand exchange process, dsDNA bound to RecA is
extended significantly beyond the B-form length Stasiak et al. (1981). Recent
theoretical work proposed that the free energy penalty associated with
extension may promote rapid unbinding of non-homologous sequences, but the
free energy penalty was assumed to be a linear function of the number of bound
triplets and the kinetic trapping due to near homologs was not considered
Savir and Tlusty (2010).Earlier work had also suggested that the dsDNA
extension promotes base-flipping Mazin and Kowalczykowski (1999) and reduces
kinetic trapping since the lattice mismatch between extended dsDNA and B-form
dsDNA presents a steric barrier to interactions between unbound dsDNA and
bound ssDNA which implies that the dsDNA must bind to the filament in order to
interact with the ssDNA Klapstein et al. (2004). These studies assumed the
dsDNA in the DNA/RecA filament is uniformly extended; however, the X-ray
crystal structures of the dsDNA in the final post-strand exchange state and
the ssDNA in the homology searching state both consist of base pair triplets
in a nearly B-form conformation separated by large rises as illustrated in
Figure 1a. The rises occur at the interfaces between adjoining RecA monomers
Chen et al. (2008), as illustrated in Fig. 2. The functional role of the non-
uniform extension has been unclear.
In this paper we present a simple model that calculates the extension of each
base pair triplet in a dsDNA. Using this model, we calculate the free energy
changes associated with progression through the homology recognition/strand
exchange process. The results of that calculation suggest a resolution to the
long standing question of why strand exchange is free energetically favorable
even though the Watson-Crick pairing in the initial and final states is the
same and the DNA/protein contacts in the ssDNA-RecA filament and final post-
strand exchange state are nearly the same.Chen et al. (2008) The model also
makes several significant qualitative predictions, the most significant being
the suggestion that the collective behavior of the triplets due to their
attachment to the phosphate backbones leads to a free energy that is a non-
linear function of the number of consecutive bound triplets. As a result of
this non-linearity, total binding energy has a minimum as a function of the
number bound triplets in a given conformation. After that minimum is reached,
adding more triplets that given conformation becomes free energetically
unfavorable.
Such a change in sign in the binding energy as a function of the number of
bound triplets can never occur in a theory where the energy is a purely linear
function of the number of bound triplets since the binding of any base pair
anywhere in the system is equally likely regardless of the state of any of the
other triplets in the system. In a system with a binding energy that is a
linear function of the number of correctly paired bound triplets, if
homologous triplets can initially bind to the system, then additional
homologous triplets will always continue to bind. Thus, binding will readily
progress across a non-homologous triplet. As we will discuss in detail in this
work, in a system with a linear energy and more than $\sim 4$ binding sites,
either homologs will be too unstable or near homologs will be too stable.
In contrast, the non-linearity may provide more rapid and accurate homology
recognition than is available for systems using linear energies because the
non-linearity requires that dsDNA binding to the ssDNA-RecA proceed
iteratively triplet by triplet through a series of checkpoints which inhibit
the progression of strand exchange past a non-homologous triplet. At each
checkpoint, the progression of strand exchange to more stable binding
conformations is only free energetically favorable if a sufficient number of
contiguous homologous base pair triplets are bound to the ssDNA/RecA filament
in the appropriate conformations. The general qualitative features of the
homology recognition based on the non-linear energy follow from basic
properties of the simple model and are insensitive to the parameters chosen.
These features include the following: 1. that homology recognition will
proceed iteratively through consecutive triplets 2. that strand exchange
reversal is much more favorable at the ends of the filament than at the center
3. that there will be two checkpoints that cannot be passed unless the bound
dsDNA contains a sufficient number of contiguous homologous base pairs in the
appropriate conformations. Though the general features of the model are very
robust, the exact number of contiguous homologous bp required to progress past
a particular checkpoint depends strongly on the choice of model parameters.
Analytical modeling and numerical simulations suggest that there are a small
range of parameters that allow the free energies predicted by the model
presented in this paper to provide homology recognition which is both fast and
accurate Kates-Harbeck et al. ; Feinstein and Prentiss .
Simulation results suggest that though the initial binding of $\sim 9$ base
pairs (bp) is free energetically accessible, adding more bound triplets is not
favorable, and adding more than 15 bp is enormously unlikely unless the first
checkpoint is passed. Feinstein and Prentiss The first major checkpoint
requires that $\sim 9$ of the $\sim 15$ bp that initially bind to the filament
are contiguous and homologous. If the initial $\sim 15$ bp do not contain
$\sim 9$ contiguous homologous base pairs, the dsDNA cannot make a transition
to the more stably bound intermediate state; therefore, the weakly bound dsDNA
will almost immediately unbind from the filament. This checkpoint rapidly
rejects all but 50 of the $\sim$ 10,000,000 possible binding positions in a
bacterial genome. If the initial $\sim 15$ bp do include $\sim 9$ contiguous
homologous base pairs, the system can make a transition to a metastable
intermediate state, which allows more base pairs to be added to the filament.
The second major checkpoint occurs when $\sim 18$ contiguous bp are bound to
the filament in the metastable intermediate state. If all of the bp are
contiguous and homologous, the system can make a transition to the final post-
strand exchange state. Otherwise, the long regions of accidental homology will
slowly reverse strand exchange and unbind. Given the statistics of bacterial
genomes, passing the homology requirement for the second checkpoint would
guarantee that the correct match had been found. These predictions are in good
agreement with known experimental results that measure the stability of strand
exchange products as a function of the number of contiguous bound base pairs
Hsieh et al. (1992).
In this work, we will not attempt to optimize the parameters of the model in
order to provide rapid and accurate homology recognition. Rather, we will
consider why homology recognition systems in which the energies are a linear
function of the number of bound base pairs can either provide rapid unbinding
of non-homologs or stable binding of complete homologs, but not both if the
number of binding sites in the system is $>\sim 4$ . We will then discuss how
qualitative features of the non-linear free energy predicted by the model
allow strand exchange to avoid kinetic trapping in near mismatches while also
permitting homologs to progress completely to strand exchange in systems where
the number of binding sites is $>4$.
### I.1 General Issues in Self-Assembly Based on the Pairing of Arrays of
Matching Binding Sites
In efficient self-assembly/recognition systems that create correct assembly by
matching linear arrays of binding sites, correctly paired arrays of binding
sites must remain stably bound, whereas incorrectly paired arrays must rapidly
unbind even if the incorrect pairing contains only one single mismatched
binding site. For a system in thermodynamic equilibrium, the populations in
different binding configurations are determined simply by their binding
energies. Thus, in a system where every array consists of $N$ binding sites if
$U(m,N)$ is the binding energy when $m$ of the binding sites are correctly
paired, accurate recognition requires that $\exp[-(U(m,N)-U(N,N))/(kT)]\ll
1\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ m<N$, where $k$ is Boltzmann’s constant and $T$ is
the Kelvin temperature. Furthermore, in a system with a temperature $T$, the
requirement that mismatched pairings rapidly unbind implies that
$U(m,N)>-kT\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ m<N$, whereas if the correctly bound ones are to
remain stably bound $U(N,N)$ must be $\ll-kT$. If the energy is a linear
function of the number bound and a mismatch contributes zero free energy then
$U(N,N)=\leavevmode\nobreak\ -\leavevmode\nobreak\ N\epsilon kT$ and
$U(N-1,N)=\leavevmode\nobreak\ -\leavevmode\nobreak\ (N-1)\epsilon kT$, so the
condition $\exp[-(U(N-1,N)-U(N,N))/(kT)]\ll 1$ requires $\epsilon\gg 1$, which
also implies that the homolog will remain stably bound. For simplicity
consider $\epsilon=3$ , which implies $U(N,N)=-N\epsilon kT=-3NkT$ .
Substitution into the requirement for the unbinding of near homolog requires
that and $U(N-1,N)=-(3N-3)kT>-kT\leavevmode\nobreak\ \implies N<4/3$ Thus, the
requirements for rapid and accurate recognition can only be met if $N=1$. The
requirements become more stringent if the specificity ratio is more strict
than 1/20. As we will discuss below, accurate homology recognition in a
bacterial genome requires accurate recognition over a length of more than 12
bp, which implies $N>4$ since 12 base pairs is 4 triplets.
Some of the problems with realizing accurate recognition in systems at
thermodynamic equilibrium were recognized by John Hopfield in the 1970’s,
inducing him to propose a kinetic proofreading system that requires an
irreversible process. Hopfield (1974) In such systems, the energy of the bound
state can be very deep without making the energy of the searching state deep
because of the irreversible step that transfers the system from the searching
state to the bound state. Even in Hopfield’s system there is a tradeoff
between search speed and accuracy since greater sequence discrimination
requires greater unbinding probabilities for homologs. The increased binding
probability for homologs increases the searching time because the correct
binding site must be revisited many times before the homolog makes the
irreversible transition to the bound state. Earlier work had proposed that
RecA based homology recognition could proceed via kinetic proofreading, Xiao
et al. (2006), but homology recognition in vitro is known to proceed without
an irreversible step. Mazin and Kowalczykowski (1996); Menetski et al. (1990);
Rosselli and Stasiak (1990) In this work, we will consider the non-linearities
in the free energy as a function of the number of contiguous bound base pairs
that arise as a function of the differential extension of dsDNA bound to RecA
and how those non-linearities can promote rapid and accurate homology
recognition without requiring irreversibility by making transitions between
successive bound states contingent on the state and relative position of the
bound base pairs.
### I.2 Homology Recognition and Strand Exchange are Likely to Proceed in
Units of Base pair Triplets
Previous experimental results suggest that homology recognition occurs via
base flipping of the complementary strand bases between their initial pairing
with the outgoing strand and their final pairing with incoming strand. Xiao et
al. (2006); Bazemore et al. (1997); Peacock-Villada et al. (2012).
Experimental results have already shown that strand exchange does progress in
triplets. Ragunathan et al. (2011) We propose that within each triplet the
nearly B-form structure preserves sufficient stacking to allow homology
discrimination to exploit the energy difference between the Watson-Crick
pairing of homologs and heterologs, while the large rises between triplets
result in mechanical stress that plays several important functional roles in
homology recognition and strand exchange. Furthermore, if the stacking makes
it free energetically favorable for the triplets to flip as a group, the lost
of Watson-Crick pairing due to a single mismatch in a triplet may make strand
exchange unfavorable for the entire triplet, as would be the case if Watson-
Crick pairing alone determined the free energy of the strand exchanged state.
If a single mismatch makes strand exchange of a triplet unfavorable, then
testing in triplets implies that in a random sequence the possibility of an
accidental mismatch is 1/64, whereas if the search were done by comparing
single base pairs the probability of an accidental match would be 1/4.
Homology stringency and searching speed would be greatly reduced if the search
were not done in triplets. Once a sufficient number of base pair triplets have
undergone strand exchange, the complementary strand backbone relocates to the
position shown in the X-ray structure of the final state Xiao et al. (2006);
Peacock-Villada et al. (2012), as illustrated in Figures 1c and 2.
### I.3 General Model for dsDNA Bound to a RecA Filament
Certain underlying assumptions and qualitative features are important to the
model proposed in this paper: 1. that in the absence of hydrolysis the
extension of the protein filament is unaffected by dsDNA binding Karplus 2\.
that homology recognition and strand exchange occur in quantized units of base
pair triplets Chen et al. (2008); Ragunathan et al. (2011) 3\. that the
incoming and outgoing strands consist of nearly B-form triplets separated by
large rises where the strand is bound to the filament due to strong charged
interactions between the backbone phosphates and positive residues in the
protein Chen et al. (2008); Danilowicz et al. (2011); Peacock-Villada et al.
(2012) 4\. that the complementary strand is bound to the filament dominantly
via Watson-Crick pairing, resulting in large stress on the base pairs unless
the dsDNA is in the final post-strand exchanges state where the L1 and L2
loops provide significant mechanical support. Chen et al. (2008); van der
Heijden et al. (2008)
The model utilizes work presented by deGennes that calculated the force
required to shear dsDNA Gennes (2001). We extend deGennes’ model to the
triplet structures in the initial, intermediate, and final dsDNA
conformations. In the model the actual three dimensional helical structure is
converted into a one dimensional system. In the simple one dimensional model,
$L_{R}$ is rise between the triplets in the incoming and outgoing strands and
a single variable, $\gamma$, characterizes the equilibrium spacing between
phosphates when the complementary strand is in a particular state. Thus, for a
particular state, the difference between the equilibrium spacing is given by
$(1-\gamma)L_{R}$ .
### I.4 Predicted Extension and Energy
As shown in Fig. 3, the extensions of rises between base pair triplets in a
strand bound directly to the protein are given by $v_{N,i}$ and for a system
with $N$ triplets bound RecA,
$v_{N,i}=iL_{R}.$ (1)
The $u_{N,i}$ specify the extensions of the rises in the complementary strand.
At equilibrium, the net force on each $u_{N,i}$ must be zero; therefore, for
$j=2$ to $N$
$Q((u_{N,j+1}-u{N,j}-\gamma L_{R})-(u_{N,j}-u_{N,j-1}-\gamma
L_{R}))+R(u_{N,j}-v_{N,j})=0,$ (2)
where $R$ and $Q$ are the spring constants for the base pairs and the
backbones, respectively. These values for $R$ and $Q$ may be substantially
different from those for individual dsDNA base pairs when the dsDNA is not
bound to RecA because of the interactions between the charged phosphates and
the protein and because dsDNA is grouped in triplets where the stacking
between the triplets is strongly disrupted; however, it is still likely that
$R\ll Q$ as it is in naked dsDNA since the interactions between the bases on
opposite strand is significantly weaker than the interaction between the
phosophates in the backbone of the same strand.
The boundary condition on the last triplet $u_{N,1}$ requires that
$+Q(u_{N,2}-u_{N,1}-\gamma L_{R})+R(u_{N,1}-v_{N,1})=0$ (3)
The values of $u_{N,i}$ can be found using Equations 1-3. The angle between
base pairs and the DNA helical axis is shown as $\theta_{bp}$ in Fig. 3.
In the continuous limit where the discrete subscript $i$ is replaced by a
continuous variable $x$, the equations have an analytical solution
$v_{N,x}=x+A_{N}(x)\sinh(\chi x/L_{R})$ (4)
where $\chi=Sqrt[R/(2Q)]$ is the deGennes length for RecA bound dsDNA and the
constant $A$ is found by using the boundary conditions for the ends, yielding
$A_{N}(x)\Big{[}R\sinh(\chi x/L_{R})+Q\chi\cosh(\chi
x/L_{R})\Big{]}=Q(\gamma-1)L_{R}.$ (5)
In the limit where $1/(2\chi)\gg 1$
$A_{N}(x)={(\gamma-1)L_{R}\over{\cosh(\chi N/2)}}.$ (6)
These assumptions and features lead to a nuanced picture of the distribution
of tension during strand exchange. The lattice mismatch between the
complementary strand and its pairing partners is largest at the ends of the
filament as shown in Fig. 4; therefore, the base pair tension is largest at
the end of the filament. Furthermore, the lattice mismatch at the ends
increases significantly with the number of bound triplets as shown in Fig. 4
and 5. The angle $\theta_{bp}$ is greatest at the ends of the molecule. When
fewer than $\sim 30$ bp are bound to the filament, the tension on the base
pairs at the ends increases rapidly as more RecA bound triplets are added, as
shown in Fig. 4 and 5; however, $\theta_{bp}$ and the tension on the base
pairs near the center decreases as more triplets are added, as shown in Fig. 4
and 5. In contrast with the tension on the base pairs, which is largest at the
ends of the complementary strand, the tension on the rises in the
complementary strand is largest at the center and smallest at the ends. These
general qualitative features are not sensitive to $R$, $Q$ or $\gamma$ as long
as $R\ll Q$.
Given the values of $u_{N,i}$ the mechanical energy of the system, which is of
the form $1/2k(x-x_{0})^{2}$, may be calculated from:
$E_{mech}(N)=[\sum\limits_{i=1}^{N}\frac{1}{2}R(u_{N,i}-v_{N,i})^{2}+\sum\limits_{i=2}^{N}\frac{1}{2}Q(u_{N,i}-u_{N,i-1}-\gamma
L_{R})^{2}]$ (7)
When $R\ll Q$ this expression simplifies to:
$E_{mech}(2)=2[\frac{1}{2}R\frac{(L_{R}(1-\gamma)}{2}^{2}]$ (8)
$E_{mech}(3)=2[\frac{1}{2}R(L_{R}(1-\gamma)^{2}]$ (9)
These energy terms represent the stress on base pairs at the ends of the
molecule due to the lattice mismatch. When $R\ll Q$, for moderate numbers of
bound base pairs, this energy is stored primarily in the extended base pairs.
This idea is particularly important because it implies that an increase in the
extension of the rises in the complementary strand can reduce the free energy
by reducing the tension on the base pairs even if the increase in extension of
the rises requires some energy. Similarly, in the RecA structure the
transition from the intermediate state to the final state may reduce the
tension on the base pairs because the interactions between the base pairs and
the L1 and L2 loops may increase the equilibrium extension of the rises in the
complementary strand.
Using the continuous limit allows us to generate scaling laws for the
extension as a function of the total number of bound base pairs. In the
continuous limit, the non-linear contribution to the free energy is given by
$E_{non-
linear}(N)=\frac{1}{2}\frac{(1-\gamma)^{2}L_{R}^{2}}{\chi^{2}\cosh^{2}[\chi
N/2]}\sum_{i=-N/2}^{i=N/2}\sinh^{2}[\chi i].$ (10)
Thus, when $\chi\ll 1$ the non linear energy term has the following scaling
$E_{non-linear}[N]\alpha(1-\gamma)^{2}L_{R}^{2}\frac{\sinh(\chi N)-\chi
N}{1+\cosh^{2}[\chi N/2]}.$ (11)
In the limit where the number of base pairs bound is much less than the
deGennes length so $\chi N\ll 1$ the energy the non-linear energy scaling is
$E_{non-linear}[N]\alpha(1-\gamma)^{2}L_{R}^{2}\chi^{2}N^{2}.$ (12)
Thus, when $N$ is small the energy increases as the square of the number of
bound triplets, consistent with the exact results for the discrete case given
in equations 7,8 and 9. In contrast, when $N$ is larger than the deGennes
length, the non-linear energy term approaches zero and the mechanical energy
increases linearly with increasing $N$. In this limit, the base pairs at the
center of the filament are no longer under tension. Thus, adding a triplet to
the end of the filament effectively adds another triplet to the unstressed
center rather than increasing the stress on all of the bound triplets.
The total energy of the system includes the mechanical energy calculated above
and the non-mechanical binding energy per RecA monomer, $E_{bind}$. Assuming
the free energy of an unbound dsDNA is zero and the free energy gain upon
binding a triplet is independent of $N$ and $i$ then the non-mechanical
contribution to the binding energy for $N$ triplets is $NE_{bind}$. When the
first RecA monomer binds, $E_{total}[1]=E_{bind}$, which is a constant
negative value. In contrast, when $N>1$, the stress on the molecule yields a
total energy of
$E_{total}[N]=E_{mech}[N]+NE_{bind}$ (13)
which changes in sign and magnitude depending on $L_{R}$, $R$, $Q$ and
$\gamma$.
### I.5 Modeling the Homology Search and Strand Exchange Process
Recent experimental results suggest that the homology recognition/strand
exchange process uses four major dsDNA conformations: 1. B-form dsDNA, which
is the structure of the dsDNA when it is not bound to the protein 2. an
initial sequence independent searching state with the dsDNA bound to the RecA
where the complementary strand bases paired with outgoing strand 3. an
intermediate sequence dependent strand exchanged state in which the
complementary strand bases are paired with the incoming strand where the
complementary strand backbone is in a position where its bases can flip
between pairing with the incoming and complementary strands 4. the final state
known from the X-ray structure where the complementary strand bases are paired
with the incoming strand bases and the phosphate backbone is in the position
shown in the x-Ray structure. Peacock-Villada et al. (2012)
Consistent with strand exchange proceeding through base flipping of triplets,
recent experimental results have suggested that outgoing strand bases are
arranged in B-form triplets separated by large rises such that the
complementary strand bases can readily rotate between pairing with the
outgoing strand and pairing with the incoming strand. Danilowicz et al.
(2011). The crystal structure shows that the incoming strand is located near
the center of the helical DNA/protein structure whereas the residues
associated with the binding of the outgoing strand are much farther away from
the center, as illustrated in Fig. 1; however, if strand exchange occurs via
the base flipping of base pair triplets, the spacing within triplets must be
approximately the same for all three strands. Thus, given that the total
extension of the outgoing strand backbone is be much larger than the total
extension of the incoming strand backbone, the rises in the outgoing strand
must be much larger than the rises in the incoming strand, as illustrated in
Fig. 2.
In the one dimensional model considered here where $\gamma$ is the only
parameter characterizing each dsDNA conformation, $\gamma$ is smallest for the
initial searching state, where the complementary strand is paired with the
very highly extended outgoing strand, consistent with experimental results
that the dsDNA in the initial bound state has a large differential extension
between the outgoing and complementary strands that prevents more base pairs
from binding to the filament unless the dsDNA undergoes strand exchange.
Danilowicz et al. (2011) The $\gamma$ for the intermediate state is slightly
larger because the complementary strand is paired with the less extended
incoming strand. This value is consistent with experimental results that
suggest that the strand exchange of homologous triplets is favorable because
it reduces the differential tension on the dsDNA. Peacock-Villada et al.
(2012) The $\gamma$ for the final state is almost 1, because of the support
provided for the rises by the L1 and L2 loops of the protein that occupy the
rises in the complementary strand, consistent with the known experimental
result that the binding of at least $\sim 80$ bp in the final state is free
energetically favorable. van der Heijden et al. (2008) In order to probe the
role of tension during homology recognition strand exchange we apply the model
to the various stages of strand exchange using the following parameters:
$\gamma={0.4,0.75,0.8,0.9875}$ and $E_{bind}$ equal to $0,-0.75kT$, $-0.8kT$
and $-0.125kT$ per homologous triplet in the unbound, initial bound,
intermediate and final states, respectively. These $\gamma$ values are
inspired by X-ray structure DNA RecA filaments, the known properties of B-form
dsDNA, experimental results on the stability of strand exchange products Hsieh
et al. (1992), and results of numerical simulations that optimize homology
recognition. Feinstein and Prentiss
## II Results and Discussion
### II.1 Changes in Mechanical Tension Allows Strand Exchange to be Free
Energetically Favorable if and only if 6 Contiguous Homologous bp Undergo
Strand Exchange Insensitive to Model Parameters
Early RecA recognition systems assumed that strand exchange is always free
energetically favorable, just as Watson-Crick pairing of unpaired ssDNA is
always favorable, but this assumption is not correct. In self-assembly based
on ssDNA/ssDNA pairing, the pairing of matched bases is free energetically
favorable and the pairing of mismatched bases is approximately neutral. Thus,
in ssDNA/ssDNA pairing systems, such as DNA origami, assembly of matching
sites is always free energetically favorable because the correct Watson-Crick
pairing reduces the free energy below that for the system where no bases were
bound. In contrast, if only Watson-Crick pairing is considered, self-assembly
based on strand exchange is free energetically neutral for matched bases and
free energetically unfavorable for mismatched bases because the system begins
in a state with all of the bases in the complementary strand correctly paired
with the corresponding bases in the outgoing strand. Thus, previous models of
strand exchange that considered only the Watson-Crick pairing were faced with
a paradox: at thermal equilibrium strand exchange is only free energetically
neutral, so at best only $50\%$ of correct pairings would end up in the strand
exchanged state. The other $50\%$ would end up paired with their initial
partners; however, in vivo strand exchange proceeds to completion. Including
protein contacts did not solve the problem since the protein contacts in the
initial ssDNA-RecA filament are almost identical to the contacts in the final
post-strand exchange state Chen et al. (2008); however, the model presented in
this paper resolves the problem, as discussed below.
In the model considered here, the coupling along the dsDNA backbone makes any
change in the dsDNA conformation (such as adding or base-flipping a triplet)
alter the positions and extensions of all other bound triplets. This effect is
illustrated in Fig. 4, which shows the rises and extensions of the base pairs
calculated from the model when an additional triplet binds in site II of the
protein when all of the bound triplets are bound in site II. The triplet
highlighted with the arrow experiences increasing tension as triplets are
added to the end of the sequence. Furthermore, as shown in Fig. 5, the lattice
mismatch at the ends of the filament increases monotonically as more triplets
are added. In contrast, the lattice mismatch for the first triplet out from
the center of the sequence decreases as base pairs are added (solid black
line) because the added triplets extend the inner rises. As the number of
bound triplets approaches the deGennes length for RecA bound dsDNA, the
lattice mismatch at the ends of the filament approaches a constant and the
central rises in the complementary strand approach $L_{R}$.
The mechanical stress model considered here suggests strand exchange is free
energetically favorable if and only if strand exchange transfers a rise
because the reduction in free energy is the result of the reduction in
mechanical stress due to a reduction in the stress on the base pairs due to
decrease in the lattice mismatch between the complementary strand and its
Watson-Crick pairing partner. Thus, when the dsDNA first binds to the
searching filament and all of the triplets are in the initial state, base
flipping of a single triplet does not lower the energy of the system since a
rise is not transferred even if that triplet is perfectly homologous; however,
once one homologous triplet is strand exchanged, it becomes favorable to
transfer a contiguous homologous triplet. Thus, the transfer of the Watson-
Crick pairing of the complementary strand from the outgoing strand to the
incoming strand is not free energetically favorable except in sequence regions
containing at least six base pairs of contiguous homology. This minimum number
of contiguous homologous bases required to pass the first checkpoint is a
basic qualitative feature of the model that is extremely insensitive to the
model parameters chosen. This first checkpoint can provide very rapid
unbinding of almost all initial dsDNA pairings, which is highly advantageous
for rapid searching. Though the model specifies that 6 contiguous bases is the
minimum number required to pass the checkpoint insensitive to the parameters
used in the model,the actual number of base pairs required to pass the first
checkpoint is sensitive to parameters.
Experimental results suggest that strand exchange is only marginally stable
when $\sim 9$ contiguous homologous bp undergo strand exchange,Hsieh et al.
(1992) suggesting that rapid unbinding will occur for all tested sequences
except for $1/4^{9}\sim 2\times 10^{-6}$ which represents only $\sim$ 50
possible positions for a bacterial genome with a length of 10,000,000 bp. All
other sequences will rapidly unbind from the RecA filament because the binding
energy for sequence independent searching state is very weak when only $\sim
9$ bp are bound and adding more base pairs to the searching state would
increase rather than decrease the binding energy.
### II.2 Calculations of the Free Energy as Function of the Number of Bound
Triplets in Each dsDNA Conformation
In order to understand the progression through homology recognition and strand
exchange, it is important to consider the free energies for all of the dsDNA
conformations involved, not just the initial bound state and the intermediate
state. Of course the process is kinetic, so the energies of the transition
states play a vital role; however, for simplicity we will only show the
energies of the various bound conformations using $\gamma$ values based on
numerical simulations. Feinstein and Prentiss Fig. 6 shows $E_{total}$ as a
function of number of base pairs in the initial bound state, the intermediate
state and the final post-strand exchange state when all of the triplets are in
the same conformation. Except for the initial binding of $\sim 9$ to 15 bp,
all of the triplets are rarely in the same conformation. Thus, the free energy
curves for states with all of the triplets same conformation rarely represent
the free energy of the system; however, it is clear that reductions in the
mechanical energy of the bound dsDNA can drive strand exchange for homologs,
as we discuss below.
### II.3 Non-linearities in the Free Energy Allows Homologs to Progress to
Complete Strand exchange
While it is energetically favorable to add base pairs to the initial bound
state for small numbers of base pairs, the quadratic term of $E_{total}$ from
Equation 7 rapidly increases as a function of increasing number of base pairs,
making the binding of a large number of base pairs in the initial state
unfavorable. This is because of the significant tension due to the lattice
mismatch between the complementary strand and the outgoing strand;
consequently, for the parameters based on simulation, no more than $\sim 15$
bp can bind to the filament in the initial searching state because the energy
required greatly exceeds $kT$. Thus, once $\sim$ 15 bp are bound to the
filament, the system is in a highly free energetically unfavorable state which
will force it to choose between the following: 1. unbinding from the filament
2. strand exchange, which is unfavorable for non-homologs. Again, the
prediction that there will be an checkpoint in the progression of strand
exchange from the initial bound state to the intermediate state is insensitive
to the model parameters. The parameters only determine whether more than 6
contiguous homologous base pairs are required to progress past the checkpoint.
Homologs can rapidly progress to complete strand exchange if the weak initial
binding holds long enough for $\sim 9$ homologous contiguous base pairs to
undergo strand exchange, which stabilizes the binding for those homologous bp
and allows strand exchange to progress. Kates-Harbeck et al. ; Feinstein and
Prentiss The non-linearity in the free energy makes the strand exchange of
consecutive homologous triplets increasingly favorable as long as the number
of bound base pairs is $<\sim$ 30; consequently, the non-linearity in the free
energy makes strand exchange reversal more improbable as a more contiguous
homologous base pairs are strand exchanged. Furthermore, the non-linearity
makes strand exchange at the center of the filament increasingly unfavorable
as the number of bound triplets increases as shown in Fig. 8, while still
allowing strand exchange reversal to remain possible at the ends of the
filament. Again, these qualitative features are basic properties of the model
that are highly insensitive to the model parameters. These qualitative
features allow true homologs to progress to complete strand exchange even
though non-homologs readily unbind. In contrast, for a system with a linear
free the probability that strand exchange will be reversed for a given triplet
is independent of the number of other triplets bound and of the position of
the particular triplet in the filament. As a result, such systems either
suffer rapid unbinding of homologs or strong kinetic trapping in near
homologs, as discussed above for the general case of a system with a linear
binding energy and greater than 4 binding sites.
### II.4 dsDNA Tension Drives the Transition from the Intermediate State to
the Final State
The free energy penalty due to binding in the intermediate state is so large
that even a perfect homolog cannot bind more than $\sim 18$ bp in the
intermediate state. The only way to continue to add base pairs to the filament
is to make a transition to the final post-strand exchange state where the L1
and L2 loops provide significant mechanical support for the rises. This
transition reduces the tension sufficiently to allow more triplets to bind in
the initial bounds state. If the transition does not occur, strand exchange
will reverse because the intermediate state energy is unfavorable. Thus, the
collectivity in the behavior of the dsDNA can have a significant effect in
enforcing homology stringency at this final step in the strand exchange
process by enforcing the following: 1. that the number of base pairs that can
be bound in the intermediate state is limited 2. that transfer of a single
triplet from the intermediate state to the final state is never favorable 3.
that transferring non- contiguous triplets from the intermediate state to the
final state is unfavorable even if they are homologs 4. that the transfer of
pairs of triplets is not favorable until $\sim 18$ bp are bound in the
intermediate state. Properties 1-3 are shared with the transition from the
initial bound state to the intermediate state, but the fourth property is
different. It may arise from some combination of two features: 1. the linear
energy in the final state may be less favorable than the linear energy in the
intermediate state 2. there is a significant boundary penalty associated with
the deformation of the backbone that occurs when the system is partially in
the intermediate state and partially in the final state. In either case, then
the transition from the intermediate state to the final state will only become
free energetically favorable when the favorable non-linear term becomes
dominant over the other terms. If, the final state only becomes free
energetically favorable when $\sim 15$ contiguous base pairs are present in
the final state, then the non-linearity in the energies of the intermediate
and final states in the energy may provide additional discrimination against
regions of accidental homology since for a particular given searching ssDNA
sequence the odds of this occurring with a given searching sequence are $\sim
1.4x10^{-11}$/base pair. Thus, even in a 10 million bp genome, the probability
that such an accidental homology is present in a given bacterial genome is
$\sim 1/10^{4}$. We have considered a few bacterial genomes and found that the
sequences are indeed random in this sense, with the exception of repeated
genes. Thus, many bacterial genomes contain no accidental mismatches
consisting of 18 contiguous bp; therefore, no further homology checking is
required if 18 contiguous bp exactly match. The important statistical property
is consistent with experimental results that strand exchange products to do
not become stable until more than $\sim 18$ bp have undergone strand exchange.
### II.5 For dsDNA in the final state the free energy has a minimum as a
function of $N$
If the final state has a much higher $\gamma$ than the intermediate state,
then adding base pairs will remain free energetically favorable for much
larger $N$; however, as long as the mechanical contribution remains a non-
linear function of $N$, the free energy as a function of $N$ will achieve a
minimum for some $N$, after which adding more base pairs becomes free
energetically unfavorable. Eventually, as $N$ approaches the deGennes length,
the mechanical energy will become a linear function of $N$. In this case, the
energy cost of adding an additional triplet remains constant as a function of
$N$. If this cost is small in comparison with $kT$ and unbinding is forbidden,
the model suggests that the length of the strand exchange product can increase
without limit.
### II.6 The Effect of the Non-Linearity on the Strand Exchange of a
Mismatched Triplet
It has previously been assumed that the free energy penalty for strand
exchange of a triplet is approximately equal to the loss of Watson-Crick
pairing for that triplet, with a possible additional factor due to the effect
of the mismatch on the pairing of the two neighboring bases which ranges from
$\sim 1.5$ to $\sim 4kT$ . SantaLucia, Jr. (1998) In contrast, for a system
with the non-linearity considered here if the initially bound base pairs
contain a single mismatch, then strand exchange may be significantly more
unfavorable because the unfavorable free energy contribution due to this
mismatch must include not only the Watson-Crick pairing energy for that base
pair and its neighbors, but also the the increased mechanical stress on the
two matched base pairs. This stress not only makes a direct contribution to
the free energy penalty, but it can also increase the stacking penalty by
distorting the bonds between the two homologous base pairs which lowers their
Watson-Crick pairing energy. A detailed structural calculation would be
required to correctly assess all of these factors. In what follows, we will
assume that the free energy penalty for the strand exchange of a mismatched
base is approximately equal to the Watson-Crick pairing loss as long as the
number of $<18$ bp are bound to the filament.
### II.7 dsDNA Tension Inhibits Progression of Strand Exchange Past a
Mismatch
In a system with a linear free energy as a function of the number of bound
homologous triplets, adding more homologous triplets is always favorable even
if the last triplet added were non-homologous, resulting in enormous kinetic
trapping. In contrast, the non-linear energy inhibits binding of additional
base pairs after a non-homologous triplet has bound, as illustrated in Fig. 7.
The dashed black line shows the curve for a perfect homolog adding a triplet
to the initial bound state if all of the other bound triplets have undergone
strand exchange. For up to $\sim 18$ bp thermal energy is sufficient to bind
additional base pairs. The dashed gray line shows the free energy penalty for
adding a homologous triplet if the last triplet added was non-homologous. The
free energy penalty is only slightly larger than the penalty for a homolog;
however, the solid gray line shows that the penalty for adding a second
triplet is very large, even though both triplets added after the non-homolog
were in fact homologous. For comparison, the solid black line shows the
energetic favorability of strand exchange of a homologous triplet from the
initial binding state. This graph shows that the non-linearity makes adding
additional triplets to the initial state is unfavorable once a mismatched
triplet has bound, even when the subsequent base pairs are homologous.
### II.8 Possible Explanations of Biological Results
We have already discussed the proposal that the energetic non-linearity
explains why strand exchange is free energetically favorable even though the
sequences of the incoming and outgoing strands are the same and the protein
contacts in the initial searching ssDNA-RecA filament are similar to those in
the final post-strand exchange state.
In addition, experimental results have shown that a rapid initial interaction
incorporating $\sim 15$ base pairs is followed by a slower progression of
strand exchange that occurs in triplets Ragunathan et al. (2011). Figure 6
suggests that the binding of dsDNA to site II is favorable for fewer than 9
bases and requires only a few $kT$ of energy for fewer than 15 bases, whereas
for more bases the binding is highly free energetically unfavorable.
Furthermore, FRET based studies indicate that homology recognition may be
accurate for short sequences, but inaccurate for longer sequences Bazemore et
al. (1997). A separate study showed that strand exchange pauses at sequence
mismatches Lee et al. (2006); D. Sagi and? T. Tlusty (2006), and we argue that
such pauses lead to the unbinding of shorter non-homologous sequences because
the binding of the dsDNA to the filament occurs sequentially In the model
presented here the pause in strand exchange at a mismatch results from the
free energy cost of transferring the non-homologous triplet to the
intermediate state as well as the cost of progressing past a mismatched
triplet. Spontaneous unbinding of the entire strand exchange product becomes
unlikely as the sequence lengthens because so many free energetically
unfavorable transitions are required. If the strand exchange product becomes
too long, the unbinding time exceeds the recognition time available to the
organism; however, as discussed above, accidental mismatches that extend
beyond 18 bp rarely exist in vivo. In vivo, strand exchange does progress
through regions of non-homology once a sufficiently long stand exchange
product is formed, but ATP hydrolysis is required. Rosselli and Stasiak
(1991); Kim et al. (1992)
Finally, it is also well known that in the presence of ATP hydrolysis the size
of the strand exchange product increases monotonically until it reaches a
limit of $M\sim 80$ bp van der Heijden et al. (2008), where $M$ is the number
of bound dsDNA base pairs. Strand exchange then continues to progress, but $M$
remains constant because the heteroduplex dsDNA unbinds from the lagging edge
of the filament at the same average rate that new dsDNA binds to site II van
der Heijden et al. (2008); Fan et al. (2011). Since the dsDNA can freely
unbind from the filament, free energy minimization implies $M$ will remain
$\sim M_{freemin}$. Additional effects associated with dynamics may explain
why the strand exchange window moves along the dsDNA with $M\sim M_{freemin}$
rather than remaining stationary Conover et al. (2011). In contrast with the
experimental results obtained in the presence of hydrolysis, experimental
results obtained in the absence of hydrolysis show that the length of the
strand exchange window can continue without bound.van der Heijden et al.
(2008) In the model presented here, if the number of base pairs bound is small
the mechanical energy penalty associated with adding base pairs is a quadratic
function of energy for small numbers already bound; however, the base pairs
redistribute the stress between the backbones, so that eventually the base
pairs in the center of the filament are not under stress. In this case, the
penalty for adding another base pair triplet to the end becomes a linear
function of the number bound rather than a quadratic function. Thus, when a
sufficient number of base pairs have undergone strand exchange, the energy
required to add an additional triplet to the filament is constant, independent
of the number bound. If, as suggested by this model, the constant energy
decrease due to additional DNA protein contacts is approximately equal to the
constant energy increase associated with the added mechanical stress, then the
filament can extend forever because both energies are independent of the
number of base pairs already bound.
### II.9 Additional Features in Three Dimensions
In the real RecA system steric factors are associated with the mismatch
between the 150 bp persistence length of dsDNA and the strong bending of dsDNA
in the 18 bp/turn helical RecA filament. The local rigidity of the dsDNA may
play a role in limiting the initial binding length to $\sim 9$ bp since that
many base pairs can interact with the ssDNA-RecA without significant bending.
After some dsDNA triplets are bound, the rigidity may also play a role in
preventing non-contiguous triplets from being added to the filament. The
nearest unbound triplet is already very near to the filament because it is
attached to the bound triplets by the phosphate backbones which cannot extend
much more than 0.5 nm/bp. Thus, the phosphates are in a position to interact
strongly with positively charged residues on the protein which can provide
sufficient free energy for the required bending. In contrast, the second
neighboring triplet will be separated by a larger distance which reduces the
interaction with the protein and requires more bending. A detailed structural
calculation would be required to correctly evaluate these effects, but both
effects would further support rapid and accurate homology recognition. In the
simple one dimensional model discussed here, the free energy effects of the
bending can be included in the $\gamma$ for the initial bound state, but the
additional degrees of freedom would alter the coupling between the initial
bound state $\gamma$ and the $\gamma$ for the intermediate state.
In addition, in the final post-strand exchange state interactions with the L1
and L2 loops may be more favorable for homologous triplets than non-homologous
triplets due to steric factors. Thus, the final state could have a sequence
dependent linear contribution to the free energy that was not considered in
this model, but may provide additional homology stringency.
## III Conclusion
We have proposed a simple mechanical model for the stress distribution on
dsDNA bound to ssDNA-RecA filaments. The model suggests that a change in the
conformation of one bound triplet can change the conformation of all of the
other bound triplets; consequently, the total energy is a non-linear function
of the number of bound base pairs. The model makes several significant
qualitative and quantitative predictions. The most important qualitative
prediction is that a change in the configuration of one bound triplet changes
configuration of all the other bound triplets. The collective behavior of the
triplets leads to a free energy that is a non-linear function of the number of
consecutive bound triplets, where the binding of additional triplets becomes
increasingly unfavorable as the number of bound triplets increases. This non-
linearity is important because in systems with more than $\sim 4$ binding
sites, neither thermodynamic equilibrium nor kinetic proofreading can combine
accurate and efficient homology recognition when the energy is a linear
function of the number of correct pairings. In contrast, an unfavorable non-
linear energy combined with the a favorable linear energy due to DNA/protein
contacts can promote rapid and accurate homology recognition by making initial
sequence independent binding interactions favorable for up to $\sim 9$ base
pairs, while preventing any additional base pairs from binding unless the
bound base pairs include 9 contiguous homologous base pairs. If the initially
bound base pairs do not contain 9 contiguous homologous base pairs, adding
more base pairs to the filament is highly improbable regardless of whether or
not the additional base pairs are homologous. This effect combined with the
statistics of the sequence distribution of bacterial genomes implies that of
all but the $\sim$ 50 out of 10,000,000 possible pairings will rapidly unbind.
In addition , the non-linearity forces the addition of triplets to the
filament to proceed sequentially from the initially binding, where adding more
than two base pair triplets after a mismatch is highly unlikely, even if the
additional base pairs are sequence matched. Furthermore, the model suggests
true homologs can proceed to complete strand exchange because the strand
exchange of contiguous homologous base pair triplets reduces the tension on
the dsDNA. The tension reduction associated with the strand exchange of the
initial $\sim 9$ contiguous homologous base pairs allows more base pairs to
bind to the filament due to two effects: 1. the reduction in dsDNA tension
reduces the free energy penalty for adding more triplets to the initial bound
state 2. the decrease in the energy of the bound dsDNA due to strand exchange
reduces the unbinding rate of the dsDNA by making unbinding more unfavorable.
As strand exchange progresses from the $\sim 9$ initial contiguous base pairs,
the binding of the strand exchanged state still remains weak enough to be
reversed unless $\sim 18$ contiguous bp make the transition to the final post-
strand exchange state, a transition which is not favorable for $<\sim 18$ bp.
These effects could provide exact sequence recognition for bacterial genomes
except for repeated genes. The actual speed and accuracy of homology
recognition depends on detailed values of parameters as well as additional
factors associated with the three dimensional geometry of the filament, so we
do not provide detailed estimates here; however, the simple model presented
here provides mechanisms for overcoming fundamental limitations encountered in
systems with more than 4 binding sites where the binding energy is a linear
function of the number of correctly paired binding sites.
In sum, the energy non-linearity produces three crucial advantages that are
unavailable to systems with a strictly linear energy: 1. the initial
interaction is limited to $\sim 15$ bp beyond which binding of dsDNA triplets
to the filament cannot progress without a sequence dependent transition of 9
contiguous homologous base pairs to the strand exchanged state 2. nearly
immediate unbinding of any sequence that does not contain at least 9
contiguous homologous base pairs 3. a large free energy penalty that prevents
strand exchange from progressing past a sequence mismatch even if the mismatch
is followed by homologous triplets 4. a large free energy penalty that makes
the transition from the intermediate state to the final state unfavorable
until $\sim 18$ contiguous base pairs make the transition to the final state.
These features provide much more rapid and accurate homology recognition than
systems using linear energies: in systems with linear energies addition and
strand exchange of a homologous triplet is always favorable; therefore, in
systems with linear energies even short regions of accidental homology can
produce substantial trapping times, as demonstrated by both analytical
modeling and numerical simulations. Feinstein and Prentiss Qualitative
features of the model provide possible explanations for well known but
previously unexplained features for homology recognition and strand exchange
and suggest that the bond rotations that appear in the overstretching of naked
dsDNA may have a role in strand exchange.
###### Acknowledgements.
We would like to thank Douglas Bishop, Yuen-Ling Chan, and Chantal Prévost for
helpful conversations about the X-ray structure of DNA bound to RecA. We would
also like to thank Chantal Prévost for the PDB file of a complete RecA helix.
## References
* Roca and Cox (1990) A. Roca and M. Cox, Mol. Biol. 25, 415 (1990).
* Kowalczykowski and Eggleston (1994) S. Kowalczykowski and A. Eggleston, Annu. Rev. Biochem. 63, 991 (1994).
* Chen et al. (2008) Z. Chen, H. Yang, and N. Pavletich, Nature 453, 489 (2008).
* Folta-Stogniew et al. (2004) E. Folta-Stogniew, S. O Malley, R. Gupta, K. Anderson, and C. Radding, Mol. Cell 15, 965 (2004).
* Xiao et al. (2006) J. Xiao, A. Lee, and S. Singleton, ChemBioChem 7, 1265 (2006).
* Mazin and Kowalczykowski (1996) A. Mazin and S. Kowalczykowski, Proc. Natl. Acad. Sci. USA 93, 10673 (1996).
* Menetski et al. (1990) J. Menetski, D. Bear, and S. Kowalczykowski, Proc. Natl. Acad. Sci. USA 87, 21 (1990).
* Rosselli and Stasiak (1990) W. Rosselli and A. Stasiak, J. Mol. Biol. 216, 335 (1990).
* Stasiak et al. (1981) A. Stasiak, E. D. Capua, and T. Koller, J. Mol. Biol. 151, 557 (1981).
* Savir and Tlusty (2010) Y. Savir and T. Tlusty, Mol. Cell 40, 388 (2010).
* Mazin and Kowalczykowski (1999) A. Mazin and S. Kowalczykowski, Genes Dev. 13, 2005 (1999).
* Klapstein et al. (2004) K. Klapstein, T. Chou, and R. Bruinsma, Biophys J. 87, 1466 (2004).
* (13) J. Kates-Harbeck, A. Tilloy, and M. Prentiss, to be published.
* (14) E. Feinstein and M. Prentiss, to be published.
* Hsieh et al. (1992) P. Hsieh, C. Camerini-Otero, and R. Camerini-Otero, Proc. Natl Acad. Sci. USA 89, 6492 (1992).
* Hopfield (1974) J. Hopfield, Proc. Natl. Acad. Sci. USA 71, 4135 (1974).
* Bazemore et al. (1997) L. Bazemore, E. Folta-Stogniew, M. Takahashi, and C. Radding, Proc. Natl. Acad. Sci. USA 94, 11863 (1997).
* Peacock-Villada et al. (2012) A. Peacock-Villada, D. Yang, C. Danilowicz, E. Feinstein, N. Pollack, S. McShan, V. Coljee, and M. Prentiss, Nucl. Acids Res. 40, 10441 (2012).
* Ragunathan et al. (2011) K. Ragunathan, C. Joo, and T. Ha, Structure 19, 1064 (2011).
* (20) M. Karplus, private communication.
* Danilowicz et al. (2011) C. Danilowicz, E. Feinstein, A. Conover, V. Coljee, J. Vlassakis, Y. Chan, D. Bishop, and M. Prentiss, Nucl. Acids Res. 40, 1717 (2011).
* van der Heijden et al. (2008) T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
* Gennes (2001) P. G. D. Gennes, CR. Acad. Sci. IV-Phys. 2, 1505 (2001).
* SantaLucia, Jr. (1998) J. SantaLucia, Jr., Proc. Natl Acad. Sci. USA 95, 1460 (1998).
* Lee et al. (2006) A. Lee, J. Xiao, and S. Singleton, J. Mol. Biol. 360, 343 (2006).
* D. Sagi and? T. Tlusty (2006) a. S. D. Sagi and? T. Tlusty, Nucl. Acids Res. 34, 5021 (2006).
* Rosselli and Stasiak (1991) W. Rosselli and A. Stasiak, EMBO J. 10, 4391 (1991).
* Kim et al. (1992) J. Kim, M. Cox, and R. Inman, J. Biol. Chem. 267, 16444 (1992).
* Fan et al. (2011) H.-F. Fan, M. Cox, and H.-W. Li, PLoS ONE 6, e21359 (2011).
* Conover et al. (2011) A. Conover, C. Danilowicz, R. Gunaratne, V. Coljee, N. Kleckner, and M. Prentiss, Nucl. Acids Res. 39, 8833 (2011).
Figure 1: (Color Online) a. Side view of the X-ray structure of the dsDNA
post-strand exchange (final) state with complementary (on the right at the
arrow) and incoming strands shown as green and red stick renderings. The arrow
points to a few rises between two triplets in the dsDNA structure. The VMD
(32) renderings of RecA crystal structure 3CMX (8) show site II residues
Arg226 (pink), Lys227 (cyan), Arg243 (yellow), and Lys245 (magenta) with
charged nitrogen atoms (blue). The cyan triangle indicates the approximate
position of an outgoing strand phosphate
b. top view of the same structure with circles indicating the radii occupied
by the incoming (green), outgoing (blue), and complementary strands (red for
final state and gray for intermediate state). c. top view showing the base
pairing in the homology recognition/strand exchange process superimposed on
the actual X-ray structure.
Figure 2: (Color Online) Schematic of Interactions between dsDNA and the
ssDNA-RecA filament as strand exchange progresses for a homolog with side
views shown in the central part of each panel and top views shown in the right
part of each panel where the grey region indicates the space occupied by the
protein DNA/protein and the circles indicate the radii occupied by the three
strands in their final post-strand exchange positions. In the schematic, the
filament consisting of three RecA proteins, showing site I (pale red), site II
(pale orange), and the support for the rises provided by the L1 and L2 loops
(cyan), with (i) unbound ssDNA ( (ii) ssDNA bound in site I and unbound B-form
dsDNA, (iii) ssDNA bound in site I and dsDNA bound in site II; The outgoing
strand (far left ), complementary strand (bound to the outgoing strand in
purple ), and incoming strand (bound to the protein on the right) are shown in
blue, red, and green, respectively. (iv) central triplet has undergone strand
exchange, resulting in a decrease in lattice mismatch and a decrease in the
stress on the bp (v) all three triplets shown have undergone strand exchange
(vi) ssDNA bound in site II and dsDNA bound in site I in the final assembled
state which has even less bp stress than the strand exchanged state due to
increase mechanical support for the rises. Figure 3: Schematic of dsDNA bound
to site II of the protein showing the model parameters. Figure 4: Effects of
adding triplets to the initial bound state as calculated by the model. (A)
Three triplets shown in the bound state with an arrow pointing to the second
one. (B) Adding a triplet to the initial state changes the tension in the
other triplets. (C) Again tension on the triplets change as a fifth triplet is
added to the initial state. Figure 5: The dashed, solid gray, and solid black
lines show the absolute value of $v_{i}-u_{i}$ for the outer triplet, central
triplet, and the first triplet out from the center respectively when the
number of bound triplets is odd. Figure 6: Total energy as a function of bound
base pairs all in the initial bound state (dashed line), the intermediate
state (solid gray line) and the final post-strand exchange state (solid black
line). Figure 7: The free energy penalty for adding another triplet in the
initial bound state as a function of the number of base pairs in the strand
exchanged state. The dashed black line shows the penalty when all of the
triplets are homologous. The dashed gray line shows the penalty for adding an
additional triplet to the initial bound state after a triplet that is non-
homologous. The solid gray line shows the penalty for adding a second triplet
after a non-homolog where the first triplet after the non-homolog was
homologous. The solid black line shows the energetic favorability of strand
exchanging a homologous triplet in the initial bound state. Figure 8: For a
completely homologous dsDNA with all triplets in the intermediate strand
exchanged state, the free energy penalty for a single triplet reversing strand
exchange by making a transition from the intermediate strand exchanged state
to the initial bound state as a function of then number of base pairs when the
base flipping triplet is in the center of the filament (solid line) or at the
end of the filament (dashed line)
|
arxiv-papers
| 2011-08-24T20:04:39 |
2024-09-04T02:49:21.763969
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Julea Vlassakis, Efraim Feinstein, Darren Yang, Antoine Tilloy,\n Dominic Weiller, Julian Kates-Harbeck, Vincent Coljee and Mara Prentiss",
"submitter": "Efraim Feinstein",
"url": "https://arxiv.org/abs/1108.4936"
}
|
1108.5014
|
# Multifractality in complex networks
Dan-Ling Wang1, Zu-Guo Yu1,2 and Vo Anh1
1Discipline of Mathematical Sciences, Faculty of Science and Technology,
Queensland University of Technology, Brisbane, Q4001, Australia.
2School of Mathematics and Computational Science, Xiangtan University, Hunan
411105, China. Corresponding author, e-mail: yuzg@hotmail.com
###### Abstract
Complex networks have recently attracted much attention in diverse areas of
science and technology. Many networks such as the WWW and biological networks
are known to display spatial heterogeneity which can be characterized by their
fractal dimensions. Multifractal analysis is a useful way to systematically
describe the spatial heterogeneity of both theoretical and experimental
fractal patterns. In this paper, we propose a new box covering algorithm for
multifractal analysis of complex networks. This algorithm is used to calculate
the generalized fractal dimensions $D_{q}$ of some theoretical networks,
namely scale-free networks, small world networks and random networks, and one
kind of real networks, namely protein-protein interaction (PPI) networks of
different species. Our numerical results indicate the existence of
multifractality in scale-free networks and PPI networks, while the
multifractal behavior is not clear-cut for small world networks and random
networks. The possible variation of $D_{q}$ due to changes in the parameters
of the theoretical network models is also discussed.
Key words: Complex networks; multifractality; box covering algorithm.
PACS numbers: 05.45.Df, 47.53.+n, 89.75.Hc
## 1 Introduction
Complex networks have been studied extensively due to their relevance to many
real-world systems such as the world-wide web, the internet, energy
landscapes, and biological and social systems [1].
It has been shown that many real complex networks share distinct
characteristics that differ in many ways from random and regular networks [2,
3]. Three fundamental properties of real complex networks have attracted much
attention recently: the small-world property [4, 5], the scale-free property
[6-8], and the self-similarity [1]. The small-world property means that the
average shortest path length between vertices in the network is short, usually
scaling logarithmically with the size $N$ of the network [3]. A famous example
is the so-called _six degrees of separation_ in social networks [5]. A large
number of real networks are referred to as _scale-free_ because the
probability distribution $P(k)$ of the number of links per node (also known as
the degree distribution) satisfies a power law $P(k)\sim k^{-\gamma}$ with the
degree exponent $\gamma$ varying in the range $2<\gamma<3$ [6]. In view of
their small-world property, it was believed that complex networks are not
self-similar under a length-scale transformation. After analyzing a variety of
real complex networks, Song et al. [1] found that they consist of self-
repeating patterns on all length scales, i.e., they have _self-similar
structures_. In order to unfold the self-similar property of complex networks,
Song et al. [1] calculated their fractal dimension, a known useful
characteristic of complex fractal sets [9-11], and found that the box-counting
method is a proper tool for further investigations of network properties.
Because a concept of metric on graphs is not as straightforward as the
Euclidean metric on Euclidean spaces, the computation of the fractal dimension
of networks via a box-counting approach is much more complicated than the
traditional box-counting algorithm for fractal sets in Euclidean spaces. Song
et al. [12] developed a more involved algorithm to calculate the fractal
dimension of complex networks. Then Kim et al. [13] proposed an improved
algorithm by considering the skeleton of networks. Zhou et al. [14] proposed
an alternative algorithm, based on the edge-covering box counting, to explore
the self-similarity of complex cellular networks. Later on, a ball-covering
approach [15] and an approach defined by the scaling property of the volume
[3, 16] were proposed for calculating the fractal dimension of complex
networks.
The tools of fractal analysis provide a global description of the
heterogeneity of an object, such as its fractal dimension. This approach is
not adequate when the object may exhibit a multifractal behavior. Multifractal
analysis is a useful way to systematically characterize the spatial
heterogeneity of both theoretical and experimental fractal patterns [17, 18].
It was initially proposed to treat turbulence data, and has recently been
applied successfully in many different fields including time series analysis
[19], financial modelling [20], biological systems [21-28] and geophysical
systems [29-34]. For complex networks, Lee and Jung [2] found that their
behaviour is best described by a multifractal approach. As mentioned above,
through the recent works by Song et al. [1], Guo and Cai [3], Kim et al. [13],
Zhou et al. [14], Gao et al. [15], it was already a big step to go from the
computation of the fractal dimension of a geometrical object to that of a
network via the box-counting approach of fractal analysis. In this paper, we
propose a new box-covering algorithm to compute the generalised fractal
dimensions of a network. This is the next step to move from fractal analysis
to multifractal analysis of complex networks.
We first adapt the random sequential box covering algorithm [13] to calculate
the fractal dimension of the human protein-protein interaction network as well
as that of its skeleton. We next propose a box covering algorithm for
multifractal analysis of networks in Section 2. This algorithm is then used to
calculate in Section 3 the generalized fractal dimensions $D_{q}$ of generated
examples of three classes of theoretical networks, namely scale-free networks,
small-world networks and random networks, and one kind of real networks,
namely protein-protein interaction networks of different species. The methods
to generate the theoretical networks are described. The multifractal behaviour
of these networks based on the computed generalised fractal dimensions $D_{q}$
is then discussed. The possible variation of $D_{q}$ due to changes in the
parameters of the theoretical network models is also investigated. Some
conclusions are then drawn in Section 4.
## 2 Methods
In this section, we first introduce the box covering methods for calculating
the fractal dimension of complex networks and the traditional fixed-size box
counting algorithms used for multifractal analysis. We then present our new
approach for multifractal analysis of complex networks in detail.
### 2.1 The box covering methods for calculation of fractal dimension
Box covering is a basic tool to estimate the fractal dimension of conventional
fractal objects embedded in the Euclidean space. The Euclidean metric is not
relevant for complex networks. A more natural metric is the shortest path
length between two nodes, which is defined as the number of edges in a
shortest path connecting them. Shortest paths play an important role in the
transport and communication within a network. It is useful to represent all
the shortest path lengths of a network as a matrix $D$ in which the entry
$d_{ij}$ is the length of the shortest path from node $i$ to node $j$. The
maximum value in the matrix $D$ is called the network diameter, which is the
longest path between any two nodes in the network. Song et al. [1] studied the
fractality and self-similarity of complex networks by using box covering
techniques. They proposed several possible box covering algorithms [1] and
applied them to a number of models and real-world networks. Kim et al. [13]
introduced another method called the random sequential box covering method,
which can be described as follows:
For a given network, let $N_{B}$ be the number of boxes of radius $r_{B}$
which are needed to cover the entire network. The fractal dimension $d_{B}$ is
then given by
$N_{B}\sim{r_{B}}^{-d_{B}}.$
By measuring the distribution of $N_{B}$ for different box sizes, the fractal
dimension $d_{B}$ can be obtained by power law fitting of the distribution.
This algorithm has the following steps [13]:
1. (i)
Select a node randomly at each step; this node serves as a seed which will be
the center of a box.
2. (ii)
Search the network by distance $r_{B}$ from the seed and cover all nodes which
are found but have not been covered yet. Assign the newly covered nodes to the
new box. If no newly covered nodes have been found, then this box is
discarded.
3. (iii)
Repeat (i) and (ii) until all nodes in the network have been assigned to their
respective boxes.
To obtain the _skeleton_ of a complex network, we firstly need to calculate
the edge betweenness of all the edges in this network. The betweenness
$b_{i},$ also referred to as load [13], is defined as
$b_{i}=\sum_{j,k\in N,j\neq k}\frac{n_{jk}(i)}{n_{jk}},$
where $N$ is the number of nodes, $n_{jk}$ is the number of shortest paths
connecting nodes $j$ and $k$, while $n_{jk}{(i)}$ is the number of shortest
paths connecting nodes $j$ and $k$ and passing through edge $i$. Similar to a
minimum spanning tree, a skeleton is constructed so that edges which have the
highest betweenness and do not form loops are selected [13]. The remaining
edges in the original network are referred to as shortcuts that contribute to
loop formation. In other words, the distance between any two nodes in the
original network may increase in the skeleton. For example, in the human
protein-protein interaction network, the largest distance between any two
nodes in the original network is 21 while the largest distance between any two
nodes in its skeleton is 27.
As an example, we used the above algorithm to estimate the fractal dimension
of the human protein-protein interaction network as well as that of its
skeleton. The result is shown in Fig. 1. When we applied the box covering
algorithm on the skeleton, more boxes were needed for each fixed box radius
$r_{B}$. The increasing rate of the number $N_{B}$ of boxes varies when the
size $r_{B}$ of the box increases. More specifically, when $r_{B}$ is smaller,
the number of boxes needed is not much different for both the original network
and its skeleton; but when $r_{B}$ is larger, many more boxes are needed to
cover the skeleton than the original network.
### 2.2 Algorithms for multifractal analysis of networks
Most well-known fractals such as the Cantor set, the Koch curve and the
Sierpinski triangle are homogeneous since they consist of a geometrical figure
repeated on an ever-reduced scale. For these objects, the fractal dimension is
the same on all scales. However, real-world fractals may not be homogeneous;
there is rarely an identical motif repeated on all scales. Two objects might
have the same fractal dimension and yet look completely different. Real-world
fractals possess rich scaling and self-similarity properties that can change
from point to point, thus can have different dimensions at different scales.
The present paper investigates these properties on complex networks.
Especially we develop tools to determine whether they are simple fractals or
multifractals, and how different two networks could be even though they have
the same fractal dimension. In other words, we aim to develop an approach for
multifractal analysis of complex networks.
The most common algorithm of multifractal analysis is the fixed-size box-
counting algorithm [18, 22, 25]. For a given probability measure $0\leq\mu\leq
1$ with support set $E$ in a metric space, we consider the partition sum
$Z_{\epsilon}(q)=\Sigma_{\mu(B)\neq 0}[\mu(B)]^{q},\,$ (1)
where $q$ is a real number and the sum runs over all different non-overlapping
boxes $B$ of a given size $\epsilon$ in a covering of the support $E$. It
follows that $Z_{\epsilon}(q)\geqslant 0$ and $Z_{\epsilon}(0)=1.$ The mass
exponent function $\tau(q)$ of the measure $\mu$ is defined by
$\tau(q)=\lim_{\epsilon\rightarrow 0}\frac{\ln Z_{\epsilon}(q)}{\ln\epsilon}.$
(2)
###### Proposition 1
The mass exponent $\tau(q)$is an increasing function of $q$.
Proof. For $q_{1}<q_{2}$, it follows from $\mu$ being a probability measure
that $\mu(B_{i})^{q_{1}}>\mu(B_{i})^{q_{2}}$; thus
$Z_{\epsilon}(q_{1})>Z_{\epsilon}(q_{2}).$ Since $\ln\epsilon<0$ when
$\epsilon\rightarrow 0$, the increasing property of $\tau\left(q\right)$
follows. $\Box$
The generalized fractal dimensions of the measure $\mu$ are defined as
$D_{q}=\frac{\tau(q)}{q-1},~{}q\neq 1,$ (3)
and
$D_{q}=\lim_{\epsilon\rightarrow 0}\frac{Z_{1,\epsilon}}{\ln\epsilon},$ (4)
for $q=1$, where $Z_{(1,\epsilon)}=\Sigma_{\mu(B)\neq 0}\mu(B)\ln\mu(B)$.
###### Proposition 2
$D_{q}$ is a decreasing function of $q$ for $q\neq 1$.
Proof. Combining Eqs. (2) and (3) yields, for $q\neq 1$,
$D_{q}=\lim_{\epsilon\rightarrow 0}\frac{\frac{1}{q-1}\ln
Z_{\epsilon}(q)}{\ln\epsilon}.$ (5)
We need to consider 3 cases:
(i) For $1<q_{1}\leqslant q_{2}<\infty,$ we have
$0<\frac{1}{q_{2}-1}\leq\frac{1}{q_{1}-1}<\infty$ (6)
and
$0<Z_{\epsilon}(q_{2})\leqslant Z_{\epsilon}(q_{1})\leqslant 1,$
that is,
$\ln Z_{\epsilon}(q_{2})\leqslant\ln Z_{\epsilon}(q_{1})<0.$ (7)
From Eqs. (5) - (7), it is seen that $\frac{1}{q-1}\ln Z_{\epsilon}(q)$
increases as a function of $q$ . Thus $D_{q}$ decreases as a function of $q$
since $\ln\epsilon<0$ as $\epsilon\rightarrow 0.$
(ii) For $0<q_{1}\leqslant q_{2}<1,$ we have
$-\infty<\frac{1}{q_{2}-1}\leqslant\frac{1}{q_{1}-1}<-1$
and
$\frac{1}{q_{2}-1}\ln Z_{\epsilon}(q_{2})\geqslant\frac{1}{q_{1}-1}\ln
Z_{\epsilon}(q_{1}).$
Thus $D_{q}$ decreases as a function of $q$ in this case.
(iii) For $-\infty<q_{1}\leqslant q_{2}<0,$ we have
$-1<\frac{1}{q_{2}-1}\leqslant\frac{1}{q_{1}-1}<0$
and also
$\frac{1}{q_{2}-1}\ln Z_{\epsilon}(q_{2})\geqslant\frac{1}{q_{1}-1}\ln
Z_{\epsilon}(q_{1}).$
Thus $D_{q}$ also decreases as a function of $q$ in this case. $\Box$
For every box size $\epsilon$, the number
$\alpha=\frac{\log\mu\left(\epsilon\right)}{\log\epsilon},$ also referred to
as the Hölder exponent, is the singularity strength of the box. This exponent
may be interpreted as a crowding index of a measure of concentration: the
greater $\alpha$ is, the smaller is the concentration of the measure, and vice
versa. For every box size $\epsilon$, the numbers of cells
$N_{\alpha}(\epsilon)$ in which the Hölder exponent $\alpha$ has a value
within the range $[\alpha,\alpha+d\alpha]$ behave like
$N_{\alpha}(\epsilon)\sim\epsilon^{-f\left(\alpha\right)}.$
The function $f\left(\alpha\right)$ signifies the Hausdorff dimension of the
subset which has singularity $\alpha$; that is, $f(\alpha)$ characterizes the
abundance of cells with Hölder exponent $\alpha$ and is called the
_singularity spectrum_ of the measure. The measure $\mu$ is said to be a
_multifractal measure_ if its singularity spectrum $f\left(\alpha\right)\neq
0$ for a range of values of $\alpha$. The singularity spectrum
$f\left(\alpha\right)$ and the mass exponent function $\tau(q)$ are connected
via the Legendre transform: ([9])
$\alpha\left(q\right)=\frac{d\tau\left(q\right)}{dq}$ (8)
and
$f(\alpha\left(q\right))=q\alpha\left(q\right)-\tau\left(q\right),~{}q\in\mathbb{R}.$
Considering the relationship between the mass exponent function $\tau(q)$ and
the generalized dimension function $D_{q}$, the singularity spectrum
$f(\alpha)$ contains exactly the same information as $\tau(q)$ and $D_{q}$.
Lau and Ngai [35] showed in their Proposition 3.4 (page 57) that
(i) $\underset{q\rightarrow\infty}{\lim}$ $D_{q}=\alpha_{\min};$
(ii) $\underset{q\rightarrow-\infty}{\lim}$ $D_{q}=\alpha_{\max}.$
This result together with Proposition 2 and the definition of a multifractal
measure given above lead to a method to determine the multifractality of a
probability measure $\mu$:
When $\alpha_{\min}=\alpha_{\max}$, the function $D_{q}$ is constant for
$q\neq 1$ and the measure $\mu$ is monofractal.
When $\alpha_{\min}\neq\alpha_{\max}$, $D_{q}$ is a decreasing function of
$q\neq 1$ and the measure $\mu$ is multifractal.
This method is the key element in the next section when we investigate the
multifractality of a variety of networks.
The generalized fractal dimensions are estimated through a linear regression
of $[\ln Z_{\epsilon}(q)]/(q-1)$ against $\ln\epsilon$ for $q\neq 1$, and
similarly through a linear regression of $Z_{1,\epsilon}$ against
$\ln\epsilon$ for $q=1$. The value $D_{1}$ is called the information dimension
and $D_{2}$ the correlation dimension, while $D_{0}$ is equal to the Hausdorff
dimension.
For a network, the measure $\mu$ of each box is defined as the ratio of the
number of nodes covered by the box and the total number of nodes in the
network. The fixed-size box-counting algorithm of Kim et al. [13] described
above could not be used to analyze the multifractal behavior of networks
directly. Because the method contains a random process of selecting the
position of the center of each box, this will affect the number of boxes with
a fixed size. Especially, if a node with large degree (a hub) is randomly
chosen, a lot more nodes could be covered, and it is an efficient way when we
produce box covering. However, if a node with small degree is randomly chosen
first, few nodes could be covered. As a result, the partition sum defined by
Eq. (1) will change each time we proceed with box counting. We illustrate this
situation in Fig. 2: We consider a network of eight nodes. In Fig. 2A, for a
fixed box size $r_{B}=1$, firstly node _a_ is chosen as the center of a box
and both nodes _a_ and _b_ are covered in the same box colored in black. Next,
node _f_ is chosen as a center of a box, and nodes _b, c, d, e_ , _g_ are all
within a distance $r_{B}=1.$ Since node _b_ has already been covered in the
previous step, so nodes _c, d, e, g, f_ are covered in the same box colored in
blue. In the last step, node _g_ is chosen as the center of a box and its
neighboring node _h_ is the only one found within a distance $r_{B}=1$ not
covered yet, so _h_ is the only one covered in a box colored in red. In
summary, three boxes are needed to cover the entire network. In Fig 2B, for
the same fixed box size $r_{B}=1$, firstly node _h_ is chosen as the center of
a box and both nodes _h_ and _g_ are covered in the same box colored in red.
Next, node _f_ is chosen as a center of a box, and nodes _e, g_ are all within
a distance $r_{B}=1.$ Since node _g_ has already been covered in the previous
step, so nodes _e, f_ are covered in the same box colored in blue. Next, node
_d_ is then chosen as the center of a box and since its two neighbors _f, g_
have already been covered, so _d_ is the only one covered in a box colored in
brown; likewise, node _c_ is chosen and covered alone in the box colored in
green. In the last step, node _a_ is chosen as a center and both nodes _a_ and
_b_ are covered within one box colored in black. In summary, five boxes are
needed to cover the entire network. In these two cases of Figs. 2A and 2B, the
partition sums are different. To avoid this effect, we propose to take the
average of the partition sums over a large number of times and accordingly
modify the original fixed-size box-counting algorithm into a new method. To
our knowledge, this improvement is the first introduced in this approach to
analyze the multifractal behavior of complex networks.
We need to calculate the shortest-path distance matrix for each network and
these matrices are the input data for fractal and multifractal analyses. We
describe the procedure as follows:
1. (i)
Transform the pairs of edges and nodes in a network into a matrix $A_{N\times
N}$, where $N$ is the number of nodes of the network. The matrix $A_{N\times
N}$ is a symmetric matrix where the elements $a_{ij}=0$ or $1$ with $a_{ij}=1$
when there is an edge between node $i$ and node $j$, while $a_{ij}=0$ when
there is no edge between them. We define that each node has no edge with
itself and accordingly $a_{ii}=0$.
Remark: $A_{N\times N}$ could be the input data for calculating the degree
distribution and characteristic path length to determine whether the network
possesses the properties of scale-free degree distribution and small-world
effect.
2. (ii)
Compute the shortest path length among all the linked pairs and save these
pairs into another matrix $B_{N\times N}$ .
Remark: In graph theory, calculation of the shortest path is a significant
problem and there are many algorithms for solving this problem. Here, in our
approach, we use Dijkstra’s algorithm [36] of the Matlab toolbox.
After the above steps we could use the matrix $B_{N\times N}$ as input data
for multifractal analysis based on our _modified fixed-size box counting
algorithm_ as follows:
1. (i)
Initially, all the nodes in the network are marked as uncovered and no node
has been chosen as a seed or center of a box.
2. (ii)
According to the number of nodes in the network, set $t=1,2,...,T$
appropriately. Group the nodes into $T$ different ordered random sequences.
More specifically, in each sequence, nodes which will be chosen as seed or
center of a box are randomly arrayed.
Remark: $T$ is the number of random sequences and is also the value over which
we take the average of the partition sum $\overline{Z_{r}(q)}$. Here in our
study, we set $T=200$ for all the networks in order to compare.
3. (iii)
Set the size of the box in the range $r\in[1,d]$, where $d$ is the diameter of
the network.
Remark: When $r=1$, the nodes covered within the same box must be connected to
each other directly. When $r=d$, the entire network could be covered in only
one box no matter which node was chosen as the center of the box.
4. (iv)
For each center of a box, search all the neighbors within distance $r$ and
cover all nodes which are found but have not been covered yet.
5. (v)
If no newly covered nodes have been found, then this box is discarded.
6. (vi)
For the nonempty boxes $B$, we define their measure as $\mu(B)=N_{B}/N,\,$
where $N_{B}$ is the number of nodes covered by the box $B$, and $N$ is the
number of nodes of the entire network.
7. (vii)
Repeat (iv) until all nodes are assigned to their respective boxes.
8. (viii)
When the process of box counting is finished, we calculate the partition sum
as $Z_{r}(q)=\Sigma_{\mu(B)\neq 0}[\mu(B)]^{q}\,$ for each value of $r$.
9. (ix)
Repeat (iii) and (iv) for all the random sequences, and take the average of
the partition sums $\overline{Z_{r}(q)}=(\sum^{t}Z_{r}(q))/T,$ and then use
$\overline{Z_{r}(q)}$ for linear regression.
Linear regression is an essential step to get the appropriate range of
$r\in[r_{min},r_{max}]$ and to get the generalized fractal dimensions $D_{q}$.
In our approach, we run the linear regression of
$[\ln\overline{Z_{r}(q)}]/(q-1)$ against $\ln(r/d)$ for $q\neq 1$, and
similarly the linear regression of $\overline{Z_{1,r}}$ against $\ln(r/d)$ for
$q=1$, where $\overline{Z_{1,r}}=\Sigma_{\mu(B)\neq 0}\mu(B)\ln\mu(B)$ and $d$
is the diameter of the network. An example of linear regression for the
Arabidopsis thaliana PPI network is shown in Fig. 3. The numerical results
show that the best fit occurs in the range $r\in(1,9)$, hence we select this
range to perform multifractal analysis and get the spectrum of generalized
dimensions $D_{q}$.
After this spectrum has been obtained, we use $\Delta D(q)=\max D(q)-\lim
D(q)$ to verify how $D_{q}$ changes along each curve. The quantity $\Delta
D(q)$ has been used in the literature to describe the density of an object. In
this paper, based on our modified fixed-size box covering method, $\Delta
D(q)$ can help to understand how the edge density changes in the complex
network. In other words, a larger value of $\Delta D(q)$ means the edge
distribution is more uneven. More specifically, for a network, edge
distribution could vary from an area of hubs where edges are dense to an area
where nodes are just connected with a few links.
In the following sections, we calculate the generalized fractal dimensions
$D_{q}$. From the shape of $D_{q}$, we determine the multifractality of the
network using the method described above. We then calculate $\Delta D(q)$ to
verify how $D_{q}$ changes along each curve.
## 3 Results and discussions
In recent years, with the development of technology, the research on networks
has shifted away from the analysis of single small graphs and the properties
of individual vertices or edges within such graphs to consideration of large-
scale statistical properties of complex networks. Newman [37] reviewed some
latest works on the structure and function of networked systems such as the
Internet, the World Wide Web, social networks and a variety of biological
networks. Besides reviewing empirical studies, the author also focused on a
number of statistical properties of networks including path lengths, degree
distributions, clustering and resilience. In this paper, we pay attention to
another aspect of networks, namely their multifractality. We aim to develop a
tool based on this property to characterize and classify real-world networks.
It has been shown that many real complex networks share distinctive
characteristics that differ in many ways from random and regular networks [2,
3, 37]. Fundamental properties of complex networks such small-world effect and
the scale-free degree distribution have attracted much attention recently.
These properties have in fact been found in many naturally occurring networks.
In Subsections 3.1, 3.2 and 3.3, we generate scale-free networks using the BA
model of Barabasi and Albert [38], small-world networks using the NW model of
Newman and Watts [39], then random networks using the ER model of Erdös and
Rényi [4] respectively. We then apply our modified fixed-size box counting
algorithm to analyze the multifractal behavior of these networks.
### 3.1 Scale-free networks
We use the elegant and simple BA model of Barabasi and Albert [38] to generate
scale-free networks. The origin of the scale-free behavior in many systems can
be traced back to this BA model, which correctly predicts the emergence of
scaling exponent. The BA model consists of two mechanisms : Initially, the
network begins with a seed network of $n$ nodes, where $n\geq 2$ and the
degree of each node in the initial network should be at least 1, otherwise it
will always remain disconnected from the rest of the network. For example,
here we start with an initial network of 5 nodes. Its interaction matrix is
$\left(\begin{array}[]{ccccc}0&1&0&0&1\\\ 1&0&0&1&0\\\ 0&0&0&1&0\\\
0&1&1&0&0\\\ 1&0&0&0&0\end{array}\right).$
We then add one node to this initial network at a time. Each new node is
connected to $n$ existing nodes with a probability that is proportional to the
number of links that the existing nodes already have. Formally, the
probability $p_{i}$ that the new node is connected to node $i$ is
$p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},$ (9)
where $k_{i}$ is the degree of node $i$. So hubs tend to quickly accumulate
even more links, while nodes with only a few links are unlikely to be chosen
as destination for a new link.
In this paper, these scale-free networks are generated based on the same seed
which is the initial network of 5 nodes. For better comparison, in each step,
one node will be added into the network with one link. Then we apply the
modified fixed-size box counting method on them to detect their multifractal
behavior.
In Fig. 4. we can see that scale-free networks are multifractal by the shape
of the $D_{q}$ curves. The $D_{q}$ functions of these networks decrease
sharply after the peak. An explanation is that, in a scale-free network, there
are several nodes which are known as hubs that have a large number of edges
connected to them, so the edge density around the areas near the hubs is
larger than the remaining parts of the network.
We summarize the numerical results in Table 1 including the number of nodes,
number of edges, diameter, power law exponent $\gamma$, maximum value of
$D_{q}$, limit of $D_{q}$, and $\Delta D_{q}$. From these results we could see
that scale free networks with larger size (more nodes and more edges) are
likely to have larger values of the maximum and limit of $D_{q}$. In other
words, the function $D_{q}$ increases with the size of a scale-free network.
An explanation for this situation is that larger scale-free networks usually
have more hubs which make the structure of the network more complex.
Scale-free networks show a power-law degree distribution of $P(k)\sim
k^{-\gamma}$, where $P(k)$ is the probability of a node randomly chosen with
degree $k$. It was shown in [6, 7] that when $\gamma<2,$ the average degree
diverges; while for $\gamma>3,$ the standard deviation of the degree diverges.
It has been found that the degree exponent $\gamma$ usually varies in the
range of $2<\gamma<3$ [6] for most scale-free networks. Accordingly, we
computed the power-law exponent of these generated scale-free networks. The
results show that there doesn’t seem to be any clear relationship between
power law and the maximum of $D_{q},$ limit of $D_{q}$ or $\Delta D_{q}$.
### 3.2 Small-world networks
In 1998, Watts and Strogatz [40] proposed a single-parameter small-world
network model that bridges the gap between a regular network and a random
graph. With the WS small-world model, one can link a regular lattice with pure
random network by a semirandom network with high clustering coefficient and
short average path length. Later on, Newman and Watts [39] modified the
original WS model. In the NW model, instead of rewiring links between nodes,
extra links called shortcuts are added between pairs of nodes chosen at
random, but no links are removed from the existing network. The NW model is
equivalent to the WS model for small p and sufficiently large N, but easier to
proceed.
In this paper, we use the NW model as follows. Firstly, we should select three
parameters: the dimension $n,$ which is the number of nodes in a graph; the
mean degree $k$ (assumed to be an even integer), which is the number of
nearest-neighbors to connect; and the probability $p$ of adding a shortcut in
a given row, where $0\leq p\leq 1$ and $n\gg k\gg\ln(n)\gg 1$. Secondly, we
follow two steps:
1. (i)
Construct a regular ring lattice. For example, if the nodes are named
$N_{0},...,N_{n-1}$, there is an edge $e_{ij}$ between node $N_{i}$ and
$N_{j}$ if and only if $|i-j|\equiv K$ for $K\in[0,k/2]$;
2. (ii)
Add a new edge between nodes $N_{i}$ and $N_{j}$ with probability $p$.
An illustration of this generating process is given in Fig. 5. The upper left
figure corresponds to the probability $p=0$. It is a regular network
containing 20 nodes and each node has two neighbors on both sides. In other
words, in this regular network, each node has four edges. All the nodes and
edges are shown in blue. Then we start generating small-world networks based
on this regular network. The upper right figure of Fig. 5 corresponds to the
probability $p=0.1;$ one edge is added into the network which is colored in
black. The network then becomes a small-world network. The bottom left figure
corresponds to the probability $p=0.5;$ seven black edges are added into the
original regular network and it is also a small-world network. The bottom
right figure corresponds to the probability $p=1$; 10 black edges are added
into the original small-world network and this time it becomes a random
network.
In this paper, we firstly generated a regular network which contains 5000
nodes and 250,000 edges. Each node has 50 edges on each side. Then we apply
the modified fixed-size box counting method on this regular network. The
numerical results are shown in the last row of Table 2. Both the maximum value
of $D_{q}$ and the limit of $D_{q}$ are equal to one, thus $\Delta D_{q}=0$.
This is because regular networks are not fractal, and they have dimension one.
Secondly, for better comparison, we generated ten small-world networks based
on a regular network of 5000 nodes with 5 edges on each side of a node. During
the generation, when the probability $p$ increases, more edges are added into
the original regular network. Then we apply the modified fixed-size box
counting method on them to detect their multifractal behavior. We summarize
the numerical results in Table 2, which includes the number of nodes, number
of edges, diameter, probability p (the generating parameter), maximum value of
$D_{q}$ and $\Delta D_{q}$. These results indicate that, when $p$ increases,
more edges are added and accordingly both the maximum and limit values of
$D_{q}$ increase.
In Fig. 6 we can see that the $D_{q}$ curve of a regular network whose
probability $p=0$ during generation is a straight line with the value of $1$.
The $D_{q}$ curves of the other small-world networks are also approximately
straight lines but with different $D_{q}$ values. So these networks are not
multifractal. Another interesting property is apparent when $0.03<p<0.2$, in
which case $D_{q}$ increases along with the value of $p$. More specifically,
when $p$ increases, more edges are added to the network, and both the maximum
and limit values of $D_{q}$ and limit of $D_{q}$ increase. The values of
$\Delta D_{q}$ are all within the error range, confirming that the $D_{q}$
curves are straight lines.
### 3.3 Random networks
The Erdös-Rényi random graph model [4] is the oldest and one of the most
studied techniques to generate complex networks.
We generate random networks based on the ER model [4]:
1. (i)
Start with $N$ isolated nodes;
2. (ii)
Pick up every pair of nodes and connect them by an edge with probability $p.$
Usually, the results of this generation are separated subnetworks. In this
work, we just consider the largest connected part as the network to work on
and apply the modified fixed-size box counting method to detect their
multifractal behaviors. We then summarize the numerical results in Table 3
including the number of nodes, number of edges, diameter, probability $p$ (the
generating parameter), maximum value of $D_{q}$, limit of $D_{q}$, and $\Delta
D_{q}$. These results indicate that there is no clear relationship between
$D_{q}$ and the size of the random network.
In Fig. 7, we can see that the $D_{q}$ curves of random networks decrease
slowly after the peak and the changes could be seen by the values of $\Delta
D_{q}$. This pattern occurs bcause, during the generating process, nodes are
randomly connected with probability $p$, and few hubs may exist. Compared with
scale-free networks, this decrease supports the claim that, in random
networks, edges are distributed more symmetrically.
Remark: In the present study, we consider the generalized fractal dimensions
$D_{q}$ to determine whether the object is multifactal from the shape of
$D_{q}$. For a monofractal system, which has the same scaling behavior at any
point, $D_{q}$ should be a constant independent of $q$, while for a
multifractal, the $D_{q}$ should be a non-increasing nonlinear curve as $q$
increases. However, in our results, an anomalous behavior is observed: the
$D_{q}$ curves increase at the beginning when $q<0$. This anomalous behavior
has also been observed in Bos et al.[41], Smith and Lange [42], Fernández et
al. [43]. Some reasons for this behavior have been suggested, including that
the boxes contain few elements [43], or the small scaling regime covers less
than a decade so that we cannot extrapolate the box counting results for the
partition function to zero box size [41]. In encountering the anomalous
spectra of $D_{q}$, we tried another method of multifractal analysis called
the sand-box method, but the linear regression fittings are not satisfactory.
We therefore used the modified fixed-size box counting algorithm in this
research. For the purpose of detecting the multifractality of complex
networks, we adopt the anomalous spectra of $D_{q}$ and focus on the
decreasing parts which are presented in Figs. 4 to 8.
### 3.4 Protein-protein interaction networks
Our fractal and multifractal analyses are based on connected networks which do
not have separated parts or isolated nodes. In order to apply them to protein-
protein interaction (PPI) networks, some preparation is needed in advance.
Firstly, we need to find the largest connected part of each data set. For this
purpose many tools and methods could be used. In our study, we adopt the
Cytoscape [44] which is an open bioinformatics software platform for
visualizing molecular interaction networks and analyzing network graphs of any
kind involving nodes and edges. In using Cytoscape, we could get the largest
connected part of each interacting PPI data set and this connected part is the
network on which fractal and multifractal analyses are performed.
The protein-protein interaction data we used here are mainly downloaded from
two databases: The PPI networks of Drosophila melanogaster (fruit fly), C.
elegans, Arabidopsis thaliana and Schizosaccharomyces pombe are downloaded
from BioGRID [45]. The PPI networks of S. cerevisiae (baker’s yeast), E. coli
and H. pylori are download from DIP [46]. We also use the same human PPI
network data as in [47].
We calculated the $D_{q}$ spectra for eight PPI networks of different
organisms as shown in Fig. 8. From these $D_{q}$ curves, we see that all PPI
networks are multifractal and there are two clear groupings of organisms based
on the peak values of their $D_{q}$ curves. The first group includes human,
Drosophila melanogaster, S. cerevisiae, and C. elegans. The second group just
includes two bacteria E.coli and H. pylori. We also see that the PPI networks
of the eight organisms have similar shape for the $D_{q}$ curves. They all
increase when $q\in[0,1]$, and reach their peak values around $q=2$, then
decrease sharply as $q>2$ and finally reach their limit value when $q>10$. So
we can take $\lim D(q)=D(20)$ and use $\Delta D(q)=\max D(q)-\lim D(q)$ to
verify how the $D_{q}$ function changes along each curve. We summarize the
corresponding numerical results in Table 4.
## 4 Conclusions
After analyzing a variety of real complex networks, Song et al. [1] found that
they consist of self-repeating patterns on all length scales, i.e., complex
networks have self-similar structures. They found that the box-counting method
is a proper tool to unfold the self-similar properties of complex networks and
to further investigate network properties.
However, describing objects by a single fractal dimension is a limitation of
fractal analysis, especially when the networks exhibit a multifractal
behavior. Multifractal analysis is a useful way to characterize the spatial
heterogeneity of both theoretical and experimental fractal patterns. It allows
the computation of a set of fractal dimensions, especially the generalized
fractal dimensions $D_{q}$.
A modified algorithm for analyzing the multifractal behavior of complex
networks is proposed in this paper. This algorithm is applied on generated
scale-free networks, small-world networks and random networks as well as
protein-protein interaction networks. The numerical results indicate that
multifractality exists in scale-free networks and PPI networks, while for
small-world networks and random networks their multifractality is not clear-
cut, particularly for small-world networks generated by the NW model.
Furthermore, for scale-free networks, the values of $D_{q}$ increase when the
size of the network increases because larger scale-free networks usually have
more hubs which make the structure of the network more complex. However, for
random networks there is no clear relationship between $D_{q}$ and the size of
the network. The quantity $\Delta D(q)=\max D(q)-\lim D(q)$ has been used to
investigate how $D_{q}$ changes. Larger $\Delta D(q)$ means the network’s edge
distribution is more uneven; while smaller $\Delta D(q)$ means the network’s
edge distribution is more symmetrical, which is the case for random networks.
These results support that the algorithm proposed in this paper is a suitable
and effective tool to perform multifractal analysis of complex networks.
Especially, in conjunction with the derived quantities from $D_{q}$, the
method and algorithm provide a needed tool to cluster and classify real
networks such as the protein-protein interaction networks of organisms.
## 5 Acknowledgement
This project was supported by the Australian Research Council (Grant No.
DP0559807), the Natural Science Foundation of China (Grant No. 11071282), the
Chinese Program for New Century Excellent Talents in University (Grant No.
NCET-08-06867), the Lotus Scholars Program of Hunan province of China, the Aid
Program for Science and Technology Innovative Research Team in Higher
Education Institutions of Hunan Province of China, and a China Scholarship
Council–Queensland University of Technology Joint Scholarship. The authors
wish to thank the Editor and the referees for their comments and suggestions
to improve the paper.
## References
* [1] C. Song, S. Havlin and H.A. Makse, Self-similarity of complex networks. _Nature_ , 433 (2005) 392-395.
* [2] C.Y. Lee and S. Jung, Statistical self-similar properties of complex networks. _Phys. Rev. E_ , 73 (2006) 066102\.
* [3] L. Guo and X. Cai, The fractal dimensions of complex networks. _Chinese Phys. Lett._ , 26 (2009) 088901.
* [4] P. Erdos and A. Renyi, On the evolution of random graphs. _Publ. Math. Inst. Hung. Acad. Sci._ , 5 (1960) 17-61.
* [5] S. Milgram, The small-world problem. _Psychol. Today_ , 2 (1967) 60-67.
* [6] R. Albert, H. Jeong and A.L. Barabasi, Diameter of the World Wide Web. _Nature_ , 401 (1999) 130-131.
* [7] R. Albert and A.L. Barabasi, Statistical mechanics of complex networks. _Rev. Mod. Phys._ , 74 (2002) 47-97.
* [8] M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology. _Comput. Commun. Rev._ , 29 (1999) 251-262.
* [9] B.B. Mandelbrot, _The fractal Geometry of Nature_ (Freeman, New York, 1983).
* [10] J. Feder, _Fractals_ (Plenum, New York, 1988).
* [11] K. Falconer, _Techniques in Fractal Geometry_ (Wiley, New York, 1997).
* [12] C. Song, L.K. Gallos, S. Havlin and H.A Makse, How to calculate the fractal dimension of a complex network: the box covering algorithm. _J. Stat. Mech.: Theor. Exper._ , 3 (2007) P03006.
* [13] J.S. Kim, K.I. Goh, G. Salvi, E. Oh, B. Kahng, and D. Kim, Fractality in complex networks: Critical and supercritical skeletons. _Phys. Rev. E_ , 75 (2007) 016110.
* [14] W.X. Zhou, Z.Q. Jiang and D. Sornette, Exploring self-similarity of complex cellular networks: The edge-covering method with simulated annealing and log-periodic sampling. _Physica A_ , 375 (2007) 741-752.
* [15] L. Gao, Y. Hu and Z. Di, Accuracy of the ball-covering approach for fractal dimensions of complex networks and a rank-driven algorithm. _Phys. Rev. E_ , 78 (2008) 046109.
* [16] O. Shanker, Defining dimension of a complex network. _Mod. Phys. Lett.B_ , 21 (2007) 321-326.
* [17] P. Grassberger and I. Procaccia, Characterization of strange attractors. _Phys. Rev. Lett._ , 50 (1983) 346-349.
* [18] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, and B.I. Shraiman, Fractal measures and their singularities: the characterization of strange sets. _Phys. Rev. A_ , 33 (1986) 1141-1151.
* [19] E. Canessa, Multifractality in time series. _J. Phys. A: Math. Gen._ 33 (2000) 3637-3651.
* [20] V.V. Anh, Q.M. Tieng and Y.K. Tse, Cointegration of stochastic multifractals with application to foreign exchange rates. _Int. Trans. Oper. Res._ , 7 (2000) 349-363.
* [21] Z.G. Yu, V. Anh and K.S. Lau, Multifractal characterisation of length sequences of coding and noncoding segments in a complete genome. _Physica A_ , 301 (2001) 351-361.
* [22] Z.G. Yu, V. Anh and K.S. Lau, Measure representation and multifractal analysis of complete genome. _Phys. Rev. E_ , 64 (2001) 31903.
* [23] Z.G. Yu, V.V. Anh and B. Wang, Correlation property of length sequences based on global structure of complete genome. _Phys. Rev. E_ , 63 (2001) 11903.
* [24] V.V. Anh, K.S. Lau and Z.G. Yu, Recognition of an organism from fragments of its complete genome. _Phys. Rev. E_ , 66 (2002) 031910.
* [25] Z.G. Yu, V. Anh and K.S. Lau, Multifractal and correlation analysis of protein sequences from complete genome. _Phys. Rev. E_ , 68 (2003) 021913.
* [26] Z.G. Yu, V. Anh and K.S. Lau, Chaos game representation of protein sequences based on the detailed HP model and their multifractal and correlation analyses. _J. Theor. Biol._ , 226 (2004) 341-348.
* [27] L.Q. Zhou, Z.G. Yu, J.Q. Deng, V. Anh and S.C. Long, A fractal method to distinguish coding and noncoding sequences in a complete genome based on a number sequence representation. _J. Theor. Biol._ , 232 (2005) 559-567.
* [28] Z.G. Yu, V.V. Anh, K.S. Lau and L.Q. Zhou, Fractal and multifractal analysis of hydrophobic free energies and solvent accessibilities in proteins. _Phys. Rev. E_ , 73 (2006) 031920.
* [29] J.W. Kantelhardt, E. Koscielny-Bunde, D. Rybski, P. Braun, A. Bunde and S. Havlin, Long-term persistence and multifractality of precipitation and river runoff records. _J. Geophys. Res._ , 111 (2006) D01106.
* [30] D. Veneziano, A. Langousis and P. Furcolo, Multifractality and rainfall extremes: A review. _Water Resour. Res._ , 42 (2006) W06D15.
* [31] V. Venugopal, S.G Roux, E. Foufoula-Georgiou and A. Arneodo, Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism. _Water Resour. Res._ , 42 (2006) W06D14.
* [32] Z.G. Yu, V.V. Anh, J.A. Wanliss and S.M. Watson, Chaos game representation of the Dst index and prediction of geomagnetic storm events, _Chaos, Solitons and Fractals_ , 31 (2007) 736-746.
* [33] Z.G. Yu, V. Anh and R. Eastes, Multifractal analysis of geomagnetic storm and solar flare indices and their class dependence. _J. Geophys. Res._ , 114 (2009) A05214.
* [34] Z.G. Yu, V. Anh, Y. Wang, D. Mao and J. Wanliss, Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition. _J. Geophys. Res._ , 115 (2010) A10219,
* [35] K. S. Lau and S. M. Ngai, Multifractal measures and a weak separation condition. _Advances in Mathematics_ , 141 (1999) 45-96.
* [36] E.W. Dijkstra, A note on two problems in connexion with graphs. _Numerische Mathematik_ , 1 (1959) 269-271.
* [37] M.E.J. Newman, The structure and function of complex networks. _SIAM Review_ , 45 (2003) 167-256
* [38] A.L. Barabsi and R. Albert, Emergence of scaling in random networks. _Science_ , 286 (1999) 509-512.
* [39] M.E.J Newman and D.J. Watts, Renormalization group analysis of the small-world network model. _Phys. Lett._ , 263 (1999) 341-346.
* [40] D.J. Watts and S.H. Strogatz, Collective dynamics of ’small-world’ networks. _Nature_ , 393 (1998) 440-442.
* [41] J.H.H. Opheusden, M.T.A. Bos, and G. van der Kaaden, Anomalous multifractal spectrum of aggregating Lennard-Jones particles with Brownian dynamics. _Physica A_ , 227 (1996) 183-196.
* [42] T.G. Smith and G.D. Lange, Biological cellular morphometry-fractal dimensions, lacunarity and multifractals. in _Fractal in Biology and Medicine_ (Birkhäuser, Basel, 1998).
* [43] E. Fern$\acute{a}$ndez, J.A. Bolea, G. Ortega and E. Louis, Are neurons multifractals? in _Journal of Neuroscience Methods_ , 89 (1999) 151-157.
* [44] Cytoscape software: http://cytoscapeweb.cytoscape.org/
* [45] BioGRID: http://thebiogrid.org/download.php
* [46] DIP: http://dip.doe-mbi.ucla.edu/dip/Download.cgi
* [47] E. Lee, H. Jung, P. Radivojac, J.W. Kim and D. Lee, Analysis of AML genes in dysregulated molecular networks. _BMC Bioinformatics_ , 10 (Suppl 2) (2009) S2.
Table 1: Comparison of different scale-free networks Number of nodes | Number of edges | Diameter | $\gamma$ | Max(Dq) | Lim(Dq) | $\Delta Dq$
---|---|---|---|---|---|---
500 | 499 | 13 | 1.94 $\pm$ 0.02 | 2.67 | 1.36 | 1.31
1000 | 999 | 16 | 2.02 $\pm$ 0.07 | 2.93 | 1.47 | 1.46
1500 | 1499 | 17 | 2.09 $\pm$ 0.04 | 2.96 | 1.65 | 1.30
2000 | 1999 | 20 | 1.99 $\pm$ 0.08 | 3.05 | 1.76 | 1.29
3000 | 2999 | 20 | 2.06 $\pm$ 0.04 | 3.26 | 1.83 | 1.44
4000 | 3999 | 23 | 2.09 $\pm$ 0.03 | 3.32 | 1.80 | 1.52
5000 | 4999 | 23 | 2.08 $\pm$ 0.04 | 3.26 | 1.75 | 1.51
6000 | 5999 | 22 | 2.06 $\pm$ 0.04 | 3.39 | 1.88 | 1.51
7000 | 5999 | 28 | 2.08 $\pm$ 0.04 | 3.39 | 2.10 | 1.29
8000 | 5999 | 25 | 1.91 $\pm$ 0.12 | 3.33 | 2.11 | 1.22
Table 2: Comparison of different small-world networks and regular networks with 5000 nodes Number of nodes | Number of edges | Diameter | p | Max(Dq) | Lim(Dq) | $\Delta Dq$
---|---|---|---|---|---|---
5000 | 25159 | 33 | 0.03 | 2.31 | 2.28 | 0.03
5000 | 25207 | 29 | 0.04 | 2.43 | 2.37 | 0.06
5000 | 25290 | 23 | 0.06 | 2.56 | 2.53 | 0.03
5000 | 25358 | 23 | 0.08 | 2.66 | 2.63 | 0.03
5000 | 25513 | 18 | 0.1 | 2.81 | 2.75 | 0.06
5000 | 25621 | 15 | 0.13 | 2.89 | 2.83 | 0.06
5000 | 25792 | 15 | 0.15 | 2.99 | 2.93 | 0.06
5000 | 26017 | 12 | 0.2 | 3.08 | 3.04 | 0.04
regular network 5000 | 250000 | 50 | 0 | 1 | 1 | 0.00
Table 3: Comparison of different random networks Number of nodes | Number of edges | Diameter | p | Max(Dq) | Lim(Dq) | $\Delta Dq$
---|---|---|---|---|---|---
449 | 610 | 15 | 0.005 | 2.42 | 2.14 | 0.28
994 | 2502 | 8 | 0.005 | 3.32 | 2.87 | 0.45
1991 | 5939 | 9 | 0.003 | 3.73 | 3.41 | 0.32
2484 | 6310 | 11 | 0.002 | 3.70 | 3.33 | 0.37
2790 | 4374 | 18 | 0.001 | 3.29 | 2.95 | 0.34
3373 | 5978 | 15 | 0.001 | 3.47 | 3.15 | 0.32
3931 | 8125 | 13 | 0.001 | 3.67 | 3.35 | 0.31
4919 | 10179 | 13 | 0.0008 | 3.78 | 3.39 | 0.39
5620 | 8804 | 16 | 0.00058 | 3.54 | 3.21 | 0.33
Table 4: Comparison of different PPI networks Networks | Number of nodes | Number of edges | Diameter | $D_{0}$ | Max(Dq) | Lim(Dq) | $\Delta Dq$
---|---|---|---|---|---|---|---
Human | 8934 | 41341 | 14 | 2.34 | 4.89 | 2.65 | 2.24
Drosophila Melanogaster | 7476 | 26534 | 11 | 2.34 | 4.84 | 2.87 | 1.97
S. cerevisiae | 4976 | 21875 | 10 | 2.36 | 4.62 | 2.48 | 2.14
E.coli | 2516 | 11465 | 12 | 2.14 | 4.15 | 2.10 | 2.05
H.pylori | 686 | 1351 | 9 | 2.27 | 3.47 | 1.91 | 1.56
C.elegans | 3343 | 6437 | 13 | 2.28 | 4.47 | 1.49 | 2.98
Arabidopsis Thaliana | 1298 | 2767 | 25 | 1.83 | 2.51 | 1.62 | 0.89
Figure 1: Fractal scaling of the human PPI network (o) and its skeleton (.).
The fractal dimension is the absolute value of the slope of each linear fit,
which is 2.20 $\pm$ 0.09 for the original network and 2.07 $\pm$ 0.09 for its
skeleton. Figure 2: The traditional box-counting algorithm may result in
different numbers of boxes needed to cover the entire network. Figure 3:
Linear regressions of the Arabidopsis Thaliana PPI network. Figure 4: The
$D_{q}$ curves for theoretically generated scale-free networks. Figure 5:
Generation of small-world networks by the NW model. Figure 6: The $D_{q}$
curves for theoretically generated small-world networks. Figure 7: The $Dq$
curves for different random networks. Figure 8: The $D_{q}$ curves for
different PPI networks.
|
arxiv-papers
| 2011-08-25T06:03:09 |
2024-09-04T02:49:21.772376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dan-Ling Wang, Zu-Guo Yu and Vo Anh",
"submitter": "Zu-Guo Yu",
"url": "https://arxiv.org/abs/1108.5014"
}
|
1108.5124
|
11institutetext: Michael J. Way 22institutetext: NASA Goddard Institute for
Space Studies, 2880 Broadway, New York, NY, USA, 22email:
michael.j.way@nasa.gov
# Challenges for LSST scale data sets
Michael J. Way
###### Abstract
The Large Synoptic Survey Telescope (LSST) simulator being built by Andy
Connolly and collaborators is an impressive undertaking and should make
working with LSST in the beginning stages far more easy than it was initially
with the Sloan Digital Sky Survey (SDSS). However, I would like to focus on an
equally important problem that has not yet been discussed here, but in the
coming years the community will need to address – can we deal with the flood
of data from LSST and will we need to rethink the way we work?
## 1 Changing the way we work: From 2MASS to SDSS
Perhaps the best way to start things is to compare two large area sky surveys
implementing the “standard way” of distributing data in their own time: The
1990s era Two Micron All Sky Survey (2MASS; Skrutskie et al., 2006) and the
2000s era Sloan Digital Sky Survey (SDSS; York et al., 2000).
Initially if a researcher wanted to access the
2MASS111http://www.ipac.caltech.edu/2mass survey data one could obtain a 5 DVD
set (double-sided) of the catalog data. The data were bar-delimited ascii text
which could be easily read by everything from legacy scripting programs like
awk to SQL databases like MySQL or Postgres. The ascii source catalog was
about 43GB in size if copied from the DVDs to local disk. The full-fidelity
Atlas Images ($\sim$10 terabytes in size and not available via DVD) were later
accessible via the 2MASS Image Services
website222http://irsa.ipac.caltech.edu/applications/2MASS/IM.
In essence, the average astronomer had to change almost nothing about the way
that they or their Ph.D. advisor had worked over the previous 30 years. For
example, instead of ordering 9-track (1/2 inch = 12.7mm), exabyte (8mm), or
DDS/DAT (4mm) tapes from the observatory (or bringing them along after an
observing run) one simply ordered the 2MASS DVDs. This was possible due to
increases in computer cpu speed and memory capacity combined with modest
input/output (IO) improvements over the previous 3-4 decades.
All of this changed with the SDSS. It may have been possible to distribute DVD
copies of the SDSS in a similar way to that of the 2MASS, but the scale had
moved from gigabytes to terabytes. Having a few terabytes on a local computer
in the early 2000s was not common, so the SDSS took a different route. Working
with top-notch computer scientists such as Jim Gray of Microsoft they decided
that much of the SDSS should be accessible via SQL query. There was certainly
some anxiety amongst much of the community when they realized that their mode
of obtaining data from the SDSS was going to be markedly different than in the
past. Hence, it took the community some time to learn this new way of working,
and certainly some early publications with SDSS data not published by the SDSS
team were problematic because, for example, they did not realize that they
could decide the quality of the photometry at a fine level, unlike that of the
2MASS which was relatively straightforward.
The SDSS is probably the most similar data set today compared with what the
LSST will look like and how we will interact with it. Currently most users of
SDSS use the casjobs333http://casjobs.sdsss.org interface to obtain their
data. The back end is a SQL database tied to a front end presented to the user
as a web interface where SQL queries are entered. The database comes with a
Schema Browser444http://casjobs.sdss.org/dr7/en/help/browser/browser.asp that
allows one to explore items of possible interest. There are also a host of on-
line tutorials that one can go through to understand how make the queries, and
many authors also publish their SDSS queries in the appendices of their
publications. However, not enough authors do the latter in my opinion, and
hence it is often impossible to replicate the data that people are using if
the original author does not publish or cannot recall their original query.
In the ten years since the creation of the SDSS disk data storage density has
continued its inexorable rise (see Sect. 2) and today one could in fact host
the entire SDSS database relatively easily and cheaply on a modestly sized
desktop computer (e.g. an off-the-shelf workstation with 4 $\times$ 2TB disks
would do the trick). Again, one could (in theory) dump the entire casjobs
catalog to a giant ascii file akin to that of the 2MASS and use awk or your
favorite fortran program on it. One would need a system that can use 64bit
addressing, but that is fairly standard today (2011). Of course the IO will
make things slow ($\sim$ 4 hours to read a 1TB disk sequentially), but
nonetheless it is in theory quite possible. The question then arises, will one
be able to work with LSST in the same manner as the SDSS given the inexorable
rise of faster CPUs, Memory, and IO?
## 2 Changing the way we work: From SDSS to LSST
As we consider the move from SDSS sized data sets to LSST the questions that
people like us might ask at this stage are fairly straightforward:
1. 1.
Will one be able to have a copy of the LSST data-set on a local desktop? This
will allow researchers to continue their pre-SDSS era methods of data
interrogation. This is what we like to call the 2MASS mode.
2. 2.
Will one still utilize a casjobs type web-query interface to obtain LSST data
of interest and then use legacy home-grown tools to work on the data? This is
what we call the SDSS mode.
3. 3.
Must one completely change the way one works with LSST scale data sets? Will
“data locality” be required – will one have to do all of the operations to
obtain a project’s scientific goals on the database directly? This may be
called the LSST mode.
To attempt to answer these questions we have to look at several other factors
discussed in the following subsections.
### 2.1 LSST Database Size and possible architecture
We heard from Andy Connolly and Kirk Borne at this conference that the LSST
query database will be of order 10 petabytes (PB) in size, while there will be
around 60PB of images available after 10 years of operation. It turns out that
query scales almost linearly with the size of the database. Given historical
trends in CPU, memory, storage and IO this means one should be able to derive
catalogs and do joins on tables in a future LSST database as we do today with
SDSS casjobs. However, there are caveats related to IO that will be discussed
later.
While query scales linearly with database size, the same cannot be said of the
kinds of operations that scientists would prefer to do on the data. For
example, classfication, clustering, density estimation are all normally O(N2)
or worse. However, earlier today Alex Gray showed us that his group has
managed to make a host of algorithms O(N) that are normally considered to be
O(N2).
Regardless, this points to some interesting issues. Assuming petabyte database
sizes, the needs to do operations that are O(N2), and the need to look at a
large fraction of the stored data (that will not fit in RAM) how are
researchers going to do these things on the LSST database of tomorrow? Let’s
touch on the possible need for “data locality”. Normally one should only
consider moving the data from the source of the data if one needs more than
100,000 CPU cycles per byte of data (Bell et al., 2006). The kinds of
applications this brings to mind are Seti@HOME or cryptography. Thankfully
most science applications are more information intensive with CPU to IO ratios
less than 10,000 to 1. This means that we may have two reasons to consider the
possibility that we will not actually download the data to our local
machine/data-center: The size of the database is too large (petabyte scale)
and we have CPU to IO ration of less than 100,000 to 1. We will address these
issues in some detail in the next section, but for now let’s assume we will
need to do some calculations at the site of the database itself.
The LSST has teamed up with several industry partners to develop a new
database to host the LSST called SciDB555http://www.scidb.org. The current
plan is to host this database in several different geographic locations
(obviously to avoid a single point of failure and to handle the anticipated
load), but they also currently plan to have an R interface for “expert users”
As I mentioned during my talk, I think this is an excellent idea, but I hope
the designers will consider adding additional languages such as Python which
currently has wrappers to support a host of useful tools commonly used by the
current generation of younger astronomers666For example, numpy, scipy, Rpy (R
interface), mlabwrap (MatLab), etc.
### 2.2 Moving the data around – can I have a local copy and make use of it?
Will one be able to download and store the LSST database to a desktop computer
in 10 years time? If one wishes to download 1 petabyte over a dedicated 1
gigabit/second line (in common use today) it will take $\sim$100 days. In 9
years let’s assume everyone will have 10 gigabit/second connections (the
growth in desktop network speed has not grown at the same accelerated rate of
storage or CPU) so that means it will only take about 10 days. That doesn’t
sound unreasonable. Now one has to ask, how much will it cost to own 1
petabyte of storage? One can look at historical trends documented in several
places on the internet to get some idea777e.g. http://www.mkomo.com/cost-per-
gigabyte. In Table 2.2 you can see what disks costs were 10 years ago, today
and by extrapolation in 10 years time.
Table 1: Storage cost historical trendsa
Year | Cost/GB | Cost/TB | Cost/PB
---|---|---|---
2000 | $19.00000 | $19,000 | $19 million
2010 | $00.06000 | $62 | $62,000
2020 | $00.00002 | $0.2 | $200
a Extrapolated from http://www.mkomo.com/cost-per-gigabyte
Looking at Table 2.2 one comes to the conclusion that if one wants to keep a
copy of the LSST data locally it should not be a problem given the drop in
price over time. After all, who who would have imagined 15 years ago that they
would be able to purchase a 1 terabyte drive for their desktop computer for
under $100?
Unfortunately things are not this simple. To illustrate my point I want to
recall some more of Amdahl’s rules of thumb for a balanced system.
1. 1.
Bandwidth (BW): One bit of sequential IO/second per instruction/second
2. 2.
Memory: $\alpha$=1=MB/MIPS888Million Instructions Per Second: one byte of
memory per one instruction/sec
3. 3.
One IO operation per 50,000 instructions
Looking at Table 2.2 in the context of Amdahl’s ROT perhaps the biggest
problem with high performance computer centers today and into the near future
is that they are CPU rich, but IO poor. The cpus may spend a lot of time
sitting idle while waiting for IO to send them more to work on because not
everything can be stored in RAM. This problem is not going to go away, but
there are (thankfully) people aware of the issue who believe that it is
currently possible to keep power consumption low while increasing sequential
read IO throughput by an order of magnitude using what are called Amdahl
blades (Szalay et al., 2010). Note that power consumption is becoming an issue
for mid-level data centers found at Universities and some goverment research
labs. Most of these don’t have Google’s electricity budget for powering them
and in fact many government data centers are even being shutdown to save
money999http://www.nytimes.com/2011/07/20/technology/us-to-close-800-computer-
data-centers.html.
Table 2: Conclusions from Amdahl’s rules of thumb for a balanced system today
Operations | RAM | Disk I/O | | No. of Disks for that
---|---|---|---|---
per second | | bytes/s | | BW at 100MB/s/disk
Giga/109 | GB | 108 | $\rightarrow$ | 1
Tera/1012 | TB | 1011 | $\rightarrow$ | 1000
Peta/1015 | PB | 1014 | $\rightarrow$ | 1,000,000
Table 2.2 tells one a couple of other interesting things.101010This table is a
modified version of one given in a talk by Alex Szalay that the author became
aware of recently. First, for a Peta-scale balanced system 100TB/s of IO
bandwidth (last row of column three = 1014) would be required. It will take
approximately 1,000,000 disks to deliver this bandwidth today assuming they
are capable of 100MB/s/disk. Note that the rate of disk IO growth has not been
remarkable in the past 10 years (see Szalay et al., 2010).
## 3 Conclusions
In the beginning of Section 2 I posed three questions and I would like to pose
some answers:
1. 1.
Will one be able to have a copy of the LSST data-set on a local desktop? Yes,
I think the average researcher will be able to have a copy of the data on
their local system assuming disk storage density continues its historical
trend (note that there are a number of technical arguments against this, as
there are for continuing Moore’s law into the future (Esmaeilzadeh et al.,
2011)111111http://www.nytimes.com/2011/08/01/science/01chips.html. However,
even if one has a copy of the LSST it is unlikely one will be able to make
much use of it using traditional 2MASS mode tools given the issues with
sequential IO that were outlined above.
2. 2.
Will one still utilize a casjobs type web-query interface to obtain LSST data
of interest and then use legacy home-grown tools to work on the data (The SDSS
mode.)? Yes, the LSST team seems interested in continuing the use of a casjobs
type interface with an SQL backend. Whether a researcher will then be able to
use their traditional home-grown tools will depend on the data sizes they
download as discussed above.
3. 3.
Must one completely change the way one works with LSST scale data sets? I
believe that many of the scientific goals will only be achievable by running
on the database locally as an “expert user”. This points to the need to have a
multitude of robust data/computational centers hosting the LSST data. Today
the best place for these (in the United States) would probably be at the
national level supercomputing centers such as PSC121212Pittsburg
Supercomputing Center, NSCA131313National Center for Supercomputer
Applications in Illinois or NAS141414NASA Advanced Supercomputing center at
NASA/Ames in California to name a few in the USA.
###### Acknowledgements.
Thanks to Andy Connolly for taking the time to discus his LSST simulator with
me prior to the conference and for encouraging me in my belief that a
commentary focused on computational challenges would be appropriate. I would
also like to thank the Astrophysics Department at Uppsala University in Sweden
for their gracious hospitality while part of this manuscript was being
completed.
## References
* Bell et al. (2006) Bell, G., Gray, J., & Szalay, A. (2006) Petascale Computational Systems. In: Computer, vol 39, no 1, pp. 110-112, doi:10.1109/MC.2006.29
* Esmaeilzadeh et al. (2011) Esmaeilzadeh, H., Blem, E., St. Amant, R., Sankaralingam, K. & Burger, D. (2011) Dark Silicon and the end of Multicore Scaling. In: Proc. of the 38th International Symposium on Computer Architecture, doi:10.1145/2000064.2000108
* Skrutskie et al. (2006) Skrutskie, M.F. et al. (2006) The Two Micron All Sky Survey (2MASS). In: The Astronomical Journal, 131, 1163, doi:10.1086/498708
* York et al. (2000) York, D.G., et al. (2000) The Sloan Digital Sky Survey: Technical Summary. In: The Astronomical Journal, 120, 1579, doi:10.1086/301513
* Szalay et al. (2010) Szalay, A., Bell, G.C, Huang, H.H., Terzis, A. & White, A. (2009) Low-Power Amdahl-Balanced Blades for Data Intensive Computing. In: ACM SIGOPS Operating Systems Review archive, vol 44, issue 1, ACM New York, NY, USA doi:10.1145/1740390.1740407
|
arxiv-papers
| 2011-08-25T16:04:45 |
2024-09-04T02:49:21.780163
|
{
"license": "Public Domain",
"authors": "M.J. Way",
"submitter": "Michael Way",
"url": "https://arxiv.org/abs/1108.5124"
}
|
1108.5573
|
LHCb-PROC-2011-043
$\gamma/\phi_{3}$ at hadron colliders
Marie-Hélène Schune (LHCb) for the CDF and LHCb collaborations
LAL, Université Paris-Sud, CNRS/IN2P3, 91898 Orsay, France
Abstract
> In the first part of the document I describe in a general way the
> $\gamma/\phi_{3}$ extraction and compare the experimental environments. I
> then switch to the available results from the CDF experiment. In the third
> part I present early results from the LHCb experiment, which are promising
> first steps on the way to a future $\gamma/\phi_{3}$ measurement.
> PRESENTED AT
>
>
>
>
> The Ninth International Conference on
> Flavor Physics and CP Violation
> (FPCP 2011)
> Maale Hachamisha, Israel, May 23–27, 2011
## 1 Introduction
Since its first observation in $K^{0}_{\rm\scriptscriptstyle L}$ decays [1],
CPV has been observed and studied in several K and B meson decay modes [2]. In
the standard model of particle physics (SM) CP violation in the quark sector
of the weak interactions arises from a single irreducible phase in the
Cabibbo-Kobayashi-Maskawa (CKM) matrix [3] that describes the mixing of the
quarks.
$V_{\rm{CKM}}=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\\\ \end{array}\right)$ (1)
There is no theoretical predictions for the different values of the matrix
elements which thus need to be determined experimentally. In fact the various
CKM matrix elements are rather different in size. This strong hierarchy is
clearly visible using the improved Wolfenstein parametrization which is an
expansion in power of $\lambda$ :
$V_{\rm{CKM}}=\left(\begin{array}[]{ccc}1-\frac{\lambda^{2}}{2}-\frac{\lambda^{4}}{8}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda+\frac{A\lambda^{5}}{2}(1-2\rho)-iA^{2}\lambda^{5}\eta&1-\frac{\lambda^{2}}{2}-\lambda^{4}(\frac{1}{8}+\frac{A^{2}}{2})&A\lambda^{2}\\\
A\lambda^{3}(1-(1-\frac{\lambda^{2}}{2})(\rho+i\eta))&-A\lambda^{2}(1-\frac{\lambda^{2}}{2})(1+\lambda^{2}(\rho+i\eta))&1-\frac{A^{2}\lambda^{2}}{2}\\\
\end{array}\right)+{\cal O}(\lambda^{6})$ (2)
with $\lambda=s_{12}\simeq 0.22$. The $\lambda,A,\rho$ and $\eta$ parameters
are known with different accuracies : $\lambda$ is known to about 0.5 %, $A$
to about 2%, $\eta$ to about 5% and $\rho$ only to 15 to 20 %***In several
cases one uses the variables $\overline{\rho}$ and $\overline{\eta}$
$\overline{\rho}=\rho(1-\lambda^{2}/2)$ and
$\overline{\eta}=\eta((1-\lambda^{2}/2)$.
The unitarity of the CKM matrix imposes a set of relations among its elements,
including the condition $V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0$
which defines a triangle in the complex plane, shown in Fig. 1. Many
measurements can be conveniently displayed and compared as constraints on
sides and angles of this triangle. CP violation is proportional to the area of
the unitarity triangle and therefore it requires that all sides and angles be
different from zero. The angle
$\gamma\equiv\phi_{3}=arg(-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*})$ is the least
precisely known [4].
Figure 1: Left: Unitarity triangle in the $(\overline{\rho},\overline{\eta})$
plane. $(1-\overline{\rho}-i\overline{\eta})$ corresponds to
$\frac{V_{td}}{\lambda V_{cb}}$ and $(\overline{\rho}+i\overline{\eta})$ to
$\frac{V^{*}_{ub}}{\lambda V_{cb}}$. Right: experimental constraints in the
$(\overline{\rho},\overline{\eta})$ plane. The closed contours show the most
probable region at 68 % and 95 % CL. The individual constraints (given at 95 %
CL) are shown by the coloured regions.
For accurate amplitude predictions from the Standard Model, the four
parameters of the CKM mass mixing matrix, in particular $\overline{\rho}$ and
$\overline{\eta}$, must be measured precisely. Furthermore, this must be done
in a way that the extracted parameters are not affected by possible new
physics contributions, i.e. done only with the quantities related to tree
diagrams. One of the most promising ways is to combine the measurement of
$|V_{ub}/V_{cb}|$ via semileptonic decays of the $B$ meson, which depends on
$\sqrt{\overline{\rho}^{2}+\overline{\eta}^{2}}$, and CP violation in
$B\rightarrow DK$ decays, which gives
$\arctan(\overline{\eta}/\overline{\rho})$ (the angle $\gamma/\phi_{3}$ of the
unitarity triangle.
Different approaches have been used to measure the angle $\gamma$ (or
$\phi_{3}$) of the unitarity triangle. They exploit the interference between
$b\rightarrow c$ and $b\rightarrow u$ transitions. Up to now, this has been
achieved using decays of type $B\\!\to D^{0}K$ and $B\\!\to\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K$ with subsequent decays into final
states accessible to both charmed meson and anti-meson†††In all this paper the
mention of a decay will refer also to its charge-conjugate state, except
explicitely stated.. They are classified in three main types:
The GLW method
[5]: the $D^{0}$ meson decays into a CP final state
The ADS method
[6]: the $D^{0}$ meson is reconstructed into the $K\pi$ final state, for the
$b\rightarrow c$ (resp. $b\rightarrow u$) transitions the $D^{0}$ decay mode
will be the Cabibbo suppressed: $D^{0}\rightarrow K^{+}\pi^{-}$ (resp. Cabibbo
allowed: $D^{0}\rightarrow K^{-}\pi^{+}$). In this way the magnitude of the
two interfering amplitudes will not be too different.
The GGSZ method
[7]: the $D^{0}$ final state is $K^{0}_{\rm\scriptscriptstyle S}\pi\pi$ which
is accessible to both $D^{0}$and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$. This requires analysis of the
$D^{0}$ Dalitz plot. This strategy can be seen as a mixture of the two
previous ones, depending on the position in the Dalitz plot.
If one writes $A(B^{-}\\!\to D^{0}K^{-})=A_{c}e^{i\delta_{c}}$, $A(B^{-}\\!\to
D^{0}K^{-})=A_{u}e^{i\delta_{u}-\gamma}$, $A(D^{0}\rightarrow
f)=A_{f}e^{i\delta_{f}}$ and $A(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
f)=A_{\overline{f}}e^{i\delta_{{\overline{f}}}}$ ($f$ is a generic final state
of the $D$ meson) then it can be shown that :
$\Gamma\left(B^{-}\rightarrow
f_{D}K^{-}\right)=A_{c}^{2}A_{\overline{f}}^{2}\left(\frac{A_{f}^{2}}{A_{\overline{f}}^{2}}+r_{B}^{2}+\frac{2r_{B}A_{f}}{A_{\overline{f}}}\Re\left({e^{i(\delta_{B}+\delta_{D}-\gamma)}}\right)\right)$
(3)
where $r_{B}=A_{u}/A_{c}$, $\delta_{B}=\delta_{u}-\delta_{c}$ and
$\delta_{D}=\delta_{f}-\delta_{{\overline{f}}}$. The rate for the charge
conjugate process is obtained from Eq.3 replacing $\gamma$ by $-\gamma$. The
first measurements have been done by the B factories (BaBar and BELLE) which
still have the most precise measurements. However, in both cases, the data
taking period has ended and the $\gamma/\phi_{3}$ angle is still not precisely
measured. An important milestone has been achieved by the CDF experiment,
operating at the Tevatron proton-antiproton collider, which has demonstrated
that GLW or ADS analyses (see section 2) are possible in an hadronic
environment. A summary of the experimental conditions and event yields in a
representative decay mode is given in Tab. 1.
Experiment | $\sigma_{b\overline{}b}$ | $\sigma_{\mathrm{inel}}/\sigma_{b\overline{}b}$ | Integrated luminosity | $B^{-}\rightarrow(K^{-}\pi^{+})_{D}K^{-}$
---|---|---|---|---
| | | | yield
BaBar | $\sim 1$ nb | $\sim 4$ | 425 fb-1 | $\sim 1940$
BELLE | $\sim 1$ nb | $\sim 4$ | 700 fb-1 | $\sim 3400$
CDF | $\sim 100\mu$b | $\sim 1000$ | $\sim$ 10 fb-1 full dataset | $\sim 1500$ (5 fb-1)
LHCb | $\sim 290\mu$b | $\sim 300$ | $\sim$ 1 fb-1 (2011 expected) | $\sim 440$ (0.035 fb-1)
Table 1: Summary of the various experimental conditions. For the Tevatron and
LHCb the yields correspond to those of the analyses so far performed. The
Cabibbo-allowed decay $B^{-}\rightarrow(K^{-}\pi^{+})_{D}K^{-}$ provides an
estimate of the selection and trigger efficiencies for the various detectors.
Note also that with the LHCb experiment it will also be possible to measure
$\gamma/\phi_{3}$ in a $B^{0}_{s}\\!\to D^{+}_{s}K$ time dependent analysis,
thanks to the large $B^{0}_{s}$ sample, excellent proper time resulution and
particle identification capabilities of the experiment. These proceedings are
organized as follow: after having shown the available results from CDF, which
will allow me to present in more details the GLW and ADS methods, I will
switch to LHCb results. Indeed, even if LHCb is not yet able to perform
$\gamma/\phi_{3}$ measurements with the very little integrated lumi collected
in 2010, some encouraging results are already available.
## 2 CDF results
### 2.1 GLW measurements
In the method proposed by GLW the $D^{0}$ meson is reconstructed to CP
eigenstate final states. Due to the high background level at hadronic
colliders the CP=-1 (e.g. $K^{0}_{\rm\scriptscriptstyle S}$$\pi^{0}$) modes
are not accessible and in fact only the CP=+1 $K^{+}K^{-}$ $D^{0}$ final state
has been used. Using a CP=+1 final state for the $D$ meson leads to a
simplified form for Eq.3 using $\delta_{D}=0$ and
$\frac{A_{f}}{A_{\overline{f}}}=1$:
$\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)=A_{c}^{2}A_{f_{CP+}}^{2}\left(1+r_{B}^{2}+r_{B}\cos\left(\delta_{B}-\gamma\right)\right)$
(4)
If CP=-1 final states could be reconstructed the formula would be similar
except a - sign in front of the cosine term. From Eq.4 one can derive the
asymmetries which are experimentally measured:
$A_{CP+}=\frac{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)-\Gamma\left(B^{+}\rightarrow
f_{CP+}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)+\Gamma\left(B^{+}\rightarrow
f_{CP+}K^{+}\right)}=\frac{2r_{B}\sin\delta_{B}\sin\gamma}{1+r_{B}^{2}+2r_{B}\cos\delta_{B}\cos\gamma}$
(5) $R_{CP+}=\frac{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)+\Gamma\left(B^{+}\rightarrow
f_{CP+}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D^{0}K^{-}\right)+\Gamma\left(B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\right)}=1+r_{B}^{2}+2r_{B}\cos\delta_{B}\cos\gamma\\\
$ (6)
in case of a CP=-1 $D$ meson final state one would have:
$A_{CP-}=\frac{-2r_{B}\sin\delta_{B}\sin\gamma}{1+r_{B}^{2}-2r_{B}\cos\delta_{B}\cos\gamma}\\\
R_{CP-}=1+r_{B}^{2}-2r_{B}\cos\delta_{B}\cos\gamma\\\ $ (7)
It should be noted that only 3 of these 4 ratios are independent while there
are 3 unknowns. In addition, due to the formulae themselves there is an 8-fold
ambiguity. In summary, the GLW measurements have a poor constraining power on
$\gamma/\phi_{3}$ when they are considered alone. However, they generally
improve the knowledge of $r_{B}$, $\gamma/\phi_{3}$ and $\delta_{B}$ when
combined with other methods (for example the ADS method). The CDF
collaboration has performed an analysis [8] on a sample collected by the
upgraded CDF detector corresponding to an integrated luminosity of $\sim 1$
fb-1. Both $B^{-}\rightarrow D^{0}K^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}$
decay modes are reconstructed and the $D^{0}$ in the $K^{-}\pi^{+}$ flavour
final state and the two CP=+1 final states $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$.
The $B^{-}\rightarrow D^{0}\pi^{-}$ modes are much more abundant than the
$B^{-}\rightarrow D^{0}K^{-}$ and provide useful constraints in the fit. An
unbinned maximum likelihood fit which combines invariant $B$ mass assuming
that the bachelor track is a pion, momenta, and Particle IDentification (PID)
information obtained from dE/dx measurement for all three modes of interest is
performed simultaneously. The measured yields are given in Tab. 2 and the
final ratios are:
$\displaystyle\frac{\Gamma\left(B^{-}\rightarrow
D^{0}K^{-}\right)+\Gamma\left(B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D^{0}\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\right)}=0.0745\pm
0.0043\mathrm{(stat.)}\pm 0.0045\mathrm{(syst.)}$ (8) $\displaystyle
R_{CP+}=\frac{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)+\Gamma\left(B^{+}\rightarrow
f_{CP+}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow D^{0}K^{-}\right)}=1.30\pm
0.24\mathrm{(stat.)}\pm 0.12\mathrm{(syst.)}$ (9) $\displaystyle
A_{CP+}=\frac{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)-\Gamma\left(B^{+}\rightarrow
f_{CP+}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
f_{CP+}K^{-}\right)+\Gamma\left(B^{+}\rightarrow f_{CP+}K^{+}\right)}=0.39\pm
0.17\mathrm{(stat.)}\pm 0.04\mathrm{(syst.)}$ (10)
| $B^{+}\rightarrow DK^{+}$ | $B^{-}\rightarrow DK^{-}$
---|---|---
$D\rightarrow K\pi$ | $250\pm 26$ | $266\pm 27$
$D\rightarrow KK$ | $22\pm 8$ | $49\pm 11$
$D\rightarrow\pi\pi$ | $6\pm 6$ | $14\pm 6$
Table 2: Summary of the measured yields in the CDF GLW analysis.
For all these three ratios, one of the main source of systematical uncertainty
is the PID. Indeed the kaon pion separation is only at 1.5 standard deviation
for a 2 GeV momentum track. Another source of systematical uncertainty is the
handling of the combinatorial background and this will improve with a larger
data sample. The CDF collaboration has now recorded more than 10 times more
data ; this should allow them to perform a more accurate measurement in the
future.
### 2.2 ADS measurements
The ADS measurement performed at the Tevatron collider by the CDF experiment
[9] uses about 5 fb-1. The $D^{0}$ meson is reconstructed using both the
Cabibbo Favoured (CF) $D^{0}\rightarrow K^{-}\pi^{+}$ ($D_{CF}$) and the
Cabibbo Suppressed (CS) $D^{0}\rightarrow K^{+}\pi^{-}$ ($D_{CS}$) modes. The
analysis strategy is similar to the one developed for the GLW measurement: not
only the $B^{-}\rightarrow DK^{-}$ is reconstructed but also the
$B^{-}\rightarrow D\pi^{-}$ decay mode. The cuts optimization is focused on
finding an evidence of the $B^{-}\rightarrow D_{CS}\pi^{-}$ mode. Since the
$B^{-}\rightarrow D_{CF}\pi^{-}$ mode has the same topology of the CS one, the
cuts optimization is performed on the CF mode. The optimized cuts are those
related to the flight of the $D$ and $B$ meson, the isolation of the $B$ meson
and PID. The fit is an unbinned maximum likelihood fit simultaneously
performed on the CF and DCS modes and spliting the samples according to the
$B$ charge. It uses two discriminating variables: the $B$ mass (asumming a
pion mass for the bachelor track) and the PID information for the bachelor
track. The projection on the $B$ mass for (CS) decays is shown in Figs. 2.
Figure 2: Fit projection onto the $D^{0}$$\pi$ variable for the doubly
Cabibbo-suppressed mode, positive charges (left) and negative charges (right).
The fitted yields can be translated into asymmetries both for the
$B^{-}\rightarrow D^{0}K^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}$ decay modes.
$\displaystyle{}R_{ADS}(\pi)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(B^{-}\rightarrow
D_{CS}\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{CS}\pi^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D_{CF}\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{CF}\pi^{+}\right)}=\left(4.1\pm 0.8{\mathrm{(}stat.)}\pm
0.4{\mathrm{(}syst.)}\right).10^{-3}$ $\displaystyle R_{ADS}(K)$
$\displaystyle=$ $\displaystyle\frac{\Gamma\left(B^{-}\rightarrow
D_{CS}K^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{CS}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D_{CF}K^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{CF}K^{+}\right)}=\left(22.5\pm 8.4{\mathrm{(}stat.)}\pm
7.9{\mathrm{(}syst.)}\right).10^{-3}$ $\displaystyle A_{ADS}(\pi)$
$\displaystyle=$ $\displaystyle\frac{\Gamma\left(B^{-}\rightarrow
D_{CS}\pi^{-}\right)-\Gamma\left(B^{+}\rightarrow
D_{CS}\pi^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D_{CS}\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{CS}\pi^{+}\right)}=0.22\pm 0.18{\mathrm{(}stat.)}\pm
0.06{\mathrm{(}syst.)}$ $\displaystyle A_{ADS}(K)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(B^{-}\rightarrow
D_{CS}K^{-}\right)-\Gamma\left(B^{+}\rightarrow
D_{CS}K^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D_{CS}K^{-}\right)+\Gamma\left(B^{+}\rightarrow D_{CS}K^{+}\right)}=-0.63\pm
0.40{\mathrm{(}stat.)}\pm 0.23{\mathrm{(}syst.)}$
These results are in agreement with B-factories measurements [10] and the
precision is similar to the one obtained by the BaBar collaboration.
## 3 LHCb results
From the two measurements described above, the CDF collaboration has proven
that $B$ physics with pure hadronic decays is also accessible to hadron
colliders experiments. At the time of FPCP conference the LHCb collboration
had only presented results based on 2010 data ($\sim 36$ pb-1)‡‡‡More than 10
times more data is already on tape at the time of writing these proceedings. I
will present some measurements made using decays similar to the ones used for
$\gamma/\phi_{3}$.
### 3.1 LHCb performances
Despite the fact that the LHCb detector has been taking data for only one
year, the experimental attributes critical for the $\gamma/\phi_{3}$ analysis
are already demonstrating a performance adequate for the measurement. These
attributes include the hadronic trigger, the displaced vertices reconstruction
and the PID for charged hadrons. LHCb operates a two-level trigger system: a
hardware trigger (L0) and a software implemented High Level Trigger (HLT). The
L0 trigger provided by the ECAL, MUON, and HCAL information reduces the
visible interaction rate from 10 MHz to 1 MHz ; the HLT reduces the trigger
rate further to 2 kHz. During the 2010 data taking period, several trigger
configurations were used both for the L0 and the HLT in order to cope with the
varying beam conditions. For the hadronic trigger the threshold was set to 3.6
GeV of transverse momentum for most of the data. The HLT confirms the L0
information using tracking data and adds some transverse momentum and impact
parameter requirements. The L0 efficiency was of the order of 50% for our
channels while the HLT one was more of the order of 80%. Note however that
triggering is also possible on tracks from the other $B$ decay present in the
event. The impact parameter resolution for tracks with large transverse
momentum is of the order of 15 $\mu m$ and the resolution on the primary
vertex reconstruction is of the order of 15 $\mu m$ in the transverse plane
and 75 $\mu m$ along the beam axis. At LHC, due to the large center of mass
energy the $B$ flies on average 1 cm before decaying. Finally, the PID
performances as a function of the track momentum, are measured on data using
the K and $\pi$ from the $D^{0}$decay originating from the decay of a $D^{*}$
with kinematical properties similar to our signal. On average, the kaon
efficiency is of the order of 95% for a pion contamination of 7%.
### 3.2 First signal yields for $B^{-}\rightarrow D^{0}K^{-}$ decays
Using 2010 data ($\sim 36$ pb-1), the LHCb experiment has been able to extract
clean samples of both $B^{-}\rightarrow D^{0}K^{-}$ and $B^{-}\rightarrow
D^{0}\pi^{-}$ decay modes. The good PID performances and impact parameter
resolution are of critical importance for the signal extraction from the large
hadronic background. The rescaled yields per fb-1 are summarized in Tab 3 for
LHCb and CDF. From these numbers, it is clear that the results expected for
the summer conferences from LHCb should be competitive with those from CDF
presented at this conference.
| CDF | LHCb
---|---|---
$B^{\pm}\rightarrow(K\pi)_{D}\pi^{\pm}$ | $\sim$ 3.5k | $\sim$ 223k
$B^{\pm}\rightarrow(K\pi)_{D}K^{\pm}$ | $\sim$ 0.3k | $\sim$ 12.6k
$B^{\pm}\rightarrow(KK)_{D}\pi^{\pm}$ | $\sim$ 0.8k | $\sim$ 28.6k
Table 3: Comparison of the measured yields rescaled to an integrated
luminosity of 1 fb-1. The choice is made to give numbers only for the modes
studied in the 2010 data set by LHCb.
### 3.3 First observation of the decay $\overline{B}_{s}\rightarrow
D^{0}K^{*0}$
Another promising way to measure $\gamma/\phi_{3}$ is to use the
$B_{d}\rightarrow DK^{*0}$ decay mode where $D$ represents an admixture of
$D^{0}$ and $\overline{D}^{0}$ mesons. Although the channel involves the decay
of a neutral B meson, the final state is self-tagging so that a time-dependent
analysis is not required. Since both the CKM favoured and suppressed decays
are also colour suppressed, the branching fractions are smaller than the
equivalent charged $B^{\pm}\rightarrow DK^{\pm}$ decays, but exhibit an
enhanced interference. The Cabibbo-allowed $B_{s}$ decays,
$\overline{B}_{s}\rightarrow D^{0}K^{*0}$ and $\overline{B}_{s}\rightarrow
D^{*0}K^{*0}$, potentially cause a significant background to the Cabibbo-
suppressed $B_{d}\rightarrow D^{0}K^{*0}$ decay. Moreover, the expected size
of this background is unknown, since the $\overline{B}_{s}\rightarrow
D^{(*)0}K^{*0}$ decay has not yet been observed. In addition, a measurement of
the branching fraction of $\overline{B}_{s}\rightarrow D^{0}K^{*0}$ is of
interest as a probe of SU(3) breaking in colour suppressed $B\rightarrow
D^{0}V^{0}$ decays. The detailed study of $\overline{B}_{s}\rightarrow
D^{0}K^{*0}$ is thus an important and interesting milestone towards the
measurement of $\gamma/\phi_{3}$. The strategy of the analysis is to measure a
ratio of branching fractions in which most of the potentially large systematic
uncertainties cancel. The decay $B_{d}\rightarrow D^{0}\rho^{0}$ for which
about two to eight times more events are expected than for the
$\overline{B}_{s}\rightarrow D^{0}K^{*0}$ decay, is used as the normalisation
channel. In both decay channels, the $D^{0}$ is reconstructed in the Cabibbo-
favoured decay mode $D^{0}\rightarrow K^{-}\pi^{+}$; the contribution from the
doubly Cabibbo-suppressed decay is negligible. The main systematic
uncertainties arise from the different particle identification requirements
and the pollution of the $B_{d}\rightarrow D^{0}\rho^{0}$ peak by
$B_{d}\rightarrow D^{0}\pi^{+}\pi^{-}$ decays where the $\pi^{+}\pi^{-}$ pairs
do not originate from a $\rho^{0}$ resonance. In addition, the normalization
of the $B_{s}$ decay to a $B_{d}$ decay suffers from a systematic uncertainty
of the order of about 13 % due to the ratio of the fragmentation fractions
$f_{d}/f_{s}=3.71\pm 0.47$. The $B_{s}\rightarrow D^{0}K^{*0}$ signal peak
obtained on 2010 data is shown on Fig. 3.
Figure 3: Invariant mass distribution for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\to D^{0}K^{*0}$ decay mode
with the fit superimposed. The black points correspond to the data and the fit
result is represented as a solid line. The signal is fitted with a double
Gaussian (dashed line), the partially reconstructed with an exponential
function (light grey area), the combinatorial background with a flat
distribution (dark grey area) and the cross-feed from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\to D^{0}\rho^{0}$ (intermediate
grey area).
A clear signal of $35.5\pm 6.9$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\to D^{0}K^{*0}$ events is
observed for the first time. The probability of the background fluctuating to
form the $B^{0}_{s}$ signal corresponds to approximately nine standard
deviations, as determined from the change in twice the natural logarithm of
the likelihood of the fit without signal. Although this significance includes
the statistical uncertainty only, the conclusion is unchanged if the small
sources of systematic error that affect the yields are included. The branching
ratio for this decay is measured relative to that for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\to D^{0}\rho^{0}$to be
$\frac{\mbox{\rm BR}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\to
D^{0}K^{*0}\right)}{\mbox{\rm BR}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\to D^{0}\rho^{0}\right)}=1.39\pm
0.31\pm 0.17\pm 0.18$
where the first uncertainty is statistical, the second systematic and the
third is due to the hadronisation fraction ($f_{d}/f_{s}$).
### 3.4 First observation of the decay $B_{d}\rightarrow D^{0}K\pi\pi$
In order to better understand $B$ decays and eventually add new modes for the
$\gamma/\phi_{3}$ measurement, the LHCb collaboration has studied various
$B\rightarrow D3h$ decays [12]. Among other signals, a clear peak of $122\pm
18$ events is obtained for the $B_{d}\rightarrow D^{0}K\pi\pi$ mode leading to
a measurement of ratio of branching ratios :
$\frac{\mbox{\rm BR}\left(B_{d}\rightarrow D^{0}K\pi\pi\right)}{\mbox{\rm
BR}\left(B_{d}\rightarrow D^{0}\pi\pi\pi\right)}=9.6\pm 1.5\pm 0.8$
The signal plot is shown on Fig. 4.
Figure 4: Invariant mass distribution for the $B_{d}\rightarrow D^{0}K\pi\pi$
mode obtained on 2010 data. The black points correspond to the data and the
fit result is represented as a solid line. The background is due to partially
reconstructed events (the shape in light grey is taken from Monte-Carlo) and
to combinatorial background (drak grey) modelled with a second order
polynomial.
### 3.5 Time dependent measurements
Information about $\gamma/\phi_{3}$ can also be obtained by other means: time
dependent measurements using the $B_{s}\rightarrow D_{s}K$ decay mode or
2-body charmless $B$ events. The data collected in 2010 do not allow for such
measurements. However the $B_{s}$ mixing frequency has been measured as well
its lifetime using the charmless decay $B_{s}\rightarrow KK$. More details are
given in [13] and [14] and in Marta Calvi’s contribution to these proceedings.
ACKNOWLEDGEMENTS
I would like to thank the FPCP organizers for this very nice conference with
many interesting discussions, my LHCb colleagues as well as the CDF
collaboration for providing the material discussed here. I am grateful to
Aurélien Martens and Guy Wilkinson for their careful reading of these
proceedings.
## References
* [1] J. H. Christenson et al., Phys. Rev. Lett. 13, 138 (1964).
* [2] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010)
* [3] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963);M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49,652 (1973).
* [4] CKMFitter collaboration :
http://www.slac.stanford.edu/xorg/ckmfitter/ckm_welcome.html
and UTFit collaboration :
http://www.utfit.org/UTfit/
* [5] M. Gronau and D. London, Phys. Lett. B 253, 483 (1991), M. Gronau and D. Wyler, Phys. Lett. B 265, 172 (1991).
* [6] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78, 3257 (1997), M. Gronau Phys. Lett. B 557, 198 (2003).
* [7] A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D 68, 054018 (2003).
* [8] The CDF collaboration, Phys. Rev. D 81, 031105 (2010) .
* [9] The CDF collaboration, CDF public note 10309.
* [10]
http://www.slac.stanford.edu/xorg/hfag/
* [11] The LHCb collaboration, LHCb-CONF-2011-008.
* [12] The LHCb collaboration, LHCb-CONF-2011-024.
* [13] The LHCb collaboration, LHCb-CONF-2011-005.
* [14] The LHCb collaboration, LHCb-CONF-2011-018.
|
arxiv-papers
| 2011-08-29T14:35:26 |
2024-09-04T02:49:21.791617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marie-H\\'el\\`ene Schune (LHCb) (for the LHCb and CDF collaborations)",
"submitter": "Marie-Helene Schune",
"url": "https://arxiv.org/abs/1108.5573"
}
|
1108.5748
|
Cusp geometry of fibered 3–manifolds
David Futer
[]Department of Mathematics, Temple University,
Philadelphia PA 19122, USA
Saul Schleimer
[]Department of Mathematics,
University of Warwick, Coventry CV4 7AL, UK
Futer is supported in part by NSF grant DMS–1007221.
Schleimer is supported in part by EPSRC grant EP/I028870/1.
This work is in the public domain.
[2000]57M50, 57M60, 30F40
Let $F$ be a surface and suppose that $\mon \from F \to F$ is a
pseudo-Anosov homeomorphism, fixing a puncture $p$ of $F$. The
mapping torus $M = M_\mon$ is hyperbolic and contains a maximal cusp
$C$ about the puncture $p$.
We show that the area (and height) of the cusp torus $\bdy C$ is equal
to the stable translation distance of $\mon$ acting on the arc complex
$\arc(F,p)$, up to an explicitly bounded multiplicative error. Our
proof relies on elementary facts about the hyperbolic geometry of
pleated surfaces. In particular, the proof of this theorem does not
use any deep results from Teichmüller theory, Kleinian group theory,
or the coarse geometry of $\arc(F,p)$.
A similar result holds for quasi-Fuchsian manifolds $N \homeo F \times
\RR$. In that setting, we find a combinatorial estimate for the area
(and height) of the cusp annulus in the convex core of $N$, up to
explicitly bounded multiplicative and additive error. As an
application, we show that covers of punctured surfaces induce
quasi-isometric embeddings of arc complexes.
§ INTRODUCTION
Following the work of Thurston, Mostow, and Prasad, it has been known
for over three decades that almost every $3$–manifold with torus
boundary admits a hyperbolic structure <cit.>, which
is necessarily unique up to isometry <cit.>. Thus, in principle, it is possible to translate
combinatorial data about a $3$–manifold into a detailed description
of its geometry — and conversely, to use geometry to identify
topological features. Indeed, given a triangulated manifold (up to
over 100 tetrahedra) the computer program SnapPy can typically
approximate the manifold's hyperbolic metric to a high degree of
precision <cit.>. However, building an effective dictionary
between combinatorial and geometric features, for all but the most
special families of manifolds, has proven elusive. The prevalence of
hyperbolic geometry makes this one of the central open problems in
low-dimensional topology.
§.§ Fibered $3$–manifolds
In this paper, we attack this problem for the class of hyperbolic
$3$–manifolds that fiber over the circle. Let $F$ be a connected,
orientable surface with $\chi(F) < 0$, and (for this paper) with at
least one puncture. Given an orientation-preserving homeomorphism
$\mon \from F \to F$ we construct the mapping torus
\[
M_\mon := \,\quotient{F \times [0,1]}{(x, 1) \sim (\mon(x), 0)}.
\]
Thus $M_\mon$ fibers over $S^1$, with fiber $F$ and monodromy $\mon$.
Thurston showed that $M_\mon$ is hyperbolic if and only if $\mon$ is
pseudo-Anosov: equivalently, if and only if $\mon^n(\gamma)$ is
not homotopic to $\gamma$ for any $n \neq 0$ and any essential simple
closed curve $\gamma \subset F$
<cit.>. See also Otal
<cit.>. In addition to the connection with dynamics,
fibered $3$–manifolds are of central importance in low-dimensional
topology because every finite-volume, non-positively curved
$3$–manifold has a finite–sheeted cover that fibers
For those fibered $3$–manifolds that are hyperbolic, the work of
Minsky, Brock, and Canary on Kleinian surface groups provides a
combinatorial, bi-Lipschitz model of the hyperbolic metric
<cit.>. The bi-Lipschitz constants depend only on
the fiber $F$ <cit.>. However, the existence
of these constants is proved using compactness arguments; as a result
the constants are unknown.
Using related ideas, Brock established the following notable entry in
the dictionary between combinatorics and geometry.
Let $F$ be an orientable surface with $\chi(F)<0$. Then there exist
positive constants $K_1$ and $K_2$, depending only on $F$, such that
the following holds. For every orientation-preserving, pseudo-Anosov
homeomorphism $\mon \from F \to F$, the mapping torus $M_\mon$ is a
hyperbolic $3$–manifold satisfying
\[
K_1 \, \dbar_\pants(\mon) \: \leq \: \vol(M_\mon) \:
\leq \: K_2 \, \dbar_\pants(\mon).
\]
Here $\pants(F)$ is the adjacency graph of pants decompositions of
$F$. Also $\dbar_\pants(\mon)$ is the stable translation distance of
$\mon$ in $\pants(F)$, defined in Equation (<ref>)
The constant $K_2$ in the upper bound can be made explicit. Agol
showed that the sharpest possible value for $K_2$ is $2v_8$, where
$v_8 = 3.6638...$ is the volume of a regular ideal octahedron
<cit.>. On the other hand, the constant $K_1$ is only
known in the special case when $F$ is a punctured torus or
$4$–puncture sphere; see Guéritaud and Futer <cit.>. For all other surfaces, it remains an open
problem to give an explicit estimate for $K_1$.
Brock's theorem is a template for obtaining combinatorial information;
the pants graph $\pants(F)$ is just one of many complexes naturally
associated to a surface $F$. Others include the curve complex
$\curve(F)$ and the arc complex $\arc(F)$; the latter is the main
focus of this paper. Using $\arc(F)$ we give effective
two-sided estimates for the geometry of maximal cusps in $M_\mon$.
Suppose $F$ is a surface of negative Euler characteristic, connected
and orientable, without boundary and with at least one puncture. The
arc complex $\arc(F)$ is the simplicial complex whose vertices
are proper isotopy classes of essential arcs from puncture to
puncture. Simplices are collections of vertices admitting pairwise
disjoint representatives. We engage in the standard abuse of notation
by using the same symbol for an arc and its isotopy class.
The $1$–skeleton $\arc^{(1)}(F)$ has a combinatorial metric. For a
pair of vertices $v, w \in \arc^{(0)}(F)$, the distance $d(v,w)$ is
the minimal number of edges required to connect $v$ to $w$. This is
well-defined, because $\arc(F)$ is
connected <cit.>.
When $F$ has a preferred puncture $p$, we define the subcomplex
$\arc(F,p) \subset \arc(F)$ whose vertices are arcs with at least one
endpoint at $p$. The $1$–skeleton $\arc^{(1)}(F,p)$ is again
connected. The distance $d_\arc(v,w)$ is the minimal number of edges
required to connect $v$ to $w$ inside of $\arc^{(1)}(F,p)$.
The mapping class group $\MCG(F)$ acts on $\arc(F)$ by isometries. In
fact, Irmak and McCarthy showed <cit.>
that, apart from a few low-complexity exceptions, $\MCG(F) \isom
\operatorname{Isom} \arc(F)$. Similarly, the subgroup of $\MCG(F)$
that fixes the puncture $p$ acts on $\arc(F,p)$ by isometries. We are
interested in the geometric implications of this action.
Let $\mon \from F \to F$ be a homeomorphism fixing $p$. Define the
translation distance of $\mon$ in $\arc(F,p)$ to be
\begin{equation}\label{eq:trans-distance}
d_\arc(\mon) = \min \{ d_\arc(v, \mon(v)) \st v \in \arc^{(0)}(F,p) \}.
\end{equation}
The same definition applies in any simplicial complex where $\MCG(F)$
acts by isometries.
We also define the stable translation distance of $\mon$ to be
\begin{equation}\label{eq:stable-distance}
\dbar_\arc(\mon) = \lim_{n \to \infty} \frac{ d_\arc(v, \mon^n(v))
}{n}, \quad \mbox{for an arbitrary vertex } v \in \arc^{(0)}(F,p),
\end{equation}
and similarly for other $\MCG$–complexes. It is a general property of
isometries of metric spaces that the limit in
(<ref>) exists and does not depend on the base
vertex $v$ <cit.>. In addition, the
triangle inequality implies that $\dbar_\arc(\mon) \leq d_\arc(\mon)$.
Note that applying equation (<ref>) to the pants
graph $\pants(F)$ gives the stable translation distance $\dbar_\pants(\mon)$
that estimates volume in Theorem <ref>.
In the same spirit, one may ask the following.
Let $\calS(F)$ be a simplicial complex associated to a surface $F$, on
which the mapping class group $\MCG(F)$ acts by isometries. How are the
dynamics of the action of $\mon \in \MCG(F)$ on $\calS(F)$ reflected
in the geometry of the mapping torus $M_\mon$?
We answer this question for the arc complex of a once-punctured
surface $F$ or, more generally, for the sub-complex $\arc(F,p)$ of a
surface with many punctures. Here, the stable distance
$\dbar_\arc(\mon)$ predicts the cusp geometry of $M_\mon$.
§.§ Cusp area from the arc complex
Let $M$ be a $3$–manifold whose boundary is a non-empty union of
tori, such that the interior of $M$ supports a complete hyperbolic
metric. In this metric, every non-compact end of $M$ is a cusp,
homeomorphic to $T^2 \times [0, \infty)$. Geometrically, each cusp is
a quotient of a horoball in $\HH^3$ by a $\ZZ \times \ZZ$ group of
deck transformations. We call this geometrically standard end a
horospherical cusp neighborhood or horocusp.
Associated to each torus $T \subset \bdy M$ is a maximal cusp
$C = C_T$. That is, $C$ is the closure of $C^\circ \subset M$, where
$C^\circ$ is the largest embedded open horocusp about $T$.
The same construction works in dimension $2$: every punctured
hyperbolic surface has a maximal cusp about each puncture.
In dimension $3$, Mostow–Prasad rigidity implies that the geometry of
a maximal cusp $C \subset M$ is completely determined by the topology
of $M$. One may compute that $\area(\bdy C) = \frac{1}{2}
\vol(C)$. The Euclidean geometry of $\bdy C$ is an important invariant
that carries a wealth of information about Dehn fillings of $M$. For
example, if a slope $s$ (an isotopy class of simple closed
curve on $\bdy C$) is sufficiently long, then Dehn filling $M$ along
$s$ produces a hyperbolic manifold <cit.>, whose volume can be estimated in terms of the
length $\ell(s)$ <cit.>.
In our setting, $M_\mon$ is a fibered hyperbolic $3$–manifold, with
fiber a punctured surface $F$. The maximal cusp torus $\bdy C$
contains a canonical slope, called the longitude of $C$, which
encircles a puncture of $F$. The Euclidean length of the longitude is
denoted $\lambda$. Any other, non-longitude slope on $\bdy C$ must
have length at least
\begin{equation}
\label{eq:height}
\height(\bdy C) := \area(\bdy C) / \lambda.
\end{equation}
As discussed in the previous paragraph, lower bounds on $\height(\bdy
C)$ imply geometric control over Dehn fillings of $M_\mon$.
Our main result in this paper uses the action of $\mon$ on $\arc(F,p)$
to give explicit estimates on the area and height of the cusp torus
$\bdy C$.
Let $F$ be a surface with a preferred puncture $p$, and let $\mon
\from F \to F$ be any orientation-preserving, pseudo-Anosov
homeomorphism. In the mapping torus $M_\mon$, let $C$ be the maximal
cusp that corresponds to $p$. Let
$\psi = \mon^n$ be the smallest positive power of $\mon$ with the
property that $\psi(p)=p$. Then
\[
\frac{ \dbar_{\arc} (\psi)}{450\, \chi(F)^4 } \: < \:
\area(\bdy C) \: \leq \: 9 \, \chi(F)^2 \, \dbar_{\arc} (\psi).
\]
Similarly, the height of the cusp relative to a longitude satisfies
\[
\frac{ \dbar_{\arc} (\psi)}{536\, \chi(F)^4 } \: < \:
\height(\bdy C) \: < \: -3 \, \chi(F) \, \dbar_{\arc} (\psi).
\]
If the surface $F$ has only one puncture $p$, the statement of Theorem
<ref> becomes simpler in several ways. In this special case,
we have $n = 1$, hence $\psi = \mon$. There is only one cusp in
$M_\mon$, and $\arc(F,p) = \arc(F)$. In this special case, the area
and height of the maximal cusp are estimated by the stable translation
distance $\dbar_\arc(\mon)$, acting on $\arc(F)$.
In the special case where $F$ is a once-punctured torus or
$4$–punctured sphere, Futer, Kalfagianni, and Purcell proved a
similar estimate, with sharper constants. See <cit.>. Theorem <ref> generalizes those
results to all punctured hyperbolic surfaces.
We note that a non-effective version of Theorem <ref> can be
derived from Minsky's a priori bounds theorem for the length of
curves appearing in a hierarchy <cit.>. In fact, this line of argument was our
original approach to estimating cusp area. In the process of studying
this problem, we came to realize that arguments using the geometry and
hierarchical structure of the curve complex $\curve(F)$ can be
replaced by elementary arguments focusing on the geometry of pleated
surfaces. See Section <ref> below for an outline of
this effective argument.
§.§ Quasi-Fuchsian $3$–manifolds
The methods used to prove Theorem <ref> also apply to
quasi-Fuchsian manifolds. We recall the core definitions; see
Marden <cit.> or
Thurston <cit.> for more details. A
hyperbolic manifold $N = \HH^3 / \Gamma$ is called
quasi-Fuchsian if the limit set $\Lambda(\Gamma)$ of $\Gamma$
is a Jordan curve on $\bdy \HH^3$, and each component of $\bdy \HH^3
\setminus \Lambda(\Gamma)$ is invariant under $\Gamma$. In this case,
$N$ is homeomorphic to $F \times \RR$ for a surface $F$. The
convex core of $N$, denoted $\core(N)$, is defined to be the
quotient, by $\Gamma$, of the convex hull of the limit set
When $N$ is quasi-Fuchsian but not Fuchsian, $\core(N) \cong F \times
[0,1]$, and its boundary is the disjoint union of two surfaces
$\bdy_+\core(N)$ and $\bdy_-\core(N)$, each intrinsically hyperbolic,
and each pleated along a lamination. See Definition <ref>.
Although the quasi-Fuchsian manifold $N$ has infinite volume, the
volume of $\core(N)$ is finite. Each puncture of $F$ corresponds to a
rank one maximal cusp $C$ (the quotient of a horoball by
$\mathbb{Z}$), such that $C \cap \core(N)$ has finite volume and $\bdy
C \cap \core(N) \cong S^1 \times [0,1]$ has finite area. Thus we may
attempt to estimate the area and height of $\bdy C \cap \core(N)$
Let $N \cong F \times \RR$ be a quasi-Fuchsian $3$–manifold, and let
$p$ be a puncture of $F$. Define $\Delta_+(N)$ to be the collection of
all shortest arcs from $p$ to $p$ in $\bdy_+ \core(N)$. By
Lemma <ref>, the arcs in $\Delta_+(N)$ are
pairwise disjoint, so $\Delta_+(N)$ is a simplex in $\arc(F, p)$.
Similarly, let $\Delta_-(N)$ be the simplex of shortest arcs from $p$
to $p$ in $\bdy_- \core(N)$.
We define the arc distance of $N$ relative to the puncture $p$
to be
\[
d_\arc(N,p) =
\min \{d_\arc(v,w) \st v \in \Delta_-(N), \, w \in \Delta_+(N) \}.
\]
In words, $d_\arc(N,p)$ is the length of the shortest path in $\arc(F,
p)$ from a shortest arc in the lower convex core boundary to a
shortest arc in the upper boundary.
Let $F$ be a surface with a preferred puncture $p$, and let $N \cong F
\times \RR$ be a quasi-Fuchsian $3$–manifold. Let $C$ be the maximal
cusp corresponding to $p$. Then
\begin{eqnarray*}
\frac{ d_\arc(N,p)}{450\, \chi(F)^4 } \,-\, \frac{1}{23\, \chi(F)^2 }
& < & \area( \bdy C \cap \core(N)) \\
& < & 9 \, \chi(F)^2 \, d_\arc(N,p) +
\Big| 12 \chi(F) \ln \abs{ \chi(F) } + 26 \chi(F) \Big|
\end{eqnarray*}
Similarly, the height of the cusp relative to a longitude satisfies
\[
\frac{ d_\arc(N,p)}{536\, \chi(F)^4 } \,-\, \frac{1}{27\,
\chi(F)^2 }\: < \: \height( \bdy C \cap \core(N)) \: < \: -3 \,
\chi(F) \, d_\arc(N,p) + 2\ln \abs{ \chi(F) } + 5.
\]
We note that the multiplicative constants in Theorem <ref>
are exactly the same as in Theorem <ref>. However, in
addition to multiplicative error, the estimates in
Theorem <ref> contain explicit additive error. This
additive error is necessary: for example, if the limit set of $N$ is
sufficiently close to a round circle, one may have $d_\arc(N,p) =
0$. (See Lemma <ref> for a constructive argument.)
On the other hand, $\area(\bdy C \cap \core(N)) > 0$ whenever $N$ is
not Fuchsian.
Theorem <ref> has an interesting relation to the work of
Akiyoshi, Miyachi, and Sakuma <cit.>. For a
quasi-Fuchian manifold $N$, they study the scale-invariant quantity
\[
\width(\bdy C) \: := \: \height(\bdy C) / \lambda \: = \:
\area(\bdy C) / \lambda^2
\]
where $\lambda$ is the Euclidean length of the longitude of cusp torus
$\bdy C$. Generalizing McShane's identity, they give an exact
expression for $\width(\bdy C)$ as the sum of an infinite series
involving the complex lengths of closed curves created by joining the
endpoints of an arc. It seems reasonable that most of the
contribution in this infinite sum should come from the finitely many
arcs in $F$ that are shortest in $N$. Theorem <ref>
matches this intuition, and indeed its lower bound is proved by
summing the contributions of finitely many short arcs.
§.§ Covers and the arc complex
Theorem <ref> has an interesting application to the
geometry of arc complexes, whose statement does not involve
$3$–manifolds in any way.
Suppose $f \from \Sigma \to S$ is an $n$–sheeted covering map of
surfaces. We define a relation $\pi \from \arc(S) \to \arc(\Sigma)$
as follows: $\alpha \in \pi(a)$ if and only if $\alpha$ is a component
of $f^{-1}(a)$. In other words, $\pi(a) \subset \arc(\Sigma)$ is the
set of all $n$ lifts of $a$, which span an $(n-1)$–simplex.
Definition <ref> also applies to curve complexes, with
the (inessential) difference that the number of lifts of a curve is
not determined by the degree of the cover. In this context, Rafi and
Schleimer proved that $\pi \from \curve(S) \to \curve(\Sigma)$ is a
quasi-isometric embedding <cit.>. That is, there
exist constants $K \geq 1$ and $C \geq 0$, such that for all $a,b \in
\curve^{(0)}(S)$ and for all $\alpha \in \pi(a)$, $\beta \in \pi(b)$,
we have
\[
d(a,b) \leq K \, d(\alpha, \beta) + C
\quad \mbox{and} \quad
d(\alpha,\beta) \leq K \, d(a, b) + C.
%\tfrac{1}{K}\, d(v, w) - C \: \leq \: d(\pi(v), \pi(w) ) \: \leq \: K
%\, d(v, w) + C.
\]
The constants $K$ and $C$ depend only on $S$ and the degree of the
cover, but are not explicit.
As a consequence of Theorem <ref>, we prove a version of
the Rafi–Schleimer theorem for arc complexes, with explicit
Let $\Sigma$ and $S$ be surfaces with one puncture, and $f \from
\Sigma \to S$ a covering map of degree $n$. Let $\pi \from \arc(S) \to
\arc(\Sigma)$ be the lifting relation. Then, for all $a,b \in
\arc^{(0)}(S)$, we have
\[
\frac{d(a,b)}{4050 \, n \, \chi(S)^6} \, - 2 \: < \: d(\alpha, \beta) \: \leq \: d(a, b)
\]
where $\alpha \in \pi(a)$ and $\beta \in \pi(b)$.
§.§ An outline of the arguments
The proofs of Theorems <ref> and <ref> have a
decidedly elementary flavor. The primary tool that we use repeatedly
is the geometry of pleated surfaces, as developed by
Thurston <cit.>. (See Bonahon <cit.>
or Canary, Epstein, and Green <cit.> for a detailed
description.) In our context, a pleated surface is typically a copy
of the fiber $F$ with a prescribed hyperbolic metric, immersed into
$M$ in a piecewise geodesic fashion, and bent along an ideal
triangulation of $F$. In Sections <ref>
and <ref> below, we give a detailed definition of
pleated surfaces and discuss the geometry of cusp neighborhoods in
such a surface. We also study a mild generalization of pleated
surfaces, called simplicial hyperbolic surfaces, that are hyperbolic
everywhere except for a single cone point with angle at least $2\pi$.
The upper bounds of Theorems <ref> and <ref> are
proved in Section <ref> and <ref>,
respectively. To sketch the argument in the fibered case, let $\tau$
be an ideal triangulation of the fiber $F$. Then $F$ can be homotoped
to a pleated surface, $F_\tau$, in which every ideal triangle is
totally geodesic. Using lemmas in Sections <ref>
and <ref>, we show that the intersection $F_\tau \cap
\bdy C$ gives a closed polygonal curve about the puncture $p$, whose
length is bounded by $-6 \chi(F)$. As a result, the pleated surface
$F_\tau$ makes a bounded contribution to the area and height of $\bdy
C$. Summing up the contributions from a sequence of triangulations
that “realize” the monodromy $\mon$ gives the desired upper bound of
Theorem <ref>. The upper bound of Theorem <ref>
uses very similar ideas; the one added ingredient is a bound on how
far a short arc in $\bdy_\pm \core(N)$ drifts when it is pulled tight,
making it geodesic in $N$.
The lower bounds on cusp area and height rely on the idea of a
geometrically controlled sweepout. This is a degree-one map
$\Psi \from F \times [0,1]/ \mon \to M_\mon$, in which every fiber $F
\times \{ t \}$ in the domain is mapped to a piecewise geodesic
surface $F_t \subset M$, which is either pleated or simplicial
hyperbolic. The elementary construction of such a sweepout, which is
due to Thurston <cit.> and Canary
<cit.>, is recalled in Section <ref>.
The lower bound of Theorem <ref> is proved in Section
<ref>. We show that every piecewise geodesic surface $F_t$
in the sweepout of Section <ref> must contain an an arc
from cusp to cusp whose length is explicitly bounded above. As the
parameter $t$ moves around the sweepout, we obtain a sequence of arcs,
representing a walk through the $1$–skeleton $\arc^{(1)}(F,p)$, such
that each arc encountered has bounded length in $M$. This sequence of
somewhat-short arcs in the fiber leads to a packing of the cusp torus
$\bdy C$ by shadows of somewhat-large horoballs, implying a lower
bound on $\area(\bdy C)$ and $\height(\bdy C)$.
It is worth emphasizing that the entire proof of Theorem
<ref> is elementary in nature. In particular, this proof does
not rely on any deep results from Teichmüller theory, Kleinian groups,
or the coarse geometry of the curve or arc complexes.
The lower bound of Theorem <ref> is proved in Section
<ref>, using very similar ideas to those of Theorem
<ref>. Once again, we have a sweepout $\Psi \from F \times
[0,r] \to \core(N)$ by simplicial hyperbolic surfaces. Once again,
each surface $F_t$ in the sweepout contains a somewhat-short arc from
cusp to cusp, corresponding to a horoball whose shadow contributes
area to $\bdy C$. However, we also need to know that the pleated
surfaces at the start and end of the sweepout can be chosen
arbitrarily close to $\bdy_\pm \core(N)$. This fact, written down as
Theorem <ref> in the appendix, is the one place in
the paper where we need to reach into the non-elementary toolbox of
Kleinian groups.
Finally, in Section <ref>, we prove
Theorem <ref>. Given a cover $\Sigma \to S$, vertices $a,
b$ of $\arc(S)$, and vertices $\alpha \in \pi(a)$, $\beta \in \pi(b)$,
the upper bound on the distance $d(\alpha, \beta)$ is immediate
because disjoint arcs lift to disjoint multi-arcs. To prove a lower
bound, we construct a quasi-Fuchsian manifold $M \cong S \times \RR$,
so that $a$ and $b$ are the unique shortest arcs on its convex core
boundaries. The hyperbolic metric on $M \cong S \times \RR$ lifts to a
quasi-Fuchsian structure on $N \cong \Sigma \times \RR$. By applying
Theorem <ref> to both $M$ and $N$, we will bound
$d(\alpha, \beta)$ from below.
§.§ Acknowledgments
This project began at the University of Warwick symposium on Low
Dimensional Geometry and Topology, in honor of David Epstein, and
continued at the MSRI program in Teichmüller Theory and
Kleinian Groups. We thank the organizers of both events for creating
such a fertile ground for collaboration.
We thank Ian Agol for numerous helpful conversations, and in
particular for contributing the key idea of Lemma
<ref>. We thank Dick Canary and Yair Minsky for
clarifying a number of points about pleated surfaces, and for helping
us sort out the proof of Theorem <ref>. We thank
Marc Lackenby for continually encouraging us to make our estimates
§ PLEATED SURFACES AND CUSPS
To prove the upper and lower bounds in our main theorems, we need a
detailed understanding of the geometry of pleated surfaces in a
hyperbolic $3$–manifold. In this section, we survey several known
results about pleated surfaces. We also describe the somewhat subtle
geometry of the intersection between a pleated surface and a cusp
neighborhood in a $3$–manifold $N$. The study of pleated surfaces is
continued in Section <ref>, where we obtain several
geometric estimates.
References for this material include Bonahon <cit.>
and Canary, Epstein, and Green <cit.>.
Let $S$ be a surface, as in Definition <ref>. A
lamination $\lamm \subset S$ is a $1$–dimensional foliation of
a closed subset of $S$.
A special case of a lamination is the union of the edges of an ideal
triangulation; this special case appears frequently in our setting.
Let $N$ be a hyperbolic $3$–manifold, and let $S$ be a surface. Fix
a proper map $f_0 \from S \to N$, sending punctures to cusps. For a
lamination $\lamm \subset S$ a pleating of $f_0$ along $\lamm$,
or a pleating map for short, is a map $f \from S \to N$,
properly homotopic to $f_0$, such that
* $f$ maps every leaf of $\lamm$ to a hyperbolic geodesic, and
* $f$ maps every component of $S \setminus \lamm$ to a totally geodesic
surface in $N$.
We say that $f$ realizes the lamination $\lamm$, and call its
image $f(S)$ a pleated surface.
Note the existence of a pleating map places restrictions on $\lamm$;
for instance, every closed leaf of $\lamm$ must be essential and
non-peripheral in $S$. The hyperbolic metric on $N$, viewed as a
path-metric, pulls back via $f$ to induce a complete hyperbolic metric
on $S$. In this induced hyperbolic metric, every leaf of $\lamm$
becomes a geodesic, and the map $f \from S \to N$ becomes a piecewise
isometry, which is bent along the geodesic leaves of $f(\lamm)$.
Every pleated surface $f(S)$ is contained in the convex core of $N$.
Adding leaves as needed to subdivide the totally geodesic regions, we
can arrange for the complement $S \setminus \lamm$ to consist of ideal
triangles. Fix $g$, a side of some ideal triangle of $S \setminus
\lamm$. Thus $g$ is a bi-infinite geodesic; each end of the image
geodesic $f(g)$ either runs out a cusp of $N$ or meets a small metric
ball infinitely many times. In either case, the lift of $f(g)$ to the
universal cover $\widetilde{N} = \HH^3$ has both endpoints at limit
points of $N$. Since an ideal triangle in $\HH^3$ is the convex hull
of its vertices, the surface $f(S)$ is contained in $\core(N)$.
In a quasi-Fuchsian manifold $N$, the components of $\bdy_\pm
\core(N)$ are themselves pleated surfaces. In this paper, the convex
core boundaries are the only examples of pleated surfaces where the
pleating laminations are not ideal triangulations.
A foundational result is that every essential surface $S \subset N$
can be pleated along an arbitrary ideal triangulation. This was first
observed by Thurston <cit.>. For a more
detailed account of the proof, see Canary, Epstein, and Green
<cit.> or Lackenby <cit.>.
Let $N$ be a cusped orientable hyperbolic $3$–manifold. Let $f_0
\from S \to N$ be a proper, essential map, sending punctures to cusps.
Then, for any ideal triangulation $\tau$ of $S$, the map $f_0$ is
homotopic to a pleating map along $\tau$.
In other words, every ideal triangulation $\tau$ is realized by a
pleating map $f_\tau \from S \to N$.
Suppose that $C \subset N$ is a horospherical cusp neighborhood in
$N$, and $f \from S \to N$ is a pleating map. Our goal is to describe
the geometry of $f(S) \cap C$. We offer Figure <ref> as a
preview of the geometric picture. The figure depicts a lift
$\widetilde{S}$ of a pleated surface to $\HH^3$. For a sufficiently
small horocusp $C_0 \subset C$, which lifts to horoball $H_0$ in the
figure, the intersection $f(S) \cap C_0$ is standard, meaning
that $f^{-1}(C_0)$ is a union of horospherical cusp neighborhoods in
$S$. The intersection $f(S) \cap C$ is more complicated, because the
surface is bent along certain geodesics whose interior intersects $C
\setminus C_0$. Nevertheless, we can use the geometry of $f(S) \cap
C_0$ to find certain cusp neighborhoods contained in $f^{-1}(C)$ (in
Lemma <ref>), and certain geometrically meaningful
closed curves in $\bdy C$ (in Lemma <ref>).
The intersection between a pleated surface and a
horocusp. The intersection with horoball $H_0$ is standard, whereas
the intersection with $H$ may contain portions of the surface bent
along geodesics whose endpoints are not in $H$. Graphic based
on a design of Agol <cit.>.
Let $N$ be a cusped orientable hyperbolic $3$–manifold, with a
horocusp $C$. Let $f \from S \to N$ be a pleating map, such that $n$
punctures of $S$ are mapped to $C$. Suppose that a loop about a
puncture of $S$ is represented by a geodesic of length $\lambda$ on
$\bdy C$. Then, in the induced hyperbolic metric on $S$, the preimage
$f^{-1} (C) \subset S$ contains horospherical cusp neighborhoods $R_1,
\ldots, R_n$ with disjoint interiors, such that
\[
\ell(\bdy R_i) \: = \: \area(R_i) \: \geq \:
\lambda \quad \mbox{for each $i$.}
\]
Our proof is inspired by an argument of
Agol <cit.>.
Without loss of generality, assume that the pleating lamination
$\lamm$ cuts $S$ into ideal triangles. (Otherwise, add more leaves to
$\lamm$.) Let $C_0 \subset C$ be a horocusp chosen sufficiently small
so that $C_0 \cap f(\lamm)$ is a union of non-compact rays into the
cusp. Then $f^{-1}(C_0) $ is a union of tips of ideal triangles in $S$
and consists of disjoint horospherical neighborhoods $R_1^0, \ldots,
R_n^0$, each mapped into $C$.
Lift $N$ to its universal cover $\HH^3$, so that $C_0$ lifts to a
horoball $H_0$ about $\infty$ in the upper half-space model. Then
$\widetilde{f(S)}$ intersects $H_0$ in vertical bands, as shown in
Figure <ref>.
Let $d$ be the distance in $N$ between $\bdy C_0$ and $\bdy C$. Since
the interior of $C$ is embedded, this means that the shortest geodesic
in $N$ from $C_0$ to $C_0$ has length at least $2d$. Since the
pleating map $f \from S \to N$ is distance-decreasing, the shortest
geodesic in $S$ from $f^{-1} (C_0)$ to itself also has length at least
$2d$. In other words, we may take a closed $d$–neighborhood of each $
R_i^0 $ and obtain a cusp neighborhood $R_i$, such that $R_1, \ldots,
R_n$ have disjoint interiors.
Consider the areas of these neighborhoods, along with their boundary
lengths. A standard calculation in the upper half-plane model of
$\HH^2$ implies that the length of a horocycle in $S$ equals the area
of the associated cusp neighborhood. Furthermore, both quantities grow
exponentially with $d$.
On $\bdy C_0$, a Euclidean geodesic about a puncture of $S$ has length
$e^{-d} \lambda$. Since $f(S) \cap C_0$ may not be totally geodesic
(in general, it is bent, as in Figure <ref>), each curve
of $\bdy R_i^0$ has length bounded below by $e^{-d} \lambda$. These
lengths grow by $e^d$ as we take a $d$–neighborhood of $\cup_i
R_i^0$. We conclude that each component $R_i$ satisfies
\begin{equation}\label{eq:distance-d}
\ell(\bdy R_i) \: = \: \area(R_i) \: \geq \:
e^d \cdot e^{-d} \cdot \lambda \: = \: \lambda.
\end{equation}
It remains to show that $f(R_i) \subset C$ for each $i$. Suppose,
without loss of generality, that $R_1^0$ is the component of
$f^{-1}(C_0)$ whose lift is mapped to the horoball $H_0$. Then, by
construction, the lift of $R_1$ is mapped into the $d$–neighborhood
of $H_0$, which is a horoball $H$ covering $C$. Thus $f(R_1) \subset
C$. Note that the containment might be strict, because $f(S)$ might be
bent along some geodesics in the region $C \setminus C_0$, as in the
middle of Figure <ref>.
The argument of Lemma <ref> also permits the
following construction, which is also important for Section
Let $N$ be a cusped orientable hyperbolic $3$–manifold, with a
horocusp $C$. Let $f \from S \to N$ be a pleating map that realizes an
ideal triangulation $\tau$. Then, for each puncture $p$ of $S$ that is
mapped to $C$, there is an immersed closed curve $\gamma = \gamma(f,
p, C)$, piecewise geodesic in the Euclidean metric on $\bdy C$, with
the following properties:
* The loop $\gamma$ is homotopic in $C$ to a loop in $f(S)$ about $p$.
* The vertices of $\gamma$ lie in $\bdy C \cap f(\tau)$, and correspond
to the endpoints of edges of $\tau$ at puncture $p$.
* $\ell(\gamma) = \ell(\bdy R_i) $, where $R_i \subset S$ is one of the
cusp neighborhoods of Lemma <ref>.
A lift of $\gamma$ to a horoball $H$ covering $C$ is shown, dotted, in
Figure <ref>.
We may construct $\gamma$ as follows. Recall, from the proof of Lemma
<ref>, that there is a horocusp $C_0 \subset C$
such that the intersection $f(S) \cap C_0$ is standard, consisting of
tips of ideal triangles. Then $f^{-1}(C_0)$ is a disjoint union of
horospherical cusp neighborhoods. Let $R_i^0$ be the component of
$f^{-1}(C_0)$ that contains puncture $p$, and let $\gamma_0 = f(\bdy
R_i^0) \subset \bdy C_0$.
Note that the curve $\gamma_0$ is piecewise geodesic in the Euclidean
metric on $\bdy C_0$, and that it is bent precisely at the
intersection points $\bdy C_0 \cap f(\tau)$, where the triangulation
$\tau$ enters the cusp. See Figure <ref>.
We define $\gamma$ to be the projection of $\gamma_0$ to the
horospherical torus $\bdy C$. Note that if $C_0$ and $C$ are lifted to
horoballs about $\infty$ in $\HH^3$, as in Figure <ref>,
this projection is just vertical projection in the upper half-space
Observe that while $\gamma_0 \subset f(S)$, its projection $\gamma$
might not be contained in the pleated surface. Nevertheless, $\gamma$
is completely defined by $\gamma_0$. The vertices where $\gamma$ is
bent are contained in $f(\tau)$.
Let $d$ be the distance between $\bdy C_0$ and $\bdy C$. Then, as in
Lemma <ref>, lengths grow by a factor of $e^d$ as
we pass from $\bdy C_0$ to $\bdy C$. Thus, by the same calculation as
in (<ref>),
\[
\ell(\gamma) \: = \: e^d \cdot \ell(\gamma_0) \: = \: e^d \cdot
\area(R_i^0) \: = \: \area(R_i) \: = \: \ell(\bdy R_i),
\]
where $R_i \supset R_i^0$ is the cusp neighborhood in $S$ that is
mapped into $C$, as in Lemma <ref>.
Our final goal in this section is to provide a universal lower bound
on the size of the cusp neighborhoods $R_i$. We do this using the
following result of Adams <cit.>.
Let $N$ be a non-elementary, orientable hyperbolic $3$–manifold, and
let $C$ be a maximal horocusp in $N$. (This neighborhood may
correspond to either a rank one or rank two cusp.) Let $s$ be a simple
closed curve on $\bdy C$, which forms part of the boundary of an
essential surface in $N$. Then $\ell(s) > 2^{1/4}$.
This is a consequence of a theorem of Adams
<cit.>. He proved that every parabolic
translation of the maximal cusp of any non-elementary hyperbolic
$3$–manifold has length greater than $2^{1/4}$, with exactly three
exceptions: one parabolic each in the three SnapPea census manifolds
m004, m009, and m015.
Each of the manifolds m004, m009, and m015 is either
a punctured torus bundle or a two-bridge knot complement. Hence the
boundary slopes of incompressible surfaces in these manifolds are
classified <cit.>. In
particular, none of the three slopes shorter than $2^{1/4}$ bounds an
essential surface.
As a result, we obtain
Let $N$ be a cusped orientable hyperbolic $3$–manifold, with a
maximal cusp $C$. Let $f(S) \subset N$ be a pleated surface, homotopic
to a properly embedded essential surface, such that $n$ punctures of
$S$ are mapped to $C$. Then $f^{-1}(C) \subset S$ contains $n$
disjoint horospherical cusp neighborhoods $R_1, \ldots, R_n$, such
\[
\ell(\bdy R_i) \: = \: \area(R_i) \: > \: 2^{1/4}
\quad \mbox{for each $i$.}
\]
This is immediate from Lemmas <ref> and
§ HYPERBOLIC SURFACES WITH ONE CONE POINT
Recall from Definition <ref> that every pleated surface
carries an intrinsic hyperbolic metric. In this section, we prove
several lemmas about the geometry of cusp neighborhoods and geodesic
arcs in these surfaces. These estimates are used throughout the proofs
of Theorems <ref> and <ref>.
In fact, we work in a slightly more general setting: namely,
hyperbolic surfaces with a cone point, whose cone angle is at least
$2\pi$. These singular surfaces arise in sweepouts of a hyperbolic
$3$–manifold: see Section <ref>. Therefore, we derive
length and area estimates for these singular surfaces, as well as
non-singular ones.
A hyperbolic cone surface is a complete metric space $S$,
homeomorphic to a surface of finite type. We require that $S$ admits a
triangulation into finitely many simplices, such that each simplex is
isometric to a totally geodesic triangle in $\HH^2$. The triangles are
allowed to have any combination of ideal vertices (which correspond to
punctures of $S$) and material vertices (which correspond to points in
$S$). The triangles are glued by isometries along their edges.
Every point of $S$ that is not a material vertex of the triangulation
thus has a neighborhood isometric to a disk in $\HH^2$. Every material
vertex $v \in S$ has a neighborhood where where the metric (in polar
coordinates) takes the form
\begin{equation}\label{eq:cone}
ds^2 = dr^2 + \sinh^2(r) \, d\theta^2,
\end{equation}
where $0 \leq r < r_v$ and $0 \leq \theta \leq \theta_v$. Here
$\theta_v$ is called the cone angle at $v$, and can be
computed as the sum of the interior angles at $v$ over all the
triangles that meet $v$. Note that if $\theta_v=2\pi$, equation
(<ref>) becomes the standard polar equation for the hyperbolic
metric in a disk. The vertices of $S$ whose cone angles are not
equal to $2\pi$ are called the cone points or singular
points of $S$; all remaining points are called non-singular.
If all singular points of $S$ have cone angles $\theta_v > 2\pi$,
another common name for $S$ is a simplicial hyperbolic
surface. Simplicial hyperbolic surfaces have played an important role
in the study of geometrically infinite Kleinian groups
Just as with non-singular hyperbolic surfaces, cone surfaces have a
natural geometric notion of a cusp neighborhood.
Let $S$ be a hyperbolic cone surface, with one or more punctures, and
let $R \subset S$ be a closed set. Then $R$ is called an
equidistant cusp neighborhood of a puncture of $S$ if the
following conditions are satisfied:
* The interior of $R$ is homeomorphic to $S^1 \times (0, \infty)$.
* There is a closed subset $Q \subset R$, whose universal cover
$\widetilde{Q}$ is isometric to a horoball in $\HH^2$. This implies
that the interior of $Q$ is disjoint from all cone points.
* There is a distance $d>0$, such that $R$ is the closed
$d$–neighborhood of $Q$.
$R$ is called a maximal cusp if it is not a proper subset of
any larger equidistant cusp neighborhood. Equivalently, $R$ is maximal
if and only if it is not homeomorphic to $S^1 \times [0, \infty)$.
Let $S$ be a hyperbolic surface with one cone point $v$, of angle
$\theta_v \geq 2 \pi$. Let $R \subset S$ be a non-maximal equidistant
neighborhood of a puncture of $S$. Then
* There is a geodesic $\alpha$ that is
shortest among all essential paths from $R$ to $R$.
* The arc $\alpha$ is either embedded, or is
the union of a segment and a loop based at $v$. In the latter case,
there is an arbitrarily small homotopy in $S$ making $\alpha$
* If $2\pi \leq \theta_v < 4 \pi$, and
$\beta$ is another shortest arc from $R$ to $R$, there is an
arbitrarily small homotopy making $\alpha$ and $\beta$ disjoint.
One way to interpret Lemma <ref> is as follows.
Let $p$ be a puncture of $S$. Then any shortest arc relative to a cusp
neighborhood about $p$ gives a vertex of the arc complex $\arc(S,p)$.
If there are two distinct shortest arcs, they span an edge of
$\arc(S,p)$; more generally, if there are $n$ distinct shortest arcs,
they span an $(n-1)$–simplex. This is used in
Sections <ref> and <ref> to construct a path in
Let $d$ be the infimal distance in the universal cover $\widetilde{S}$ between two
different lifts of $\bdy R$; since $R$ is not a maximal cusp,
$d>0$. Furthermore, since distance to the nearest translate is an
equivariant function on $\bdy \widetilde{R}$, it achieves a
minimum. Thus there are points $x,y$ on distinct lifts of $\bdy R$,
whose distance is exactly $d$. By the Hopf–Rinow theorem for cone
manifolds <cit.>, the distance between $x$ and
$y$ is realized by a geodesic path $\alpha$, and every
distance-realizing path is a geodesic. This
proves (<ref>).
Suppose that $\alpha \from [0,d] \to S$ is a unit-speed
parametrization. The image of $\alpha$ is a graph $\Gamma =
\Image(\alpha)$. If $\alpha$ is not embedded, then $\Gamma$ has at
least one vertex of valence larger than $2$. In this case, we will
show that $\Gamma$ is the union of a segment and a loop based at $v$.
Let $w$ be a vertex of $\Gamma$ that has valence larger than
$2$. Consider preimages $x, y \in \alpha^{-1}(w)$, where $x < y$. Then
$[0,d]$ splits into sub-intervals
\[
I_1 = [0,x], \quad I_2 = [x,y], \quad I_3 = [y,d].
\]
Let $\alpha_i$ be the restriction of $\alpha$ to the sub-interval $I_i$.
We claim that $\alpha_1$ must be homotopic to $\overline{\alpha_3}$
(the reverse of $\alpha_3$): otherwise, cutting out the middle segment
$\alpha_2$ would produce a shorter essential path. We also claim that
$\ell(\alpha_1) = \ell(\alpha_3)$: for, if $\ell(\alpha_1) <
\ell(\alpha_3)$, we could homotope $\alpha_3$ to $\overline{\alpha_1}$
while shortening the length of $\alpha$. In particular, the last claim
implies that $x$ and $y$ are the only preimages of $w$, and $w$ has
valence $3$ or $4$.
If $w \neq v$, then it is a non-singular point of $S$, hence a
$4$–valent vertex. This means that $\alpha_1$ meets $\alpha_3$ at a
nonzero angle. Then, exchanging $\alpha_1$ and $\overline{\alpha_3}$
by homotopy and rounding off the corner at $w$, as in Figure
<ref>, produces an essential arc shorter than
$\alpha$. This is a contradiction.
Exchanging and rounding off the arcs $\alpha_1$ and
$\alpha_3$ produces a shorter path.
We may now assume that the only vertex in the interior of $\Gamma =
\Image(\alpha)$ occurs at the cone point $v$. This implies that $v$ is
not in the cusp neighborhood $R$, hence $R$ contains no singular
points and its universal cover $\widetilde{R}$ is isometric to a
horoball. Thus, since horoballs are convex, there is a unique
shortest path from $v$ to $R$, in each homotopy class.
In particular, the homotopic arcs $\alpha_1$ and $\overline{\alpha_3}$
must coincide. Thus $w=v$ is $3$–valent, and is the only vertex of
$\Gamma$. Hence $\alpha_2$ is an embedded loop based at $v$, and
$\alpha$ must be an “eyeglass” that follows $\alpha_1$ from $R$ to
$v$, runs around the loop $\alpha_2$, and returns to $R$ by retracing
$\alpha_1$. In this case, even though $\alpha$ is not embedded, one
component of the frontier of an $\varepsilon$–neighborhood of
$\Gamma$ is an embedded arc homotopic to $\alpha$. This proves
For future reference, we note an important feature of eyeglass
geodesics. Suppose that $\alpha$ consists of an arc $\alpha_1$ from
$v$ to $R$ and a loop $\alpha_2$ based at $v$. Then $\alpha_1$ must be
the unique shortest path from $v$ to $R$. For if another
geodesic $\alpha_3$ from $v$ to $R$ has length $\ell(\alpha_3) \leq
\ell(\alpha_1)$, then $\alpha_1$ and $\alpha_3$ must be in different
homotopy classes. This means $\alpha_1 \cup \alpha_3$ is an essential
arc from $R$ to $R$, whose length is
\[
\ell(\alpha_1) + \ell(\alpha_3) \: \leq\: 2 \ell(\alpha_1) \: <\:
2\ell(\alpha_1) + \ell(\alpha_2) \: = \: \ell(a),
\]
contradicting the fact that $\alpha$ is shortest.
For part (<ref>), suppose that $\alpha$ and $\beta$ are
two distinct shortest arcs from $R$ to $R$. By statement
(<ref>), each of $\alpha$ and $\beta$ is either an
embedded arc or an eyeglass with a loop based at $v$. Suppose that
$\alpha$ and $\beta$ intersect, and let $\Gamma = \Image(\alpha) \cup
\Image(\beta)$.
If $\Gamma$ has a non-singular vertex $w$, then $w$ cuts $\alpha$ into
sub-arcs $\alpha_1, \alpha_2$ that run from $w$ to $R$. Similarly, $w$
cuts $\beta$ into sub-arcs $\beta_3, \beta_4$ from $w$ to $R$. Without
loss of generality, say that $\alpha_1$ is shortest among these four
arcs. Then at least one of $\alpha_1 \cup \beta_1$ or $\alpha_1 \cup
\beta_2$ is an essential arc from $R$ to $R$, and both of these arcs
are no longer than $\beta$. By rounding off the corner at $w$, we can
make $\alpha_1 \cup \beta_1$ or $\alpha_1 \cup \beta_2$ into an
essential arc shorter than $\beta$. This is a contradiction.
For the rest of the proof, we assume that the only vertex of $\Gamma$
is at $v$. One consequence of this assumption is that $R$ contains no
singular points. Hence, as in the proof of (<ref>),
there is a unique shortest path from $v$ to $R$ in each homotopy
class. There are three cases: $(i)$ neither $\alpha$ nor $\beta$ is
an eyeglass, $(ii)$, $\alpha$ is an eyeglass but $\beta$ is not, and
$(iii)$ both $\alpha$ and $\beta$ are eyeglasses.
If $\alpha$ and $\beta$ are embedded arcs that intersect at $v$,
consider the valence of $v$, which must be $3$ or $4$. If $v$ is
$3$–valent, $\alpha$ and $\beta$ must share the same path $\alpha_1 =
\beta_1$ from $R$ to $v$, then diverge. In this case, the
$\varepsilon$–neighborhood of $\Gamma$ contains disjointly embedded
arcs homotopic to $\alpha$ and $\beta$.
If $v$ is $4$–valent, let $\gamma_1, \ldots, \gamma_4$ be the four
geodesic sub-arcs of $\Gamma$ from $v$ to $R$. Since each $\gamma_i$
is the unique shortest path in its homotopy class, any combination
$\gamma_i \cup \gamma_j$ is an essential arc. Since $\alpha = \gamma_1
\cup \gamma_2$ and $\beta = \gamma_3 \cup \gamma_4$ are both shortest
arcs in $S$, every $\gamma_i$ must have the same length. But since $v$
has cone angle $\theta_v < 4\pi$, there must be two sub-arcs
$\gamma_i, \gamma_j$ that meet at an angle less than $\pi$. Thus
$\gamma_i \cup \gamma_j$ can be shortened by smoothing the corner at
$v$, contradicting the assumption that $\alpha$ and $\beta$ are
(-2.5,10)$\geq \! \pi$
(6,10)$\geq \! \pi$
(62,10)$\geq \! \pi$
(70,10)$\geq \! \pi$
An eyeglass path and an embedded arc can be made disjointly
embedded after a short homotopy. The dashed sections of $\alpha$ and
$\beta$ are schematics meant to indicate that the arcs are traveling
through a distant part of the surface, while staying disjoint.
If $\alpha$ is an eyeglass, but $\beta$ is not, let $\alpha_1$ be the
sub-arc of $\alpha$ from $v$ to $R$. By the observation at the end of
part (<ref>), $\alpha_1$ is the unique shortest geodesic
from $v$ to $R$. Let $\beta_1$, $\beta_2$ be the sub-arcs of $\beta$
from $v$ to $R$, where $\ell(\beta_1) \leq \ell(\beta_2)$. If
$\alpha_1$ is distinct from $\beta_1$, then $\alpha_1 \cup \beta_1$
would be an essential path that is shorter than $\beta$ —
contradiction. Thus $\alpha_1 = \beta_1$. In this case, homotoping
$\alpha$ to an embedded arc makes it disjoint from $\beta$. See Figure
(-2,12)$\geq \! \pi$
(11,12)$\geq \! \pi$
(49.5,12)$\geq \! \pi$
(37,12)$\geq \! \pi$
(49.5,12)$\geq \! \pi$
(81,12)$\geq \! \pi$
(93.5,12)$\geq \! \pi$
Two eyeglass paths that share the same stem can be made
disjointly embedded after a short homotopy. Shown are the embedded
versions of $\alpha$ and $\beta$, in the three possible interleaving
configurations at vertex $v$.
Finally, if each of $\alpha$ and $\beta$ is an eyeglass, the
observation at the end of part (<ref>) implies that each
of $\alpha$ and $\beta$ must contain the unique shortest geodesic from
$v$ to $R$. Thus each of $\alpha$ and $\beta$ consists of the same
geodesic arc $\gamma_1$ from $v$ to $R$, as well as a loop based at
$v$. Figure <ref> shows that the
$\varepsilon$–neighborhood of $\Gamma = \Image(\alpha) \cup
\Image(\beta)$ contains disjointly embedded paths representing
$\alpha$ and $\beta$.
In the case where $S$ is a non-singular surface, we have a stronger
version of Lemma <ref>: not only are shortest arcs
disjoint, but nearly-shortest arcs must be disjoint as well.
Let $S$ be a punctured hyperbolic surface, and let $R$ be a
horospherical neighborhood about one puncture. Let $\alpha$ and
$\beta$ be distinct geodesic arcs from $R$ to $R$. If $\alpha$ and
$\beta$ intersect, then there is a third geodesic arc $\gamma$,
\[
\ell(\gamma) \: \leq \: \max \{ \ell(\alpha), \, \ell(\beta) \}
- \ln(2).
\]
Here, all lengths are measured relative to the cusp neighborhood $R$.
In practice, we use the contrapositive statement: if both $\alpha$ and
$\beta$ are at most $\ln (2)$ longer than the shortest geodesic from
$R$ to $R$, then they must be disjoint.
If we change the size of the cusp neighborhood $R$, then all geodesic
arcs from $R$ to $R$ have their lengths changed by the same additive
constant. Thus, without loss of generality, we may assume that $R$ is
small enough so that all intersections between $\alpha$ and $\beta$
happen outside $R$. As $S$ has no cone points, $\alpha$ and $\beta$
meet transversely.
Orient both $\alpha$ and $\beta$. If we cut the geodesic $\alpha$
along its intersection points with $\beta$, we obtain a collection of
segments. Let $\alpha_1$ and $\alpha_2$ be the first and last such
segments, respectively, along an orientation of $\alpha$. That is,
$\alpha_1$ (respectively $\alpha_2$) is the sub-arc of $\alpha$ from
$\bdy R$ to the first (last) point of intersection with $\beta$.
Similarly, let $\beta_1$ and $\beta_2$ be the first and last segments
of $\beta$, along an orientation of $\beta$.
Assume, without loss of generality, that $\alpha_1$ is shortest among
the segments $\alpha_1, \alpha_2, \beta_1, \beta_2$. Set $v =
\alpha_1 \cap \beta$. Then the vertex $v$ cuts $\beta$ into sub-arcs
$\beta_3$ and $\beta_4$, such that $\beta_1 \subset \beta_3$ and
$\beta_2 \subset \beta_4$. Without loss of generality, we may also
assume that $\beta_3$ (rather than $\beta_4$) is the sub-arc of
$\beta$ that meets $\alpha_1$ at an angle of at most $\pi/2$.
Note that $\alpha_1 \cup \beta_3$ is an embedded arc, because (by
construction) $\alpha_1$ only intersects $\beta$ at the vertex
$v$. Furthermore, $\alpha_1 \cup \beta_3$ is topologically essential
(otherwise, one could homotope $\beta$ to reduce its length). The
hypothesis that $\alpha_1$ is shortest among $\alpha_1, \alpha_2,
\beta_1, \beta_2$ implies that
\[
\ell(\alpha_1 \cup \beta_3) \: \leq \: \ell(\beta_2 \cup \beta_3) \:
\leq \: \ell(\beta_4 \cup \beta_3) \: = \: \ell(\beta).
\]
Let $\gamma$ denote the geodesic from $R$ to $R$ in the homotopy class
of $\alpha_1 \cup \beta_3$. Then the geodesic extensions of
$\alpha_1$, $\beta_3$, and $\gamma$ form a $2/3$ ideal triangle, with
angle $\theta \leq \pi/2$ at the material vertex $v$. Then,
<cit.> gives
\[
\ell(\gamma) \: = \: \ell(\alpha_1 \cup \beta_3) + \ln \left(
\tfrac{1 - \cos \theta}{2} \right) \: \leq \: \ell(\beta) + \ln
\left( \tfrac{1}{2} \right),
\]
as desired.
Next, we consider the area of equidistant cusp neighborhoods in $S$.
Let $S$ be a simplicial hyperbolic surface with at most one singular
point. Let $R_1$ and $R_2$ be embedded equidistant neighborhoods of
the same puncture of $S$, such that $R_1 \subset R_2 \subset S$, and
$d$ is the distance between $\bdy R_1$ and $\bdy R_2$. Then
\[
\area(R_2) \geq e^d \, \area(R_1).
\]
Let $Q \subset R_1$ be a cusp neighborhood isometric to the quotient
of a horoball. Then there is a number $ m>0$, such that for all $x \in
[0,m]$, the closed $x$–neighborhood of $Q$ is an equidistant cusp
neighborhood $R(x)$. In particular, $R_1 = R(x_1)$ and $R_2 =
R(x_2)$, where $x_2 = x_1 + d$. We shall explore the dependence of
$\area(R(x))$ on the parameter $x$.
Let $v$ be the singular point of $S$. (If $S$ is non-singular, let $v$
be an arbitrary point of $S \setminus Q$, and consider it a cone point
of cone angle $2\pi$.) Let $x_v$ be the distance from $v$ to $\bdy
Q$. Then, for $x < x_v$, $R(x)$ is a non-singular neighborhood of a
cusp, itself the quotient of a horoball. As mentioned in the proof
of Lemma <ref>, the area of a horospherical cusp
grows exponentially with distance. In symbols,
\[
\area(R(x)) = e^x\, \area(Q) \quad \mbox{if} \quad x \leq x_v.
\]
For $x > x_v$, the cusp neighborhood $R(x)$ can be constructed from a
horoball and a cone. More precisely: take a horospherical cusp, and
cut it along a vertical slit of length $r = x-x_v$. Then, take a cone
of radius $r$ and angle $\theta = \theta_v - 2\pi$, and cut it open
along a radius. Gluing these pieces together along the slits produces
$R(x)$. The area of a hyperbolic cone with radius $r$ and angle
$\theta$ can be computed as $2\theta \sinh^2(r/2)$. Adding this to the
area of a horospherical cusp, we obtain
\[
\area(R(x)) = \left\{
\begin{array}{l l}
e^x\, \area(Q) & \mbox{if} \quad x < x_v, \\
e^x\, \area(Q) + 2 \theta \sinh^2((x-x_v)/2) & \mbox{if} \quad x \geq x_v.
\end{array} \right.
\]
To complete the proof, it suffices to check that the function $f(x) =
\sinh^2 (x/2)$ grows super-exponentially for $x\geq 0$:
\begin{eqnarray*}
\left( \sinh \frac{x+d}{2} \right)^2 &=& \left( \sinh \frac{x}{2}\,
\cosh \frac{d}{2} + \cosh \frac{x}{2} \, \sinh \frac{d}{2} \right)^2\\
&>& \left( \sinh \frac{x}{2}\, \cosh \frac{d}{2} + \sinh \frac{x}{2} \, \sinh \frac{d}{2} \right)^2\\
&=& \left(e^{d/2} \sinh \frac{x}{2} \right)^2 \\
&=& e^d \sinh^2 \left( \frac{x}{2} \right).
\end{eqnarray*}
Thus, since the area of $R(x)$ is the sum of two functions, each of
which grows at least exponentially with $x$, it follows that
$\area(R(x+d)) \geq e^d \, \area(R(x))$.
Let $S$ be a simplicial hyperbolic surface with at most one singular
point. Let $R_{\max} \subset S$ be a maximal cusp neighborhood of a
puncture of $S$. Then
$$\area(R_{\max}) \leq -2\pi \chi(S).$$
Furthermore, if $S$ is non-singular, then
$$\area(R_{\max}) \leq -6 \chi(S).$$
Let $\theta_v \geq 2\pi$ be the cone angle at the singular point. (As
above, we take $\theta_v = 2\pi$ if the surface $S$ is non-singular.)
Then, by the Gauss–Bonnet theorem <cit.>,
\begin{equation}\label{eq:gauss-bonnet}
\area(R_{\max}) \: \leq \: \area(S) \: = \: - 2\pi \chi(S) + [2\pi - \theta_v] \: \leq \: - 2\pi \chi(S).
\end{equation}
If $S$ is a non-singular surface, then horosphere packing estimates of
Böröczky <cit.> imply that at most $3/\pi$ of the area of
$S$ can be contained in the cusp neighborhood $R_{\max}$. Thus the
above estimate improves to $\area(R_{\max}) \leq -6 \chi(S)$.
By the Gauss–Bonnet theorem expressed in equation
(<ref>), the area of $S$ decreases as the cone angle
$\theta_v$ increases from $2\pi$. Thus it seems reasonable that the
area of a maximal cusp $R_{\max}$ would also decrease as $\theta_v$
increases from $2\pi$. If this conjecture is true, then the estimate
$\area(R_{\max}) \leq -6 \chi(S)$ would hold for singular hyperbolic
surfaces as well as non-singular ones.
Let $S$ be a simplicial hyperbolic surface with at most one singular
point. Let $R \subset S$ be an embedded equidistant neighborhood of a
puncture of $S$. Then there exists a geodesic arc $\alpha$ from $R$ to
$R$, satisfying
\[
\ell(\alpha) \leq 2 \ln \abs{ 2\pi \, \chi(S) / \area(R) }.
\]
Furthermore, if $S$ is non-singular, then
\[
\ell(\alpha) \leq 2 \ln \abs{ 6 \, \chi(S) / \area(R) }.
\]
By Lemma <ref>, there is a geodesic arc $\alpha$
that is shortest among all essential arcs from $R$ to $R$. Let
$R_{\max}$ be the maximal cusp neighborhood containing $R$. Then, by
construction, $R_{\max}$ meets itself at the midpoint of
$\alpha$. Thus the distance from $\bdy R$ to $\bdy R_{\max}$ is $d =
\ell(\alpha)/2$. By Lemma <ref>, this implies
\[
e^{d}\, \area(R) \: \leq \: \area(R_{\max}),
\]
which simplifies to
\[
\ell(\alpha) \: = \: 2d \: \leq \: 2 \ln \left( \area(R_{\max} )/
\area(R) \right) .
\]
Substituting the bound on $\area(R_{\max})$ from Lemma
<ref> completes the proof.
§ UPPER BOUND: FIBERED MANIFOLDS
In this section, we prove the upper bounds of Theorem
<ref>. We begin with a slightly simpler statement:
Let $F$ be an orientable hyperbolic surface with a preferred puncture
$p$, and let $\psi \from F \to F$ be an orientation-preserving,
pseudo-Anosov homeomorphism such that $\psi(p)=p$. In the mapping
torus $M_\psi$, let $C$ be the maximal
cusp that corresponds to $p$. Then
\[
\area(\bdy C) \: \leq \: 9 \, \chi(F)^2 \, d_{\arc} (\psi)
\quad \mbox{ and } \quad
\height(\bdy C) \: < \: -3 \, \chi(F) \, d_{\arc} (\psi).
\]
Theorem <ref> differs from the upper bound of Theorem
<ref> in two relatively small ways. First, Theorem
<ref> restricts attention to monodromies that fix the
puncture $p$. (Given an arbitrary pseudo-Anosov $\mon$, one can let
$\psi$ be the smallest power of $\mon$ such that $\psi(p)=p$.)
Second, Theorem <ref> estimates cusp area and height in
terms of the translation distance $d_\arc(\psi)$, rather than the
stable translation distance $\dbar_\arc(\psi)$. We shall see at the
end of the section that this simpler statement quickly implies the
upper bound of Theorem <ref>.
The proof involves a direct construction. Suppose that $d_{\arc}
(\psi) = k$. Then, by Definition <ref>, there is a
vertex $a_0 \in \arc^{(0)}(F,p)$, that is an isotopy class of arc
in $F$ meeting the puncture $p$, so that $d_\arc(a_0, \psi(a_0)) = k$. Fix
a geodesic segment in $\arc^{(1)}(F,p)$ with vertices
\[
a_0, a_1, \ldots, a_k = \psi(a_0).
\]
By Definition <ref>, the arcs representing $a_{i-1}$
and $a_i$ are disjoint. Thus, for every $i=1,\ldots,k$, we can choose
an ideal triangulation $\tau_i$ of $F$ that contains $a_{i-1}$ and
$a_i$. In the arc complex $\arc(F,p)$, the geodesic segment from
$a_0$ to $a_k$ extends to a bi-infinite, $\psi$–invariant, piecewise
geodesic. Similarly, the sequence of ideal triangulations $\tau_1,
\ldots, \tau_k$ extends to a bi-infinite sequence of triangulations in
which $\tau_{i+k} = \psi(\tau_i)$.
To prove the upper bounds on cusp area and height, it is convenient to
work with the infinite cyclic cover of $M_\psi$. This is a hyperbolic
manifold $N_{\psi} \cong F \times \RR$, in which the torus cusps of
$M$ lift to annular, rank one cusps. Let $A \subset N_{\psi}$ be the
lift of $\bdy C$ that corresponds to the puncture $p$ of $F$.
We choose geodesic coordinates for the Euclidean metric on $A \cong
S^1 \times \RR$, in which the non-trivial circle (a longitude about
$p$) is horizontal, and the $\RR$ direction is vertical. We also
choose an orientation for every arc $a_i$, such that the oriented edge
$a_i$ points into the preferred puncture $p$. In the $3$–manifold
$N_{\psi}$, the edge $a_i$ is homotopic to a unique oriented
geodesic. Given our choices, every arc $a_i$ has an associated
height $h(a_i)$, namely the vertical coordinate of the point of
$A$ where the oriented geodesic representing $a_i$ enters the
cusp. For simplicity, we may assume that $h(a_0) = 0$ and $h(a_k) =
h(\psi(a_0)) > 0$. See Figure <ref>.
The edges $a_i$ enter the cusp annulus $A \subset N_{\psi}$
at well-defined heights.
To estimate distances on the annulus $A$, we place many copies of the
fiber $F$ into pleated form. That is, for each ideal triangulation
$\tau_i$, $i \in \ZZ$, Proposition <ref> implies that
the fiber $F \times \{0 \} \subset N_\psi = F \times \RR$ can be
homotoped in $N_\psi$ to a pleated surface $F_i$ realizing the
triangulation $\tau_i$. Recall that every ideal triangle of $\tau_i$
is totally geodesic in $F_i$.
For each pleated surface $F_i$, Lemma <ref>
gives a possibly self-intersecting, piecewise geodesic closed curve
$\gamma_i \subset A$, which is homotopic to a loop about the puncture
$p$. The curve $\gamma_i$ is not necessarily contained in $F_i \cap
A$, but we do know that the vertices where it bends are the endpoints
of edges of $\tau_i$ meeting the annulus $A$. See Figure
<ref> for a review.
Each piecewise linear closed curve $\gamma_i \subset A$, determined by
the pleated surface $F_i$, has length $\: \ell(\gamma_i) \leq -6 \,
\chi(F)$.
Lemma <ref> states that $\ell(\gamma_i) =
\ell(\bdy R_i) = \area(R_i)$, where $R_i \subset F_i$ is an embedded
horospherical neighborhood of the puncture $p$. By Lemma
<ref>, $\area(R_i) \leq -6 \, \chi(F)$.
See Agol <cit.> or Lackenby <cit.> for a very similar statement, on which Lemma
<ref> is based.
Applying Lemma <ref> to the pleated surface
$F_i$ gives a height estimate.
The heights of consecutive arcs satisfy
$\: \abs{h(a_i) - h(a_{i-1})} < -3 \, \chi(F)$.
By construction, the arcs $a_{i-1}$ and $a_i$ have endpoints at the
puncture $p$. Additionally, the geodesic representatives of both arcs
are contained in the triangulation $\tau_i$ along which $F_i$ is bent.
Thus the piecewise geodesic closed curve $\gamma_i$, containing the
vertices at the forward endpoints of $a_{i-1}$ and $a_i$, must visit
heights $h(a_{i-1})$ and $h(a_i)$. See Figure <ref>,
left. Since $\ell(\gamma_i) \leq -6 \, \chi(F)$, and this closed
curve covers the distance between heights $h(a_{i-1})$ and $h(a_i)$ at
least twice, we conclude that
\[
\abs{h(a_i) - h(a_{i-1})} < -3 \, \chi(F).
\]
The inequality is strict because $\gamma_i$ must also travel around a
horizontal loop in $A$.
Left: the polygonal closed curve $\gamma_i \subset A
$. Right: the shape of $\gamma_i$ that maximizes the area of the band
$B_i$ between heights $h(a_{i-1})$ and $h(a_i)$.
Lemma <ref> also leads to an area estimate.
Let $B_i \subset A$ be the band whose boundary consists of horizontal
circles at heights $h(a_{i-1})$ and $h(a_i)$. Then $\area(B_i) \leq 9
\, \chi(F)^2$.
As in the proof of Lemma <ref>, we study the
piecewise geodesic closed curve $\gamma_i \subset A$. Since $\gamma_i$
meets both $a_{i-1}$ and $a_i$, it must meet both boundary components
of $B_i$. The goal is to determine the shape of $\gamma_i$ that allows
the largest possible area for $B_i$.
Without loss of generality, we may assume that $\gamma_i$ contains
exactly two geodesic segments connecting the two boundary circles of
$B_i$: otherwise, one can straighten $\gamma_i$ while stretching
$B_i$. Such a piecewise-linear loop consisting of two segments splits
$B_i$ into two isometric triangles: one triangle below $\gamma_i$, and
the other triangle above $\gamma_i$. See Figure <ref>,
At this point, we have reduced to the classical calculus problem of
building a triangular corral adjacent to a river. As is well-known,
the optimal shape for $\gamma_i$ is one where the two segments have
the same length and meet at right angles. By
Lemma <ref>, the total length of these two
equal segments is at most $-6 \, \chi(F)$. Therefore, the maximum
possible area for $B_i$ is $9 \, \chi(F)^2$.
We can now complete the proof of Theorem <ref>. A
fundamental domain for the torus $\bdy C$ is the portion of $A$
between height $h(a_0) = 0$ and height $h(a_k) = h(\psi(a_0))$. This
fundamental domain is contained in $B_1 \cup \ldots \cup B_k$. (The
containment might be strict, since there is no guarantee that the
sequence $h(a_i)$ is monotonically increasing; see Figure
<ref>.) Thus, by Lemma <ref>,
\[
\area(\bdy C) \: \leq \: \sum_{i=1}^{k} \area(B_i) \: \leq \: 9 k \,
\chi(F)^2.
\]
Similarly, by Lemma <ref>,
\[
\height(\bdy C) \: = \: h(a_k) - h(a_0) \: \leq \:
\sum_{i=1}^{k} \abs{h(a_i) - h(a_{i-1})} \:< \: -3k \, \chi(F).
\]
Recalling that $d_{\arc}(\psi)= k$ completes the proof.
Let $F$ be an orientable hyperbolic surface with a preferred puncture
$p$, and let $\psi \from F \to F$ be an orientation-preserving,
pseudo-Anosov homeomorphism such that $\psi(p)=p$. In the mapping
torus $M_\psi$, let $C$ be the maximal
cusp that corresponds to $p$. Then
\[
\area(\bdy C) \: \leq \: 9 \, \chi(F)^2 \, \dbar_{\arc} (\psi)
\quad \mbox{ and } \quad
\height(\bdy C) \: < \: -3 \, \chi(F) \, \dbar_{\arc} (\psi).
\]
Corollary <ref> differs from Theorem <ref> in
that $d_\arc$ has been replaced by $\dbar_\arc$. Since
$\dbar_\arc(\psi) \leq d_\arc(\psi)$ by triangle inequalities, the
statement of Corollary <ref> is slightly sharper.
Let $n \geq 1$. The maximal cusp $C$ of $M_\psi$ lifts to an embedded
horocusp in $M_{\psi^n}$, whose area is $n \cdot \area(\bdy
C)$. Applying Theorem <ref> to $M_{\psi^n}$, we obtain
\[
n \cdot \area(\bdy C) \: \leq \: 9 \, \chi(F)^2 \, d_{\arc} (\psi^n).
\]
\begin{eqnarray*}
\area(\bdy C)
&\leq& 9 \, \chi(F)^2 \, \inf_{n \geq 1} \frac{ d_{\arc} (\psi^n)}{n} \\
&\leq& 9 \, \chi(F)^2 \, \liminf_{n \to \infty} \frac{ d_{\arc} (\psi^n)}{n} \\
% &\leq& \lim_{n \to \infty} \frac{9 \, \chi(F)^2 \, d_{\arc} (\psi^n)}{n} \\
&\leq& 9 \, \chi(F)^2 \, \, \dbar_{\arc} (\psi).
\end{eqnarray*}
The identical calculation goes through for $\height(\bdy C)$.
Let $\mon \from F \to F$ be a pseudo-Anosov homeomorphism, and let
$\psi = \mon^n$ be the smallest power of $\mon$ that fixes the
puncture $p$. Let $C$ be the maximal cusp of $M_\mon$ corresponding to
$p$. Then $C$ lifts to a (not necessarily maximal) horocusp $C'
\subset M_\psi$, which is a one-sheeted cover of $C$.
Corollary <ref> gives upper bounds on the area and
height of the maximal cusp of $M_\psi$, which implies upper bounds on
the (possibly smaller) area and height of $C$.
§ UPPER BOUND: QUASI-FUCHSIAN MANIFOLDS
In this section, we prove the upper bounds of Theorem
<ref>. The proof strategy is nearly the same as the proof
of Theorem <ref>, with the quasi-Fuchsian manifold $N \cong
F \times \RR$ playing the same role as $N_\psi $ in
the previous section. The main geometric difference is that
$\core(N_\psi)$ is the whole manifold, whereas $\core(N)$ has finite
volume and finite cusp area.
Let $C \subset N$ be the maximal cusp corresponding to the puncture
$p$ of $F$. As in Section <ref>, we choose geodesic
coordinates on $A = \bdy C \cong S^1 \times \RR$, in which the $\RR$
direction is vertical. Since $\core(N)$ is convex, the intersection
$\core(N) \cap A$ must be a compact annulus whose boundary is a pair
of horizontal circles. We choose the orientation on $\RR$ so that
$\bdy_+ \core(N)$ is higher than $\bdy_- \core(N)$. Then every
oriented essential arc $a_i \subset N$ whose forward endpoint is at
$C$ has a well-defined height $h(a_i)$, namely the vertical coordinate
of the point on $A$ where the geodesic homotopic to $a_i$ enters the
cusp. See Figure <ref>.
Following Definition <ref>, let $\Delta_\pm(N)$ be the
simplex in $\arc(F,p)$ consisting of all shortest arcs from $p$ to $p$
in $\bdy_\pm \core (N)$. Let $a_0 \in \Delta_-(N)$ and $a_k \in
\Delta_+(N)$ realize the distance between these simplices, so that
\[
k \: = \: d_\arc(a_0, a_k) \: = \: d_\arc(N,p).
\]
Since $a_0$ has both of its endpoints at $p$, we may choose the
orientation on $a_0$ so that the point where $a_0$ enters the cusp is
the lower of the two endpoints. Similarly, we choose the orientation
on $a_k$ so that the point where $a_k$ enters the cusp is the higher
of the two endpoints.
Let $B \subset A$ be the compact annular band whose boundary consists
of horizontal circles at heights $h(a_0)$ and $h(a_k)$. Then
\[
\area(B) \: \leq \: 9 \, \chi(F)^2 \, d_\arc(N, p)
\quad \mbox{ and } \quad
|h(a_k) - h(a_0)| \: < \: -3 \, \chi(F) \, d_\arc(N, p).
\]
This follows from the results of Section <ref>. Let $a_0,
a_1, \ldots, a_k$ be the vertices of a geodesic in $\arc^{(1)}(F,p)$
between $a_0$ and $a_k$. Then, for every $i$, let $B_i \subset A$ be
the annular band whose boundary consists of horizontal circles at
heights $h(a_{i-1})$ and $h(a_i)$. By
Lemmas <ref> and <ref>,
\[
\area(B_i) \: \leq \: 9 \, \chi(F)^2
\quad \mbox{ and } \quad
| h(a_i) - h(a_{i-1})| \: < \: -3 \, \chi(F) .
\]
Adding up these estimates as $i$ ranges from $1$ to $k$ gives the
To prove the upper bounds of Theorem <ref>, it remains to
estimate the area and height of the part of $\core(N) \cap A$ that is
not contained in the band $B$. To make this region more
precise, define $h_\pm(N)$ to be the vertical coordinate of the circle
$\bdy_\pm \core(N) \cap A$. Then we may define $B_- = B_-(N)$ to be
the band whose boundary consists of horizontal circles at heights
$h_-(N)$ and $h(a_0)$, and similarly $B_+ = B_+(N)$ to be the band
between heights $h(a_k)$ and $h_+(N)$. Note that the orientations of
$a_0$ and $a_k$ were chosen precisely so as to minimize the size of
$B_-(N)$ and $B_+(N)$, respectively.
Recall that $\bdy_- \core(N)$ is an intrinsically hyperbolic surface,
pleated along a lamination. The arc $a_0$ has a geodesic
representative in $\bdy_- \core(N)$; in fact, by definition this
geodesic is shortest in $\bdy_- \core(N)$ among all arcs from $p$ to
$p$. Then $h_-(N)$ is the height at which the geodesic representative
of $a_0$ in $\bdy_- \core(N)$ enters the cusp $C$, and $h(a_0)$ is the
height at which the geodesic representative of $a_0$ in $N$ enters the
cusp $C$. The difference $\abs{h(a_0) - h_-(N)}$ is the height of
$B_-(N)$. We control $\abs{h(a_0) - h_-(N)}$ via the following
Let $\gamma \subset N$ be an oriented, essential arc from cusp $C$ back to $C$,
which is disjoint from the interior of $C$. Let $g
\subset N$ be the geodesic in the homotopy class of $\gamma$. Let
$h(\gamma)$ be the height at which $\gamma$ enters $C$, and $h(g)$ be
the height at which $g$ enters $C$.
Assume that the orientation of $\gamma$ has been chosen to minimize
$\abs{h(g) - h(\gamma)}$. Then either $\abs{h(g) - h(\gamma)} \leq \sqrt{2}$, or
\begin{equation}\label{eq:drifting-distance}
\abs{h(g) - h(\gamma)} \: \leq \: \frac{\ell(\gamma) - \ln \left(3 + 2 \sqrt{2} \right) + 2 \sqrt{2}}{2}
\: = \: \frac{\ell(\gamma)}{2} + 0.5328...,
\end{equation}
where $\ell(\gamma)$ is the arclength of $\gamma$.
Lift $\gamma$ to an arc $\widetilde{\gamma} \subset \HH^3$. The
oriented arc $\widetilde{\gamma}$ runs from horoball $H'$ to horoball
$H$. Let $\widetilde{g}$ be the corresponding lift of $g$, namely the
oriented geodesic from $H'$ to $H$. Let $d_+ = d_+(g,\gamma)$ be the
distance along $\bdy H$ between the endpoints of $\widetilde{g}$ and
$\widetilde{\gamma}$ on $H$, and similarly let $d_- = d_-(g,\gamma)$
be the distance along $\bdy H'$ between the endpoints of
$\widetilde{g}$ and $\widetilde{\gamma}$ on $H'$. Since the
orientation of $\gamma$ has been chosen to minimize $\abs{h(g) -
h(\gamma)}$, we have
\begin{equation}
\label{eq:h-gamma}
\abs{h(g) - h(\gamma)} \: \leq \: \min\{ d_-(g,\gamma), \, d_+(g,\gamma) \}
\: \leq \: \frac{ d_- + d_+}{2}.
\end{equation}
Thus, to bound $\abs{h(g) - h(\gamma)}$, it will suffice to bound the
average of $d_-$ and $d_+$.
Next, we reduce the problem from three to two dimensions, as follows.
Consider cylindrical coordinates $(r, \theta, z)$ on $\HH^3$, with the
geodesic $\widetilde{g}$ at the core of the cylinder. Thus $r$
measures distance from $\widetilde{g}$, while $\theta$ is the
rotational parameter, and $z$ measures distance along $\widetilde{g}$.
With these coordinates, the hyperbolic metric becomes
\begin{equation}\label{eq:cylindrical}
ds^2 \: = \: dr^2 + \sinh^2(r) \, d\theta^2 + \cosh^2(r) \, dz^2.
\end{equation}
We claim that no generality is lost by assuming $\widetilde{\gamma}$
lies in the half-plane corresponding to $\theta = 0$. This is because
the expression for the metric in (<ref>) is
diagonalized, hence the map $(r,\theta,z) \mapsto (r, 0,z)$ is
distance–decreasing. Thus replacing $\widetilde{\gamma}$ by its image
in this half-plane only makes it shorter. Furthermore, horoballs $H$
and $H'$ are rotationally symmetric about $\widetilde{g}$, hence the
new planar curve is still disjoint from their interiors. Finally,
observe that rotation about $\widetilde{g}$ keeps the endpoints of
$\widetilde{\gamma}$ a constant distance from $\widetilde{g} \cap H'$
and $\widetilde{g} \cap H$, respectively. Thus the quantities
$d_-(g,\gamma)$ and $d_+(g, \gamma)$ are unchanged when we replace
$\widetilde{\gamma}$ by its rotated image. From now on, we will
assume that $\widetilde{\gamma}$ lies in the copy of $\HH^2$ for which
$\theta \in \{0, \pi\}$.
Recall that $\widetilde{\gamma}$ is disjoint from the interiors of $H$
and $H'$. When its endpoints are sufficiently far apart from $\widetilde{g}$, the geodesic between those points would pass through the interiors of the horoballs. Instead, the shortest path that stays outside $H$ and $H'$ follows the boundary of $H'$, then tracks a
hyperbolic geodesic tangent to $H'$ and $H$, then follows the boundary
of $H$. (See Figure <ref>.) The following lemma estimates
the length of the geodesic segment in the middle of
this path.
(44,19.5)$\widetilde{g} \subset \beta$
(12,20)$\ell_0 - \ell_2$
(73,20)$\ell_3 - \ell_2$
The setup of Proposition <ref>. When the endpoints of $\widetilde{\gamma}$ are sufficiently far apart, the
shortest path that stays
disjoint from the interiors of $H$ and $H'$ is the three-piece blue
arc, of length $(\ell_0 - \ell_2) + \ell_1 + (\ell_3-\ell_2)$.
The setup of Lemma <ref>. Left: making
geodesic $\alpha$ vertical helps estimate $\ell_1$. Right: making
geodesic $\beta$ vertical helps estimate $\ell_2$. In both panels, the
horoball $H'$ has Euclidean radius $r$.
Let $H$ and $H'$ be horoballs in $\HH^2$ with disjoint interiors. Let
$\alpha \subset \HH^2$ be a hyperbolic geodesic such that $H$ and $H'$
are both tangent to $\alpha$, on the same side of $\alpha$. Let
$\beta\subset \HH^2$ be a hyperbolic geodesic perpendicular to both
$H$ and $H'$. If $\ell_1$ denotes the length along $\alpha$ from $H
\cap \alpha$ to $H' \cap \alpha$, and $\ell_2$ denotes the length
along $\bdy H$ from $H \cap \alpha$ to $H \cap \beta$, then
\[
\ell_1 \: \geq \: \ln \left( 3 + 2 \sqrt{2} \right) \qquad \mbox{and}
\qquad \ell_2 \: \leq \: \sqrt{2}.
\]
Each inequality becomes equal if and only if $H$ is tangent to $H'$.
Let $g$ be the geodesic segment of $\alpha$ whose length is $\ell_1$.
For the first inequality, apply an isometry of $\HH^2$ so that
$g \subset \alpha$ is vertical in the upper half-plane model, so that $H$ is the
larger horoball, and so that the Euclidean radius of $H$ is $1$. Then
the point of tangency $\alpha \cap H$ is at Euclidean height $1$.
(See Figure <ref>, left.) A calculation with the
Pythagorean theorem then implies that the Euclidean radius of $H'$
must be $r \leq 1/(3 + 2 \sqrt{2} )$, with equality if and only if $H$
is tangent to $H'$. Since one endpoint of $g$ is at height $1$ and the
other endpoint is at height $r$, we have $\ell_1 = \ln(1/r) \geq \ln(3
+ 2 \sqrt{2} )$.
For the second inequality, apply an isometry of $\HH^2$ so that $\bdy
H$ is a horizontal line at Euclidean height $1$. Then $\alpha$ is a
Euclidean semicircle of radius $1$, and the horoball $H'$ must have
Euclidean radius $r \leq 1/2$. (See Figure <ref>,
right.) Again, a calculation with the Pythagorean theorem implies that
$\ell_2 \leq \sqrt{2}$, with equality if and only if $r= 1/2$.
Returning to the proof of Proposition <ref>, we
import the notation of Lemma <ref>. That is: the
geodesic $\alpha$ is tangent to $H$ and $H'$, while the geodesic
$\beta$ contains $\widetilde{g}$. Let $\ell_1$ and $\ell_2$ be as in
the lemma. Let $\ell_0 = d_-(g,\gamma)$ be the distance along $\bdy
H'$ between the $\widetilde{g}\cap \bdy H'$ and $\widetilde{\gamma}
\cap \bdy H'$, and let $\ell_3 = d_+(g,\gamma)$.
If one of $\ell_0 = d_-$ or $\ell_3 = d_+$ is no longer than
$\sqrt{2}$, then equation (<ref>) gives $\abs{h(g) -
h(\gamma)} \leq \sqrt{2}$ as well, and the proof is
complete. Otherwise, if $\ell_0$ and $\ell_3$ are both longer than
$\sqrt{2}$, then Lemma <ref> implies that they
are longer than $\ell_2$. As a consequence, the shortest path between
the endpoints of $\widetilde{\gamma}$ that stays disjoint from $H$ and
$H'$ will need to track horoball $H'$ for distance $(\ell_0 -
\ell_2)$, then follow geodesic $\alpha$ for distance $\ell_1$, then
track horoball $H$ for distance $(\ell_3 - \ell_2)$. See
Figure <ref>.
Thus we may compute:
\begin{array}{r c l c l}
\ell(\gamma) & \geq & (\ell_0 - \ell_2) + \ell_1 + (\ell_3 - \ell_2) & & \mbox{by construction of $\widetilde{\gamma}$} \\
& = & (d_- + d_+) - 2 \ell_2 + \ell_1 & & \mbox{by the definition of $\ell_0$ and $\ell_3$} \\
& \geq & (d_- + d_+) - 2 \sqrt{2} + \ln (3 + 2 \sqrt{2}) & & \mbox{by Lemma~\ref{lemma:tangent-length},} \\
& \geq & 2 \abs{h(g) - h(\gamma)} - 2 \sqrt{2} + \ln (3 + 2 \sqrt{2}) & & \mbox{by \eqref{eq:h-gamma},} \end{array}
implying (<ref>).
We can now prove the upper bounds of Theorem <ref>.
Let $C \subset N$ be a maximal cusp corresponding to the puncture $p$
of $F$. As above, $\bdy C \cap \core(N)$ decomposes into three compact
annular bands: the band $B_-$ between heights $h_-(N)$ and $h(a_0)$,
the band $B$ between heights $h(a_0)$ and $h(a_k)$, and the band $B_+$
between heights $h(a_k)$ and $h_+(N)$.
The area and height of $B$ were bounded in Lemma
<ref>. As for $B_-$, let $a_0 \in \Delta_-(N)$ be
one of the arcs from $C$ to $C$ that is shortest on $\bdy_-
\core(N)$. Let $\gamma$ be the geodesic in $\bdy_-\core(N)$ in the
homotopy class of $a_0$. By Lemmas <ref> and
\begin{equation}\label{eq:gamma-length}
\ell(\gamma) \: \leq \: 2 \ln \abs{ 6 \chi (F) \, / \, 2 ^{1/4 }} \: =
\: 2 \ln \abs{ \chi(F) } + \ln \left( 18 \sqrt{2} \right).
\end{equation}
Note that $\ln \left( 18 \sqrt{2} \right) \approx 3.2369 > \sqrt{2}$, hence the larger upper bound in Proposition <ref> is the one in equation (<ref>).
Thus, by plugging estimate (<ref>) into (<ref>), we obtain
\[
\height(B_-) \: = \: \abs{ h(a_0) - h(\gamma) } \: \leq \: \ln \abs{
\chi(F) } + \frac{ \ln \left( 18 \, \sqrt{2} \right) }{2} + 0.54
\: < \: \ln \abs{ \chi(F) } + 2.16.
\]
By Lemma <ref>, the circumference of $B_-$
(which is a longitude of the cusp $C$) satisfies $\lambda \leq - 6
\chi(F)$. Thus
\[
\area(B_-) \: = \: \lambda \cdot \height(B_-)
\: < \: \big| 6 \chi(F) \ln \abs{ \chi(F) } + 13 \chi(F) \big|.
\]
The top band $B_+ = B_+(N)$ satisfies the same estimates.
Combining these estimates with Lemma <ref>, we
\[
\area(\bdy C \cap \core(N)) \: = \: \area(B_- \cup B \cup B_+) \: <
\: 9 \chi(F)^2 \, d_\arc(N,p) + \Big| 12 \chi(F) \ln \abs{ \chi(F) }
+ 26 \chi(F) \Big|.
\]
\[
\height(\bdy C \cap \core(N)) \: = \: \height(B_- \cup B \cup B_+)
\: < \: -3 \, \chi(F) \, d_\arc(N, p) + 2 \ln \abs{ \chi(F) } + 5,
\]
completing the proof.
§ SWEEPOUTS
In this section, we describe an important geometric and topological
construction needed for the lower bounds in Theorems <ref>
and <ref>.
Let $N$ be a hyperbolic $3$–manifold, $F$ a surface, and $f_0 \from F
\to N$ a map sending punctures to cusps. Fix a connected set $J
\subset \RR$. A sweepout through $f_0$ is a map $\Psi \from F
\times J \to N$, thought of as a one-parameter family of maps $\Psi_t
\from F \to N$, each homotopic to $f_0$.
A sweepout $\Psi$ is called geometric if every $\Psi_t$ is a
simplicial hyperbolic map: that is, for every $t \in J$, the
image $F_t = \Psi_t(F)$ is a hyperbolic cone surface with at most one
cone point of angle $2\pi \leq \theta_t < 4\pi$. Note that a pleating
map along an ideal triangulation is a special case of a simplicial
hyperbolic map.
Let $g_t$ be the hyperbolic cone metric on $F$ induced by $\Psi_t$.
Then the continuity of $\Psi$ implies that $g_t$ varies continuously
with $t$. In particular, the lengths of geodesic realizations of
homotopy classes of arcs and curves (with respect to $g_t$) vary
continuously with $t$.
Let $M = M_\psi$ be a fibered hyperbolic $3$–manifold with fiber $F$
and monodromy $\psi$. Let $N = N_\psi$ be the infinite cyclic cover
of $M$, with deck transformation $Z \from N \to N$. Fix $r > 0$ and
define $z \from F \cross \RR \to F \cross \RR$ by $z(x, t) = (\psi(x),
t + r)$. We say a sweepout $\Psi \from F \times \RR \to N$ is equivariant if each $\Psi_t$ is properly homotopic to the fiber,
\[
Z \circ \Psi = \Psi \circ z.
\]
Note that equivariance implies that $\Psi$ descends to a degree-one
sweepout of $M$.
Let $M = M_\psi$ be a fibered hyperbolic $3$–manifold with fiber $F$
and monodromy $\psi$. Let $N = N_\psi$ be the infinite cyclic cover.
Then there is a geometric, equivariant sweepout $\Psi \from F \times
\RR \to N$.
This result originates in the work of Thurston <cit.>; see page 9.47 in particular. A careful
account of the proof was also written down by Canary <cit.>. What follows below is a review of their
argument, adapted to ideal triangulations.
Let $\tau_0$ be an ideal triangulation of $F$. Hatcher
proved <cit.> that the triangulations $\tau_0$
and $\psi(\tau_0)$ can be connected by a sequence of diagonal
exchanges, as in Figure <ref>. Thus we have a
sequence of ideal triangulations, $\tau_0, \tau_1, \ldots, \tau_r =
\psi(\tau_0)$, where each $\tau_i$ differs from $\tau_{i-1}$ by a
diagonal exchange. We extend the sequence of triangulations $\tau_i$
to a bi-infinite sequence $\{\tau_i \st i \in \ZZ \}$, such that
$\tau_{i+r} = \psi(\tau_i)$.
Left: a diagonal exchange in a quadrilateral. Center right: a
diagonal exchange between two pleated surfaces creates a
$3$–dimensional tetrahedron $\Delta_i$. Right: a singular
quadrilateral $Q_t$ interpolates between the top and bottom pleated
sides of the tetrahedron.
Fix an embedding $f_0 \from F \to N$ isotopic to the fiber. By
Proposition <ref>, for each $i \in \ZZ$ we may take
$\Psi_i \from F \to N$ to be a pleating of $f_0$ along $\tau_i$. We
choose these pleating maps so that $Z \circ \Psi_i = \Psi_{i+r} \circ
\psi$. Define $F_i = \Image(\Psi_i)$ and notice that, since $\tau_i$
differs from $\tau_{i-1}$ by a diagonal exchange, $F_i$ differs from
$F_{i-1}$ by an ideal tetrahedron $\Delta_i$.
Fix $i \in \ZZ$. Let $\eps_{i-1}$ be the edge of $\tau_{i-1}$ that is
exchanged for the edge $e_i$ of $\tau_i$. Because all six edges of the
tetrahedron $\Delta_i$ lift to hyperbolic geodesics, the edges
$\eps_{i-1}$ and $e_i$ lift to hyperbolic geodesics with no shared
endpoints at infinity. In $\HH^3$, this pair of geodesics is joined by
a unique geodesic segment $\gamma$ that meets $\eps_{i-1}, e_i$
perpendicularly. (In the special case where $\eps_{i-1}$ and $e_i$
intersect, $\gamma$ has length $0$.)
Now, for every $t \in [i-1,i]$, let $x_t$ be the point on
$\gamma$ that is distance $\ell(\gamma)(t-i+1)$ from $\eps_{i-1}$ and
distance $\ell(\gamma)(i-t)$ from $e_i$. In other words, $x_t$ is
obtained by linear interpolation between the points where $\gamma$
meets $\eps_{i-1}$ and $e_i$. We construct a (singular) ideal
quadrilateral $Q_t$ by coning $x_t$ to the four edges of $\Delta_i
\setminus ( e_i \cup \eps_{i-1} ) $. (See Figure
<ref>, right.) Finally, let $F_t$ be the surface
that includes the quadrilateral $Q_t$ inside the tetrahedron
$\Delta_i$, and agrees with $F_i$ everywhere else. Recall that
$F_{i-1}$ agrees with $F_i$ outside $\Delta_i$.
For $t \in (i-1,i)$, we take $\Psi_t \from F \to N_\psi$ to be a
simplicial hyperbolic map with image $F_t$. We choose the maps
$\Psi_t$ so that $\Psi|_{F \cross [i-1,i]}$ is continuous and so that
$Z \circ \Psi_t = \Psi_{t+r} \circ \psi$.
Now consider the geometry of $F_t = \Psi_t(F)$. Wherever this surface
agrees with $F_i$, it is built out of ideal triangles, and inherits an
intrinsically hyperbolic metric from $N_\psi$. Meanwhile, the
quadrilateral $Q_t$ where $F_t$ disagrees with $F_i$ is constructed
out of four $(2/3)$–ideal triangles that share a vertex at
$x_t$. Thus the surface $F_t$ has a smooth hyperbolic metric
everywhere except at $x_t$. At this cone point, observe that the
singular quadrilateral $Q_t$ is not contained in any hyperbolic
half-space through $x_t$. As a result, a lemma of Canary <cit.> implies that the cone angle at $x_t$ is
$\theta_t \geq 2\pi$. Also, because each of the four triangles meeting
at $x_t$ has an interior angle less than $\pi$, we have $\theta_t <
Since the triangulations $\tau_i$ satisfy $\tau_{i+r} =
\psi(\tau_i)$, we have
\[
Z \circ \Psi_{t} = \Psi_{t+r} \circ \psi,
\quad \mbox{for all $t \in \RR$.}
\]
Therefore, $\Psi$ is the desired equivariant, geometric sweepout of
The above construction extends nicely to quasi-Fuchsian manifolds.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian
$3$–manifold. Let $\tau, \tau'$ be ideal triangulations of $F$. Then
there exists a geometric sweepout $\Psi \from F \times [0, r] \to N,$
such that $\Psi_0$ is the pleating map along $\tau$ and $\Psi_r$ is
the pleating map along $\tau'$.
We repeat the proof of Proposition <ref>, without
needing to worry about equivariance. Let $\tau = \tau_0, \tau_1,
\ldots, \tau_r = \tau'$ be a sequence of ideal triangulations of $F$,
with each $\tau_i$ differing from $\tau_{i-1}$ by a diagonal
exchange. Then each $\tau_i$ can be realized by a pleated surface
$F_i$, and we may interpolate from $F_{i-1}$ to $F_i$ by a
$1$–parameter family of simplicial hyperbolic surfaces, as in Figure
Let $\Psi \from F \times [0, r] \to N$ be a geometric sweepout in a
hyperbolic $3$–manifold $N$. Let $C$ be an embedded horocusp in $N$,
with longitude of length $\lambda$. Then, for every surface $F_t =
\Psi_t(F)$ in the sweepout, $C \cap F_t$ contains an equidistant cusp
neighborhood whose area is at least $\lambda$.
Recall that an equidistant cusp of a hyperbolic cone surface is
defined in Definition <ref>.
This is identical to the proof of Lemma <ref>, with
pleated surfaces replaced by simplicial hyperbolic surfaces. First,
take a horocusp $C_0 \subset C$ small enough so that $F_t \cap C_0$ is
a non-singular, horospherical cusp $R_t^0$. If $d$ denotes the
distance between $\bdy C_0$ and $\bdy C$, then $\area(R_t^0) \geq
e^{-d} \lambda$. Then $F_t \cap C$ contains an equidistant
$d$–neighborhood of $R_t^0$. By Lemma <ref>,
this equidistant neighborhood $R_t$ satisfies
\[
\area(R_t) \: \geq \: e^d \, \area(R_t^0) \: \geq \: \lambda.
\qedhere
\]
§ LOWER BOUND: FIBERED MANIFOLDS
In this section, we prove the lower bound of Theorem <ref>.
We begin with a slightly more restricted statement:
Let $F$ be an orientable hyperbolic surface with a preferred puncture
$p$, and let $\psi \from F \to F$ be an orientation-preserving,
pseudo-Anosov homeomorphism such that $\psi(p)=p$. In the mapping
torus $M_\psi$, let $C$ be a horocusp corresponding to $p$, whose
longitude has length $\lambda = 2^{1/4}$. Then there exists an integer
$n \geq 1$, such that
\[
\area(\bdy C) \: > \: \frac{ d_{\arc} (\psi^n)}{450\, \chi(F)^4 }
\quad \mbox{and} \quad
\height(\bdy C) \: > \: \frac{ d_{\arc} (\psi^n)}{536 \,
\chi(F)^4 } .
\]
Theorem <ref> differs from the lower bound of Theorem
<ref> in several small ways. First, it restricts attention to
horocusps that have longitude of length $2^{1/4}$. (This choice of
longitude is justified by Lemma <ref>, which represents
the best available lower bound on the longitude.) Second, Theorem
<ref> restricts attention to monodromies that fix the
puncture $p$. (Given an arbitrary pseudo-Anosov $\mon$, one can let
$\psi$ be the smallest power of $\mon$ such that $\psi(p)=p$.)
Finally, Theorem <ref> estimates cusp area and height in
terms of the translation distance $d_\arc(\psi^n)$ for an unspecified
integer $n$, rather than the stable translation distance
$\dbar_\arc(\psi)$. We shall see at the end of the section that this
restricted statement quickly implies the lower bound of Theorem
As in Sections <ref> and <ref>, it is
convenient to work with the infinite cyclic cover of $M_\psi$, namely
$N_{\psi} \cong F \times \RR$. The horocusps of $M$ lift to annular,
rank one cusps. Let $A \subset N_{\psi}$ be the lift of $\bdy C$ that
corresponds to the puncture $p$ of $F$. Then $A$ is an annulus, with
longitude of length $\lambda = 2^{1/4}$, which covers the torus $\bdy
By Proposition <ref>, there is a geometric, equivariant
sweepout $\Psi \from F \times \RR \to N_\psi$. In particular, for
every $t$, we have $F_t = \Psi_t(F)$ is a hyperbolic cone surface with
at most one singular point of cone angle $2\pi \leq \theta_t < 4\pi$.
Let $F_t$ be a simplicial hyperbolic surface in $N$ (such as one
occurring in the sweepout). We say that an arc $a \in \arc^{(0)}(F,p)$
is short on $F_t$ if $a$ runs from $p$ to $p$, and if its
geodesic representative in the hyperbolic metric of $F_t$ is shortest
among all such arcs.
Note that a given surface $F_t$ can have multiple short arcs. However,
all short arcs on $F_t$ have disjoint representatives, by
Lemma <ref>.
For each surface $F_t$, Lemma <ref> gives an
explicit upper bound on the length of a short arc in $F_t$ (hence,
also on its length in $N_\psi$). On the cusp annulus $A$, each short
arc gives a shadow of a horoball, with area bounded below. More
precisely, we obtain the following quantitative estimate.
Let $a \in \arc^{(0)}(F,p)$ be an arc that is short on $F_t$ for some
$t$. Then, on the cusp annulus $A \subset N_\psi$, the arc $a$
corresponds to a pair of disjoint disks, each of radius
\[
r = \frac{\sqrt{2}}{8 \pi^2 \, \chi(F)^2} \, .
\]
Furthermore, if an arc $b$ is short on $F_{t'}$ for some $t'$, and $a
\neq b \in \arc^{(0)}(F,p)$, then the disks corresponding to $a$ and
$b$ are disjoint on $A$.
By convention (and by Lemma <ref>), the longitude of $A$
has length $\lambda = 2^{1/4}$. Thus, by Lemma <ref>, the
intersection between $F_t$ and the horocusp of $N_\psi$ contains an
equidistant cusp neighborhood $R_t$, of area at least $2^{1/4}$.
Therefore, by Lemma <ref>, the length of a short
arc $a$ in $F_t \setminus R_t$ is
\[
\ell(a) < 2 \ln \abs{ 2\pi \, \chi(F) / 2^{1/4} }.
\]
Since $F_t$ is immersed in $N_\psi$ as a piecewise geodesic union of
hyperbolic triangles, this immersion is distance-decreasing. Thus, in
$N_\psi$, the geodesic $g_a$ in the homotopy class of $a$ must also be
shorter than the above estimate.
Lift $N_\psi$ to the universal cover $\HH^3$ in the upper half-space
model, so that the cusp annulus $A$ lifts to a horizontal horosphere
at Euclidean height $1$. Let $H_\infty$ be the horoball above this
horosphere. This means that $g_a$ lifts to a vertical geodesic that
starts at height $1$ and ends at the top of a horoball $H_a$, of
\begin{equation}
\label{eq:horoball-height}
e^{-\ell(a)} \geq \frac{\sqrt{2}}{4 \pi^2 \, \chi(F)^2} \, .
\end{equation}
In $\HH^3$, there is a covering transformation for $N_\psi$ that maps
$H_a$ to $H_\infty$, and maps $H_\infty$ to another horoball
$H'_a$. The diameter of $H'_a$ must be the same as that of $H_a$
because they lie at the same distance from $H_\infty$ (namely, the
length of $g_a$). This new horoball $H'_a$ cannot belong to the same
orbit as $H_a$ under the parabolic subgroup $\pi_1(A) = \ZZ$
preserving $H_\infty$: for, this parabolic subgroup preserves the
orientation on all the lifts of $g_a$, but one lift of $g_a$ is
oriented downward toward $H_a$ whereas the other is oriented upward
from $H'_a$ toward $H_\infty$. Thus the shadows of $H_a$ and $H'_a$
are disjoint disks on the horosphere at height $1$, which project to
disjoint disks $D_a$ and $D'_a$ on $A$ because $H_a$ and $H'_a$ are in
different orbits. After shrinking $D_a$ and $D'_a$ if necessary, we
obtain a pair of disjoint disks of radius $\sqrt{2} / 8 \pi^2 \,
\chi(F)^2$.
Now, suppose that $b$ is another arc from $p$ to $p$, not isotopic to
$a$, such that $b$ is short on $F_{t'}$ for some $t'$. Then,
performing the same construction as for $a$, we obtain a pair of
horoballs $H_b$ and $H'_b$, whose heights also satisfy equation
(<ref>). The four horoballs $H_a, H'_a, H_b, H'_b$
must lie in distinct orbits of the parabolic $\ZZ$ subgroup preserving
$\infty$, because $a$ and $b$ are in distinct homotopy classes on
$F$. Thus the four horoballs are disjoint in $\HH^3$. Their shadows on
$H_\infty$ are not necessarily disjoint. However, if we shrink all
four horoballs until their diameter is exactly $\sqrt{2} / 4\pi^2 \,
\chi(F)^2$, then the shadows of disjoint horoballs of the same size are
themselves disjoint.
Since $H_a, H'_a, H_b, H'_b$ lie in distinct orbits under $\pi_1(A)$,
we obtain four disjoint disks $D_a, D'_a, D_b, D'_b$ in $A$, each of
radius $\sqrt{2} / 8 \pi^2 \, \chi(F)^2$.
To obtain a lower bound on the area and height of $\bdy C$, we need to
find a sequence of arcs in $F$, each of which is short in a surface
$F_t$ for some $t$.
There is a sequence of real numbers $0 = t_0, t_1, \ldots, t_k = r$
and an associated sequence $a_0, a_1, \ldots, a_k$ of arcs embedded in
$F$, with the following properties:
* Each $a_i$ is short on $F_{t_i}$.
* The first and last arcs satisfy $a_k = \psi(a_0)$.
* Each $a_i$ is disjoint from $a_{i-1}$. In other words, $[a_{i-1},
a_i]$ is an edge of $\arc(F,p)$.
Let $a$ be an embedded arc in $F$ from $p$ to $p$. Define
\[
S(a) := \{ t \in \RR
\st \mbox{$a$ is shortest on $F_t$ among all arcs from $p$ to $p$} \}.
\]
In other words, $S(a)$ consists of those values of $t$ for which $a$
is short (as in Definition <ref>). According to the
definition of a geometric sweepout (Definition <ref>) the
length of any arc varies continuously with $t$. Since being shortest
is a closed condition it follows that the set $S(a)$ is closed. Also,
since every surface $F_t$ in the sweepout has a short arc, the line
$\RR$ is covered by the sets $S(a)$, as $a$ varies over the vertices
of $\arc(F,p)$.
We claim that the arcs $a$ for which $S(a) \neq \emptyset$ belong to
finitely many $\psi$–orbits. This is because every $\psi$–orbit of
arcs in $F$ descends to a single arc in $M_\psi$, with distinct orbits
descending to distinct arcs. The two endpoints of a geodesic $g_a
\subset M_\psi$ representing the arc $a$ must be distinct in $\bdy C$
(otherwise, a deck transformation of $M_\psi$ would reverse the
orientation of a lift of $g_a$, fixing a point in the middle). The
two endpoints of $g_a$ on $\bdy C$ are the centers of disjoint disks
guaranteed by Lemma <ref>. Thus, by
Lemma <ref>, every orbit of arcs that is
shortest on some $F_t$ makes a definite contribution to $\area(\bdy
C)$. On the other hand, $\area(\bdy C)$ is bounded above (e.g. by
Theorem <ref>), hence there can be only finitely many
$\psi$–orbits of arcs for which $S(a) \neq \emptyset$.
Next, we claim that each $S(a)$ is compact. Fix an arc $a$, and let
$g_a$ be the geodesic in $N_\psi$ that is homotopic to $a$. Note that
the sweepout of $N_\psi$ must eventually exit: that is, as
$\abs{t} \to \infty$, $F_t$ must leave any compact set in
$N_\psi$. Thus, as $\abs{t} \to \infty$, the distance from $F_t$ to
the geodesic $g_a$ becomes unbounded (outside the horocusp
$C$). However, any path in $F_t$ that is homotopic to $g_a$ but
remains outside an $s$–neighborhood of $g_a$ must be extremely long
(with length growing exponentially in $s$). Therefore, when $\abs{t}
\gg 0$, the geodesic representative of $a$ on $F_t$ must be very long,
and in particular cannot be the shortest arc on $F_t$. This means that
$S(a)$ is bounded, hence compact.
We can now conclude that there are only finitely many arcs $a$ for
which $S(a) \cap [0,r] \neq \emptyset$. By the first claim above,
these arcs belong to finitely many $\psi$–orbits. Within each orbit,
the compact sets $S(a)$ and $S(\psi(a))$ differ by a translation by
$r$. Thus only finitely many sets in each $\psi$–orbit can intersect
Let $a_0$ be an arc that is short on $F_0$; that is, $0 \in
S(a_0)$. Then $\psi(a_0)$ is short on $F_r$. Since the connected
interval $[0,r]$ is covered by finitely many closed sets $S(a)$, a
lemma in point-set topology (Lemma <ref> in the
Appendix) implies that one may “walk” from $S(a_0)$ to
$S(\psi(a_0))$ by intersecting sets: there is a real number $t_1 \in
S(a_0) \cap S(a_1)$, a number $t_2 \in S(a_1) \cap S(a_2)$, and so on,
for arcs $a_0, \ldots, a_k = \psi(a_0)$.
By definition, $t_i \in S(a_{i-1}) \cap S(a_i)$ means that both
$a_{i-1}$ and $a_i$ are short on $F_{t_i}$. Thus, by Lemma
<ref>, $[a_{i-1}, a_i]$ is an edge of
$\arc(F,p)$. We have therefore constructed a walk through the
$1$–skeleton of $\arc(F,p)$, from $a_0$ to $a_k = \psi(a_0)$, through
arcs $a_i$ that are each short on some simplicial hyperbolic surface.
The sequence of arcs $a_0, a_1, \ldots, a_k$ in Lemma
<ref> contains a subsequence
$b_0, b_1,
\ldots, b_m$ with the following properties:
* Each $b_i$ is short on $F_{t_i}$.
* The arcs $b_1, \ldots, b_{m}$ are all in distinct $\psi$–orbits.
* The first and last arcs satisfy $b_m = \psi^n(b_0)$, for some integer
$n \neq 0$.
* Each $b_i$ is disjoint from $b_{i-1}$. In other words, $[b_{i-1},
b_i]$ is an edge of $\arc(F,p)$.
The arcs $a_0, a_1, \ldots, a_k$ in Lemma <ref>
constitute a walk through the $1$–skeleton of $\arc(F,p)$, from $a_0$
to $a_k = \psi(a_0)$. Given that this walk exists, one can excise some
of the $a_i$ if necessary to form a loop-erased walk $a_0$ to $a_k =
\psi(a_0)$. That is, one may walk from $a_0$ to $a_k = \psi(a_0)$
through some subcollection of the $a_i$, without visiting the same
isotopy class more than once.
Next, suppose that there are indices $i<j$, such that $a_i$ and $a_j$
belong to the same $\psi$–orbit. Without loss of generality, assume
that $i,j$ are an innermost pair with this property. This means that
$a_j = \psi^n(a_i)$ for some $n \neq 0$, and $a_{i+1}, \ldots, a_j$
are all in distinct $\psi$–orbits. Now, we simply restrict attention
to the subsequence from $i$ to $j$. That is, let $b_0 = a_i$, $b_1 =
a_{i+1}$, and so on, until $b_m = a_j$ for $m = j-i$. This subsequence
satisfies the lemma.
We can now complete the proof of Theorem <ref>. Notice that
in Lemma <ref>, $b_0, b_1, \ldots, b_m$ are the
vertices of a path through $\arc^{(1)}(F,p)$ from $b_0$ to
$\psi^n(b_0)$. Thus $m \geq d_{\arc} (\psi^n)$. By Lemma
<ref>, each arc $b_i$ corresponds to two
disjoint disks $D_i, D'_i \subset A$, each of radius $\sqrt{2} / 8
\pi^2 \, \chi(F)^2$. Furthermore, for $j \neq i$, the disks $D_i,
D'_i, D_j, D'_j$ are all disjoint. Thus we have at least $2d_{\arc}
(\psi^n)$ disjoint disks altogether. Since the arcs $b_1, \ldots, b_m$
are all in different $\psi$–orbits, these disks project to $2d_{\arc}
(\psi^n)$ disjoint disks on the cusp torus $\bdy C \subset M_\psi$.
To obtain a lower bound on $\area(\bdy C)$, we sum up the areas of
these disjoint disks, and multiply by the circle packing constant of
$2\sqrt{3}/\pi$ (see <cit.>). Thus
\[
\area(\bdy C) \: \geq \:
\frac{2 \sqrt{3}}{\pi} \cdot 2d_{\arc} (\psi^n) \cdot \pi \left( \frac{\sqrt{2}}{ 8 \pi^2 \, \chi(F)^2} \right)^{\!\! 2}
\: = \: \frac{ \sqrt{3} \, d_{\arc} (\psi^n) }{8 \pi^4 \, \chi(F)^4}
\: > \: \frac{ d_{\arc} (\psi^n) }{450 \, \chi(F)^4} \, .
\]
Finally, since $\area(\bdy C) = \lambda \cdot \height(\bdy C)$, and
we have normalized the horocusp so that $\lambda = 2^{1/4}$, we have
\[
\height(\bdy C)
\: \geq \: \frac{ \sqrt{3} \, d_{\arc} (\psi^n) }{2^{1/4} \, 8 \pi^4 \, \chi(F)^4}
\: > \: \frac{ d_{\arc} (\psi^n) }{536 \, \chi(F)^4} \, .
\qedhere
\]
Let $\mon \from F \to F$ be a pseudo-Anosov homeomorphism, and let
$\psi = \mon^n$ be the smallest power of $\mon$ that fixes the
puncture $p$. Let $C$ be an embedded cusp of $M_\mon$ corresponding to
$p$, whose longitude has length $\lambda = 2^{1/4}$. By Lemma
<ref>, an embedded cusp of this size exists, and is
smaller than the maximal cusp of $M_\mon$. Thus all lower bounds on
$C$ also apply to the maximal cusp.
In the cover $M_\psi$, $C$ lifts to an embedded horocusp $C_1 \subset
M_\psi$, which is a one-sheeted cover of $C$. Furthermore, for every
integer $m \geq 1$, the mapping torus $M_{\psi^m}$ contains an
embedded horocusp $C_m$ whose longitude has length $2^{1/4}$ and which
forms an $m$–fold cover of $C$. Consider what Theorem <ref>
says about the geometry of $C_m$.
By Theorem <ref>, there exists an integer $n(m) \geq m$, such that
\[
\area(\bdy C_m) \: > \: \frac{ d_{\arc} (\psi^{n(m)})}{450\, \chi(F)^4 }
\quad \mbox{and} \quad
\height(\bdy C_m) \: > \: \frac{ d_{\arc} (\psi^{n(m)})}{536 \, \chi(F)^4 } .
\]
Because $C_m$ is an $m$–fold cover of $C$, both its area and its
height are $m$ times larger than those of $C$. Thus, for all $m \geq
\[
\area(\bdy C)
\: > \: \frac{ d_{\arc} (\psi^{n(m)})}{m \cdot 450\, \chi(F)^4 }
\: \geq \: \frac{ d_{\arc} (\psi^{n(m)})}{n(m) \cdot 450\, \chi(F)^4 }
\: \geq \: \inf_{r \geq n(m)} \, \frac{ d_{\arc} (\psi^r)}{r \cdot 450\, \chi(F)^4 }.
\]
Since every $m \geq 1$ gives rise to an integer $n(m) \geq m$
satisfying the above inequality, we have
\[
\area(\bdy C)
\: \geq \: \liminf_{n \to \infty} \, \frac{ d_{\arc} (\psi^n)}{n \cdot 450\, \chi(F)^4 }
\: = \: \frac{ \dbar_{\arc} (\psi)}{ 450\, \chi(F)^4 }.
\]
The last equality holds because the limit in equation
(<ref>), which defines the stable translation
distance, always exists <cit.>. An
identical calculation goes through for $\height(\bdy C)$.
§ LOWER BOUND: QUASI-FUCHSIAN MANIFOLDS
In this section, we prove the lower bounds of Theorem
<ref>. The argument uses almost exactly the same
ingredients as the proof of Theorem <ref>. The one
additional ingredient is the ability to approximate the convex core
boundary by surfaces pleated along triangulations.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian $3$–manifold,
and $p$ a puncture of $F$. As in Definition <ref>, let
$\Delta_-(N)$ be the simplex of $\arc(F,p)$ whose vertices are the
shortest arcs from $p$ to $p$ in the lower core boundary $\bdy_-
\core(N)$. Then there is an ideal triangulation $\tau$ of $F$, such
that in the pleated surface $F_\tau$ pleated along $\tau$, each
shortest arc from $p$ to $p$ is distance at most $1$ from
The same statement holds for the simplex $\Delta_+ (N)$.
Let $R_0 \subset \bdy_- \core(N)$ be a horospherical cusp neighborhood
about puncture $p$, such that $\area(R_0) = 1$. Note that by Lemma
<ref>, the neighborhood $R_0$ is embedded. For
any arc $a$ from puncture $p$ to $p$ in $F$, let $\ell_0(a)$ be the
length of the geodesic representing $a$, in the hyperbolic metric of
$\bdy_- \core(N)$, outside the cusp neighborhood $R_0$. Let $\ell_N(a)$ be
the length of the geodesic representing $a$ in the $3$–manifold $N$, outside a cusp neighborhood of longitude $1$.
Define the set
\begin{equation}
\label{eq:s-define}
T(N) \: = \: \left\{ a \in \arc^{(0)}(F,p) \st \mbox{both ends of $a$ are at $p$, } \:
\ell_N(a) < 2 \ln \abs{6 \chi(F) }+ \ln(2) \right\}.
\end{equation}
Note that by Lemmas <ref> and <ref>, every arc $a \in \Delta_-(N)$ has length
\[
\ell_N(a) \: \leq \: \ell_0(a) \: \leq \: 2 \ln \abs{6 \chi(F) } .
\]
Thus the simplex $\Delta_-(N)$
of shortest arcs in $\bdy_- \core(N)$ must be contained in
$T(N)$. Also note that $T(N)$ must be a finite set: one way to see
this is to recall (e.g. from Lemma <ref>)
that every arc of bounded
length makes a definite contribution to $\area (\bdy C \cap \core(N))$.
Now, we apply Theorem <ref>: there exists a
sequence of triangulations $\tau_i$ of $F$, such that the hyperbolic
metrics on the pleated surfaces $F_{\tau_i}$ converge in the
Teichmüller space $\teich(F)$ to $\bdy_- \core(N)$. For each $i$,
let $R_i \subset F_{\tau_i}$ be an embedded cusp neighborhood of area
$1$. For each arc $a \subset F$, let $\ell_i(a)$ be the length of the
geodesic representing $a$, in the induced hyperbolic metric on
$F_{\tau_i}$, relative to the cusp neighborhood $R_i$. Note that
$\ell_N(a) \leq \ell_i(a)$ by Lemma <ref>. Then,
because the metrics on $F_{\tau_i}$ converge to that on $\bdy_-
\core(N)$, the length of any arc also converges. In particular,
because $T(N)$ is a finite set of arcs, there is some $k \gg 0$ such
\begin{equation}\label{eq:length-change-teich}
\abs{ \ell_k(a) - \ell_0(a) } < \ln(2)/3, \qquad \forall a \in T(N).
\end{equation}
Let $\tau = \tau_k$, and let $b$ be any shortest arc on $F_\tau =
F_{\tau_k}$. By Lemma <ref>, $\ell_k(b) \leq 2 \ln
\abs{6 \chi(F) }.$ Furthermore, $\ell_N(b) \leq \ell_k(b)$, since the pleating
map that produces $F_{\tau_k}$ is $1$–Lipschitz.
Then, by equation (<ref>), it follows that $b \in T(N)$. Thus,
for any arc $a$ that is shortest on $\bdy_- \core(N)$ (that is, $a \in
\Delta_-(N)$), we obtain
\begin{equation}\label{eq:double-change}
\ell_0(b) - \ln(2)/3 \: < \: \ell_k(b)
\: \leq \: \ell_k(a) \: < \: \ell_0(a) + \ln(2)/3,
\end{equation}
where the outer inequalities follow by (<ref>)
and the middle inequality holds because $b$ is shortest on $F_\tau$. A
simpler way to state the conclusion of (<ref>) is
that on the hyperbolic surface $\bdy_- \core(N)$,
\[
\ell_0(b) \: < \: \ell_0(a) + 2 \ln(2)/3.
\]
Since $a$ is shortest on $\bdy_- \core(N)$, and $b$ is nearly
shortest, Lemma <ref> implies that $a$
and $b$ are disjoint (or the same arc) for any $a \in \Delta_-(N)$.
Thus $b$ is distance at most $1$ from $\Delta_-(N)$.
We can now begin proving the lower bound of
Theorem <ref>. Applying the same ideas as in
Section <ref> gives the following analogue of
Lemma <ref>.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian $3$–manifold,
and $p$ a puncture of $F$. There is a sequence $a_0, a_1, \ldots, a_k$
of arcs embedded in $F$, and an associated sequence of simplicial
hyperbolic surfaces $F_{t(a_i)} \subset N$ with at most one singular
point of cone angle $2\pi \leq \theta_t < 4\pi$, such that the
following hold:
* Each $a_i$ is short on $F_{t(a_i)}$, in the sense of Definition
The arcs $a_0, \ldots, a_k$ are distinct up to isotopy.
* $F_{t(a_0)}$ is the lower core boundary $\bdy_- \core(N)$, and
$F_{t(a_k)} = \bdy_+ \core(N)$.
* Each $a_i$ is disjoint from $a_{i-1}$. In other words, $[a_{i-1},
a_i]$ is an edge of $\arc(F,p)$.
As in Definition <ref>, let $\Delta_-(N)$ be the
simplex of $\arc(F,p)$ whose vertices are the short arcs on the lower
boundary $\bdy_- \core(N)$. By Proposition <ref>,
there is a triangulation $\tau$ of $F$, and a pleated surface $F_\tau$
pleated along $\tau$, such that a short arc on this pleated surface
either belongs to the simplex $\Delta_-(N)$, or is at distance $1$
from some vertex of $\Delta_-(N)$. Similarly, there is a
triangulation $\tau'$ of $F$, and a pleated surface $F_{\tau'}$
pleated along $\tau'$, whose short arc either belongs to the simplex
$\Delta_+(N)$, or is at distance $1$ from some vertex of
By Proposition <ref>, there is a geometric sweepout
$\Psi \from F \times [0, r] \to N$, where each $F_t = \Psi(F \times \{
t\})$ is a hyperbolic cone surface with at most one singular point of
cone angle $2\pi \leq \theta_t < 4\pi$. Furthermore, $F_0 = F_\tau$
and $F_r = F_{\tau'}$, for the given triangulations $\tau$ and
Next, we apply the argument of Lemma <ref>. In the
quasi-Fuchsian setting, the proof simplifies in several ways. There
is no need to worry about equivariance, and the set of arcs $\{a :
S(a) \neq \emptyset \}$ is finite because it is contained in $T(N)$
from equation (<ref>). We thus obtain a sequence of arcs
$a_1, \ldots, a_{k-1}$ with the following properties.
* Each $a_i$ is short on some surface $F_{t(a_i)}$ in the sweepout.
* Arc $a_1$ is short on $F_0 = F_\tau$, and $a_{k-1}$ is short on $F_r =
* Each $a_i$ is disjoint from $a_{i-1}$. In other words, $[a_{i-1},
a_i]$ is an edge of $\arc(F,p)$.
Next, we extend this sequence of arcs to the convex core boundary. If
$a_1 \notin \Delta_-(N)$, then
there is an arc $a_0 \in \Delta_-(N)$,
which is shortest on $\bdy_- \core(N)$ by definition, and such that
$[a_0, a_1]$ is an edge of $\arc(F,p)$. Otherwise, if $a_1 \in
\Delta_-(N)$, then we simply shift indices by $1$, so that $a_1$
becomes $a_0$. Similarly, if $a_{k-1} \notin \Delta_+(N)$, then we add
an arc $a_k \in \Delta_+(N)$, which is shortest on $\bdy_+
\core(N)$. Otherwise, if $a_{k-1} \in \Delta_+(N)$, then we simply
redefine $k := k-1$, and stop the sequence there.
We now have a sequence of arcs $a_0, \ldots, a_k$, with associated
simplicial hyperbolic surfaces $F_{t(a_i)}$, so that this sequence
satisfies all the conclusions of the lemma except possibly
(<ref>). That is, some of the $a_i$ might be in the
same isotopy class. But if the arcs $a_0, \ldots, a_k$ are the
vertices of a path in $\arc^{(1)}(F,p)$, then some subcollection of
the $a_i$ give an embedded path. This means that
(<ref>) is satisfied, and the proof is complete.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian manifold, and
$p$ a puncture of $F$. Let $C \subset N$ be a horocusp corresponding
to the puncture $p$, whose longitude is $\lambda = 2^{1/4}$. Then, for
some $k \geq d_\arc(N,p)$, the annulus $A = \bdy C$ contains $2k+2$
disjoint disks, each of radius
$$r = \frac{\sqrt{2}}{8 \pi^2 \, \chi(F)^2},$$
such that the center of each disk is in the convex core $\core(N)$.
Recall that by Lemma <ref>, there is indeed an embedded
horocusp of longitude $\lambda = 2^{1/4}$. Recall as well, from
Definition <ref>, that $d_\arc(N,p)$ is defined to be
the shortest distance in $\arc(F,p)$ between a vertex of $\Delta_-(N)$
and a vertex of $\Delta_+(N)$.
The sequence of arcs $a_0, \ldots, a_k$, constructed in Lemma
<ref>, is a walk through the $1$–skeleton of
$\arc(F,p)$ from a vertex of $\Delta_-(N)$ to a vertex of
$\Delta_+(N)$. Thus, by Definition <ref>, $k \geq
For each $i \in \{0, \ldots, k \}$, let $g_{a_i}$ be the geodesic in
$N$ in the homotopy class of $a_i$. Then, Lemma
<ref> guarantees that there is a pair of
disjoint disks $D_{a_i}$ and $D'_{a_i}$, of radius $r = \sqrt{2} / 8
\pi^2 \, \chi(F)^2$, whose centers are the endpoints of $g_{a_i}$ on
the cusp annulus $A$. Since the geodesic $g_{a_i}$ is contained in
the convex core of $N$, the centers of $D_{a_i}$ and $D'_{a_i}$ lie in
$\core(N)$ as well.
Finally, Lemma <ref> also implies that if $j
\neq i$, the disks of $a_i$ are disjoint from those of $a_j$. Thus we
have at least $2k+2$ disks in total.
Let $p$ be a puncture of $F$, and let $C \subset N$ be a horospherical
cusp corresponding to the puncture $p$, whose longitude is $\lambda =
2^{1/4}$. Note that by Lemma <ref>, $C$ is contained in
the maximal cusp about puncture $p$. Thus lower bounds on the area and
height of $\bdy C \cap \core(N)$ also apply to the maximal cusp.
As in Section <ref> and <ref>, we may place
Euclidean coordinates on the annulus $A = \bdy C \cong S^1 \times
\RR$, in which the $\RR$ direction is vertical. Let $B \subset A$ be a
compact annular band, with boundary consisting of horizontal circles,
which is the smallest such band that contains all of the $2k+2$ disks
of Lemma <ref>. Since each disk has radius $r =
\sqrt{2} / 8 \pi^2 \, \chi(F)^2$, Böröczky's estimate on the
density of a circle packing <cit.> implies that
$$\area(B) \: \geq \:
\frac{2 \sqrt{3}}{\pi} \cdot (2d_{\arc} (N) + 2) \cdot \pi \left( \frac{\sqrt{2}}{ 8 \pi^2 \, \chi(F)^2} \right)^{\!\! 2}
\: = \: \frac{ \sqrt{3} \, (d_{\arc} (N) + 1) }{8 \pi^4 \, \chi(F)^4}
\: > \: \frac{ d_{\arc} (N) + 1 }{450 \, \chi(F)^4} \, .
Similarly, since $\area(B) = \lambda \cdot \height(B)$, and we have
normalized the horocusp so that $\lambda = 2^{1/4}$, we have
\: \geq \: \frac{ \sqrt{3} \, (d_{\arc} (N) + 1) }{2^{1/4} \, 8 \pi^4 \, \chi(F)^4}
\: > \: \frac{ (d_{\arc} (N) + 1) }{536 \, \chi(F)^4} \, .
To complete the proof, it remains is to bound the difference in area
(or height) between $B$ and $\bdy C \cap \core(N)$. Note that by Lemma
<ref>, each of the $(2k+2)$ disks has its center inside
$\core(N)$. Therefore, the upper boundary of $B$ is at most $r =
\sqrt{2} / 8 \pi^2 \, \chi(F)^2$ higher than $\bdy_+ \core(N)$, and
the lower boundary of $B$ is at most $r$ lower than $\bdy_-
\core(N)$. This implies that
\[
\area(\bdy C \cap \core(N)) \: \geq \: \area(B) - 2 \lambda r
\: > \: \frac{ d_{\arc} (N) }{450 \, \chi(F)^4} - \frac{2 \cdot 2^{3/4} }{8 \pi^2 \, \chi(F)^2}
\: > \: \frac{ d_{\arc} (N) }{450 \, \chi(F)^4} - \frac{1}{23 \, \chi(F)^2}.
\]
\[
\height(\bdy C \cap \core(N)) \: \geq \: \height(B) - 2 r
\: > \: \frac{ d_{\arc} (N) }{536 \, \chi(F)^4} - \frac{1}{27 \, \chi(F)^2}.
\qedhere
\]
§ COVERS AND THE ARC COMPLEX
In this section, we will apply Theorem <ref> to prove Theorem <ref>, which relates the arc complex of a surface $S$ to that of its cover $\Sigma$. The proof uses some classical results in Kleinian groups to construct a quasi-Fuchsian manifold with prescribed short arcs on its convex core boundary (see Lemma <ref>). We begin by recalling some terminology and results, while pointing the reader to Marden <cit.> for a more detailed reference.
Let $\Gamma$ be a Kleinian group with limit set $\Lambda$. The domain of discontinuity is $\Omega = \bdy \HH^3 \setminus \Lambda$. When $N = \HH^3 / \Gamma$ is quasi-Fuchsian, $\Omega$ is the disjoint union of two open disks $\Omega_+$ and $\Omega_-$, each of which admits a conformal, properly discontinuous action by $\Gamma$. The quotients $S_\pm = \Omega_\pm / \Gamma$ are Riemann surfaces, called the (top and bottom) conformal boundary of $N = \HH^3 / \Gamma$.
For each quasi-Fuchsian manifold $N$, there is a natural “nearest point retraction” map $r \from S_\pm \to \bdy_\pm \core(N)$. Sullivan proved that, if $S_\pm$ is given the unique hyperbolic metric in its conformal class, the map $r \from S_\pm \to \bdy_\pm \core(N)$ is $K$–Lipschitz, for a universal constant $K > 1$. Much more recently, Epstein, Marden, and Markovic <cit.> showed that the optimal Lipschitz constant is $2$. We will use their result for concreteness, while emphasizing that Sullivan's original $K$–Lipschitz statement is all that is truly needed.
We also recall some facts from the geometry of surfaces. Let $\eps_2$ be the $2$–dimensional Margulis constant. For any simple closed geodesic $\gamma$ in a hyperbolic surface $S$, of length $\ell = \ell(\gamma) < \eps_2$, the $\eps_2$–thin region of $S$ containing $\gamma$ is an embedded collar of radius $r(\ell)$. The function $r(\ell)$ is monotonically decreasing, and $r(\ell) \to \infty$ as $\ell \to 0$. See Buser <cit.> for explicit estimates on $\eps_2$ and $r(\ell)$.
Let $S$ be a hyperbolic surface, and $\gamma$ a simple closed geodesic on $S$. We say that $\gamma$ is sufficiently thin if its length $\ell = \ell(\gamma)$ is short enough that the $\eps_2$–thin collar about $\gamma$ has radius
\begin{equation}\label{eq:thin}
r(\ell) \: > \: \ln \abs{ 6 \chi(S) / \eps_2 }.
\end{equation}
This requirement on collar radius is motivated by Lemma <ref>.
Given this background, we can prove the following constructive lemma.
Let $F$ be a surface with a preferred puncture $p$, and let $a_-, a_+ \in \arc^{(0)}(F,p)$ be arcs from $p$ to $p$. Then there exists a quasi-Fuchsian manifold $N \cong F \times \RR$, such that $a_\pm$ is the unique shortest arc from $p$ to $p$ on $\bdy_\pm \core(N)$.
Let $R_0 \subset F$ be a neighborhood of the puncture $p$, and let $R(a_+)$ be a regular neighborhood of $R_0 \cup a_+$. This is topologically a pair of pants, whose frontier in $F$ consists of a pair of essential closed curves $c_+ ,c'_+$. (If $F$ is a once-punctured torus, then $c_+$ is isotopic to $c'_+$; this will not affect our arguments.) Similarly, let $c_+ ,c'_+$ be closed curves that form the frontier of a regular neighborhood $R(a_-)$.
Choose hyperbolic metrics $X_\pm$ on $F$, in which the geodesic representatives of $c_\pm$ and $c'_\pm$ have less than half the length required to be sufficiently thin. By Bers simultaneous uniformization <cit.>, there is a quasi-Fuchsian manifold $N \cong F \times \RR$ whose top conformal boundary is $X_+$ and whose bottom conformal boundary is $X_-$. Thus, by Epstein, Marden, and Markovic <cit.>, the geodesic representatives of $c_\pm$ and $c'_\pm$ are sufficiently thin on $\bdy_\pm \core(N)$.
We claim that $a_+$ is the unique shortest arc from $p$ to $p$ on $\bdy_+ \core(N)$. For concreteness, we will measure lengths relative to the horospherical cusp neighborhood $Q(p) \subset \bdy_+ \core(N)$ whose boundary has length $\eps_2$. Since horoballs are convex, it follows that $Q(p)$ is $\eps_2$–thin. Then, by Lemma <ref>,
there must be a geodesic $\alpha_+$ from $p$ to $p$ whose length relative to this cusp neighborhood satisfies
$$\ell(\alpha_+) \: \leq \: 2 \ln \abs{ 6 \chi(S) / \eps_2 }.$$
Now, let $b_+$ be any arc from $p$ to $p$, other than $a_+$. Since $b_+$ is not isotopic into the neighborhood $R(a_+)$, it must cross $c_+ \cup c'_+$.
By the Margulis lemma, the $\eps_2$–thin collars about those curves
are disjoint from the $\eps_2$–thin cusp neighborhood $Q(p)$. Thus,
since $c_+$ and $c'_+$ are sufficiently thin, and the width of a
collar is twice the radius, equation (<ref>) implies that
$\ell(b_+) > \ell(\alpha_+)$. Therefore $a_+ = \alpha_+$, the unique
shortest arc from $p$ to $p$.
By the same argument, $a_-$ is the unique short arc on $\bdy_- \core(N)$.
We can now complete the proof of Theorem <ref>, which we restate.
Theorem <ref>
Let $\Sigma$ and $S$ be surfaces with one puncture, and $f \from
\Sigma \to S$ a covering map of degree $n$. Let $\pi \from \arc(S)
\to \arc(\Sigma)$ be the lifting relation induced by $f$. Then, for
all $a,b \in \arc^{(0)}(S)$, we have
\[
\frac{d(a,b)}{4050 \, n \, \chi(S)^6} \, - 2 \: < \: d(\alpha, \beta) \: \leq \: d(a, b)
\]
where $\alpha \in \pi(a)$ and $\beta \in \pi(b)$.
Recall that by Definition <ref>, $\pi(a)$ is the set
of $n$ vertices in $\arc(\Sigma)$ representing arcs that project to
$a$. These vertices form a simplex in $\arc(\Sigma)$, since the arcs
that comprise $f^{-1}(a)$ are disjoint. Similarly, if $a$ and $a'$
are distance $1$ in $\arc(S)$, then all the $2n$ lifts of $a$ and $a'$
are disjoint in $\Sigma$, hence every vertex $\alpha \in \pi(a)$ is
distance $1$ in $\arc(\Sigma)$ from every vertex $\alpha' \in
\pi(a')$. Thus, by induction on distance, we have
\[
d(\alpha, \beta) \, \leq \, d(a, b)
\]
for any $\alpha \in \pi(a)$ and any $\beta \in \pi(b)$.
To prove the other inequality in Theorem <ref>, assume the
cover is non-trivial: that is, $n > 1$. Fix $a, b \in
\arc^{(0)}(\Sigma)$. By Lemma <ref>, there is a
quasi-Fuchsian manifold $M \cong S \times \RR$, such that $a$ is the
unique shortest arc on $\bdy_- \core(M)$ and $b$ is the unique
shortest arc on $\bdy_+ \core(M)$.
Since $S$ has a unique puncture $p$, we have $\arc(S) = \arc(S,p)$.
Let $C \subset M$ be the maximal horospherical cusp corresponding to
this unique puncture. By Definition <ref>,
$d_\arc(M,p) = d(a,b)$. Thus, by the lower bound of
Theorem <ref>,
\begin{equation}\label{eq:lower-m}
\frac{d(a,b)}{450 \, \chi(S)^4} - \frac{1}{23 \, \chi(S)^2}
\: < \: \area( \bdy C \cap \core(M)).
\end{equation}
Use the $n$–fold covering map $f \from \Sigma \to S$ to lift the
hyperbolic metric on $M \cong S \times \RR$ to a quasi-Fuchsian
structure on $N \cong \Sigma \times \RR$. The convex core of $N$
covers the convex core of $M$. The horocusp $C \subset M$ lifts to a
horocusp $D \subset N$, which is an $n$–sheeted cover of $C$. Thus
\begin{equation}\label{eq:m-n}
n \cdot \area (\bdy C \cap \core(M)) \: = \: \area(\bdy D \cap \core(N)).
\end{equation}
Observe that every arc $a \in \bdy_+ \core(M)$ lifts to $n$ disjoint
arcs in $\bdy_+ \core(N)$, each of which has the same length as $a$
(outside the horocusps $C$ and $D$, respectively). Thus the $n$ arcs
of $\pi(b)$ are shortest on $\bdy_+ \core(N)$, and similarly the $n$
arcs of $\pi(a)$ are shortest on $\bdy_- \core(N)$. By
Definition <ref>, this implies $d_\arc(N,p) \leq
d(\alpha, \beta)$ for any $\alpha \in \pi(a)$ and any $\beta \in
\pi(b)$. Therefore, the upper bound of Theorem <ref>
\begin{equation}\label{eq:upper-n}
\area( \bdy D \cap \core(N)) \: < \: 9 \, \chi(\Sigma)^2 \, d(\alpha,\beta) +
\Big| 12 \chi(\Sigma) \ln \abs{ \chi(\Sigma) } + 26 \chi(\Sigma) \Big|.
\end{equation}
Combining equations (<ref>), (<ref>) and
(<ref>), we obtain
\begin{equation*}
\frac{n \cdot d(a, b) }{450 \, \chi(S)^4} - \frac{n}{23 \, \chi(S)^2} \: < \: 9 \, \chi(\Sigma)^2 \, d(\alpha,\beta) +
\Big| 12 \chi(\Sigma) \ln \abs{ \chi(\Sigma) } + 26 \chi(\Sigma) \Big|,
\end{equation*}
which can be rearranged, using $\chi(\Sigma) = n \chi(S)$, to give
\begin{equation*}
\frac{d(a, b) }{4050 \, n \, \chi(S)^6} - \frac{1}{23 \cdot 9 \, n \, \chi(S)^4} - \frac{\Big| 12 \chi(\Sigma) \ln \abs{ \chi(\Sigma) } + 26 \chi(\Sigma) \Big| } {9 \chi(\Sigma)^2} \: < \: d(\alpha, \beta).
\end{equation*}
Since $\Sigma$ is a once-punctured surface that non-trivially covers
$S$, we have $\abs{\chi(\Sigma)} \geq 3$. It follows that the additive
error on the left-hand side is bounded above by 2. This completes the
§ APPROXIMATING THE CONVEX CORE BOUNDARY
The goal of this appendix is to write down a proof of the following
result, which is needed in the argument of Section <ref>.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian
$3$–manifold. Then there is a sequence $\tau_i$ of ideal
triangulations of $F$, such that the induced hyperbolic metrics on the
pleated surfaces $F_{\tau_i}$ converge in the Teichmüller space
$\teich(F)$ to the hyperbolic metric on the lower core boundary
$\bdy_- \core(N)$. Furthermore, the pleating maps for the $F_{\tau_i}$
converge in the compact–open topology to a pleating map for $\bdy_-
\core(N)$.
The same statement holds for the upper core boundary $\bdy_+ \core(N)$.
The statement of Theorem <ref> is entirely
unsurprising, and morally it should fit into the toolbox of well-known
results about laminations and pleated surfaces <cit.>. Indeed, the standard toolbox of Kleinian group
theory leads to a relatively quick proof of the theorem. However,
this short proof is also somewhat technical, as it requires passing
between several different topologies on spaces of laminations and
pleated surfaces.
The main reference for the following argument is Canary, Epstein, and
Green <cit.>. See also Thurston
<cit.>, Bonahon
<cit.>, and Ohshika <cit.>.
Let $S$ be a punctured hyperbolic surface of finite area. Let $\GL(S)$
denote the set of geodesic laminations on $S$: that is, laminations
where each leaf is a geodesic. We equip $\GL(S)$ with the
Chabauty topology. In this topology, a sequence $\{L_i \} $
converges to $L \in \GL(S)$ if and only if:
* If a subsequence $x_{n_i} \in L_{n_i}$ converges to $x \in S$, then $x
\in L$.
* For all $x \in L$, there exists a sequence $x_i \in L_i$, such that
$x_i \to x$.
The Chabauty topology is metrizable
<cit.>. In fact, when restricted
to compact laminations, the Chabauty topology reduces to be the
Hausdorff topology (induced by the Hausdorff distance between compact
sets). See <cit.> for more details.
Definition <ref> makes use of a hyperbolic metric on $S$, but
in an inessential way. If we modify a metric $d$ to a new hyperbolic
metric $d'$, each lamination $L \in \GL(S)$ that is geodesic in $d$
can be straightened to a geodesic lamination of $d'$. This
straightening does not affect convergence of laminations. Thus the
space $\GL(S)$ only depends on the topology of $S$.
We use the term curve to denote a simple closed geodesic in
$S$. The following lemma is a good example of convergence in the
Chabauty topology.
For every curve $\alpha \subset S$, there is a sequence of ideal
triangulations $\tau_i$, converging in the Chabauty topology to a
lamination $\alpha' \supset \alpha$.
Fix any ideal triangulation $\tau$. Let $D = D_\alpha$ be a Dehn
twist about $\alpha$. Then $\tau_i = D^i(\tau)$ converges to the desired
Let $S$ be a punctured hyperbolic surface of finite area. Then
$\ML(S)$ denotes the space of compact, transversely measured
laminations. Every point of $\ML(S)$ is a pair $(\lamm, \mu)$ where
$\lamm$ is a compact geodesic lamination and $\mu$ is a transverse
measure of full support. That is, for each arc $\alpha$ intersecting
$\lamm$ transversely, $\mu(\alpha)$ is a positive number that stays
invariant under an isotopy preserving the leaves of $\lamm$. The
natural topology on $\ML(S)$ is called the measure topology.
Let $\PML(S)$ denote the projectivization of $\ML(S)$, in which
nonzero measures that differ by scaling become identified. The measure
topology on $\ML(S)$ descends to $\PML(S)$.
If $\lamm$ is a disjoint union of arcs and closed curves in $S$, an
example of a transverse measure is the counting measure, where
$\mu(\alpha) = \abs{\alpha \cap \lamm}$. For another example, suppose
that $S = \bdy_+ \core(N)$ is the upper boundary of the convex core in
a quasi-Fuchsian $3$–manifold. Then the pleating lamination $\lamm$
has a bending measure, where $\mu(\alpha)$ is the integral of
the bending of $\alpha$ as it crosses leaves of $\lamm$.
The measure-forgetting map $\PML(S) \to \GL(S)$ is not continuous, but
it has the following convenient property.
Suppose that $(\lamm_i, \mu_i) \to (\lamm, \mu) \in \PML(S)$, in the
measure topology. Then, after passing to a subsequence, there is a
lamination $\lamm' \in \GL(S)$ so that $\lamm \subset \lamm'$ and
$\lamm_i \to \lamm'$ in the Chabauty topology.
With these facts in hand, we can prove
Theorem <ref>. The proof contains two steps, the
first of which deals with laminations only.
Let $\lamm \in \GL(S)$ be a measurable lamination. Then there is a
sequence of ideal triangulations $\tau_i$ converging in the Chabauty
topology to a lamination $\lamm'' \supset \lamm$.
Pick $\mu$, a measure of full support on $\lamm$, so that $(\lamm,
\mu) \in \PML(S)$.
Thurston proved that
curves, equipped with the counting measure, are dense in $\PML(S)$
<cit.>. Thus we may pick a
sequence of curves $\{ \alpha_i \}$ converging to $(\lamm, \mu)$ in
the measure topology. By Fact <ref>, we may pass to a
subsequence and reindex so that $\alpha_i$ converges, in the Chabauty
topology, to a lamination $\lamm'$ containing $\lamm$.
By Lemma <ref>, we may choose $\{\tau_{i,j}\}$, a sequence of
sequences of ideal triangulations, so that for all $i$
\[
\tau_{i,j} \to \alpha'_i \supset \alpha_i
\quad \mbox{as} \quad j \to \infty.
\]
For all $i$, choose an increasing function $j \from \NN \to \NN$, so
that $\tau_{i,j(i)}$ and $\alpha'_i$ have distance at most $1/i$, in
the metric that induces the Chabauty topology.
The sequence $\tau_{i,j(i)}$ contains a subsequence converging to
$\lamm'' \supset \lamm'$.
As $\GL(S)$ is compact <cit.>,
the sequence $\tau_{i,j(i)}$ contains a convergent subsequence. In an
abuse of notation, denote this convergent subsequence by $\tau_i =
\tau_{i,j(i)}$. Let $\lamm''$ be the limit of the $\tau_i$. Note
that $\lamm''$ is not compact.
Recall that the Chabauty distance between $\tau_i$ and $\alpha'_i$ is
at most $1/i$. Thus the sequence $\alpha'_i$ has the same limit as
$\tau_i$, namely $\lamm''$. Since $\alpha_i \subset \alpha'_i$, it
follows by <cit.> that the limit of
$\alpha_i$ must be contained in the limit of $\alpha'_i$. That is,
$\lamm' \subset \lamm''$, as desired.
Since $\lamm \subset \lamm'$, Lemma <ref> is
The second step of the argument connects the above discussion of
laminations to pleated surfaces. Following Definition
<ref>, we call a lamination $L \in \GL(S)$
realizable if it is realized by a pleating map $f \from S \to
N$ homotopic to a prescribed map $f_0$.
Let $N \cong S \times \RR$ be a cusped quasi-Fuchsian $3$–manifold,
and fix an embedding $f_0 \from S \to S \times \{ 0 \}$. Let $L_i \in
\GL(S)$ be a sequence of laminations on $S$, which are realizable by
pleating maps homotopic to $f_0$. Then, if $L_i \to L'$ in the
Chabauty topology, and $L \subset L'$ is also realizable, the
hyperbolic metrics $d_i$ induced by pleating along $L_i$ converge in
$\teich(S)$ to the hyperbolic metric $d$ induced by pleating along
In fact, the pleating maps $f_i \from S \to N$ converge to the
pleating map $f \from S \to N$ that realizes $L$, in the space
$\mathcal{MPS}(S, N)$ of marked pleated surfaces homotopic to $f_0$.
We refer the reader to <cit.>
for the full definition of $\mathcal{MPS}(S, N)$. It suffices to note
that convergence in $\mathcal{MPS}(S, N)$ involves both
convergence of metrics in $\teich(S)$ and convergence of pleating maps
in the compact-open topology.
Let $C$ be an embedded neighborhood of the cusps of $N$. Then $K
:= \core(N) \setminus C$ is a compact set, and every pleated
surface homotopic to $f_0$ must intersect $K$.
With this notation, <cit.> implies
that the space $\mathcal{MPS}(S, N) = \mathcal{MPS}(S, K)$ of marked
pleated surfaces that intersect $K$ is compact. In particular, the
pleating maps $f_i \from S \to N$, which pleat along lamination $L_i$,
have a convergent subsequence, $f_{n_i} \to f_\infty$. Let $L_\infty$
be the pleating lamination of $f_\infty$.
Now, recall that $L_i \to L'$ in the Chabauty topology, and $L \subset
L'$ is realizable by a pleating map $f \from S \to N$ homotopic to
$f_0$. Since every component of $f(S \setminus L)$ is totally geodesic
in $N$, the leaves of $L' \setminus L$ are mapped into these totally
geodesic regions, hence $L'$ is realized by the same map $f$. In this
setting, Ohshika <cit.> notes that $f_\infty = f$
is the same pleating map that realizes the limiting lamination $L'$.
Therefore, every convergent subsequence of $f_i$ must limit to the
same map $f_\infty = f$ that realizes the lamination $L$. Since
$\mathcal{MPS}(S, K)$ is compact, this means that $f_i \to f$ in the
topology on $\mathcal{MPS}(S, K)$. This means that $f_i \to f$ in the
compact-open topology, and also that the hyperbolic metrics $d_i$
induced by $f_i$ converge in $\teich(S)$ to the hyperbolic metric $d$
induced by $f$.
Let $N \cong F \times \RR$ be a cusped quasi-Fuchsian $3$–manifold,
as in the statement of the theorem. Let $L$ be the pleating lamination
on the lower core boundary $\bdy_- \core(N)$. The bending measure
$\mu$ on $\bdy_- \core(N)$ is a transverse measure of full support, so
$L$ is a measurable lamination. By Lemma <ref>,
there is a sequence of ideal triangulations $\tau_i \to L'$ in the
Chabauty topology, where $L \subset L'$. Now, by Lemma
<ref>, the pleating maps $f_i \from F \to N$ that
pleat along $\tau_i$ induce hyperbolic metrics on $F$ that converge in
$\teich(F)$ to the induced metric on $\bdy_- \core(N)$. By the definition of
$\mathcal{MPS}(S, N)$, these pleating maps also converge in the
compact–open topology to a pleating map for $\bdy_- \core(N)$.
§ A LEMMA IN POINT-SET TOPOLOGY
Let $X$ be a topological space, and let $\mathcal{S}$ be a cover of
$X$ by closed sets. Define a discrete walk of length $k$
through sets of $\mathcal{S}$ to be a sequence of points $x_0, x_1,
\ldots, x_k$, such that $x_0, x_1 \in S_1$, $x_1, x_2 \in S_2$, and so
on, for sets $S_1, \ldots, S_k$ that belong to $\mathcal{S}$. We say
this sequence is a walk from $x_0$ to $x_k$.
The following observation is needed in the proof of
Lemma <ref>.
Let $X$ be a connected topological space, and let $S_1, \ldots, S_n$
be closed sets whose union is $X$. Then, for any pair of points $x,y
\in X$, there is a discrete walk from $x$ to $y$ through sets in the
collection $\{ S_1, \ldots, S_n \}$.
Define an equivalence relation $\equiv$ on $X$, where
\[
x \equiv y \quad \Leftrightarrow \quad \mbox{there exists a discrete
walk from $x$ to $y$ through $\{ S_1, \ldots, S_n \}$.}
\]
Reversing and concatenating walks proves this is an equivalence
relation. Furthermore, each closed set $S_i$ must be entirely
contained in an equivalence class, because its points are connected by
a walk of length $1$. Thus each equivalence class is closed. Since the
connected space $X$ cannot be expressed as a disjoint union of
finitely many closed sets, all of $X$ must be in the same equivalence
|
arxiv-papers
| 2011-08-29T20:40:54 |
2024-09-04T02:49:21.803936
|
{
"license": "Public Domain",
"authors": "David Futer, Saul Schleimer",
"submitter": "David Futer",
"url": "https://arxiv.org/abs/1108.5748"
}
|
1108.5985
|
# An Oracle-based, Output-sensitive Algorithm
for Projections of Resultant Polytopes
Ioannis Z. Emiris Department of Informatics & Telecommunications, University
of Athens, Athens, 15784, Greece. emiris@di.uoa.gr, vfisikop@di.uoa.gr
Vissarion Fisikopoulos11footnotemark: 1 Christos Konaxis Archimedes Center for
Modeling, Analysis & Computation (ACMAC), University of Crete, Heraklio,
71409, Greece. ckonaxis@acmac.uoc.gr Luis Peñaranda $\dagger$
###### Abstract
We design an algorithm to compute the Newton polytope of the resultant, known
as resultant polytope, or its orthogonal projection along a given direction.
The resultant is fundamental in algebraic elimination, optimization, and
geometric modeling. Our algorithm exactly computes vertex- and halfspace-
representations of the polytope using an oracle producing resultant vertices
in a given direction, thus avoiding walking on the polytope whose dimension is
$\alpha-n-1$, where the input consists of $\alpha$ points in
${\mathbb{Z}}^{n}$. Our approach is output-sensitive as it makes one oracle
call per vertex and facet. It extends to any polytope whose oracle-based
definition is advantageous, such as the secondary and discriminant polytopes.
Our publicly available implementation uses the experimental CGAL package
triangulation. Our method computes $5$-, $6$\- and $7$-dimensional polytopes
with $35$K, $23$K and $500$ vertices, respectively, within $2$hrs, and the
Newton polytopes of many important surface equations encountered in geometric
modeling in $<1$sec, whereas the corresponding secondary polytopes are
intractable. It is faster than tropical geometry software up to dimension $5$
or $6$. Hashing determinantal predicates accelerates execution up to $100$
times. One variant computes inner and outer approximations with, respectively,
90% and 105% of the true volume, up to $25$ times faster.
### Keywords:
General Dimension, Convex Hull, Regular Triangulation, Secondary Polytope,
Resultant, CGAL Implementation, Experimental Complexity.
00footnotetext: $\dagger$ IMPA – Instituto Nacional de Matemática Pura e
Aplicada, Rio de Janeiro, 22460-320, Brazil. luisp@impa.br
## 1 Introduction
Given pointsets $A_{0},\dots,A_{n}\subset{\mathbb{Z}}^{n}$, we define the
pointset
${\mathcal{A}}:=\bigcup_{i=0}^{n}(A_{i}\times\\{e_{i}\\})\subset{\mathbb{Z}}^{2n},$
(1)
where $e_{0},\ldots,e_{n}$ form an affine basis of ${\mathbb{R}}^{n}$: $e_{0}$
is the zero vector,$e_{i}=(0,\dots,0,1,0,\dots,0),i=1,\dots,n$. Clearly,
$|{\mathcal{A}}|=|A_{0}|+\cdots+|A_{n}|$, where $|\cdot|$ denotes cardinality.
By Cayley’s trick (Proposition 2) the regular tight mixed subdivisions of the
Minkowski sum $A_{0}+\cdots+A_{n}$ are in bijection with the regular
triangulations of ${\mathcal{A}}$, which are in bijection with the vertices of
the secondary polytope $\Sigma({\mathcal{A}})$ (see Section 2).
The Newton polytope of a polynomial is the convex hull of its support, i.e.
the exponent vectors of monomials with nonzero coefficient. It subsumes the
notion of degree for sparse multivariate polynomials by providing more precise
information (see Figures 1 and 3). Given $n+1$ polynomials in $n$ variables,
with fixed supports $A_{i}$ and symbolic coefficients, their sparse (or toric)
resultant ${\mathcal{R}}$ is a polynomial in these coefficients which vanishes
exactly when the polynomials have a common root (Definition 1). The resultant
is the most fundamental tool in elimination theory, it is instrumental in
system solving and optimization, and is crucial in geometric modeling, most
notably for changing the representation of parametric hypersurfaces to
implicit.
The Newton polytope of the resultant $N({\mathcal{R}})$, or resultant
polytope, is the object of our study; it is of dimension
$|{\mathcal{A}}|-2n-1$ (Proposition 4). We further consider the case when some
of the input coefficients are not symbolic, hence we seek an orthogonal
projection of the resultant polytope. The lattice points in $N({\mathcal{R}})$
yield a superset of the support of ${\mathcal{R}}$; this reduces
implicitization 1, 2 and computation of ${\mathcal{R}}$ to sparse
interpolation (Section 2). The number of coefficients of the $n+1$ polynomials
ranges from $O(n)$ for sparse systems, to $O(n^{d}d^{n})$, where $d$ bounds
their total degree. In system solving and implicitization, one computes
${\mathcal{R}}$ when all but $O(n)$ of the coefficients are specialized to
constants, hence the need for resultant polytope projections.
The resultant polytope is a Minkowski summand of $\Sigma({\mathcal{A}})$,
which is also of dimension $|{\mathcal{A}}|-2n-1$. We consider an equivalence
relation defined on the $\Sigma({\mathcal{A}})$ vertices, where the classes
are in bijection with the vertices of the resultant polytope. This yields an
oracle producing a resultant vertex in a given direction, thus avoiding to
compute $\Sigma({\mathcal{A}})$, which typically has much more vertices than
$N({\mathcal{R}})$. This is known in the literature as an optimization oracle
since it optimizes inner product with a given vector over the (unknown)
polytope.
Figure 1: The Newton polytope of a polynomial of degree $5$ in two variables.
Every monomial corresponds to an integral point on the plane. The dashed
triangle is the corresponding polytope of the dense polynomial of degree $5$.
###### Example 1.
[The bicubic surface] A standard benchmark in geometric modeling is the
implicitization of the bicubic surface, with $n=2$, defined by $3$ polynomials
in two parameters. The input polynomials have supports
$A_{i}\subset{\mathbb{Z}}^{2},\,i=0,1,2$, with cardinalities $7,6,14$,
respectively; the total degrees are $3,3,6$, respectively. The Cayley set
${\mathcal{A}}\subset{\mathbb{Z}}^{4}$, constructed as in Equation 1, has
$7+6+14=27$ points. It is depicted in the following matrix, with coordinates
as columns, where the supports from different polynomials and the Cayley
coordinates are distinguished. By Proposition 4 it follows that
$N({\mathcal{R}})$ has dimension $|{\mathcal{A}}|-4-1=22$; it lies in
${\mathbb{R}}^{27}$.
$0$ | $0$ | $1$ | $0$ | $2$ | $0$ | $3$ | $0$ | $0$ | $1$ | $2$ | $0$ | $3$ | $0$ | $0$ | $1$ | $0$ | $1$ | $2$ | $1$ | $2$ | $1$ | $2$ | $3$ | $2$ | $3$ | $3$ | } support
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0$ | $1$ | $0$ | $2$ | $0$ | $3$ | $0$ | $0$ | $1$ | $0$ | $0$ | $3$ | $0$ | $0$ | $1$ | $0$ | $2$ | $1$ | $0$ | $2$ | $1$ | $3$ | $2$ | $1$ | $3$ | $2$ | $3$
$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | } Cayley
$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
Implicitization requires eliminating the two parameters to obtain a constraint
equation over the symbolic coefficients of the polynomials. Most of the
coefficients are specialized except for $3$ variables, hence the sought for
implicit equation of the surface is trivariate and the projection of
$N({\mathcal{R}})$ lies in ${\mathbb{R}}^{3}$.
TOPCOM 3 needs more than a day and $9$GB of RAM to compute $1,806,467$ regular
triangulations of ${\mathcal{A}}$, corresponding to $29$ of the vertices of
$N({\mathcal{R}})$, and crashes before computing the entire
$N({\mathcal{R}})$. Our algorithm yields the projected vertices
$\\{(0,0,1),(0,1,0),$ $(1,0,0),(0,0,9),(0,18,0),(18,0,0)\\}$ of the
$3$-dimensional projection of $N({\mathcal{R}})$, which is the Newton polytope
of the implicit equation, in $30$msec. Given this polytope, the implicit
equation of the bicubic surface is interpolated in 42 seconds 4. It is a
polynomial of degree $18$ containing $715$ terms which corresponds exactly to
the lattice points contained in the predicted polytope.
Our main contribution is twofold. First, we design an oracle-based algorithm
for computing the Newton polytope of $\mathcal{R}$, or of specializations of
${\mathcal{R}}$. The algorithm utilizes the Beneath-and-Beyond method to
compute both vertex (V) and halfspace (H) representations, which are required
by the algorithm and may also be relevant for the targeted applications. Its
incremental nature implies that we also obtain a triangulation of the
polytope, which may be useful for enumerating its lattice points. The
complexity is proportional to the number of output vertices and facets; in
this sense, the algorithms is output sensitive. The overall cost is
asymptotically dominated by computing as many regular triangulations of
${\mathcal{A}}$ (Theorem 1). We work in the space of the projected
$N({\mathcal{R}})$ and revert to the high-dimensional space of
$\Sigma({\mathcal{A}})$ only if needed. Our algorithm readily extends to
computing $\Sigma({\mathcal{A}})$, the Newton polytope of the discriminant
and, more generally, any polytope that can be efficiently described by a
vertex oracle or its orthogonal projection. In particular, it suffices to
replace our oracle by the oracle in Ref. 5 to obtain a method for computing
the discriminant polytope.
Second, we describe an efficient, publicly available implementation based on
CGAL 6 and its experimental package triangulation. Our method computes
instances of $5$-, $6$\- or $7$-dimensional polytopes with $35$K, $23$K or
$500$ vertices, respectively, in $<2$hr. Our code is faster up to dimensions
$5$ or $6$, compared to a method computing $N({\mathcal{R}})$ via tropical
geometry, implemented in the Gfan library 7. In higher dimensions Gfan seems
to perform better although neither implementation can compute enough instances
for a fair comparison. Our code, in the critical step of computing the convex
hull of the resultant polytope, uses triangulation. On our instances,
triangulation, compared to state-of-the-art software lrs, cdd, and polymake,
is the fastest together with polymake. We factor out repeated computation by
reducing the bulk of our work to a sequence of determinants: this is often the
case in high-dimensional geometric computing. Here, we exploit the nature of
our problem and matrix structure to capture the similarities of the
predicates, and hash the computed minors which are needed later, to speedup
subsequent determinants. A variant of our algorithm computes successively
tighter inner and outer approximations: when these polytopes have,
respectively, 90% and 105% of the true volume, runtime is reduced up to $25$
times. This may lead to an approximation algorithm.
### Previous work.
Sparse (or toric) elimination theory was introduced in Ref. 8. They show that
$N({\mathcal{R}})$, for two univariate polynomials with $k_{0}+1,k_{1}+1$
monomials, has $\binom{k_{0}+k_{1}}{k_{0}}$ vertices and, when both $k_{i}\geq
2$, it has $k_{0}k_{1}+3$ facets. In Section 6 of Ref. 9 is proven that
$N({\mathcal{R}})$ is $1$-dimensional if and only if $|A_{i}|=2$, for all $i$,
the only planar $N({\mathcal{R}})$ is the triangle, whereas the only
$3$-dimensional ones are the tetrahedron, the square-based pyramid, and the
resultant polytope of two univariate trinomials; we compute an affinely
isomorphic instance of the latter (Figure 2(b)) as the resultant polytope of
three bivariate polynomials. Following Theorem 6.2 of Ref. 9, the
$4$-dimensional polytopes include the 4-simplex, some $N({\mathcal{R}})$
obtained by pairs of univariate polynomials, and those of 3 trinomials, which
have been investigated with our code in Ref. 10. The maximal (in terms of
number of vertices) such polytope we have computed has f-vector
$(22,66,66,22)$ (Figure 2(c)). Furthermore, Table 2 presents some typical
f-vectors of $4,5,6$-dimensional projections of resultant polytopes.
A lower bound on the volume of the Newton polytope of the discriminant
polynomial that refutes a conjecture in algebraic geometry is presented in
Ref. 11.
A direct approach for computing the vertices of $N({\mathcal{R}})$ might
consider all vertices of $\Sigma({\mathcal{A}})$ since the vertices of the
former are equivalence classes over the vertices of the latter. Its complexity
grows with the number of vertices of $\Sigma({\mathcal{A}})$, hence is
impractical (Example 1).
The computation of secondary polytopes has been efficiently implemented in
TOPCOM 3, which has been the reference software for computing regular or all
triangulations. The software builds a search tree with flips as edges over the
vertices of $\Sigma(A)$. This approach is limited by space usage. To address
this, reverse search was proposed 12, but the implementation cannot compete
with TOPCOM. The approach based on computing $\Sigma({\mathcal{A}})$ is not
efficient for computing $N({\mathcal{R}})$. For instance, in implicitizing
parametric surfaces with up to $100$ terms, which includes all common
instances in geometric modeling, we compute the Newton polytope of the
equations in less than $1$sec, whereas $\Sigma({\mathcal{A}})$ is intractable
(see e.g. Example 1).
In Ref. 13 they describe all Minkowski summands of $\Sigma(A)$. In Ref. 14 is
defined an equivalence class over $\Sigma(A)$ vertices having the same mixed
cells. The classes map in a many-to-one fashion to resultant vertices; our
algorithm exploits a stronger equivalence relationship.
Tropical geometry is a polyhedral analogue of algebraic geometry and can be
viewed as generalizing sparse elimination theory. It gives alternative ways of
recovering resultant polytopes 7 and Newton polytopes of implicit equations 2.
See Section 5 for comparisons of the software in Ref. 7, called Gfan, with our
software. In Ref. 5, tropical geometry is used to define vertex oracles for
the Newton polytope of the discriminant polynomial.
In Ref. 15 there is a general implementation of a Beneath-and-Beyond based
procedure which reconstructs a polytope given by a vertex oracle. This
implementation, as reported in Ref. 7, is outperformed by Gfan, especially in
dimensions higher than $5$.
As is typical in computational geometry, the practical bottleneck is in
computing determinantal predicates. For determinants, the record bit
complexity is $O(n^{2.697})$ 16, while more specialized methods exist for the
sign of general determinants, e.g. Ref. 17. These results are relevant for
higher dimensions and do not exploit the structure of our determinantal
predicates, nor the fact that we deal with sequences of determinants whose
matrices are not very different (this is formalized and addressed in Section
4). We compared linear algebra libraries LinBox 18 and Eigen 19, which seem
most suitable in dimension greater than $100$ and medium to high dimensions,
respectively, whereas CGAL provides the most efficient determinant computation
for the dimensions to which we focus.
The roadmap of the paper follows: Section 2 describes the combinatorics of
resultants, and the following section presents our algorithm. Section 4
overcomes the bottleneck of Orientation predicates. Section 5 discusses the
implementation, experiments, and comparison with other software. We conclude
with future work.
A preliminary version containing most of the presented results appeared in
Ref. 20. This extended version contains a more detailed presentation of the
background theory of resultants, applications and examples, a more complete
account of previous work, omitted proofs, an improved description of the
approximation algorithm, an extended version of the hashing determinants
method, and more experimental results.
## 2 Resultant polytopes and their projections
We introduce tools from combinatorial geometry 21, 22 to describe resultants
8, 23. We shall denote by vol$(\cdot)\in{\mathbb{R}}$ the normalized Euclidean
volume, $({\mathbb{R}}^{m})^{\times}$ the linear $m$-dimensional functionals,
$\mbox{Aff}(\cdot)$ the affine hull, and $\mbox{CH}(\cdot)$ the convex hull.
Let ${\mathcal{A}}\subset{\mathbb{R}}^{d}$ be a pointset whose convex hull is
of dimension $d$. For any triangulation $T$ of ${\mathcal{A}}$, define vector
$\phi_{T}\in{\mathbb{R}}^{|{\mathcal{A}}|}$ with coordinate
$\phi_{T}(a)=\sum_{\sigma\in T:a\in\sigma}\mbox{vol}(\sigma),\qquad
a\in{\mathcal{A}},$ (2)
summing over all simplices $\sigma$ of $T$ having $a$ as a vertex;
$\Sigma({\mathcal{A}})$ is the convex hull of $\phi_{T}$ for all
triangulations $T$. Let ${\mathcal{A}}^{w}$ denote pointset ${\mathcal{A}}$
lifted to ${\mathbb{R}}^{d+1}$ via a generic lifting function $w$ in
$({\mathbb{R}}^{|{\mathcal{A}}|})^{\times}$. Regular triangulations of
${\mathcal{A}}$ are obtained by projecting the upper (or lower) hull of
${\mathcal{A}}^{w}$ back to ${\mathbb{R}}^{d}$.
###### Proposition 1.
[Ref. 8] The vertices of $\Sigma({\mathcal{A}})$ correspond to the regular
triangulations of ${\mathcal{A}}$, while its face lattice corresponds to the
poset of regular polyhedral subdivisions of ${\mathcal{A}}$, ordered by
refinement. A lifting vector produces a regular triangulation $T$ (resp. a
regular polyhedral subdivision of ${\mathcal{A}}$) if and only if it lies in
the normal cone of vertex $\phi_{T}$ (resp. of the corresponding face) of
$\Sigma({\mathcal{A}})$. The dimension of $\Sigma({\mathcal{A}})$ is
$|{\mathcal{A}}|-d-1$.
Let $A_{0},\ldots,A_{n}$ be subsets of ${\mathbb{Z}}^{n}$,
$P_{0},\ldots,P_{n}\subset{\mathbb{R}}^{n}$ their convex hulls, and
$P=P_{0}+\cdots+P_{n}$ their Minkowski sum. A _Minkowski (maximal) cell_ of
$P$ is any full-dimensional convex polytope $B=\sum_{i=0}^{n}B_{i}$, where
each $B_{i}$ is a convex polytope with vertices in $A_{i}$. Minkowski cells
$B,B^{\prime}=\sum_{i=0}^{n}B_{i}^{\prime}$ intersect properly when $B_{i}\cap
B_{i}^{\prime}$ is a face of both and their Minkowski sum descriptions are
compatible, i.e. coincide on the common face. A mixed subdivision of $P$ is
any family of Minkowski cells which partition $P$ and intersect properly. A
Minkowski cell is $i$-mixed or $v_{i}$-mixed, if it is the Minkowski sum of
$n$ one-dimensional segments from $P_{j},\,j\neq i$, and some vertex $v_{i}\in
P_{i}$. In the sequel we shall call a Minkowski cell, simply cell.
Mixed subdivisions contain faces of all dimensions between 0 and $n$, the
maximum dimension corresponding to cells. Every face of a mixed subdivision of
$P$ has a unique description as Minkowski sum of $B_{i}\subset P_{i}$. A mixed
subdivision is regular if it is obtained as the projection of the upper (or
lower) hull of the Minkowski sum of lifted polytopes
$P_{i}^{w_{i}}:=\\{(p_{i},w_{i}(p_{i}))~{}|~{}p_{i}\in P_{i}\\}$, for lifting
$w_{i}:P_{i}\rightarrow{\mathbb{R}}$. If the lifting function
$w:=(w_{0}\ldots,w_{n})$ is sufficiently generic, then the mixed subdivision
is _tight_ , and $\sum_{i=0}^{n}\dim B_{i}=\dim\sum_{i=0}^{n}B_{i}$, for every
cell. Given $A_{0},\ldots,A_{n}$ and the affine basis
$\\{e_{0},\ldots,e_{n}\\}$ of ${\mathbb{R}}^{n}$, we define the Cayley
pointset ${\mathcal{A}}\subset{\mathbb{Z}}^{2n}$ as in equation (1).
###### Proposition 2.
[Cayley trick, Ref. 8] There exist bijections between: the regular tight mixed
subdivisions of $P$ and the regular triangulations of ${\mathcal{A}}$; the
tight mixed subdivisions of $P$ and the triangulations of ${\mathcal{A}}$; the
mixed subdivisions of $P$ and the polyhedral subdivisions of ${\mathcal{A}}$.
The family $A_{0},\dots,A_{n}\subset{\mathbb{Z}}^{n}$ is essential if they
jointly affinely span ${\mathbb{Z}}^{n}$ and every subset of cardinality
$j,1\leq j<n$, spans a space of dimension greater than or equal to $j$. It is
straightforward to check this property algorithmically and, if it does not
hold, to find an essential subset 9. In the sequel, the input
$A_{0},\dots,A_{n}\subset{\mathbb{Z}}^{n}$ is supposed to be essential. Given
a finite $A\subset{\mathbb{Z}}^{n}$, we denote by ${\mathbb{C}}^{A}$ the space
of all Laurent polynomials of the form $\sum_{a\in A}c_{a}x^{a},c_{a}\neq
0,x=(x_{1},\ldots,x_{n})$. Similarly, given
$A_{0},\dots,A_{n}\subset{\mathbb{Z}}^{n}$ we denote by
$\prod_{i=0}^{n}{\mathbb{C}}^{A_{i}}$ the space of all systems of polynomials
$f_{0}=f_{1}=\dots=f_{n}=0,$ (3)
where $\sum_{a\in A_{i}}c_{i,a}x^{a},c_{i,a}\neq 0$. The vector of all
coefficients $(\ldots,c_{i,a},\ldots)$ of (3) defines a point in
$\prod_{i=0}^{n}{\mathbb{C}}^{A_{i}}$. Let
$Z\subset\prod_{i=0}^{n}{\mathbb{C}}^{A_{i}}$ be the set of points
corresponding to systems (3) which have a solution in
$({\mathbb{C}}^{*})^{n}$, and let $\overline{Z}$ be its closure.
$\overline{Z}$ is an irreducible variety defined over ${\mathbb{Q}}$.
###### Definition 1.
If codim$(\overline{Z})=1$, then the sparse (or toric) resultant of the system
of polynomials (3) is the unique (up to sign) polynomial ${\mathcal{R}}$ in
${\mathbb{Z}}[c_{i,a}:$ $i=0,\dots,n,$ $a\in A_{i}]$, which vanishes on
$\overline{Z}$. If codim$(\overline{Z})>2$, then ${\mathcal{R}}=1$.
The resultant offers a solvability condition from which $x$ has been
eliminated, hence is also known as the eliminant. For $n=1$, it is named after
Sylvester. For linear systems, it equals the determinant of the
$(n+1)\times(n+1)$ coefficient matrix. The discriminant of a polynomial
$F(x_{1},\dots,x_{n})$ is given by the resultant of $F,\partial F/\partial
x_{1},\dots,$ $\partial F/\partial x_{n}$.
The Newton polytope $N({\mathcal{R}})$ of the resultant is a lattice polytope
called the resultant polytope. The resultant has
$|{\mathcal{A}}|=\sum_{i=0}^{n}|A_{i}|$ variables, hence $N({\mathcal{R}})$
lies in ${\mathbb{R}}^{|{\mathcal{A}}|}$, though it is of smaller dimension
(Proposition 4). The monomials corresponding to vertices of $N({\mathcal{R}})$
are the extreme resultant monomials.
###### Proposition 3.
[Refs. 8, 9] For a sufficiently generic lifting function
$w\in({\mathbb{R}}^{|{\mathcal{A}}|})^{\times}$, the $w$-extreme monomial of
${\mathcal{R}}$, whose exponent vector maximizes the inner product with $w$,
equals
$\pm\prod_{i=0}^{n}\prod_{\sigma}c_{i,v_{i}}^{\mathrm{vol}(\sigma)},$ (4)
where $\sigma$ ranges over all $v_{i}$-mixed cells of the regular tight mixed
subdivision $S$ of $P$ induced by $w$, and $c_{i,v_{i}}$ is the coefficient of
the monomial $x^{v_{i}}$ in $f_{i}$.
Let $T$ be the regular triangulation corresponding, via the Cayley trick, to
$S$, and $\rho_{T}\in{\mathbb{N}}^{|{\mathcal{A}}|}$ the exponent of the
$w$-extreme monomial. For simplicity we shall denote by $\sigma$, both a cell
of $S$ and its corresponding simplex in $T$. Then,
$\rho_{T}(a)=\sum_{\stackrel{{\scriptstyle a-\text{mixed}}}{{\sigma\in
T:a\in\sigma}}}\mbox{vol}(\sigma)\;\in{\mathbb{N}},\qquad a\in{\mathcal{A}},$
(5)
where simplex $\sigma$ is $a$-mixed if and only if the corresponding cell is
$a$-mixed in $S$. Note that, $\rho_{T}(a)\in{\mathbb{N}}$, since it is a sum
of volumes of mixed simplices $\sigma\in T$, and each of these volumes is
equal to the _mixed volume_23 of a set of _lattice_ polytopes, the Minkowksi
summands of the corresponding $\sigma\in S$. In particular, assuming that
$\sigma\in S$ is $i$-mixed, it can be written as
$\sigma=\sigma_{0}+\dots+\sigma_{n},\;\sigma_{j}\subseteq
A_{j},\,j=0,\ldots,n$, and
$\mbox{vol}(\sigma)=MV(\sigma_{0},\ldots,\sigma_{i-1},\sigma_{i+1},\ldots,\sigma_{n}),$
where $MV$ denotes the mixed volume function which is integer valued for
lattice polytopes 23. Now, $N({\mathcal{R}})$ is the convex hull of all
$\rho_{T}$ vectors 8, 9.
Figure 2: (a) The secondary polytope $\Sigma({\mathcal{A}})$ of two triangles
(dark, light grey) and one segment
$A_{0}=\\{(0,0),(1,2),(4,1)\\},\,A_{1}=\\{(0,1),(1,0)\\},\,A_{2}=\\{(0,0),(0,1),(2,0)\\}$,
where ${\mathcal{A}}$ is defined as in Equation 1; vertices correspond to
mixed subdivisions of the Minkowski sum $A_{0}+A_{1}+A_{2}$ and edges to flips
between them (b) $N({\mathcal{R}})$, whose vertices correspond to the dashed
classes of $\Sigma({\mathcal{A}})$. Bold edges of $\Sigma({\mathcal{A}})$,
called cubical flips, map to edges of $N({\mathcal{R}})$ (c) $4$-dimensional
$N({\mathcal{R}})$ of 3 generic trinomials with f-vector $(22,66,66,22)$;
figure made with polymake.
Proposition 3 establishes a many-to-one surjection from regular triangulations
of ${\mathcal{A}}$ to regular tight mixed subdivisions of $P$, or,
equivalently, from vertices of $\Sigma({\mathcal{A}})$ to those of
$N({\mathcal{R}})$. One defines an equivalence relationship on all regular
tight mixed subdivisions, where equivalent subdivisions yield the same vertex
in $N({\mathcal{R}})$. Thus, equivalent vertices of $\Sigma({\mathcal{A}})$
correspond to the same resultant vertex. Consider
$w\in({\mathbb{R}}^{|{\mathcal{A}}|})^{\times}$ lying in the union of outer-
normal cones of equivalent vertices of $\Sigma({\mathcal{A}})$. They
correspond to a resultant vertex whose outer-normal cone contains $w$; this
defines a $w$-extremal resultant monomial. If $w$ is non-generic, it specifies
a sum of extremal monomials in ${\mathcal{R}}$, i.e. a face of
$N({\mathcal{R}})$. The above discussion is illustrated in Figure 2(a),(b).
###### Proposition 4.
[Ref. 8] $N({\mathcal{R}})$ is a Minkowski summand of $\Sigma({\mathcal{A}})$,
and both $\Sigma({\mathcal{A}})$ and $N({\mathcal{R}})$ have dimension
$|{\mathcal{A}}|-2n-1.$
Let us describe the $2n+1$ hyperplanes in whose intersection lies
$N({\mathcal{R}})$. For this, let $M$ be the $(2n+1)\times|{\mathcal{A}}|$
matrix whose columns are the points in the $A_{i}$, where each $a\in A_{i}$ is
followed by the $i$-th unit vector in ${\mathbb{N}}^{n+1}$. Then, the inner
product of any coordinate vector of $N({\mathcal{R}})$ with row $i$ of $M$ is:
constant, for $i=1,\dots,n$, and known, and depends on $i$, for
$i=n+1,\dots,2n+1$, see Prop. 7.1.11 of Ref. 8. This implies that one obtains
an isomorphic polytope when projecting $N({\mathcal{R}})$ along $2n+1$ points
in ${\mathcal{A}}$ which affinely span ${\mathbb{R}}^{2n}$; this is possible
because of the assumption of essential family. Having computed the projection,
we obtain $N({\mathcal{R}})$ by computing the missing coordinates as the
solution of a linear system: we write the aforementioned inner products as
$M[X\,V]^{T}=C$, where $C$ is a known matrix and $[X\,V]^{T}$ is a transposed
$(2n+1)\times u$ matrix, expressing the partition of the coordinates to
unknown and known values, where $u$ is the number of $N({\mathcal{R}})$
vertices. If the first $2n+1$ columns of $M$ correspond to specialized
coefficients, $M=[M_{1}\,M_{2}]$, where submatrix $M_{1}$ is of dimension
$2n+1$ and invertible, hence $X=M_{1}^{-1}(C-M_{2}B)$.
We compute some orthogonal projection of $N({\mathcal{R}})$, denoted $\varPi$,
in ${\mathbb{R}}^{m}$:
$\pi:{\mathbb{R}}^{|{\mathcal{A}}|}\rightarrow{\mathbb{R}}^{m}:N({\mathcal{R}})\rightarrow\varPi,\;\,m\leq|{\mathcal{A}}|.$
By reindexing, this is the subspace of the first $m$ coordinates,
so$\pi(\rho)=(\rho_{1},\dots,\rho_{m})$. It is possible that none of the
coefficients $c_{ij}$ is specialized, hence $m=|{\mathcal{A}}|$, $\pi$ is
trivial, and $\varPi=N({\mathcal{R}})$. Assuming the specialized coefficients
take sufficiently generic values, $\varPi$ is the Newton polytope of the
corresponding specialization of ${\mathcal{R}}$. The following is used for
preprocessing.
###### Lemma 1.
[Ref. 7 Lemma 3.20] If $a_{ij}\in A_{i}$ corresponds to a specialized
coefficient of $f_{i}$, and lies in the convex hull of the other points in
$A_{i}$ corresponding to specialized coefficients, then removing $a_{ij}$ from
$A_{i}$ does not change the Newton polytope of the specialized resultant.
We focus on three applications. First, we interpolate the resultant in all
coefficients, thus illustrating an alternative method for computing
resultants.
###### Example 2.
Let $f_{0}=a_{2}x^{2}+a_{1}x+a_{0}$, $f_{1}=b_{1}x^{2}+b_{0}$, with supports
$A_{0}=\\{2,1,0\\},A_{1}=\\{1,0\\}$. Their (Sylvester) resultant is a
polynomial in $a_{2},a_{1},a_{0},$ $b_{1},b_{0}$. Our algorithm computes its
Newton polytope with vertices $(0,2,0,1,1)$, $(0,0,2,2,0)$, $(2,0,0,0,2)$; it
contains 4 lattice points, corresponding to 4 potential resultant monomials
$a_{1}^{2}b_{1}b_{0},~{}a_{0}^{2}b_{1}^{2},~{}a_{2}a_{0}b_{1}b_{0},~{}a_{2}^{2}b_{0}^{2}$.
Knowing these potential monomials, to interpolate the resultant, we need 4
points $(a_{0},a_{1},a_{2},b_{0},b_{1})$ for which the system $f_{0}=f_{1}=0$
has a solution. For computing these points we use the parameterization of
resultants in Ref. 24, which yields: $a_{2}=(2t_{1}+t_{2})t_{3}^{2}t_{4}$,
$a_{1}=(-2t_{1}-2t_{2})t_{3}t_{4}$, $a_{0}=t_{2}t_{4}$,
$b_{1}=-t_{1}t_{3}^{2}t_{5}$, $b_{0}=t_{1}t_{5},$ where the $t_{i}$’s are
parameters. We substitute these expressions to the monomials, evaluate at 4
sufficiently random $t_{i}$’s, and obtain a matrix whose kernel vector
$(1,1,-2,1)$ yields
${\mathcal{R}}=a_{1}^{2}b_{1}b_{0}+a_{0}^{2}b_{1}^{2}-2a_{2}a_{0}b_{1}b_{0}+a_{2}^{2}b_{0}^{2}$.
Second, consider system solving by the rational univariate representation of
roots 25. Given $f_{1},\dots,f_{n}\in{\mathbb{C}}[x_{1},\dots,x_{n}]$, define
an overconstrained system by adding $f_{0}=u_{0}+u_{1}x_{1}+\cdots+u_{n}x_{n}$
with symbolic $u_{i}$’s. Let coefficients $c_{ij},i\geq 1$, take specific
values, and suppose that the roots of $f_{1}=\cdots=f_{n}=0$ are isolated,
denoted $r_{i}=(r_{i1},\dots,r_{in})$. Then the $u$-resultant is
${\mathcal{R}}_{u}=a\,\prod_{r_{i}}(u_{0}+u_{1}r_{i1}+\cdots+u_{n}r_{in})^{m_{i}}$,
$a\in{\mathbb{C}}^{*}$, where $m_{i}$ is the multiplicity of $r_{i}$.
Computing ${\mathcal{R}}_{u}$ is the bottleneck; our method computes (a
superset of) $N({\mathcal{R}}_{u})$.
###### Example 3.
Let $f_{1}=x_{1}^{2}+x_{2}^{2}-4$, $f_{2}=x_{1}-x_{2}+2$, and
$f_{0}=u_{0}+u_{1}x_{1}+u_{2}x_{2}$. Our algorithm computes a polygon with
vertices $\\{(2,0,0),(0,2,0),(0,0,2)\\}$, which contains
$N({\mathcal{R}}_{u})=\mbox{CH}(\\{(2,0,0),(1,1,0),(1,0,1),(0,1,1)\\})$. The
coefficient specialization is not generic, hence $N({\mathcal{R}}_{u})$ is
strictly contained in the computed polygon. Proceeding as in Example 2,
${\mathcal{R}}_{u}=2u_{0}^{2}+4u_{0}u_{1}-4u_{0}u_{2}-8u_{1}u_{2}$, which
factors as $2(u_{0}+2u_{1})(u_{0}-2u_{2})$.
The last application comes from geometric modeling, where $y_{i}=f_{i}(x)$,
$i=0,\dots,n$, $x=(x_{1},\dots,x_{n})\in\Omega\subset{\mathbb{R}}^{n}$,
defines a parametric hypersurface. Many applications require the equivalent
implicit representation $F(y_{1},\dots,y_{n})=0$. This amounts to eliminating
$x$, so it is crucial to compute the resultant when coefficients are
specialized except the $y_{i}$’s. Our approach computes a polytope that
contains the Newton polytope of $F$, thus reducing implicitization to
interpolation 4, 1. In particular, we compute the polytope of surface
equations within $1$sec, assuming $\leq 100$ terms in parametric polynomials,
which includes all common instances in geometric modeling.
Figure 3: The supports $A_{0},A_{1},A_{2}$ of Example 4, their Newton
polytopes (segments) and the two mixed subdivisions of their Minkowski sum.
###### Example 4.
Let us see how the above computation can serve in implicitization. Consider
the surface given by the polynomial parameterization
$(y_{1},y_{2},y_{3})=(x_{1}x_{2},x_{1}x_{2}^{2},x_{1}^{2}).$
For polynomials
$f_{0}:=c_{00}-c_{01}x_{1}x_{2},~{}f_{1}:=c_{10}-c_{11}x_{1}x_{2}^{2},~{}f_{2}:=c_{20}-c_{21}x_{1}^{2}$
with supports $A_{0}=\\{(0,0),(1,2)\\},A_{1}=\\{(0,0),(1,2)\\}$ and
$A_{2}=\\{(0,0),(2,0)\\}$. The resultant polytope is a segment in
${\mathbb{R}}^{6}$ with endpoints $(4,0,0,2,0,1)$, $(0,4,2,0,1,0)$ and,
actually,
${\mathcal{R}}=-c_{00}^{4}c_{11}^{2}c_{21}+c_{01}^{4}c_{10}^{2}c_{20}$. The
supports and the two mixed subdivisions corresponding to the vertices of
$N({\mathcal{R}})$ are illustrated in Figure 3. Specializing the symbolic
coefficients of the polynomials as:
$(c_{00},c_{01},c_{10},c_{11},c_{20},c_{21})\mapsto(y_{1},-1,y_{2},-1,y_{3},-1)$
yields the vertices of the implicit polytope: $(4,0,0),(0,2,1)$, which our
algorithm can compute directly. The implicit equation of the surface turns out
to be $-y_{1}^{4}+y_{2}^{2}y_{3}$.
## 3 Algorithms and complexity
This section analyzes our exact and approximate algorithms for computing
orthogonal projections of polytopes whose vertices are defined by an _oracle_.
This oracle computes a vertex of the polytope which is extremal in a given
direction $w$. If there are more than one such vertices the oracle returns
exactly one of these. Moreover, we define such an oracle for the vertices of
orthogonal projections $\varPi$ of $N({\mathcal{R}})$ which results in
algorithms for computing $\varPi$ while avoiding computing $N({\mathcal{R}})$.
Finally, we analyze the asymptotic complexity of these algorithms.
Given a pointset $V$, reg_subdivision($V,\omega$) computes the regular
subdivision of its convex hull by projecting the upper hull of $V$ lifted by
$\omega$, and conv($V$) computes the H-representation of the convex hull of
$V$. The oracle VTX(${\mathcal{A}},\,w,\,\pi$) computes a point in
$\varPi=\pi(N({\mathcal{R}}))$, extremal in the direction
$w\in({\mathbb{R}}^{m})^{\times}$. First it adds to $w$ an infinitesimal
symbolic perturbation vector, thus obtaining $w_{p}$. Then calls
reg_subdivision(${\mathcal{A}},\widehat{w_{p}}$),
$\widehat{w_{p}}=(w_{p},\vec{0})\in({\mathbb{R}}^{|{\mathcal{A}}|})^{\times}$
that yields a regular triangulation $T$ of ${\mathcal{A}}$, since $w_{p}$ is
generic, and finally returns $\pi(\rho_{T})$. It is clear that the
triangulation $T$ constructed by VTX$(\cdot)$ is regular and corresponds to
some secondary vertex $\phi_{T}$ which maximizes the inner product with
$\widehat{w_{p}}$. Since the perturbation is arbitrarily small, both
$\phi_{T},\rho_{T}$ also maximize the inner product with
$\widehat{w}=(w,\vec{0})\in({\mathbb{R}}^{|{\mathcal{A}}|})^{\times}$.
We use perturbation to avoid computing non-vertex points on the boundary of
$\varPi$. The perturbation can be implemented in VTX$(\cdot)$, without
affecting any other parts of the algorithm, either by case analysis or by a
method of symbolic perturbation. In practice, our implementation does avoid
computing non-vertex points on the boundary of $\varPi$ by computing a
refinement of the subdivision obtained by calling
reg_subdivision(${\mathcal{A}},\widehat{w}$). This refinement is implemented
in triangulation by computing a placing triangulation with a random insertion
order 26 (Section 5).
###### Lemma 2.
All points computed by VTX$(\cdot)$ are vertices of $\varPi$.
###### Proof.
Let $v=\pi(\rho_{T})=\mbox{VTX}({\mathcal{A}},w,\pi)$. We first prove that $v$
lies on $\partial\varPi$. The point $\rho_{T}$ of $N({\mathcal{R}})$ is a
Minkowski summand of the vertex $\phi_{T}$ of $\Sigma({\mathcal{A}})$ extremal
with respect to $\widehat{w}$, hence $\rho_{T}$ is extremal with respect to
$\widehat{w}$. Since $\widehat{w}$ is perpendicular to projection $\pi$,
$\rho_{T}$ projects to a point in $\partial\varPi$. The same argument implies
that every vertex $\phi_{T}^{\prime}$, where $T^{\prime}$ is a triangulation
refining the subdivision produced by $\widehat{w}$, corresponds to a resultant
vertex $\rho_{T^{\prime}}$ such that $\pi(\rho_{T^{\prime}})$ lies on a face
of $\varPi$. This is actually the same face on which $\pi(\rho_{T})$ lies.
Hence $\rho_{T^{\prime}}$ also lies on $\partial\varPi$.
Now we prove that $v$ is a vertex of $\varPi$ by showing that it does not lie
in the relative interior of a face of $\Pi$. Let $w$ be such that the face $f$
of $N({\mathcal{R}})$ extremal with respect to $\widehat{w}$ contains a vertex
$\rho_{T}$ which projects to $\mbox{relint}(\pi(f))$, where
$\mbox{relint}(\cdot)$ denotes relative interior. However, $f$ will not be
extremal with respect to $\widehat{w_{p}}$ and since
VTX$({\mathcal{A}},w,\pi)$ uses the perturbed vector $w_{p}$, it will never
compute a vertex of $N({\mathcal{R}})$ whose projection lies inside a face of
$\varPi$. ∎
The initialization algorithm computes an inner approximation of $\varPi$ in
both V- and H-representations (denoted $Q,\ Q^{H}$, respectively), and
triangulated. First, it calls VTX$({\mathcal{A}},w,\pi)$ for $w\in
W\subset({\mathbb{R}}^{m})^{\times}$; the set $W$ is either random or
contains, say, vectors in the $2m$ coordinate directions. Then, it updates $Q$
by adding VTX$({\mathcal{A}},w,\pi)$ and VTX$({\mathcal{A}},-w,\pi)$, where
$w$ is normal to hyperplane $H\subset{\mathbb{R}}^{m}$ containing $Q$, as long
as either of these points lies outside $H$. Since every new vertex lies
outside the affine hull of the current polytope $Q$, all polytopes produced
are simplices. We stop when these points do no longer increase $\dim(Q)$.
###### Lemma 3.
The initialization algorithm computes $Q\subseteq\varPi$ such that
$\dim(Q)=\dim(\varPi)$.
###### Proof.
Suppose that the initialization algorithm computes a polytope
$Q^{\prime}\subset\varPi$ such that $\dim(Q^{\prime})<m$. Then there exists
vertex $v\in\varPi$, $v\notin\mbox{Aff}(Q^{\prime})$ and vector
$w\in({\mathbb{R}}^{m})^{\times}$ perpendicular to $\mbox{Aff}(Q^{\prime})$,
such that $w$ belongs to the normal cone of $v$ in $\varPi$ and
$\dim(\mbox{Aff}(Q^{\prime}\cup v))>\dim Q^{\prime}$. This is a contradiction,
since such a $w$ would have been computed as VTX(${\mathcal{A}},w,\pi$) or
VTX(${\mathcal{A}},-w,\pi$), where $w$ is normal to the hyperplane $H$
containing $Q^{\prime}$. ∎
Incremental Algorithm 1 computes both V- and H-representations of $\varPi$ and
a triangulation of $\varPi$, given an inner approximation $Q,Q^{H}$ of
$\varPi$ computed at the initialization. A hyperplane $H$ is called _legal_ if
it is a supporting hyperplane to a facet of $\varPi$, otherwise it is called
_illegal_. At every step of Algorithm 1, we compute
$v=\mbox{VTX}({\mathcal{A}},w,\pi)$ for a supporting hyperplane $H$ of a facet
of $Q$ with normal $w$. If $v\notin H$, it is a new vertex thus yielding a
tighter inner approximation of $\varPi$ by inserting it to $Q$, i.e.
$Q\subset\mbox{CH}(Q\cup v)\subseteq\varPi$. This happens when the preimage
$\pi^{-1}(f)\subset N({\mathcal{R}})$ of the facet $f$ of $Q$ defined by $H$,
is not a Minkowski summand of a face of $\Sigma({\mathcal{A}})$ having normal
$\widehat{w}$. Otherwise, there are two cases: either $v\in H$ and $v\in Q$,
thus the algorithm simply decides hyperplane $H$ is legal, or $v\in H$ and
$v\notin Q$, in which case the algorithm again decides $H$ is legal but also
inserts $v$ to $Q$.
The algorithm computes $Q^{H}$ from $Q$, then iterates over the new
hyperplanes to either compute new vertices or decide they are legal, until no
increment is possible, which happens when all hyperplanes are legal. Algorithm
1 ensures that each normal $w$ to a hyperplane supporting a facet of $Q$ is
used only _once_ , by storing all used $w$’s in a set $W$. When a new normal
$w$ is created, the algorithm checks if $w\notin W$, then calls
VTX$({\mathcal{A}},w,\pi)$ and updates $W\leftarrow W\cup w$. If $w\in W$ then
the same or a parallel hyperplane has been checked in a previous step of the
algorithm. It is straightforward that $w$ can be safely ignored; Lemma 4
formalizes the latter case.
###### Lemma 4.
Let $H^{\prime}$ be a hyperplane supporting a facet constructed by Algorithm
1, and $H\neq H^{\prime}$ an illegal hyperplane at a previous step. If
$H^{\prime},H$ are parallel then $H^{\prime}$ is legal.
###### Proof.
Let $w,w^{\prime}$ be the outer normal vectors of the facets supported by
$H,H^{\prime}$ respectively. If $H,H^{\prime}$ are parallel then
$v=\mbox{VTX}({\mathcal{A}},w,\pi)$ maximizes the inner product with
$w^{\prime}$ in $Q$ which implies that hyperplane $H^{\prime}$ is legal. ∎
Input : essential $A_{0},\dots,A_{n}\subset{\mathbb{Z}}^{n}$ processed by
Lemma 1,
projection $\pi:{\mathbb{R}}^{|{\mathcal{A}}|}\rightarrow{\mathbb{R}}^{m}$,
H-, V-repres. $Q^{H},Q$; triang. ${T}_{Q}$ of $Q\subseteq\varPi$.
Output : H-, V-repres. $Q^{H},Q$; triang. ${T}_{Q}$ of $Q=\varPi$.
${\mathcal{A}}\leftarrow\bigcup_{0}^{n}(A_{i}\times{e_{i}})$ // Cayley trick
${\cal H}_{illegal}\leftarrow\emptyset$;
foreach _$H\in Q^{H}$_ do ${\cal H}_{illegal}\leftarrow{\cal
H}_{illegal}\cup\\{H\\}$ ;
while _${\cal H}_{illegal}\neq\emptyset$_ do
select $H\in{\cal H}_{illegal}$ and ${\cal H}_{illegal}\leftarrow{\cal
H}_{illegal}\setminus\\{H\\}$;
$w$ is the outer normal vector of $H$;
$v\leftarrow$ VTX(${\mathcal{A}},w,\pi$);
if _$v\notin H\cap Q$_ then
$Q^{H}_{temp}\leftarrow{\rm conv}(Q\cup\\{v\\})$ // convex hull computation
foreach _$(d-1)$ -face $f\in{T}_{Q}$ visible from $v$_ do
${T}_{Q}\leftarrow{T}_{Q}\cup\\{\text{faces of }{conv}(f,v)\\}$
foreach _$H^{\prime}\in\\{Q^{H}\setminus Q^{H}_{temp}\\}$_ do
${\cal H}_{illegal}\leftarrow{\cal H}_{illegal}\setminus\\{H^{\prime}\\}$ //
$H^{\prime}$ separates $Q,v$
foreach _$H^{\prime}\in\\{Q^{H}_{temp}\setminus Q^{H}\\}$_ do
${\cal H}_{illegal}\leftarrow{\cal H}_{illegal}\cup\\{H^{\prime}\\}$ // new
hyperplane
$Q\leftarrow Q\cup\\{v\\}$;
$Q^{H}\leftarrow Q^{H}_{temp}$;
return $Q,Q^{H},{T}_{Q}$;
Algorithm 1 Compute$\varPi$ $(A_{0},\dots,A_{n},\pi)$
The next lemma formulates the termination criterion of our algorithm.
###### Lemma 5.
Let $v=\mbox{{\em VTX}}({\mathcal{A}},w,\pi)$, where $w$ is normal to a
supporting hyperplane $H$ of $Q$, then $v\not\in H$ if and only if $H$ is not
a supporting hyperplane of $\varPi$.
###### Proof.
Let $v=\pi(\rho_{T})$, where $T$ is a triangulation refining subdivision $S$
in VTX$(\cdot)$. It is clear that, since $v\in\partial\varPi$ is extremal with
respect to $w$, if $v\not\in H$ then $H$ cannot be a supporting hyperplane of
$\varPi$. Conversely, let $v\in H$. By the proof of Lemma 2, every other
vertex $\pi(\rho{{}_{T}^{\prime}})$ on the face of $N({\mathcal{R}})$ is
extremal with respect to $w$, hence lies on $H$, thus $H$ is a supporting
hyperplane of $\varPi$. ∎
We now bound the complexity of our algorithm. Beneath-and-Beyond, given a
$k$-dimensional polytope with $l$ vertices, computes its H-representation and
a triangulation in $O(k^{5}lt^{2})$, where $t$ is the number of full-
dimensional faces (cells) Ref. 27. Let $|\varPi|,|\varPi^{H}|$ be the number
of vertices and facets of $\varPi$.
###### Lemma 6.
Algorithm 1 executes VTX$(\cdot)$ at most $|\varPi|+|\varPi^{H}|$ times.
###### Proof.
The steps of Algorithm 1 increment $Q$. At every such step, and for each
supporting hyperplane $H$ of $Q$ with normal $w$, the algorithm calls
VTX$(\cdot)$ and computes one vertex of $\varPi$, by Lemma 2. If $H$ is
illegal, this vertex is unique because $H$ separates the set of (already
computed) vertices of $Q$ from the set of vertices of $\varPi\setminus Q$
which are extremal with respect to $w$, hence, an appropriate translate of $H$
also separates the corresponding sets of vertices of $\Sigma({\mathcal{A}})$
(Figure 4). This vertex is never computed again because it now belongs to $Q$.
The number of VTX$(\cdot)$ calls yielding vertices is thus bounded by
$|\varPi|$.
For a legal hyperplane of $Q$, we compute one vertex of $\varPi$ that confirms
its legality; the VTX$(\cdot)$ call yielding this vertex is accounted for by
the legal hyperplane. The statement follows by observing that every normal to
a hyperplane of $Q$ is used only once in Algorithm 1 (by the earlier
discussion concerning the set $W$ of all used normals). ∎
Figure 4: Lemma 6: each illegal hyperplane of $Q$ with normal $w$, separates
the already computed vertices of $\varPi$ (here equal to $N({\mathcal{R}})$)
from new ones, extremal with respect to $w$. $X$ is a polytope such that
$X+N({\mathcal{R}})=\Sigma({\mathcal{A}})$.
Let the size of a triangulation be the number of its cells. Let
$s_{{\mathcal{A}}}$ denote the size of the largest triangulation of
${\mathcal{A}}$ computed by VTX$(\cdot)$, and $s_{\varPi}$ that of $\varPi$
computed by Algorithm 1. In VTX$(\cdot)$, the computation of a regular
triangulation reduces to a convex hull, computed in
$O(n^{5}|{\mathcal{A}}|s_{{\mathcal{A}}}^{2})$; for $\rho_{{T}}$ we compute
Volume for all cells of ${T}$ in $O(s_{{\mathcal{A}}}n^{3})$. The overall
complexity of VTX$(\cdot)$ becomes
$O(n^{5}|{\mathcal{A}}|s_{{\mathcal{A}}}^{2})$. Algorithm 1 calls, in every
step, VTX$(\cdot)$ to find a point on $\partial\varPi$ and insert it to $Q$,
or to conclude that a hyperplane is legal. By Lemma 6 it executes VTX$(\cdot)$
as many as $|\varPi|+|\varPi^{H}|$ times, in
$O((|\varPi|+|\varPi^{H}|)n^{5}|{\mathcal{A}}|s_{{\mathcal{A}}}^{2})$, and
computes the H-representation of $\varPi$ in $O(m^{5}|\varPi|s_{\varPi}^{2})$.
Now we have, $|{\mathcal{A}}|\leq(2n+1)s_{{\mathcal{A}}}$ and as the input
$|{\mathcal{A}}|,m,n$ grows large we can assume that
$|\varPi|\gg|{\mathcal{A}}|$ and thus $s_{\varPi}$ dominates
$s_{{\mathcal{A}}}$. Moreover, $s_{\varPi}(m+1)\geq|\varPi^{H}|$. Now, let
$\widetilde{O}(\cdot)$ imply that polylogarithmic factors are ignored.
###### Theorem 1.
The time complexity of Algorithm 1 to compute $\varPi\subset{\mathbb{R}}^{m}$
is
$O(m^{5}|\varPi|s_{\varPi}^{2}+(|\varPi|+|\varPi^{H}|)n^{5}|{\mathcal{A}}|s_{{\mathcal{A}}}^{2})$,
which becomes $\widetilde{O}(|\varPi|s_{\varPi}^{2})$ when
$|\varPi|\gg|{\mathcal{A}}|$.
This implies our algorithm is output sensitive. Its experimental performance
confirms this property, see Section 5.
We have proven that oracle VTX$(\cdot)$ (within our algorithm) has two
important properties:
1. 1.
Its output is a vertex of the target polytope (Lemma 2).
2. 2.
When the direction $w$ is normal to an illegal facet, then the vertex computed
by the oracle is computed once (Lemma 6).
The algorithm can easily be generalized to incrementally compute any polytope
$P$ if the oracle associated with the problem satisfies property (1); if it
satisfies also property (2), then the computation can be done in
$O(|P|+|P^{H}|)$ oracle calls, where $|P|$, $|P^{H}|$ denotes the number of
vertices and number of facets of $P$, respectively. For example, if the
described oracle returns $\pi(\phi_{T})$ instead of $\pi(\rho_{T})$, it can be
used to compute orthogonal projections of secondary polytopes.
The algorithm readily yields an approximate variant: for each supporting
hyperplane $H$, we use its normal $w$ to compute
$v=$VTX$({\mathcal{A}},w,\pi)$. Instead of computing a convex hull, now simply
take the hyperplane parallel to $H$ through $v$. The set of these hyperplanes
defines a polytope $Q_{o}\supseteq\varPi$, i.e. an outer approximation of
$\varPi$. In particular, at every step of the algorithm, $Q$ and $Q_{o}$ are
an inner and an outer approximation of $\varPi$, respectively. Thus, we have
an approximation algorithm by stopping Algorithm 1 when
$\text{vol}(Q)/\text{vol}(Q_{o})$ achieves a user-defined threshold. Then,
$\text{vol}(Q)/\text{vol}(\varPi)$ is bounded by the same threshold.
Implementing this algorithm yields a speedup of up to 25 times (Section 5). It
is clear that vol$(Q)$ is available by our incremental convex hull algorithm.
However, vol$(Q_{o})$ is the critical step; we plan to examine algorithms that
update (exactly or approximately) this volume.
When all hyperplanes of $Q$ are checked, knowledge of legal hyperplanes
accelerates subsequent computations of $Q^{H}$, although it does not affect
its worst-case complexity. Specifically, it allows us to avoid checking legal
facets against new vertices.
## 4 Hashing of Determinants
This section discusses methods to avoid duplication of computations by
exploiting the nature of the determinants appearing in the inner loop of our
algorithm. Our algorithm computes many regular triangulations, which are
typically dominated by the computation of determinants. A similar technique,
using dynamic determinant computations, is used to improve determinantal
predicates in incremental convex hull computations 28.
Consider the $2n\times|{\mathcal{A}}|$ matrix with the points of
${\mathcal{A}}$ as columns. Define $P$ as the extension of this matrix by
adding lifting values $\widehat{w}$ as the last row. We use the Laplace (or
cofactor) expansion along the last row for computing the determinant of the
square submatrix formed by any $2n+1$ columns of $P$; without loss of
generality, we assume these are the first $2n+1$ columns
$a_{1},\ldots,a_{2n+1}$. Let $(1,\ldots,2n+1)\setminus i$ be the vector
resulting from removing the $i$-th element from the vector $(1,\ldots,2n+1)$
and let $P_{(1,\ldots,2n+1)\setminus i}$ be the $(2n)\times(2n)$ matrix
obtained from the $2n$ elements of the columns whose indices are in
$(1,\ldots,2n+1)\setminus i$.
The Orientation predicate is the sign of the determinant of
$P^{hom}_{(1,\ldots,2n+2)}$, constructed by columns $a_{1},\ldots,a_{2n+2}$
and adding $\vec{1}\in{\mathbb{R}}^{2n+2}$ as the last row. Computing a
regular subdivision is a long sequence of such predicates, varying $a_{i}$’s
on each step. We expand along the next-to-last row, which contains the lifting
values, and compute the determinants $|P_{(1,\ldots,2n+2)\setminus i}|$ for
$i\in\\{1,\ldots,2n+2\\}$. Another predicate is Volume, used by VTX$(\cdot)$.
It equals the determinant of $P^{hom}_{(1,\ldots,2n+1)}$, constructed by
columns $a_{1},\ldots,a_{2n+1}$ and replacing the last row of the matrix by
$\vec{1}\in{\mathbb{R}}^{2n+1}$.
###### Example 5.
Consider the polynomials $f_{0}:=c_{00}-c_{01}x_{1}x_{2}+c_{02}x_{2}$,
$f_{1}:=c_{10}-c_{11}x_{1}x_{2}^{2}+c_{12}x_{2}^{2}$ and
$f_{2}:=c_{20}-c_{21}x_{1}^{2}+c_{22}x_{2}$ and the lifting vector
$\widehat{w}$ yielding the matrix $P$.
$P=\begin{tabular}[]{|m{.5mm} >{\centering}m{\w} >{\centering}m{\w}
>{\centering}m{\w} >{\centering}m{\w} >{\centering}m{\w} >{\centering}m{\w}
>{\centering}m{\w} >{\centering}m{\w} >{\centering}m{\w} m{.5mm}|
l}\cline{1-1}\cr\cline{11-11}\cr&$0$\@add@centering&$0$\@add@centering&$0$\@add@centering&$1$\@add@centering&$1$\@add@centering&$2$\@add@centering&$0$\@add@centering&$0$\@add@centering&$0$\@add@centering&&\hbox{\multirowsetup
\Big{\\}} \text{\small support coordinates}}\\\
&$0$\@add@centering&$0$\@add@centering&$0$\@add@centering&$1$\@add@centering&$2$\@add@centering&$0$\@add@centering&$1$\@add@centering&$2$\@add@centering&$1$\@add@centering&\\\
&\color[rgb]{1,0.49,0} $0$\@add@centering&\color[rgb]{1,0.49,0}
$1$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$1$\@add@centering&\color[rgb]{1,0.49,0}
$0$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$1$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&&\hbox{\multirowsetup
\Big{\\}} \text{\small Cayley trick coordinates}}\\\
&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$1$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$1$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$0$\@add@centering&\color[rgb]{1,0.49,0}$1$\@add@centering&\\\
&\color[rgb]{0,0.44,0.57}$w_{1}$\@add@centering&\color[rgb]{0,0.44,0.57}$w_{2}$\@add@centering&\color[rgb]{0,0.44,0.57}$w_{3}$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&\color[rgb]{0,0.44,0.57}$0$\@add@centering&&\hbox{\multirowsetup
\\} \text{\small$\widehat{w}$}}\\\
\cline{1-1}\cr\cline{11-11}\cr\end{tabular}$
We reduce the computations of predicates to computations of minors of the
matrix obtained from deleting the last row of $P$. Computing an Orientation
predicate using Laplace expansion consists of computing $\binom{6}{4}=15$
minors. On the other hand, if we compute $|P^{hom}_{(1,2,3,4,5,6)}|$, the
computation of $|P^{hom}_{(1,2,3,4,5,7)}|$ requires the computation of only
$\binom{6}{4}-\binom{5}{4}=10$ new minors. More interestingly, when given a
new lifting $\widehat{w^{\prime}}$, we compute
$|P^{\prime}~{}^{hom}_{(1,2,3,4,5,6)}|$ without computing any new minors.
Our contribution consists in maintaining a hash table with the computed
minors, which will be reused at subsequent steps of the algorithm. We store
all minors of sizes between $2$ and $2n$. For Orientation, they are
independent of $w$ and once computed they are stored in the hash table. The
main advantage of our scheme is that, for a new ${w}$, the only change in $P$
are $m$ (nonzero) coordinates in the last row, hence computing the new
determinants can be done by reusing hashed minors. This also saves time from
matrix constructions.
Laplace expansion computation of a matrix of size $n$ has
complexity$O(n)\sum_{i=1}^{n}L_{i}$, where $L_{i}$ is the cost of computing
the $i$-th minor. $L_{i}$ equals $1$ when the $i$-th minor was precomputed;
otherwise, it is bounded by $O\big{(}(n-1)!\big{)}$. This allows us to
formulate the following Lemma.
###### Lemma 7.
Using hashing of determinants, the complexity of the Orientation and Volume
predicates is $O(n)$ and $O(1)$, respectively, if all minors have already been
computed.
Many determinant algorithms modify the input matrix; this makes necessary to
create a new matrix and introduces a constant overhead on each minor
computation. Computing with Laplace expansion, while hashing the minors of
smaller size, performs better than state-of-the-art algorithms, in practice.
Experiments in Section 5 show that our algorithm with hashed determinants
outperforms the version without hash. For $m=3$ and $m=4$, we experimentally
observed that the speedup factor is between 18 and 100; Figure 6 illustrates
the second case.
The drawback of hashing determinants is the amount of storage, which is in
$O(n!)$. The hash table can be cleared at any moment to limit memory
consumption, at the cost of dropping all previously computed minors. Finding a
policy to clear the hash table according to the number of times each minor was
used would decrease the memory consumption, while keeping running times low.
Exploring different heuristics, such as using a LRU (least recently used)
cache, to choose which minors to drop when freeing memory will be an
interesting research subject.
It is possible to exploit the structure of the above $(2n)\times(2n)$ minor
matrices. Let $M$ be such a matrix, with columns corresponding to points of
$A_{0},\dots,A_{n}$. After column permutations, we split $M$ into four
$n\times n$ submatrices $A,B,D,I$, where $I$ is the identity matrix and $D$
has at most one $1$ in each column. This follows from the fact that the bottom
half of every column in $M$ has at most one $1$ and the last $n$ rows of $M$
contain at least one $1$ each, unless $\det M=0$, which is easily checked.
Now, $\det M=\pm\det(B-AD)$, with $AD$ constructed in $O(n)$. Hence, the
computation of $(2n)\times(2n)$ minors is asymptotically equal to computing an
$n\times n$ determinant. This only decreases the constant within the
asymptotic bound. A simple implementation of this idea is not faster than
Laplace expansion in the dimensions that we currently focus. However, this
idea should be valuable in higher dimensions.
## 5 Implementation and Experiments
We implemented Algorithm 1 in C++ to compute $\varPi$; our code can be
obtained from
http://respol.sourceforge.net.
All timings shown in this section were obtained on an Intel Core i5-2400
$3.1$GHz, with $6$MB L2 cache and $8$GB RAM, running 64-bit Debian GNU/Linux.
Our implementation, respol, relies on CGAL, using mainly a preliminary version
of package triangulation 26, for both regular triangulations, as well as for
the V- and H-representation of $\varPi$. As for hashing determinants, we
looked for a hashing function, that takes as input a vector of integers and
returns an integer, which minimizes collisions. We considered many different
hash functions, including some variations of the well-known FNV hash 29. We
obtained the best results with the implementation of Boost Hash 30, which
shows fewer collisions than the other tested functions. We clear the hash
table when it contains $10^{6}$ minors. This gives a good tradeoff between
efficiency and memory consumption. Last column of Table 1 shows that the
memory consumption of our algorithm is related to $|A|$ and $\dim(\varPi)$.
We start our experiments by comparing four state-of-the-art exact convex hull
packages: triangulation implementing Ref. 31 and beneath-and-beyond (bb) in
polymake 32; double description implemented in cdd 33; and lrs implementing
reverse search 34. We compute $\varPi$, actually extending the work in Ref. 35
for the new class of polytopes $\varPi$. The triangulation package was shown
to be faster in computing Delaunay triangulations in $\leq 6$ dimensions 26.
The other three packages are run through polymake, where we have ignored the
time to load the data. We test all packages in an offline version. We first
compute the V-representation of $\varPi$ using our implementation and then we
give this as an input to the convex hull packages that compute the
H-representation of $\varPi$. Moreover, we test triangulation by inserting
points in the order that Algorithm 1 computes them, while improving the point
location of these points since we know by the execution of Algorithm 1 one
facet to be removed (online version). The experiments show that triangulation
and bb are faster than lrs, which outperforms cdd. Furthermore, the online
version of triangulation is $2.5$ times faster than its offline counterpart
due to faster point location (Table 1, Figure 5).
$m$ | $|{\mathcal{A}}|$ | # of $\varPi$ | time (seconds) | respol
---|---|---|---|---
vertices | respol | tr/on | tr/off | bb | cdd | lrs | Mb
3 | 2490 | 318 | 85.03 | 0.07 | 0.10 | 0.07 | 1.20 | 0.10 | 37
4 | 27 | 830 | 15.92 | 0.71 | 1.08 | 0.50 | 26.85 | 3.12 | 46
4 | 37 | 2852 | 97.82 | 2.85 | 3.91 | 2.29 | 335.23 | 39.41 | 64
5 | 15 | 510 | 11.25 | 2.31 | 5.57 | 1.22 | 47.87 | 6.65 | 44
5 | 18 | 2584 | 102.46 | 13.31 | 34.25 | 9.58 | 2332.63 | 215.22 | 88
5 | 24 | 35768 | 4610.31 | 238.76 | 577.47 | 339.05 | $>1$hr | $>1$hr | 360
6 | 15 | 985 | 102.62 | 20.51 | 61.56 | 28.22 | 610.39 | 146.83 | 2868
6 | 19 | 23066 | 6556.42 | 1191.80 | 2754.30 | $>1$hr | $>1$hr | $>1$hr | 6693
7 | 12 | 249 | 18.12 | 7.55 | 23.95 | 4.99 | 6.09 | 11.95 | 114
7 | 17 | 500 | 302.61 | 267.01 | 614.34 | 603.12 | 10495.14 | 358.79 | 5258
Table 1: Total time and memory consumption of our code (respol) and time
comparison of online version of triangulation (tr/on) and offline versions of
all convex hull packages for computing the H-representation of $\varPi$.
Figure 5: Comparison of convex hull packages for $4$-dimensional (a) and
$5$-dimensional (b) $\varPi$. triang_on/triang_off are the online/offline
versions of triangulation package (y-axis is in logarithmic scale).
A placing triangulation of a set of points is a triangulation produced by the
Beneath-and-Beyond convex hull algorithm for some ordering of the points. That
is, the algorithm places the points in the triangulation with respect to the
ordering. Each point which is going to be placed, is connected to all visible
faces of the current triangulation resulting to the construction of new cells.
An advantage of triangulation is that it maintains a placing triangulation of
a polytope in ${\mathbb{R}}^{d}$ by storing the $0,1,d-1,d$-dimensional cells
of the triangulation. This is useful when the oracle
VTX$({\mathcal{A}},w,\pi)$ needs to refine the regular subdivision of
${\mathcal{A}}$ which is obtained by projecting the upper hull of the lifted
pointset ${\mathcal{A}}^{\widehat{w}}$ (Section 3). In fact this refinement is
attained by a placing triangulation, i.e., by computing the projection of the
upper hull of the placing triangulation of ${\mathcal{A}}^{\widehat{w}}$. This
is implemented in two steps:
1. Step 1.
compute the placing triangulation $T_{0}$ of the last $|{\mathcal{A}}|-m$
points with a random insertion order as described in Ref. 26 (they all have
height zero),
2. Step 2.
place the first $m$ points of ${\mathcal{A}}^{\widehat{w}}$ in $T_{0}$ with a
random insertion order 26.
Step 1 is performed only once at the beginning of the algorithm, whereas Step
2 is performed every time we check a new $w$. The order of placing the points
in Step 2 only matters if $w$ is not generic; otherwise, $w$ already produces
a triangulation of the $m$ points, so any placing order results in this
triangulation.
This is the implemented method; although different from the perturbation in
the proof of Lemma 2, it is more efficient because of the reuse of
triangulation $T_{0}$ in Step 1 above. Moreover, our experiments show that it
always validates the two conditions in Section 3.
We can formulate this 2-step construction using a single lifting. Let $c>0$ be
a sufficiently large constant, $a_{i}\in{\mathcal{A}},\,q_{i}\in{\mathbb{R}}$,
$q_{i}>c\,q_{i+1},$ for $i=1,\ldots,|{\mathcal{A}}|$. Define lifting
$h:{\mathcal{A}}\rightarrow{\mathbb{R}}^{2}$, where $h(a_{i})=(w_{i},q_{i}),$
for $i=1,\ldots,m$, and $h(a_{i})=(0,q_{i})$, for
$i=m+1,\ldots,|{\mathcal{A}}|$. Then, projecting the upper hull of
${\mathcal{A}}^{h}$ to ${\mathbb{R}}^{2n}$ yields the triangulation of
${\mathcal{A}}$ obtained by the 2-step construction.
Fixing the dimension of the triangulation at compile time results in $<1\%$
speedup. We also tested a kernel that uses the filtering technique based on
interval arithmetic from Ref. 36 with a similar time speedup. On the other
hand, triangulation is expected to implement incremental high-dimensional
regular triangulations with respect to a lifting, faster than the above method
37. Moreover, we use a modified version of triangulation in order to benefit
from our hashing scheme. Therefore, all cells of the triangulated facets of
$\varPi$ have the same normal vector and we use a structure (STL set) to
maintain the set of unique normal vectors, thus computing only one regular
triangulation per triangulated facet of $\varPi$.
Figure 6: (a) Implicitization and $u$-resultants for $n=2,m=3$; (b)
Comparison of respol (hashing and not hashing determinants) and Gfan
(traversing tropical resultants and computing normal fan from stable
intersection) for $m=4$; (c) Performance of Alg. 1 for $m=3,4,5$ as a function
of input; (d) Performance of Alg. 1 as a function of its output; y-axes in
(b), (c), (d) are in logarithmic scale.
We perform an experimental analysis of our algorithm. We design experiments
parameterized on: the total number of input points $|{\mathcal{A}}|$, the
dimension $n$ of pointsets $A_{i}$, and the dimension of projection $m$.
First, we examine our algorithm on random inputs for implicitization and
$u$-resultants, where $m=n+1$, while varying $|{\mathcal{A}}|,n$. We fix
$\delta\in{\mathbb{N}}$ and select random points on the $\delta$-simplex to
generate dense inputs, and points on the $(\delta/2)$-cube to generate sparse
inputs. For implicitization the projection coordinates correspond to point
$a_{i1}=(0,\ldots,0)\in A_{i}$. For $n=2$ the problem corresponds to
implicitizing surfaces: when $|{\mathcal{A}}|<60$, we compute the polytopes in
$<1$sec (Figure 6). When computing the $u$-resultant polytope, the projection
coordinates correspond to $A_{0}=\\{(1,\ldots,0),\ldots,(0,\dots,1)\\}$. For
$n=2$, when $|{\mathcal{A}}|<500$, we compute the polytopes in $<1$sec (Figure
6).
By using the hashing determinants scheme we gain a $18\times$ speedup when
$n=2,\ m=3$. For $m=4$ we gain a larger speedup; we computed in $<2$min an
instance where $|{\mathcal{A}}|=37$ and would take $>1$hr to compute
otherwise. Thus, when the dimension and $|{\mathcal{A}}|$ becomes larger, this
method allows our algorithm to compute instances of the problem that would be
intractable otherwise, as shown for $n=3,\ m=4$ (Figure 6).
We confirm experimentally the output-sensitivity of our algorithm. First, our
algorithm always computes vertices of $\varPi$ either to extend $\varPi$ or to
legalize a facet. We experimentally show that our algorithm has, for fixed
$m$, a subexponential behaviour with respect to both input and output (Figure
6, 6) and its output is subexponential with respect to the input.
# cells in triangulation | time (sec) | f-vector of $\varPi$
---|---|---
$\mu$ | $\sigma$ | min | max | $\mu$ | $\sigma$ | min | max
4781 | 154 | 4560 | 5087 | 0.35 | 0.01 | 0.34 | 0.38 | | | 449 | 1405 | 1438 | 482
16966 | 407 | 16223 | 17598 | 1.51 | 0.03 | 1.45 | 1.56 | | | 1412 | 4498 | 4705 | 1619
18229 | 935 | 16668 | 20058 | 1.92 | 0.10 | 1.77 | 2.11 | | 432 | 1974 | 3121 | 2082 | 505
563838 | 6325 | 548206 | 578873 | 99 | 1.62 | 93.84 | 103.07 | | 9678 | 43569 | 71004 | 50170 | 13059
289847 | 15788 | 264473 | 318976 | 69 | 4.88 | 61.67 | 77.31 | 1308 | 7576 | 16137 | 16324 | 7959 | 1504
400552 | 14424 | 374149 | 426476 | 96.5 | 4.91 | 88.86 | 107.12 | 1680 | 9740 | 21022 | 21719 | 10890 | 2133
Table 2: Typical f-vectors of projections of resultant polytopes and the size
of their triangulations. We perform 20 runs with random insertion order of
vertices for each polytope and report the minimum, maximum, average value
$\mu$ and the standard deviation $\sigma$ for the number of cells and the
runtime.
Figure 7: (a) $\mbox{vol}(Q)/\mbox{vol}(\varPi)$ as a function of the number
of random normal vectors used to compute $Q$; (b) The size of the
triangulation of $\varPi$ as a function of the output of Alg. 1.
As the complexity analysis (Theorem 1) indicates, the runtime of the algorithm
depends on the size of the constructed placing triangulation of $\varPi$. The
size of the placing triangulation depends on the ordering of the inserted
points. We perform experiments on the effect of the inserting order to the
size of the triangulation as well as the running time of the computation of
the triangulation (Table 2). These sizes as well as the runtimes vary in a
very narrow range. Thus, the insertion order is not crucial in both the
runtime and the space of our algorithm. Further experiments in $4$-dimensional
$N(R)$ show that the size of the input bounds polynomially the size of the
triangulation of the output (Figure 7) which explains the efficiency of our
algorithm in this dimension.
We explore the limits of our implementation. By bounding runtime to $<2$hr, we
compute instances of $5$-, $6$-, $7$-dimensional $\varPi$ with $35$K, $23$K,
$500$ vertices, respectively (Table 1).
We also compare with the implementation of Ref. 7, which is based on Gfan
library. They develop two algorithms to compute projections of
$N({\mathcal{R}})$. Assuming ${\mathcal{R}}$ defines a hypersurface, their
methods compute a union of (possibly overlapping) cones, along with their
multiplicities, see Theorem 2.9 of Ref. 7. From this intermediate result they
construct the normal cones to the resultant vertices.
examples in Ref. 7 | a | b | c | d | e | f | g | h | i
---|---|---|---|---|---|---|---|---|---
$|{\mathcal{A}}|$ | 12 | 12 | 15 | 12 | 12 | 16 | 27 | 16 | 20
$m$ | 12 | 12 | 15 | 6 | 7 | 9 | 3 | 4 | 5
$n$ | 3 | 2 | 4 | 2 | 2 | 3 | 2 | 3 | 4
Gfan(secs∗) | 1.40 | 6 | 55 | 0.70 | 1.30 | 798 | 0.40 | 2.60 | 184
respol(secs) | 1.40 | 18.41 | 99.90 | 0.26 | 1.24 | 934 | 0.02 | 0.96 | 292.01
Table 3: Comparison of our implementation with Gfan. ∗ Timings for Gfan as
reported in Ref. 7.
We compare with the best timings of Gfan methods using the examples and
timings of Ref. 7 (Table 3). Our method is faster in examples (d), (e), (g),
(h) where $m<7$, is competitive (up to $2$ times slower) in (a) where
$m=|{\mathcal{A}}|=12$ and (i) where $m=5,|{\mathcal{A}}|=20$ and slower in
(b), (c), (f) where $m\geq 12$. The bottleneck of our implementation, that
makes it slower when the dimension of the projection $m$ is high, is the
incremental convex hull construction in ${\mathbb{R}}^{m}$. Moreover, since
our implementation considers that $N({\mathcal{R}})$ lies in
${\mathbb{R}}^{|{\mathcal{A}}|}$ instead of
${\mathbb{R}}^{|{\mathcal{A}}|-2n-1}$, (see also the discussion on the
homogeneities of ${\mathcal{R}}$ in Section 2), it cannot take advantage of
the fact that $\dim(N({\mathcal{R}}))$ could be less than $m$ when
$|{\mathcal{A}}|-2n-1<m<|{\mathcal{A}}|$. This is the case in examples (b),
(c) and (f). On the other hand, we run extensive experiments for $n=3$,
considering implicitization, where $m=4$ and our method, with and without
using hashing, is much faster than any of the two algorithms based on Gfan
(Figure 6). However, for $n=4,\ m=5$ the beta version of Gfan used in our
experiments was not stable and always crashed when $|{\mathcal{A}}|>13$.
input | | m | 3 | 3 | 4 | 4 | 5 | 5
---|---|---|---|---|---|---|---|---
| $|{\mathcal{A}}|$ | 200 | 490 | 20 | 30 | 17 | 20
approximation | | # of $Q$ vertices | 15 | 11 | 63 | 121 | $>10$hr | $>10$hr
| $\mbox{vol}(Q)/\mbox{vol}(\varPi)$ | 0.96 | 0.95 | 0.93 | 0.94 | $>10$hr | $>10$hr
algorithm | | $\mbox{vol}(Q_{o})/\mbox{vol}(\varPi)$ | 1.02 | 1.03 | 1.04 | 1.03 | $>10$hr | $>10$hr
| time (sec) | 0.15 | 0.22 | 0.37 | 1.42 | $>10$hr | $>10$hr
uniformly | | $|Q|$ | 34 | 45 | 123 | 207 | 228 | 257
| random vectors | 606 | 576 | 613 | 646 | 977 | 924
random | | $\mbox{vol}(Q)/\mbox{vol}(\varPi)$ | 0.93 | 0.99 | 0.94 | 0.90 | 0.90 | 0.90
| time (sec) | 5.61 | 12.78 | 1.10 | 4.73 | 8.41 | 16.90
exact | | # of $\varPi$ vertices | 98 | 133 | 416 | 1296 | 1674 | 5093
algorithm | | time (sec) | 2.03 | 5.87 | 3.72 | 25.97 | 51.54 | 239.96
Table 4: Results on experiments computing $Q,Q_{o}^{H}$ using the
approximation algorithm and the random vectors procedure; we stop the
approximation algorithm when $\mbox{vol}(Q)/\mbox{vol}(Q_{o})>0.9$; the
results with random vectors are the average values over $10$ independent
experiments; “$>10$hr” indicates computation of $\mbox{vol}(Q_{o})$ was
interrupted after $10$hr.
We analyze the computation of inner and outer approximations $Q$ and
$Q_{o}^{H}$. We test the variant of Section 3 by stopping it when
$\mbox{vol}(Q)/\mbox{vol}(Q_{o}^{H})>0.9$. In the experiments, the number of
$Q$ vertices is $<15\%$ of the $\varPi$ vertices, thus there is a speedup of
up to $25$ times over the exact algorithm at the largest instances. The
approximation of the volume is very satisfactory:
$\mbox{vol}(Q_{o}^{H})/\mbox{vol}(\varPi)<1.04$ and
$\mbox{vol}(Q)/\mbox{vol}(\varPi)>0.93$ for the tested instances (Table 4).
The bottleneck here is the computation of vol$(Q_{o}^{H})$, where $Q_{o}^{H}$
is given in H-representation: the runtime explodes for $m\geq 5$. We use
polymake in every step to compute vol$(Q_{o}^{H})$ because we are lacking of
an implementation that, given a polytope $P$ in H-representation, its volume
and a halfspace $H$, computes the volume of the intersection of $P$ and $H$.
Note that we do not include this computation time in the reported time. Our
current work considers ways to extend these observations to a polynomial time
approximation algorithm for the volume and the polytope itself when the latter
is given by an optimization oracle, as is the case here.
Next, we study procedures that compute only the V-representation of $Q$. For
this, we count how many random vectors uniformly distributed on the
$m$-dimensional sphere are needed to obtain
$\mbox{vol}(Q)/\mbox{vol}(\varPi)>0.9$. This procedure runs up to $10$ times
faster than the exact algorithm (Table 4). Figure 7 illustrates the
convergence of $\mbox{vol}(Q)/\mbox{vol}(\varPi)$ to the threshold value $0.9$
in typical $3,4,5$-dimensional examples. The basic drawback of this method is
that it does not provide guarantees for $\mbox{vol}(Q)/\mbox{vol}(\varPi)$
because we do not have sufficient a priori information on $\varPi$. These
experiments also illustrate the extent in which the normal vectors required to
deterministically construct $\varPi$ are uniformly distributed over the
sphere.
## 6 Future work
One algorithm that should be experimentally evaluated is the following. We
perform a search over the vertices of $\Sigma(A)$, that is, we build a search
tree with flips as edges. We keep a set with the extreme vertices with respect
to a given projection. Each computed vertex that is not extreme in the above
set is discarded and no flips are executed on it, i.e. the search tree is
pruned in this vertex. The search procedure could be the algorithm of TOPCOM
or the one presented in Ref. 14 which builds a search tree in some equivalence
classes of $\Sigma(A)$. The main advantage of this algorithm is that it does
not involve a convex hull computation. On the other hand, it is not output-
sensitive with respect to the number of vertices of the resultant polytope;
its complexity depends on the number of vertices on the _silhouette_ of
$\Sigma(A)$, with respect to a given projection and those that are connected
by an edge with them.
As shown, polymake’s convex hull algorithm is competitive, thus one may use it
for implementing our algorithm. On the other hand, triangulation is expected
to include fast enumeration of all regular triangulations for a given (non
generic) lifting, in which case $\varPi$ may be extended by more than one
(coplanar) vertices.
Our proposed algorithm uses an incremental convex hull algorithm and it is
known that any such algorithm has a worst-case super-polynomial _total time
complexity_ 38 in the number of input points and output facets. The basic open
question that this paper raises is whether there is a polynomial total time
algorithm for $\varPi$ or even for the set of its vertices.
## 7 Acknowledgments
All authors were partially supported from project “Computational Geometric
Learning”, which acknowledges the financial support of the Future and Emerging
Technologies (FET) programme within the 7th Framework Programme for research
of the European Commission, under FET-Open grant number: 255827. Most of the
work was done while C. Konaxis and L. Peñaranda were at the University of
Athens. C. Konaxis’ research leading to these results has also received
funding from the European Union’s Seventh Framework Programme
(FP7-REGPOT-2009-1) under grant agreement no 245749\. We thank O. Devillers
and S. Hornus for discussions on triangulation, and A. Jensen and J. Yu for
discussions and for sending us a beta version of their code.
## References
* [1] I.Z. Emiris, T. Kalinka, C. Konaxis, and T. Luu Ba. Implicitization of curves and (hyper)surfaces using predicted support. Theor. Comp. Science, Special Issue on Symbolic & Numeric Computing, 479(0):81–98, 2013.
* [2] B. Sturmfels and J. Yu. Tropical implicitization and mixed fiber polytopes. In Software for Algebraic Geometry, volume 148 of IMA Volumes in Math. & its Applic., pages 111–131. Springer, New York, 2008.
* [3] J. Rambau. TOPCOM: Triangulations of point configurations and oriented matroids. In Proc. Intern. Congress Math. Software, pages 330–340, 2002.
* [4] I.Z. Emiris, T. Kalinka, C. Konaxis, and T. Luu Ba. Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants. J. Computer Aided Design, 45:252–261, 2013. Special Issue on Symposium Solid & Phys. Modeling 2012 (Dijon, France).
* [5] F. Rincón. Computing tropical linear spaces. In J. Symbolic Computation, volume 51, pages 86–98, 2013.
* [6] CGAL: Computational geometry algorithms library. http://www.cgal.org.
* [7] A. Jensen and J. Yu. Computing tropical resultants. arXiv:math.AG/1109.2368v1, 2011.
* [8] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston, 1994.
* [9] B. Sturmfels. On the Newton polytope of the resultant. J. Algebraic Combin., 3:207–236, 1994.
* [10] A. Dickenstein, I.Z. Emiris, and V. Fisikopoulos. Combinatorics of 4-dimensional resultant polytopes. In Proc. ACM Intern. Symp. on Symbolic & Algebraic Comput., 2013\. (to appear).
* [11] S.Yu. Orevkov. The volume of the Newton polytope of a discriminant. Russ. Math. Surv., 54(5):1033–1034, 1999.
* [12] H. Imai, T. Masada, F. Takeuchi, and K. Imai. Enumerating triangulations in general dimensions. Intern. J. Comput. Geom. Appl., 12(6):455–480, 2002.
* [13] T. Michiels and R. Cools. Decomposing the secondary Cayley polytope. Discr. Comput. Geometry, 23:367–380, 2000.
* [14] T. Michiels and J. Verschelde. Enumerating regular mixed-cell configurations. Discr. Comput. Geometry, 21(4):569–579, 1999.
* [15] P. Huggins. ib4e: A software framework for parametrizing specialized lp problems. In A. Iglesias and N. Takayama, editors, Mathematical Software (ICMS 2006), volume 4151 of Lecture Notes in Computer Science, pages 245–247. Springer, Berlin, 2006.
* [16] E. Kaltofen and G. Villard. On the complexity of computing determinants. Computational Complexity, 13:91–130, 2005.
* [17] H. Brönnimann, I.Z. Emiris, V. Pan, and S. Pion. Sign determination in Residue Number Systems. Theor. Comp. Science, Spec. Issue on Real Numbers & Computers, 210(1):173–197, 1999.
* [18] J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. D. Saunders, W. J. Turner, and G. Villard. Linbox: A generic library for exact linear algebra. In Proc. Intern. Congress Math. Software, pages 40–50, Beijing, 2002.
* [19] G. Guennebaud, B. Jacob, et al. Eigen v3. http://eigen.tuxfamily.org, 2010.
* [20] I.Z. Emiris, V. Fisikopoulos, C. Konaxis, and L. Peñaranda. An output-sensitive algorithm for computing projections of resultant polytopes. In Proc. Annual ACM Symp. Computational Geometry, pages 179–188, 2012.
* [21] J.A. De Loera, J. Rambau, and F. Santos. Triangulations: Structures for Algorithms and Applications, volume 25 of Algorithms and Computation in Mathematics. Springer, 2010.
* [22] G.M. Ziegler. Lectures on Polytopes. Springer, 1995.
* [23] D. Cox, J. Little, and D. O’Shea. Using Algebraic Geometry. Number 185 in GTM. Springer, New York, 2nd edition, 2005.
* [24] M.M. Kapranov. Characterization of A-discriminantal hypersurfaces in terms of logarithmic Gauss map. Math. Annalen, 290:277–285, 1991.
* [25] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry. Springer-Verlag, Berlin, 2003.
* [26] J.-D. Boissonnat, O. Devillers, and S. Hornus. Incremental construction of the Delaunay triangulation and the Delaunay graph in medium dimension. In Proc. Annual ACM Symp. Computational Geometry, pages 208–216, 2009.
* [27] M. Joswig. Beneath-and-beyond revisited. In M. Joswig and N. Takayama, editors, Algebra, Geometry, and Software Systems, Mathematics and Visualization. Springer, Berlin, 2003.
* [28] V. Fisikopoulos and L. Peñaranda. Faster geometric algorithms via dynamic determinant computation. In Proc. 20th Europ. Symp. Algorithms, pages 443–454, 2012.
* [29] G. Fowler, L.C. Noll, and P. Vo. Fowler/Noll/Vo (FNV) hash algorithm. http://www.isthe.com/chongo/tech/comp/fnv, 1991.
* [30] D. James. Boost functional library. http://www.boost.org/libs/functional/hash, 2008.
* [31] K.L. Clarkson, K. Mehlhorn, and R. Seidel. Four results on randomized incremental constructions. Comput. Geom.: Theory & Appl., 3:185–121, 1993.
* [32] E. Gawrilow and M. Joswig. Polymake: an approach to modular software design in computational geometry. In Proc. Annual ACM Symp. Computational Geometry, pages 222–231. ACM Press, 2001.
* [33] K. Fukuda. cdd and cdd+ Home Page. ETH Z”urich. http://www.ifor.math.ethz.ch/~fukuda/cdd_home, 2008.
* [34] D. Avis. lrs: A revised implementation of the reverse search vertex enumeration algorithm. In Polytopes: Combinatorics & Computation, volume 29 of Oberwolfach Seminars, pages 177–198. Birkhäuser, 2000.
* [35] D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms? Comput. Geom.: Theory & Appl., 7:265–301, 1997.
* [36] H. Brönnimann, C. Burnikel, and S. Pion. Interval arithmetic yields efficient dynamic filters for computational geometry. In Proc. Annual ACM Symp. Computational Geometry, pages 165–174, New York, 1998. ACM.
* [37] O. Devillers, 2011. Personal communication.
* [38] D. Bremner. Incremental convex hull algorithms are not output sensitive. In Proc. 7th Intern. Symp. Algorithms and Comput., pages 26–35, London, UK, 1996. Springer.
|
arxiv-papers
| 2011-08-30T15:38:54 |
2024-09-04T02:49:21.818451
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ioannis Z. Emiris, Vissarion Fisikopoulos, Christos Konaxis, Luis\n Pe\\~naranda",
"submitter": "Luis Pe\\~naranda",
"url": "https://arxiv.org/abs/1108.5985"
}
|
1108.5999
|
11institutetext: Physics Department, Syracuse University, Syracuse New York,
13244-1130, USA
Department of Mathematics, University of Bristol, University Walk, Clifton,
Bristol BS8 1TW, U.K.
Syracuse Biomaterials Institute, Syracuse University, Syracuse New York,
13244-1130, USA
Filaments, microtubules, their networks, and supramolecular assemblies General
theory and mathematical aspects
# Generic phases of cross-linked active gels: Relaxation, Oscillation and
Contractility
S. Banerjee 11 T.B. Liverpool 22 M.C. Marchetti 1133112233
###### Abstract
We study analytically and numerically a generic continuum model of an
isotropic active solid with internal stresses generated by non-equilibrium
‘active’ mechano-chemical reactions. Our analysis shows that the gel can be
tuned through three classes of dynamical states by increasing motor activity:
a constant unstrained state of homogeneous density, a state where the local
density exhibits sustained oscillations, and a steady-state which is
spontaneously contracted, with a uniform mean density.
###### pacs:
87.16.Ka
###### pacs:
87.10.-e
## 1 Introduction
The mechanical properties of living cells are largely controlled by a variety
of filament-motor networks optimized for diverse physiological processes [1].
In the presence of ATP, such networks are capable of generating controlled
contractile forces and spontaneous oscillations. These phenomena arise from
the presence of groups of motor proteins, such as myosin II, that convert the
chemical energy from ATP hydrolysis into mechanical work via a cyclic process
of attachment and detachment to associated polar protein filaments, e.g.
F-actin [2]. There is great variation in the organization of actin, myosins,
and other cross-linking proteins in the actomyosin structures found in cells.
Myofibrils in striated muscle cells are examples of highly organized
structures [1], composed of repeated subunits of actin and myosin, known as
sarcomeres, arranged in series. Each sarcomere consists of actin filament of
alternating polarity bound at their pointed end by large clusters of myosins,
known as myosin “thick filaments”. The periodic structure of the myofibril
allows it to generate forces on large length scales due to the collective
dynamics of individual units of microscopic size, giving rise to muscle
oscillation and contraction. More difficult is to understand the origin of
spontaneous oscillations and contractility in cytoskeletal filament-motor
assemblies that lack such a highly organized structure [3].
Oscillations are for instance observed in many organisms during repositioning
of the mitotic spindle from the cell center towards the cell pole when unequal
cell division occurs [3]. In vitro examples of such phenomena are sustained
cilia-like beatings in self-assembled bundles of microtubules and dyneins [4]
and spontaneous contractility in isotropic reconstituted actomyosin networks
with additional F-actin crosslinking proteins, such as filamin or
$\alpha$-actinin [5, 6].
Theoretical models have shown that spontaneous oscillations can arise from the
collective action of groups of molecular motors coupled to a single elastic
element [7]. In these models the load dependence of the binding/unbinding
kinetics of motor proteins breaks detailed balance and provides the crucial
nonlinearity that tunes the system through a Hopf bifurcation to a
spontaneously oscillating state. This simple theoretical concept has been
adapted to describe the beating of cilia and flagella [8], mitotic spindles
during asymmetric cell division [9], and spontaneous waves in muscle
sarcomeres [10, 11, 12, 13].
In a parallel development, generic continuum theories of active polar ‘gels’
have been constructed by suitable modification of the hydrodynamic equations
of equilibrium liquid crystals to incorporate the effect of activity. These
continuum models are capable of capturing some of the large-scale consequences
of the internal contractile stresses induced by active myosin crosslinkers,
including the presence of propagating actin waves in cells adhering to a
substrate [14] and the retrograde flow in the lamellipodium of crawling cells
[15]. In these studies the acto-myosin network is modeled as a Maxwell
viscoelastic fluid, with short-time elasticity and liquid response at long
times [16]. The active viscoelastic liquid cannot, however, support sustained
oscillations that require low frequency, long wavelength elastic restoring
forces.
In this letter we consider a nonlinear version of the generic continuum model
of an isotropic active solid discussed recently by two of us [17]. We go
beyond the linear stability analysis discussed in [17] and show that the
presence of nonlinearities leads to both stable contracted and oscillatory
states in different regions of parameters. The acto-myosin network is modeled
as an elastic continuum, as appropriate for a cross-linked polymer gel
embedded in a permeating viscous fluid, with elastic response at long times
and liquid-like dissipation at short times. The active solid model describes
the various phases of acto-myosin systems as a function of motor activity,
including spontaneous contractility and oscillations. It provides a unified
description of both phenomena and a minimal model relevant to many biological
systems with motor-filament assemblies that behave as solids at low
frequencies. When the active solid is isotropic, as assumed in the present
work, the coupling to a non-hydrodynamic mode provided by the binding and
unbinding kinetics of motor proteins is essential to generate spontaneous
sustained oscillations.
The main results are summarized in Fig. 1, where we sketch the steady states
of the system as we tune the activity, defined as the difference $\Delta\mu$
between the chemical potential of ATP and its hydrolysis products, and the
compressional modulus $B$ of the passive gel. For a fixed value of $B$ we find
a regime where the active gel supports sustained oscillating states and a
contracted steady state as $\Delta\mu$ is increased. For fixed $\Delta\mu$
spontaneous contractility is only observed below a critical network stiffness.
This is consistent with experiments on isotropic acto-myosin networks with
additional cross-linking $\alpha$-actinin where spontaneous contractility was
seen only in an intermediate range of $\alpha$-actinin concentration [5]. Our
model does not, however, yield a lower bound on $B$ below which the contracted
state does not exist. This may be because, in contrast to the experiments
where a minimum concentration of $\alpha$-actinin is required to provide
integrity to the network, our system is always by definition an elastic solid,
even at the lowest values of $B$. The phase diagram resulting from our model
also resembles the state space diagram of a muscle sarcomere obtained
experimentally [13].
Interestingly, we find that a simplified dynamical system obtained by a one-
mode approximation to our continuum theory corresponds to the half-sarcomere
model proposed recently by Günther and Kruse [10] for a particular set of
parameters. Our analysis shows, however, that the phase behavior described
above is generic and can be expected in a wide variety of active elastic
systems, as it relies solely on symmetry arguments. It provides a unified
description of both spontaneous and oscillatory states and predicts their
region of stability as a function of the elastic properties of the network and
motor activity.
Figure 1: (Color online) “Phases” of the nonlinear active elastic gel obtained
by varying the compressional modulus $B$ and the activity $\Delta\mu$.
## 2 Model
In Ref. [17] two of us formulated, on the basis of symmetry arguments, a
phenomenological hydrodynamic model of a cross-linked gel (a network of actin
filaments crosslinked by filamins or other ”passive” linkers) under the
influence of active forces exerted by clusters of crosslinking motor proteins
(e.g., myosin II minifilaments). Only linear terms were retained in the
continuum equations in [17] where the linear modes of the system and their
stability were analyzed. Here we consider a nonlinear continuum model and show
how nonlinearities can stabilize oscillatory and contracted states. The active
gel consists of a polymer network in a permeating fluid [17]. On length scales
large compared to the network mesh size, the gel can be described as a
continuum elastic medium, viscously coupled to a Newtonian fluid. The model
follows closely that formulated for a passive gel [18]. We focus on
compressional deformations and consider for simplicity a one-dimensional
model. We assume the permeating fluid to be incompressible and consider the
case of small volume fraction of the gel. In this limit the permeating fluid
simply provides a frictional drag to the polymer network. The hydrodynamic
description is then obtained in terms of two conserved fields: the density
$\rho(x,t)$ of the gel and the one-dimensional displacement field, $u(x,t)$.
The momentum density of the gel is not conserved due to drag exerted by the
permeating fluid. In addition, density variations are determined by the local
strain, with $\delta\rho=\rho-\rho_{0}=-\rho_{0}\partial_{x}u$ and $\rho_{0}$
the equilibrium mean density. The elastic free energy density of the gel can
be expanded in the strain $s=\partial_{x}u$ about the state $s=0$. To describe
the possibility of swelling and collapse of a gel (even a passive one) one
needs to keep terms up to fourth order in $s$ in the elastic free energy
density (dropping a linear term that can be eliminated by redefining the
ground state) [19]
$f_{e}=\frac{B}{2}s^{2}+\frac{\alpha}{3}s^{3}+\frac{\beta}{4}s^{4}\;,$ (1)
where $B$ is the longitudinal compressional modulus of the gel and
$\alpha,\beta>0$ are phenomenological parameters capturing the effects of
many-body interactions and nonlinear elasticity of the components [20].
It is straightforward to show that within mean-field theory the free energy
given in Eq. (1) yields a line of first order phase transitions at $B\equiv
B_{c}=2\alpha^{2}/(9\beta)$ between an unstrained state with $s=0$ for
$B>B_{c}$ and a strained state with finite $s$ for $B<B_{c}$. The stable
strained state is one of higher density ($s<0$) for $\alpha>0$, corresponding
to a contracted state, and one of lower density ($s>0$) for $\alpha<0$,
corresponding to an expanded state. The transition line terminates at a
critical point at $\alpha=0$.
Activity is induced by the presence of a concentration $c(x,t)$ of bound
active cross-linkers that undergo a cyclic binding/unbinding transformation
fueled by ATP, exerting forces on the gel. The dynamics of the active gel on
time scales larger than the Kelvin-Voigt viscoelastic relaxation time is
described by coupled equations for $u(x,t)$ and $c(x,t)$, given by
$\displaystyle\Gamma\partial_{t}u=\partial_{x}\sigma_{e}-\partial_{x}p_{a}\;,$
(2) $\displaystyle\partial_{t}c=-\partial_{x}(c\partial_{t}{u})-k(s)(c-c_{0})$
(3)
where $\sigma_{e}=\frac{\partial f_{e}}{\partial s}=Bs+\alpha s^{2}+\beta
s^{3}$ is the elastic stress, $\Gamma$ is a friction constant describing the
coupling of the polymer network to the permeating fluid, and $p_{a}(\rho,c)$
is the active contribution to the pressure, describing the isotropic part of
the active stresses induced by myosins. Equation (3) allows for convection of
bound motors by the gel at speed $\partial_{t}u$ and incorporates the
binding/unbinding dynamics, with $k(s)$ the strain-dependent motor unbinding
rate and $c_{0}$ the equilibrium concentration of bound motors. Since highly
non-processive motor proteins such as myosins are on average largely unbound,
we neglect the dynamics of free motors that provide an infinite motor
reservoir.
The active pressure is taken to be linear in the rate $\Delta\mu$ of ATP
consumption, with $p_{a}=\Delta\mu~{}\zeta(\rho,c)$. This is a reasonable
approximation for weakly active systems, where the number of active elements
make up a small fraction of the total mass of the gel, as is the case in most
experiments. 111We note that the effects of nonlinearities in $\Delta\mu$ can
taken account of by expanding in $\delta\mu=\Delta\mu-\Delta\mu_{0}$ about a
stationary state with finite $\Delta\mu_{0}$. We then expand
$\begin{split}\zeta(\rho,c)\simeq&\zeta_{0}-\zeta_{1}\delta\tilde{\rho}-\zeta_{2}\phi-\zeta_{3}\left(\delta\tilde{\rho}\right)^{2}+\zeta_{4}\delta\tilde{\rho}\phi\\\
&+\zeta_{5}\phi^{2}+\zeta_{6}\left(\delta\tilde{\rho}\right)^{3}...\end{split}$
(4)
with $\delta\tilde{\rho}=\delta\rho/\rho_{0}$ and $\phi=(c-c_{0})/c_{0}$ the
fluctuations in the gel and bound motor concentrations, respectively, and all
parameters $\zeta_{i}$ defined positive. The positive sign of $\zeta_{1}$ and
$\zeta_{2}$ corresponds to a contractile acto-myosin system and describes the
reduction in the longitudinal stiffness of the gel from contractile forces
exerted by motor proteins. The parameters $\zeta_{3}$, $\zeta_{4}$ and
$\zeta_{5}$ describe excluded volume effects. A positive $\zeta_{3}$ favors
contracted over expanded states. A positive $\zeta_{6}$ guarantees the
stability of the network in regions of negative effective compressional
modulus.
There are several sources of nonlinearities in Eqs. (2,3): the nonlinear
strain dependence of the elastic free energy, the nonlinear terms in the
active pressure, and the dependence of the motor unbinding rate $k$ on the
load force on bound motors, which in turn is proportional to the strain $s$ of
the elastic gel. We assume an exponential dependence of the form
$k(s)=k_{0}e^{\epsilon s}$ [21], with $k_{0}$ the unbinding rate of unloaded
motors and $\epsilon$ a dimensionless parameter determined by microscopic
properties of the motor-filament interaction. In the following we expand the
unbinding rate for small strain as $k(s)\simeq k_{0}\left[1+\epsilon
s+\frac{\epsilon^{2}}{2}s^{2}+{\cal O}(s^{3})\right].$ Keeping higher order
terms in $\epsilon$ does not change qualitatively the behavior described
below. Finally, the first term on the rhs of Eq. (3) contains a convective
nonlinearity that does not affect the key features of dynamics observed and
will be neglected in most of the following.
The equations for the active gel can then be written as
$\displaystyle\Gamma\partial_{t}u=\partial_{x}\sigma_{e}^{a}+\Delta\mu\partial_{x}\left[\zeta_{2}\phi+\zeta_{4}s\phi-\zeta_{5}\phi^{2}\right]\;,$
(5a)
$\displaystyle\partial_{t}\phi=-\partial_{x}\left[\left(1+\phi\right)\partial_{t}{u}\right]-k_{0}\left[1+\epsilon
s+\frac{\epsilon^{2}}{2}s^{2}\right]\phi\;,$ (5b)
where $\sigma_{e}^{a}=B_{a}s+\alpha_{a}s^{2}+\beta_{a}s^{3}$, with
renormalized elastic constants, $B_{a}=B-\zeta_{1}\Delta\mu$,
$\alpha_{a}=\alpha+\zeta_{3}\Delta\mu$ and
$\beta_{a}=\beta+\zeta_{6}\Delta\mu$.
We render our equations dimensionless by letting $u\rightarrow u/L$ and
$t\rightarrow tk_{0}^{-1}$. We then define dimensionless parameters as
$\tilde{B}=\frac{B}{\Gamma k_{0}L^{2}}$, $\tilde{\alpha}=\frac{\alpha}{\Gamma
k_{0}L^{2}}$, $\tilde{\beta}=\frac{\beta}{\Gamma k_{0}L^{2}}$,
$\Delta\tilde{\mu}=\frac{\Delta\mu}{\Gamma k_{0}L^{3}}$ and
$\tilde{\zeta}_{i}=\zeta_{i}L$. In the following we drop the tilde to simplify
notation and all quantities should be understood as dimensionless.
## 3 Linear stability analysis
There are three steady state solutions of Eqs. (5a,5b): an unstrained state,
with $(u_{s},\phi_{s})=(0,0)$, and two strained states, with
$(u_{s},\phi_{s})=(s_{\pm}x,0)$ and
$s_{\pm}=\left(-\alpha_{a}\pm\sqrt{\alpha_{a}^{2}-4B_{a}\beta_{a}}\right)/2\beta_{a}$,
provided $B_{a}<\alpha_{a}^{2}/4\beta_{a}$. For concreteness we choose
$\alpha_{a},\beta_{a}>0$. Then if $B_{a}>0$, $s_{\pm}<0$ and
$\delta\tilde{\rho}_{\pm}>0$, so that both strained solutions correspond to
contracted states. If $B_{a}<0$, then $s_{-}<0$ and $s_{+}>0$, corresponding
to contracted and expanded states, respectively.
In this section we examine the linear stability of each of these states.
Letting $s=s_{0}+\delta s$, where $s_{0}=(0,s_{\pm})$ represents the constant
value of strain in the steady state and $\delta s=\partial_{x}\delta u$, the
linearized equations for the displacement and motor concentration fluctuations
are given by
$\displaystyle\partial_{t}\delta u={\cal B}\partial_{x}^{2}\delta
u+{Z}\partial_{x}\phi\;,$ (6a)
$\displaystyle\partial_{t}\phi=-\partial_{x}\partial_{t}\delta
u-\kappa\phi\;,$ (6b)
where
$\displaystyle{\cal B}=B_{a}+2\alpha_{a}s_{0}+3\beta_{a}s_{0}^{2}\;,$ (7a)
$\displaystyle{Z}=\Delta\mu(\zeta_{2}+\zeta_{4}s_{0})\;,$ (7b)
$\displaystyle\kappa=1+\epsilon s_{0}+\epsilon s_{0}^{2}/2\;.$ (7c)
Looking for solution of the form $\delta u,\phi\sim e^{zt+iqx}$ we find that
the dynamics of fluctuations is controlled by two eigenvalues, with dispersion
relations
$\displaystyle
z_{u,\phi}(q)=-\frac{b(q)}{2}\pm\frac{1}{2}\sqrt{\left[b(q)\right]^{2}-4\kappa{\cal
B}q^{2}}$ (8)
where $b(q)=\kappa+({\cal B}-Z)q^{2}$ and $z_{u}(q)$ and $z_{\phi}(q)$
correspond to the root with the plus and minus signs, respectively.
If motor fluctuations are neglected ($\phi=0$), corresponding to letting $Z=0$
in Eq. (8), one finds $z_{u}=-{\cal B}q^{2}$ and the linear stability of the
steady states is entirely determined by the sign of ${\cal B}$, with ${\cal
B}>0$, corresponding to a linearly stable state. When $\phi=0$ the problem is
equivalent to an equilibrium problem, with $q^{2}{\cal B}$ the curvature of a
free energy of the form given in Eq. (1), albeit with elastic constants
renormalized by activity. In this case the unstrained state $s=0$ is the
global minimum of the free energy for $B_{a}>2\alpha_{a}^{2}/(9\beta_{a})$,
while the contracted state $s=s_{-}$ is the global minimum for
$B_{a}<2\alpha_{a}^{2}/(9\beta_{a})$. The line
$B_{a}=2\alpha_{a}^{2}/(9\beta_{a})$ defines a line of first order phase
transitions in the $(B,\Delta\mu)$ (see Fig. 2, left frame). If, on the other
hand, we consider the problem dynamically, the condition of stability of
linear fluctuations given by ${\cal B}>0$ is a necessary, but not sufficient
one to identify the stable steady states of the system, as multiple fixed
points with different basins of attraction coexist in the same region of
parameters. In particular, both the unstrained $s=0$ state and the contracted
$s=s_{-}$ state are linearly stable for $B_{a}<B<\alpha_{a}^{2}/(4\beta_{a})$,
while both the contracted and expanded states ($s=s_{\mp}$) are stable for
$B_{a}<0$. If we assume that among these linearly stable states the system
selects dynamically the steady state with the fastest decay rate $q^{2}{\cal
B}$, then we recover the linear phase diagram of Fig. 2 obtained form the
equilibrium analysis.
Figure 2: (Color online) Phase diagram obtained by analyzing the linear
stability of the modes. The left frame is for $\zeta_{2}=\zeta_{4}=0$,
corresponding to $\phi=0$. The right frame is obtained including motor
fluctuations, with $\zeta_{2}=\zeta$ and $\zeta_{4}=\eta\zeta$. The other
parameter values are $\alpha=0.1$, $\beta=0.1$, $\zeta_{1}=\zeta=1$,
$\zeta_{3}=\eta\zeta$, and $\zeta_{6}=\eta^{2}\zeta$, with $\eta=0.1$.
When the coupling to motor concentration fluctuations is included, we find a
region of parameters where $Re\left[z_{u,\phi}\right]>0$ for all three
homogeneous solutions $s=(0,s_{\pm})$. This is the white region in Fig. 2
(right frame), where the dynamical system is linearly unstable. The
instability occurs in a region where the modes are complex, describing
oscillatory states, and $Z-\kappa/q^{2}>{\cal B}>0$. The linear stability
diagram for the system is shown in Fig. 2 for the shortest wavevectors $\sim
1/L$, with $L$ the system size. The phase diagram is constructed again by
assuming that the system will relax to linearly stable states characterized by
the fastest relaxation rates. To linear order, the coupling to motor
fluctuations yields oscillatory or propagating (as opposed to purely
diffusive) fluctuations. Some of these oscillatory states are linearly
unstable in a region of parameters, as shown in the phase diagram in Fig. 2.
Next we consider the effect of nonlinearities by first examining a minimal
model that only retains the longest wavelength mode in the Fourier expansion
of the nonlinear equations, Eqs. (2,3), and then comparing the latter to the
numerical solution of the full nonlinear partial differential equations.
## 4 One-mode model
We begin by incorporating only the nonlinearities in the unbinding rate, while
neglecting convective, elastic and pressure nonlinearities. We impose boundary
conditions $[\partial_{x}u]_{x=0}=[\partial_{x}u]_{x=L}=0$ and
$[\phi]_{x=0}=[\phi]_{x=L}=0$, and seek a solution of Eqs. (2,3) in the form
of a Fourier series as, $u(x,t)=\sum_{m=0}^{\infty}u_{m}(t)\cos{(\hat{m}x)}$
and $\phi(x,t)=\sum_{m=1}^{\infty}\phi_{m}(t)\sin{(\hat{m}x)}$, where,
$\hat{m}=m\pi/L$. We perform a 1-mode Galerkin truncation, and only consider
the dynamics of the first nontrivial mode, $m=1$ (setting
$u_{m}=\phi_{m}=0,\;\forall\;m\neq 1$). This corresponds to approximating the
system by only its longest wavelength excitations, ignoring all the shorter
wavelength modes.
The coupled equations for $u_{1}$ and $\phi_{1}$ are given by
$\displaystyle\dot{u}_{1}=-B_{a}\pi^{2}u_{1}+\zeta_{2}\Delta\mu\pi\phi_{1}\;,$
(9a) $\displaystyle\dot{\phi}_{1}=\pi\dot{u}_{1}-\left(1-\frac{8}{3}\epsilon
u_{1}+\frac{3}{8}\pi^{2}\epsilon^{2}u_{1}^{2}\right)\phi_{1}\;.$ (9b)
These equations can be recast into an effective second order differential
equation for $u_{1}$ that has the structure of the equation for a Van der Pol
oscillator [22] coupled to a nonlinear spring, of the form
$\ddot{u}_{1}+\dot{u}_{1}\left[\lambda-f(u_{1})\right]+u_{1}B_{a}\left[1-f(u_{1})\right]=0$,
with
$f(u_{1})=\frac{8\epsilon}{3}u_{1}-\frac{3\epsilon^{2}\pi^{2}}{8}u_{1}^{2}$
and friction $\lambda=\pi^{2}(B_{a}-\zeta_{2}\Delta\mu)+1$. Equations (9a) and
(9b) admit one fixed point $(u_{1},\phi)=(0,0)$. Linear stability analysis
about this null fixed point shows that the fixed point is unstable (a
repeller) when $\lambda<0$ and is stable for positive $\lambda$. From a global
analysis [22] of Eqs. (9a) and (9b) with the $\epsilon$ nonlinearity, we find
that the existence of the unstable fixed point signals the appearance a stable
limit cycle as $\lambda$ crosses zero. In other words the system undergoes a
supercritical Hopf bifurcation at
$\Delta\mu=\Delta\mu_{c1}=(B+\pi^{-2})/(\zeta_{1}+\zeta_{2})$, with sustained
oscillations for $\Delta\mu>\Delta\mu_{c1}$ and stable spirals or nodes [22]
in the two-dimensional, $(u_{1},\phi_{1})$ phase space, otherwise.
We now consider the role of pressure and elastic nonlinearities, while letting
$\epsilon=0$. Within the one-mode model
Figure 3: (color online) Left frame : Time evolution of $u_{1}(t)$ for various
values of activity : $\Delta\mu=1.0$ (black), corresponding to an unstrained
steady state, $\Delta\mu=2.0$ (red) and $\Delta\mu=3.5$ (green), corresponding
to the region of sustained oscillations and $\Delta\mu=4.5$ (blue),
corresponding to the contracted steady state. Right frame: bifurcation diagram
obtained from the 1-mode model. The figure shows a plot of the value
$u_{1}^{s}$ of $u_{1}$ at long times ($t=100$) vs $\Delta\mu$. A supercritical
Hopf bifurcation occurs at $\Delta\mu=\Delta\mu_{c1}\simeq 1.5$. As one
increases $\Delta\mu$ within the range
$\Delta\mu_{c1}<\Delta\mu<\Delta\mu_{c2}$, the amplitude of oscillations grows
continuously. At $\Delta\mu=\Delta\mu_{c2}$ the limit cycle disappears and the
system settles into a contracted steady state for $\Delta\mu>\Delta\mu_{c2}$.
The dashed line indicates an unstable fixed point. Parameters: $B=3.0$,
$\epsilon=0.1$. Other parameter values are same as in Fig. 2.
we find that in this case if $\beta_{a}=\alpha_{a}=0$ and $B_{a}>0$ there is
again only one fixed point at $(u_{1},\phi)=(0,0)$ and the linear stability
analysis is identical to that of the model with only rate nonlinearity. In
other words, if $\epsilon=0$, the pressure nonlinearities do not stabilize the
system for $\lambda<0$. When $\alpha_{a}\not=0,\beta_{a}\not=0$ and
$B_{a}<16\alpha_{a}^{2}/27\beta_{a}\pi^{2}$, we have two new nonzero fixed
points $(u_{1},\phi)\equiv(u_{\pm},0)$, describing a contracted and an
expanded state of the gel, respectively, with
$u_{\pm}=\frac{8\alpha_{a}}{9\pi^{2}\beta_{a}}\pm\frac{2}{3\beta_{a}\pi^{2}}\sqrt{\frac{16\alpha_{a}^{2}}{9}-3\pi^{2}\beta_{a}B_{a}}\;.$
(10)
Since $\delta\rho\simeq\pi\rho_{0}u_{1}$, $u_{+}$ corresponds to a contracted
state whereas $u_{-}$ corresponds to an expanded state when
$B>\zeta\Delta\mu$. The contracted/expanded state is linearly unstable when
$\lambda^{*}_{\pm}=-B_{a}+\frac{8\alpha_{a}u_{\pm}}{3}-\frac{9\beta_{a}\pi^{2}u_{\pm}^{2}}{4}+\zeta_{2}-\frac{8\zeta_{4}u_{\pm}}{3}-\frac{1}{\pi^{2}}>0\;.$
For $\lambda^{*}_{\pm}<0$ the nonzero fixed points are unstable if
$B_{a}>16\alpha_{a}^{2}/27\beta_{a}\pi^{2}$. When
$B_{a}<16\alpha_{a}^{2}/27\beta_{a}\pi^{2}$ and $\lambda^{*}_{\pm}<0$ the
nonzero fixed points are stable and the orbits in the two-dimensional,
$(u_{1},\phi_{1})$ phase-space describe nodes or spirals settling at long
times to $(u_{\pm},0)$ [22]. When $\lambda^{*}_{\pm}>0$, the
contracted/expanded states are linearly unstable. The nonlinearities in the
active pressure stabilize these unstable states into stable asymmetric limit
cycles. A supercritical Hopf bifurcation occurs when
$\Delta\mu>\Delta\mu_{c1}=(B+\pi^{-2})/(\zeta_{1}+\zeta_{2})$, which
terminates to a stable contracted steady state for $\Delta\mu>\Delta\mu_{c2}$
($\Delta\mu_{c2}$ is determined by the solution of
$\lambda^{*}_{\pm}(B,\Delta\mu)=0$).
It is instructive to note that when $\alpha_{a}=\beta_{a}=0$, our 1-mode model
corresponds to that of a half-sarcomere derived in Ref. [10]. The bifurcation
diagram for the complete 1-mode model is shown in Fig. 3, with all the
nonlinearities incorporated. The diagram summarizes the steady state
crossovers of the system as we increase $\Delta\mu$ keeping $B$ fixed.
Finally, the steady states of the nonlinear gel within the 1-mode
approximation are shown in the phase diagram in Fig. 4 (indicated by dashed
lines), in the $(B,\Delta\mu)$ plane. The unstrained state of the gel is
stable when $\lambda>0$ whereas the contracted state is stable for
$\lambda^{*}_{+}>0$. When $\lambda<0$ and $\lambda_{+}^{*}<0$ the gel exhibits
sustained oscillations.
## 5 Numerical analysis of the Continuum Nonlinear Model
In this section we discuss the numerical solution of the nonlinear PDEs given
in Eqs. (5a,5b) (but with no convective nonlinearities) with boundary
conditions $\left[\partial_{x}u(t)\right]_{x=0,L}=0$ and
$\left[\phi\right]_{x=0,L}=0$ and an initial strained state. We spatially
discretize the PDEs using finite difference method and then integrate the
resulting coupled ODEs using the rkf45 method.
The behavior of the system as we vary $B$ and $\Delta\mu$, keeping all other
parameters constants, is summarized in the numerically constructed phase
diagram, shown in Fig. 4. Assuming $\zeta_{1}=\zeta_{2}=\zeta$, the system
settles into an unstrained state for $B>B_{c1}\simeq 2\zeta\Delta\mu$ and Hopf
bifurcates to sustained oscillations for $B<B_{c1}$. We note that the phase
boundary in the 1-mode model $B=2\zeta\Delta\mu-1/\pi^{2}$, although different
from the numerical phase boundary $B=B_{c1}$, lies within the error bar of the
numerics. For $B<B_{c2}\simeq 0.85\zeta\Delta\mu$ the system settles into a
stable contracted state. The basin of attraction of the expanded state is much
smaller than that of the contracted state. This is due to the choice of the
sign of the coupling constants, $\alpha_{a}$ and $\zeta_{4}$. The regimes
predicted by our continuum phenomenological model may be used to classify the
behaviour seen in a number of experimental systems [13, 5, 6]. To estimate the
parameter in real systems we assume $\Gamma\sim\eta/\xi^{2}$, with $\eta$ the
viscosity of the permeating fluid and $\xi$ the gel mesh size [18] and use
experimental values of $B$. For muscle fibers, using $B\sim 2$ kPa [23],
$\eta\sim 2\times 10^{-3}Pa~{}s$ [2], $\xi=0.5~{}\mu m$, $L\sim 100\ \mu m$
and $k_{0}^{-1}=40\ ms$, we estimate the dimensionless modulus
$\tilde{B}=B/\Gamma L^{2}k_{0}$ as $\tilde{B}\sim 1$. For isotropic cross-
linked actomyosin gels only direct measurements of the low frequency shear
modulus $G$ are available, with $G\sim 1-10Pa$ [6]. It has been argued,
however, that these networks may support much higher compressional forces, of
the order of buckling forces on the scale of the mesh-size, yielding a value
of $B$ comparable to that of muscle fibers [24]. For $B\sim 1-10^{3}Pa$ and
$\eta$ comparable to that of water, we obtain $\tilde{B}\sim 10^{-3}-1$. The
active coupling $\zeta\Delta\mu$ can be estimated as [10]
$\zeta\Delta\mu\sim\xi n_{b}k_{m}\Delta_{m}$, where $n_{b}$ is the number
density of bound motors, $k_{m}$ is the stiffness of the myosin filaments and
$\Delta_{m}$ is the steady state stretch of the bound motor tails. Using
$n_{b}\sim 10^{2}-10^{4}~{}\mu m^{-3}$, $k_{m}=4pN/nm$ and $\Delta_{m}\sim 1\
nm$, we obtain $\zeta\Delta\mu\sim 0.1-10$ in dimensionless units. In the
phase diagram in Fig. 4, these parameter values would put muscle fibers in the
regions U, SO and C respectively as we increase $n_{b}$. Isotropic actomyosin
networks may have a much lower value of $\tilde{B}$, possibly corresponding to
the contracted region. Indeed, while spontaneous contractility has been
observed in vitro in reconstituted actomyosin networks [5], no experimental
evidence has yet been put forward of spontaneous oscillations in these
systems.
Figure 4: (Color online) Phase diagram in the $(B,\Delta\mu)$ plane for the
continuum nonlinear model described by Eqs. (5a) and (5b). U : Unstrained, SO
: Sustained Oscillations, C : Contracted. The points are obtained by numerical
solution of the full PDE’s; the error bars are determined by the step size
used in the $\Delta\mu$ increments. The dashed lines indicate the 1-mode phase
boundaries. Inset : Plot of extension $\Delta\ell(t)=u(0,t)-u(L,t)$ for
$B=3.0$ and $\Delta\mu=1.0$ (black), $\Delta\mu=2.0$ (red), $\Delta\mu=3.5$
(green) and $\Delta\mu=4.5$ (blue). Parameter values are same as in Fig. 3.
## 6 Conclusion
We have presented a generic continuum model of a cross-linked active gel which
can be used to describe a wide variety of isotropic elastic active systems. We
find an elastic active gel system can, in general, be tuned through three
classes of dynamical states by increasing motor activity: an unstrained steady
state of homogeneous constant density, a state where the local density
exhibits sustained oscillations, and a spontaneously contracted steady state,
with a uniform mean density. The continuum model is not strictly hydrodynamic
due to presence of the fast variable $\phi$, describing the dynamics of motor
proteins. The motor binding/unbinding kinetics plays a crucial role in
generating oscillating states at finite wavevectors. The one-mode model is in
excellent qualitative agreement with the results described by the solutions to
the nonlinear continuum equations and is comparable to a variety of one
dimensional models for active elastic systems. Quantitative agreement in the
phase boundaries between the one-mode model and the continuum model fails due
to the non hydrodynamic nature of the model and we expect the oscillatory
and/or contractile behaviour to depend on system size [25].
###### Acknowledgements.
MCM and SB were supported by the National Science Foundation through awards
DMR-0806511 and DMR-1004789. TBL acknowledges the support of the EPSRC under
grant EP/G026440/1.
## References
* [1] Alberts B. et al. Molecular biology of the cell (Garland, New York) 2007.
* [2] Howard J. Mechanics of motor proteins and the cytoskeleton (Sinauer Associates Sunderland, MA) 2001.
* [3] Albertson D. Dev. Biol.101198461; Yeh E. et al. Mol. Bio. Cell 1120003949.
* [4] Sanchez T., Welch D., Nicastro D. Dogic Z. Science3332011456.
* [5] Bendix P. M. et al. Biophys. J.9420083126.
* [6] Koenderink G. H. et al. Proc. Nat. Acad. Sci.106200915192.
* [7] Jülicher F. Prost J. Phys. Rev. Lett.7519952618; Jülicher F. Prost J. Phys. Rev. Lett.7819974510.
* [8] Camalet S. Jülicher F. New J. Phys.2200024.
* [9] Grill S. W., Kruse K. Jülicher F. Phys. Rev. Lett.942005108104; Plaçais P.-Y., Balland M., Guérin T., Joanny J.-F. Martin P. Phys. Rev. Lett.1032009158102.
* [10] Günther S. Kruse K. New J. Phys.92007417.
* [11] Günther S. Kruse K. Chaos2020105122.
* [12] Anazawa T., Yasuda K. Ishiwata S. Biophys. J.6119921099.
* [13] Okamura N. Ishiwata S. J. Muscle Res. Cell M.91988111.
* [14] Asano Y. et al. HFSP J.32009194.
* [15] Jülicher F., Kruse K., Prost J. Joanny J.-F. Phys. Rep.44920073.
* [16] Kruse K. et al. Eur. Phys. J. E1620055.
* [17] Banerjee S. Marchetti M. C. Soft Matter72011463.
* [18] Levine A. J. Lubensky T. C. Phys. Rev. E632001041510.
* [19] Tanaka T. Phys. Rev. Lett.401978820; Golubović L. Lubensky T. C. Phys. Rev. Lett.6319891082.
* [20] Storm C. et al. Nature4352005191; Gardel M. L. et al. Science30420041310;
* [21] Parmeggiani A., Jülicher F., Peliti L. Prost J. Europhys. Lett.562001603.
* [22] Strogatz S. H. Nonlinear dynamics and chaos (Westview Press) 1994\.
* [23] Magid A. Law D. Science23019851280.
* [24] O. Chaudhuri S. H. P. Fletcher D. A. Nature4452006295.
* [25] Thoresen T. et al. Biophys. J.1002011446; Shimamato Y. et al. Biochem. Biophys. Res. Comm.3662008233; Dufresne E. R., Private Comm.
|
arxiv-papers
| 2011-08-30T16:19:01 |
2024-09-04T02:49:21.827345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shiladitya Banerjee, Tanniemola B. Liverpool and M. Cristina Marchetti",
"submitter": "Shiladitya Banerjee",
"url": "https://arxiv.org/abs/1108.5999"
}
|
1108.6024
|
††thanks: Corresponding Author
# Impact of surface roughness on diffusion of confined fluids
William P. Krekelberg william.krekelberg@nist.gov Vincent K. Shen
vincent.shen@nist.gov Chemical and Biochemical Reference Data Division,
National Institute of Standards and Technology, Gaithersburg, Maryland
02899-8380, USA Jeffrey R. Errington jerring@buffalo.edu Department of
Chemical and Biological Engineering, University at Buffalo, The State
University of New York, Buffalo, New York 14260-4200, USA Thomas M. Truskett
truskett@che.utexas.edu Department of Chemical Engineering, University of
Texas at Austin, Austin, TX 78712. Institute for Theoretical Chemistry,
University of Texas at Austin, Austin, TX 78712.
###### Abstract
Using event-driven molecular dynamics simulations, we quantify how the self
diffusivity of confined hard-sphere fluids depends on the nature of the
confining boundaries. We explore systems with featureless confining boundaries
that treat particle-boundary collisions in different ways and also various
types of physically (i.e., geometrically) rough boundaries. We show that, for
moderately dense fluids, the ratio of the self diffusivity of a rough wall
system to that of an appropriate smooth-wall reference system is a linear
function of the reciprocal wall separation, with the slope depending on the
nature of the roughness. We also discuss some simple practical ways to use
this information to predict confined hard-sphere fluid behavior in different
rough-wall systems.
## I Introduction
Predicting the dynamic properties of moderate-to-high density bulk fluids from
first principles remains an outstanding scientific challenge. This already-
difficult problem becomes considerably more formidable when the fluid is
confined to small spaces, a situation that is commonly encountered across a
broad range of technologically important settings.Bhushan et al. (1995) In
recent years, an important step toward solving this problem for confined
fluids was taken by identifying physically meaningful static (i.e.,
thermodynamic) variables (or combinations thereof) against which dynamic fluid
properties scaled independently of the degree of confinement.Mittal et al.
(2006, 2007a, 2007b, 2007c); Goel et al. (2008); Mittal et al. (2008); Goel et
al. (2009); Mittal (2009) This identification suggested that dynamic
properties such as the self diffusivity of confined fluids could be predicted
based on knowledge of their static properties and the corresponding static-
dynamics scaling relationship obtained from, e.g., bulk fluid data. To date,
tests of this scaling strategy for predicting how confinement affects dynamics
of model systems have been carried out by molecular simulation of
monatomicMittal et al. (2006, 2007a, 2007b, 2007c); Goel et al. (2009); Mittal
(2009) and a limited number of molecular Chopra et al. (2010) fluids. Although
these tests suggest that the scaling method can successfully predict the
dynamic behavior of a variety of fluids confined to pores with different
geometries, little attention has been paid to the nature of the confining
walls themselves and how it affects the relationship between static variables
and dynamic properties. In this paper, we address this issue by investigating
how surface roughness impacts the self diffusivity of model confined fluids.
Smooth, flat structureless walls represent a mathematically convenient and
idealized theoretical construct, and thus, historically, have been commonly
used in the study of model confined fluids. However, real solid surfaces do in
fact exhibit structure, which can significantly impact the thermodynamic and
dynamic properties of the fluids they contact. Wetting and lubrication are two
obvious examples of phenomena where the structure and shape of the fluid-
exposed solid surface have significant implications.Grzelak et al. (2010);
Grzelak and Errington (2010) This should not be surprising because surface
roughness, which arises from the structural arrangement of “wall” particles,
ultimately influences the fluid-wall interaction. Also, it is clear that
surface roughness greatly impacts dynamics. For low density gases, where
Knudsen diffusionKnudsen (1909) dominates, self diffusion decreases with
surface roughness.Arya et al. (2003); Malek and Coppens (2001, 2002, 2003)
Furthermore, experiments of confined colloids find that some particles stick
to walls, further slowing dynamics.Edmond et al. (2010) Simulations have shown
that position-dependent relaxation processes near rough surfaces are much
slower than those near smooth surfaces.Scheidler et al. (2002, 2004)
A question which has recieved comparatively less attention is how surface
roughness impact the self-diffusivity of moderate-to-dense confined fluids.
The goal of this paper is to answer this question systematically via molecular
simulations. The most obvious way in which roughness influences dynamic
properties is through the modification of fluid particle–wall collisions.
Here, we investigate how different representations of this collisional
modification lead to changes in the self-diffusion of a simple, confined
fluid. As a starting point, we study the monodisperse hard-sphere (HS) fluid
confined between smooth hard walls. Despite its simplicity, we choose the HS
fluid because it captures many of the important effects arising from the
dominant excluded volume interactions in liquids Errington et al. (2003);
Hansen and McDonald (2006) Obviously, a smooth hard wall does not possess any
roughness whatsoever, but it still serves as a useful reference point for
studying confined fluids, specifically the dynamic properties in this paper.
In fact, the slit pore geometry is often assumed when determining the pore
size distribution from adsorption isotherms in real porous materials.Lastoskie
et al. (1993); Olivier et al. (1994); Lastoskie et al. (1997); Rouquerol and
Rouquerol (1999); Klobes et al. (2006) In our simulations, we introduce
surface roughness in two separate ways (i) by modifying the boundary
conditions for collisions between fluid particles and a flat wall surface and
(ii) by giving the surface of the confining walls physical roughness or shape.
The former crudely represents surface roughness on a length scale much smaller
than the diameters of the fluid particles (e.g., confined colloids), while the
latter models physical roughness on a length scale comparable to the diameter
of the fluid particles (e.g., confined molecular fluids). While neither of the
above types of walls can be said to completely represent the roughness present
in real physical systems, they represent the types of roughness that can be
incorporated in molecular simulations, and serve as a good starting point to
address the impact of surface roughness on self-diffusivity.
We find that surface roughness reduces average particle mobility relative to a
smooth flat wall, where the collisions between fluid particles and the wall
surface are perfectly reflecting. Moreover, the reduction appears to be
systematic with increasing degree of confinement (i.e., decreasing pore
width). In the case of physically rough walls, we show that it is necessary to
make the distinction between the spatially homogeneous and inhomogeneous
directions because the associated self-diffusion coefficients can differ
significantly. Finally, we find that the dynamic properties of the hard-sphere
fluid confined between rough surfaces can be regarded as a perturbation on the
dynamic properties of an appropriately chosen smooth-wall reference system.
This paper is organized as follows. In Section II, we describe the model fluid
studied in this work, as well as the simulation methods used. We then discuss
how wall-surface roughness is implemented in Section III. Results are
presented and discussed in Section IV. Finally, we present conclusions and
directions for future work in Section V.
## II Model fluid
Event-driven microcanonical molecular dynamics (MD) simulationsRapaport (2004)
were used to study a fluid composed of hard spheres of diameter $\sigma$ and
mass $m$ with interaction potential
$\phi(r_{ij})=\begin{cases}\infty&r_{ij}\leq\sigma,\\\
0&r_{ij}>\sigma,\end{cases}$ (1)
where $r_{ij}$ is the distance between particles $i$ and $j$. Event-driven MD
consists of four basic steps: (1) Calculate future events (collisions) times.
(2) Sort events to determine next event to occur. (3) Advance system to next
event using Newtonian dynamics (free flight). (4) Execute event (collision) to
determine new particle velocities, and return to (1). Various methods are
available to speed up the general algorithm. The exact method used in this
study is described in Ref.Rapaport, 2004. A rectangular simulation cell of
dimensions $L_{x}\times L_{y}\times L_{z}$ with $N=4000$ particles was used
throughout this work. Periodic boundary conditions were applied in the $x$ and
$y$ directions in all cases, and in the $z$ direction for the bulk fluid. For
the confined fluid, where the $z$ direction was non-periodic, appropriate
wall-boundary conditions (see below) were employed. The dimensions of the
simulation box were chosen to correspond to a desired reduced number density
$\rho\sigma^{3}$ (or packing fraction$=\pi\rho\sigma^{3}/6$), and such that
the dimensions of the simulation box in the periodic directions were (nearly)
equal. Initial configurations for the MD simulations were generated by
randomly inserting hard spheres at low density with no overlaps, followed by
compression and (Monte-Carlo) relaxation steps until the desired density was
obtained. The systems were deemed equilibrated when the number of collisions
(particle-particle and particle-wall) per unit time reached a constant value
as a function of time. Equilibration times of
$1000\,(m\sigma^{2}/k_{\rm{B}}T)^{1/2}$ were found to be sufficient to meet
this requirement. Production simulations spanned times exceeding
$10^{5}\,(m\sigma^{2}/k_{\rm{B}}T)^{1/2}$, which allowed the slowest system to
display displacements on the order of $50\sigma$. The (pore-averaged) self-
diffusion coefficients were calculated by fitting the long-time behavior of
the mean-squared displacement in a periodic direction to the Einstein
relation, e.g., $D_{x}=\lim_{t\rightarrow\infty}{\langle{\Delta
x^{2}}\rangle}/(2t)$. Where appropriate, self-diffusion coefficients in
equivalent periodic directions were averaged together.
## III Surface Roughness and Boundary conditions
We study the hard-sphere fluid confined between parallel hard walls in a slit-
pore geometry. To avoid any confusion, we point out that none of the walls
studied here exhibits internal structure. That is, the walls are of uniform
density and can be regarded as a continuum solid. Let $H$ denote the average
distance between the two surfaces of the confining walls. In this work, we
study two general types of wall surfaces. The first involves flat walls where
the boundary conditions for the particle-wall collisions have been modified.
These walls model surface roughness on a length scale much smaller than the
diameters of fluid particles, and thus, because the surfaces are geometrically
flat, they are referred to as featureless surfaces (walls). The second type of
surface studied here possesses physical (i.e., geometric) roughness, i.e., the
height of the wall surface varies with lateral position, and thus they are
referred to as physically rough surfaces (walls). Schematics of each type of
surface are given in Fig. 1.
### III.1 Featureless Walls
We considered three different types of boundary conditions for particle
collisions with featureless walls. As a useful reference state, we first
considered flat, smooth walls (SW), where the fluid-wall collisions are
specular. The second type of featureless surface studied was a so-called
thermal wall (TW). This model boundary attempts to reproduce particle
collisions with a surface uneven on length scales considerably smaller than
the particle diameter, the essential feature being that pre- and post-wall
collision velocities are uncorrelated. More importantly, the distribution of
post- collision velocities is determined by the temperature of the wall. The
third type of featureless surface gives an alternative approach to generate
uncorrelated pre- and post-wall-collision velocities. We refer to it as a
rotational wall (RW) for reasons that will become apparent below.
In order to distinguish between the featureless walls with different boundary
conditions, consider a particle with a pre-wall-collision velocity
$\mathbf{v}=(v_{x},v_{y},v_{z})$, and a post-wall collision velocity
$\mathbf{v}^{\prime}=(v_{x}^{\prime},v_{y}^{\prime},v_{z}^{\prime})$. For the
cases considered here, the components of $\mathbf{v}^{\prime}$ are as follows.
* •
Smooth walls (SW): $v^{\prime}_{x}=v_{x}$, $v^{\prime}_{y}=v_{y}$, and
$v^{\prime}_{z}=-v_{z}$
* •
Thermal walls (TW): $v_{z}^{\prime}$ is chosen according to the following
distributionTehver et al. (1998) depending on wall temperature
$T_{\mathrm{w}}$ (set equal to fluid temperature $T$):
$\phi_{z}(v_{z}^{\prime})=\frac{m}{k_{B}T_{w}}\ \lvert v_{z}^{\prime}\rvert\
\exp{\left[\frac{-mv_{z}^{\prime 2}}{2k_{B}T_{\mathrm{w}}}\right]}$ (2)
where the sign of $v_{z}^{\prime}$ depends on the location of the wall
collision (i.e., positive for lower wall collisions and negative for upper
wall collisions). $v_{i}^{\prime}$ (where $i=x,y$) is chosen randomly from a
Gaussian distribution:
$\phi_{i}(v_{i}^{\prime})=\sqrt{\frac{m}{2\pi k_{B}T_{\mathrm{w}}}}\
\exp{\left[\frac{-mv_{i}^{\prime 2}}{2k_{B}T_{w}}\right]}$ (3)
* •
Rotational walls (RW): $v^{\prime}_{z}=-v_{z}$, and the lateral components
$v^{\prime}_{x}$ and $v^{\prime}_{y}$ are determined by rotating the
corresponding components of the pre-wall-collision velocity $v_{x}$ and
$v_{y}$ by an angle $\theta$:
$\begin{split}v_{x}^{\prime}&=v_{x}\cos\theta^{\prime}-v_{y}\sin\theta^{\prime}\\\
v_{y}^{\prime}&=v_{x}\sin\theta^{\prime}+v_{y}\cos\theta^{\prime},\end{split}$
(4)
where $\theta^{\prime}=\pm\theta$, with the sign chosen randomly. Note that
$\theta=0$ corresponds to smooth walls, while $\theta=\pi$ corresponds to
bounce-back boundary conditions.
Note that the boundary conditions described above affect the dynamics of the
system only. Because all three of the above walls have the same geometrical
form and produce the same velocity component distributions, they lead to
identical thermodynamic and structural properties Tehver et al. (1998). This
was verified by the simulation results.
We note that the featureless walls described above are not intended to model
any specific physical system. Rather, they are mathematically convenient
course-grained models for surface roughness. For example, the motivation for
the thermal walls goes back to Maxwell Maxwell (1867), who considered
collisions with a highly uneven low density granular surface. Particles that
strike this type of surface undergo a series of collisions with many different
surface molecules. The resulting outgoing velocity is expected to be
randomized, with a distribution determined by the temperature of the wall. We
also stress that the rotational walls introduced here are not intended to
mimic any type of real system. Instead, they should be considered a
mathematical construct that allows us to change the surface boundary condition
in a continuous fashion from perfectly smooth ($\theta=0$) to rough (bounce-
back, $\theta=\pi$) walls, with the system having identical thermodynamics for
all values of $\theta$. This allows us to study how different levels of
surface roughness impact the dynamics of the model fluid without changing
thermodynamics.
### III.2 Physically rough walls
For a physically rough surface, we intuitively expect that the height of the
surface should vary with lateral position. For simplicity, we use the
following expressions for the upper and lower wall surfaces in a slit- pore
geometry:
$\begin{split}f_{\rm{U}}(y)&=\frac{H}{2}+a_{w}\cos\left[\frac{2\pi
y}{\lambda}+\pi\right],\\\
f_{\rm{L}}(y)&=-\frac{H}{2}+a_{w}\cos\left[\frac{2\pi
y}{\lambda}\right]\end{split}$ (5)
where the subscripts $\rm{L}$ and $\rm{U}$ denote the lower and upper
surfaces, respectively, and $a_{w}$ and $\lambda$ are the amplitude and
wavelength of the well-behaved surface variations, respectively. Figure 1b
displays the geometry of the physically rough walls employed. As written in
Eq. (5) and depicted in Figure 1b, minima in the upper wall and maxima in the
lower wall are aligned, which imposes a maximum value upon $a_{w}$ if a fluid
particle is to have access to the entire pore length. The average surface-to-
surface distance is $H$ as long as the quantity $L_{y}/\lambda$ is an integer,
which is true throughout this work. Because we have chosen to have the height
of the surface be a function of only $y$, the system is spatially
inhomogeneous in the $y$-direction and homogeneous in the $x$-direction. This
will have important consequences for the self diffusivity of a fluid confined
between these surfaces.
The rigid character of the confining walls was maintained by requiring
perfectly reflecting specular particle-surface collisions. However, instead of
solving the set of nonlinear equations to determine particle-surface collision
times, which is computationally expensive, the curved walls were discretized
into a set of short, connected line segments of length $0.015\sigma$. Using
smaller segments does not lead to noticeable changes in the resulting
properties of the systems studied here.
We stress that the specific form of the physically rough walls studied here is
not intended to mimic a specific physical system. Rather, they allow us to
systematically determine the impact of different surface feature sizes on the
self-diffusivity of our model fluid.
## IV Results
### IV.1 Featureless Flat Walls
We first present results for the hard-sphere fluid confined between smooth,
flat hard walls. In Fig. 2, we plot the self-diffusion coefficient,
$D^{\rm{SW}}$, where the superscript SW signifies smooth wall case, as a
function of the total density $\rho=N/(AH)$. Data are shown for a number of
pore widths $H$. As previously pointed out elsewhereMittal et al. (2006); Goel
et al. (2009), all data points seem to fall approximately onto a single curve
independently of $H$ for $\rho\sigma^{3}\lesssim 0.75$. Below, we investigate
how modifying the nature of the confining surfaces changes the results
obtained in this basic reference system.
### IV.2 Thermal Walls
Figure 3 displays the self-diffusion coefficient $D^{\rm{TW}}$ for the hard-
sphere fluid confined between featureless thermal walls as a function of
$\rho$ for various values of $H$. Compared to the smooth, flat wall, roughness
due to the thermal wall has a clear and noticeable influence on particle
dynamics. Thermal walls reduce particle mobility, with the magnitude of the
reduction increasing with decreasing $H$. That average mobility of fluid
particles confined between thermal walls decreases with $\rho$ at fixed $H$
has been previously reported.Subramanian and Davis (1979)
The interesting trend we quantify in this study is the reduction of
$D^{\rm{TW}}$ with increasing degree of confinement at fixed $\rho$. This
behavior can be understood by considering the quantity $f_{w}$, the fraction
of collisions in the system involving the surfaces of the confining walls.
Intuitively, we expect non-specular particle-surface collisions, such as those
that take place in the thermal-wall systems, to slow dynamics in the
transverse direction relative to specular particle-surface collisions. We base
this expectation by considering the free flight of a single particle. In this
case, specular particle-wall collisions do not change the transverse motion,
while non-specular particle-wall collisions do. We also expect that the
fraction of wall collisions grows with the prominence of the walls (the
fraction of the fluid particles near the walls), that is, with decreasing $H$.
That is, in order to maintain a fixed $N$ and $V$, a decrease in $H$ must be
compensated by an increase in fluid-exposed wall surface area. Figure 4 shows
that this is indeed the case. Decreasing $H$ at constant $\rho$ systematically
increases the fraction of wall collisions. Clearly, the reduction in
$D^{\rm{TW}}$ with decreasing $H$ observed in Fig. 3 is directly linked to the
increasing fraction of wall collisions. Also, note that the results given in
Fig. 4 are not unique to the thermal-wall systems. In fact, all three kinds of
featureless surfaces studied in this work yield identical wall-collision
statistics. As noted above, the collision boundary conditions only affect the
dynamic properties of the fluid and not the thermodynamic properties. The
thermodynamic pressure of the system is intimately related to the wall-
collision statistics.
While there are clear differences between the smooth-wall and thermal-wall
self-diffusion coefficients, $D^{\rm{SW}}$ and $D^{\rm{TW}}$, respectively,
these differences appear to depend systematically on $H$. This suggests that
it might be possible to develop an approach to predict the self-diffusivity of
the hard-sphere fluid confined between thermal walls using a limited amount of
information. In particular, note the dependence of the fraction of wall
collisions on density. Initially, increasing density leads to a pronounced
decrease in the fraction of wall collisions. This is due to the associated
increase in particle-particle collisions compared to particle-wall collisions
(not shown). However, the fraction of wall collisions eventually becomes a
weak function of density. Since we expect the reduction in self diffusivity
due to the presence of rough walls to scale with the fraction of wall
collisions, we should likewise expect a similar density dependence of the
ratio $D^{\rm{TW}}/D^{\rm{SW}}$.
In Fig. 5a, we plot the quantity $D^{\rm{TW}}/D^{\rm{SW}}$ as a function of
density. The ratio of the diffusivities is taken at the same thermodynamic
state of the fluid, namely at the same density $\rho$ and average wall
separation $H$. Notice that, for each pore width, the ratio of self-diffusion
coefficients initially increases with density before reaching a limiting
value. Furthermore, the behavior of $D^{\rm{TW}}/D^{\rm{SW}}$ vs $\rho$
reflects that of $f_{w}$ vs $\rho$. Again, this points to the strong
connection between the self-diffusion in a system with rough walls to the
prominence of the walls (i.e., the fraction of the fluid near the walls). The
diffusivity ratio takes its smallest value at low density, conditions where
surface collisions are most influential, and thus, where the greatest
difference between the two surfaces is observed. Also, the density at which
the ratio reaches a plateau value ($\rho\sigma^{3}\approx 0.2$) is independent
of pore width. This is due to the dominant influence of particle-particle
collisions on the mobility of hard-spheres at moderate-to-high densities.
The most striking feature of Fig. 5a involves the $H$-dependent plateau values
for $\rho\sigma^{3}\gtrsim 0.2$. In Fig. 5b, we plot all of the data points
for which $\rho\sigma^{3}\gtrsim 0.2$ versus the inverse pore width $H^{-1}$.
Fig. 5b shows that the ratio of diffusivities can be well described in this
density range by a linear function of $H^{-1}$
$\frac{D^{\rm{rough}}}{D^{\rm{ref}}}=1-C\left(\frac{\sigma}{H}\right),$ (6)
where the constant $C$ is a fitting parameter, $D^{\rm{rough}}$ is the self-
diffusivity of the fluid between the rough surface of interest, and
$D^{\rm{ref}}$ is the self diffusivity of the fluid at an appropriately chosen
reference state under the same thermodynamic conditions. We note that Eq. (6)
has, to first order, the same dependence on $H$ as the self-diffusivity of a
Brownian particle in the center of a slit-pore.Happel and Brenner (1983) This,
in fact, was the inspiration for the functional form chosen in Eq. (6). For
the thermal walls, we take $D^{\rm{rough}}=D^{\rm{TW}}$ and
$D^{\rm{ref}}=D^{\rm{SW}}$, and we find that $C\approx 1.44\sigma$.
Assuming sufficient smooth-wall self-diffusivity data are available as a
function of density, this provides the basis for estimating the diffusivity of
the hard-sphere fluid confined between thermal walls for $\rho\sigma^{3}>0.2$.
To estimate the diffusivity of a fluid confined between thermal walls in a
slit pore of width $H$ at density $\rho$, one calculates two quantities. The
first quantity is the $H$-dependent ratio $D^{\rm{TW}}/D^{\rm{SW}}$ using Eq.
6, and the second quantity is the self-diffusion coefficient of the reference
system, $D^{\rm{SW}}$, which, as discussed in Section I, may be approximated
by a scaling analysis. Knowledge of these two quantities then allows for an
estimate of the self diffusivity between thermal walls. In Fig. 6, we test the
predictive ability of this approach. The ratio of the predicted to observed
thermal wall self diffusivity is plotted against the observed self diffusivity
for the thermal-wall system. If the predictions were perfect, the points would
fall on the horizontal line $D^{\rm{TW}}/D^{\rm{SW}}=1$, and this is clearly
not the case. However, the predicted data points do fall within the indicated
$5\%$ error bounds. Moreover, we emphasize that to truly make predictions, one
must still have knowledge of $C$ and the reference self-diffusivity.
### IV.3 Rotational walls
In Figure 7, we present results for the self-diffusion coefficient
$D^{\rm{RW}}$ of the hard-sphere fluid confined between rotational walls as a
function of density $\rho$ with (a) $\theta=\pi/2$ and various values of $H$,
and (b) $H/\sigma=7.0$ and various values of $\theta$. Qualitatively similar
results were observed for other choices of rotational-wall parameters and can
be found in Supplementary Materials.sup Recall that $\theta$ is the angle of
rotation that the lateral velocity of a particle undergoes after it collides
with the surface. Fig. 7a shows that self diffusivity decreases with
increasing density at fixed $H$ and fixed $\theta$. In addition, at fixed
density $\rho$, the self-diffusivity decreases with decreasing $H$. This
latter trend, which was also observed in the the thermal-wall systems, is not
surprising considering rotational walls, like thermal walls, inherently retard
particle mobility, an effect that is most apparent at small wall separations
(at the same fluid density). Therefore, recalling that the collision
statistics are independent of the boundary condition for featureless walls,
the same physics explaining the reduction in mobility due to thermal walls
mentioned in Section IV.2 also applies to rotational walls. We also expect
that the reduction in mobility should increase with $\theta$, since $\theta$
controls the effective roughness of the walls. The data presented Figure 7b
bear this out.
Figures 8a and 8b again show that rotational walls influence self diffusivity
relative to smooth walls in a manner qualitatively similar to thermal walls.
Specifically, the ratio of self diffusion coefficients
$D^{\rm{RW}}/D^{\rm{SW}}$ initially increases with density and then levels off
beyond $\rho\sigma^{3}\gtrsim 0.2$. In fact, for a given $\theta$, the plateau
value appears to be systematically dependent upon $H$, which suggests that an
approach similar to that adopted for the thermal wall system can be used to
estimate the diffusivity in the rotational-wall system. Figure 8b shows that
the ratio $D^{\rm{RW}}/D^{\rm{SW}}$ decreases with increasing $\theta$,
indicating that the constant $C$ in Eq. (6) is dependent upon $\theta$. This
just reflects the $\theta$ dependence noted in Fig. 7b. Accounting for this
$\theta$ dependence, Fig. 8c shows that diffusivity ratios with densities
$\rho\sigma^{3}>0.2$ are well described by Eq. (6). The inset to Fig. 8c
displays the values of $C(\theta)$ from the fits. For simplicity, we fit $C$
to the function $C=2.0sin(\theta/2)$, which as shown in Fig. 8c fits the data
well. specific form of the fit was based on the fact that $C=0$ for $\theta=0$
(by definition), $C$ is a maximum at $\theta=\pi$, and $C$ is periodic in
$\theta$. Using the procedure outlined above for thermal walls, the self-
diffusion of the hard-sphere fluid between rotational walls can be estimated
in an analogous fashion. Figure 9 tests these estimates, plotting the ratio
$D^{\rm{RW}}_{\mathrm{pred}}/D^{\rm{RW}}$ versus $D^{\rm{RW}}$, where
$D^{\rm{RW}}_{\mathrm{pred}}$ is the predicted value and $D^{\rm{RW}}$ is the
value observed from simulation. In the case of rotational walls, the
estimation procedure yields predictions that fall within $10\%$ of the actual
values. More accurate predictions can clearly be obtained by making use of a
more quantitative model for $C(\theta)$.
### IV.4 Physically rough walls
Here, we present results for physically rough surfaces at fixed wavelength
$\lambda/\sigma=3$ and several densities $\rho=N/V$, where $V$ is the surface
accessible volume, while (1) fixing the amplitude of the features
$a_{w}/\sigma=1.0$ and varying $H$ and (2) fixing $H/\sigma=7.5$ and varying
$a_{w}$. Since the surface height of a physically rough wall depends on
lateral position, these parameters (see Eq. 5 and Fig. 1b) allow us to study
the influence of surface height variation on mobility in a systematic way.
Results for $\lambda/\sigma=1.5$, $2.0$, and $6.0$ give results qualitatively
similar to those presented below, and can be found in Supplementary
Materials.sup
Notice that in the geometry of the physically rough wall system studied (see
Fig. 1b), the $x$ and $y$ directions, though both infinite, are not
equivalent. Because the surface height is a function of $y$ only, the $y$
direction is termed rough (inhomogeneous), while the $x$ direction is termed
smooth (homogeneous). Also, because of this height variation, we expect the
self diffusivities in the $x$ and $y$ directions at the same state point to be
significantly different, with $D_{x}>D_{y}$. Figure 10 shows these two self-
diffusion coefficients separately as a function of density for various values
of $H$ and $a_{w}$. Observe that the self diffusivity in the $x$ direction has
little dependence on $H$ or $a_{w}$. That is, the $D_{x}$–$\rho$ correlation
is almost equivalent to the bulk correlation, much like the case for hard
spheres between smooth flat walls (see Fig. 2). This seems consistent with
previous results for hard-spheres in cylindrical pores where the surface is
smooth and exhibits curvature.Goel et al. (2009)
In contrast, the self-diffusion coefficient in the $y$ direction depends
strongly on the degree of confinement (Fig. 10b) and on the amplitude $a_{w}$
of the surface variation (Fig. 10d). Specifically, decreasing $H$ at fixed
wall feature size (constant $a_{w}$ and $\lambda$) systematically decreases
$D_{y}$. This behavior resembles that for the fluid confined between thermal
walls (Fig. 3) and rotational walls (Figs. 7a). Also, at fixed $H$, $D_{y}$
systematically decreases with increasing $a_{w}$ (i.e., increasing surface
roughness). This effect is analogous to increasing surface roughness
($\theta$) in the rotational wall system (see Fig. 7b). Also, the slowing down
of dynamics due to physical roughness is consistent with previous
studies.Sofos et al. (2010)
The disparity between $D_{x}$ and $D_{y}$ grows with decreasing $H$ at fixed
$a_{w}$ and $\lambda$, and with increasing roughness ($a_{w}$) and fixed $H$.
Given the close parallels between $D_{y}$ and the self-diffusion coefficient
of the fluid between featureless rough walls, we expect the reduction in
$D_{y}$ due to surface roughness should be connected to the fraction of
collisions in the system involving the walls. In Figure 11, we plot $f_{w}$
versus density for the state points considered in Fig. 10. Fig. 11a shows that
$f_{w}$ grows with decreasing $H$ at fixed $\rho$ and $a_{w}$, while Fig. 11b
shows that the wall-collision fraction grows with $a_{w}$ at fixed $\rho$ and
$H$. In both cases, $f_{w}$ grows with the increasing prominence of the walls
(caused by decreasing $H$, or increasing $a_{w}$) Also, at fixed wall
conditions (e.g., $a_{w}$ and $\lambda$), $f_{w}$ initially decreases with
increasing density ($\lesssim 0.2$). However, this quantity changes only
moderately with further increases in density ($\gtrsim 0.2$). This behavior is
analogous to that observed in the featureless-wall systems.
Because of the qualitative similarities between the properties of the
physically rough-wall system and the featureless rough-wall systems studied
above, we now examine the diffusivity ratio $D_{y}/D^{\rm{ref}}$. However,
unlike the other surfaces encountered in this work, the choice of
$D^{\rm{ref}}$ is not obvious for the physically rough walls. For the thermal
and rotational walls, we chose the smooth (flat) wall self-diffusion
coefficient at the same $\rho$ and $H$, which corresponds to the self-
diffusion of a fluid at the same thermodynamic state without surface
roughness. Likewise, for physically rough-wall system, the reference state
should be the self-diffusion of a system at the same thermodynamic state, but
without surface roughness. For this, we choose $D^{\rm{ref}}=D_{x}$. In Figure
12a, we plot the ratio $D_{y}/D_{x}$ versus density for various values of $H$
and $a_{w}$. At fixed $H$, the ratio initially increases with $\rho$ and then
levels off beyond $\rho\sigma\gtrsim 0.2$. For a specified value of $\rho$,
the ratio $D_{y}/D_{x}$ decreases with decreasing $H$ at fixed surface
roughness (Fig. 12a) and with increasing roughness at fixed $H$ (Fig. 12b).
Comparison with Figs. 5a and 8 shows that physical roughness alters $D_{y}$
relative to $D_{x}$ in a manner similar to how featureless rough walls alter
the self-diffusion relative to smooth walls. Figure 12c shows that, for given
surface features, $D_{y}/D_{x}$ is a linear function of $H^{-1}$. That is,
$D_{y}$ follows Eq. (6) with $D^{\rm{rough}}=D_{y}$ and $D^{\rm{ref}}=D_{x}$.
Figure 12d shows the fit parameter $C$ versus $a_{w}$ for the different
surface feature wavelengths $\lambda$ studied (see Supplementary materialsup
). Clearly, $C$ is much more sensitive to $a_{w}$ than $\lambda$. We find that
a linear fit to $C$ as a function of $a_{w}$, although crude, describes that
data reasonably well.
From the above analysis, we can now formulate a means to predict the self-
diffusion in the rough $y$ direction from the self diffusion in the smooth $x$
direction. Specifically, from Eq. (6),
$D_{y}^{\mathrm{pred}}=D_{x}[1-C\,(H/\sigma)^{-1}]$. Figure 13 shows the
quality of the prediction based on this formalism for the state points in Fig.
10. In Figs. 13a,LABEL:sub@fig:DY_prediction_exact_h_7p5 we use $C$ from the
fits at a given $\lambda$ (i.e., data points in Fig. 12d), while in Figs.
13b,LABEL:sub@fig:DY_prediction_approx_h_7p5 we use the approximation
$C=2.12a_{w}/\sigma$ (linear fit in Fig. 12d). We find that using the $C$ for
a given wall configuration yields good predictions, with the majority of the
data within $10\%$ of the actual data. On the other hand, using the
approximate $C$ yields predictions within $25\%$ of the actual value. The
information necessary to estimate $D_{y}$ in this case is greatly reduced, but
one would still need knowledge of $D_{x}$ to truly make predictions of
$D_{y}$.
## V Conclusions
We have systematically studied how surface roughness affects the self-
diffusion of confined hard-sphere fluids. Specifically, for
$\rho\sigma^{3}>0.2$, we have shown that the ratio of the self-diffusion
coefficient in a rough wall system to the self-diffusion coefficient in
smooth-wall reference system is a linear function of reciprocal wall
separation [see Eq. (6)], with the precise slope depending on the specific
nature of the surface roughness. If this slope and the reference self-
diffusivity are known, accurate predictions of the self-diffusion in rough
wall systems can be obtained.
In real world applications, however, the specific nature of surface roughness
is often unknown. This study leads to two possible ways of making useful self-
diffusivity estimates without this knowledge. First, our analysis shows that a
only a limited number of measurements are needed to determine $C$. Second, if
only qualitative information is sought, our analysis shows that $C$ is
typically of $O(1)$, and that the basic physics of self-diffusion in a rough
system relative to that in a smooth system is captured by the $H^{-1}$
correction in Eq. (6). That is, the specific nature of the surface roughness
does not appear to impact radically or significantly the resulting dynamics.
This is particularly important since modeling the specific details of real
surfaces is a daunting task. However, it is not our intention to suggest that
any type of surface roughness can be modeled in any way one chooses, for
example, modeling geometric roughness with featureless rough walls. Rather,
the specific details of the surface roughness do not appear to be overly
important.
To make predictions for self-diffusion of confined fluids between rough
surfaces using Eq. (6), one also requires a prediction of the reference state
self-diffusion. As discussed in Section I, one may use a scaling method to map
the bulk system to the confined system. For the case of smooth featureless
walls, for example, this has been shown to be very effective.Goel et al.
(2009) However, for the case of the physically rough system, one must be able
to predict $D_{x}$. We plan to address this in a future publication.
Another question is whether the results presented in this paper apply to more
complex pore scenarios, such as diffusion through porous materials. Such
systems have complex geometry and connectivity, as well as rough surfaces. We
believe that the simplified geometry studied here, slit pores, is applicable,
however, when one considers that many of the properties in such complex
systems can be considered as an average over a collection of simple slit pore
(or cylindrical pore) systems. Lastoskie et al. (1993); Olivier et al. (1994);
Lastoskie et al. (1997); Rouquerol and Rouquerol (1999); Klobes et al. (2006)
The lack of particle-particle or particle-wall attractions is an obvious
shortcoming of this study. Clearly, wall attractions will greatly alter the
number of particles near the wall, and therefore the fraction of wall
collisions. As we point out, the ratio of self diffusivity in a rough-wall
system to that in a corresponding smooth-wall system appears to be closely
linked with the fraction of wall collisions. Therefore, attractions can have a
strong effect on the behavior of the self diffusivity ratio. However, we
speculate that the introduction of moderately strong wall attractions to the
rough systems studied here will result in an increased retardation of self-
diffusion and not a fundamental change to the physics. We also plan to explore
this issue in detail in a future publication.
## Acknowledgements
W.P.K. gratefully acknowledges financial support from a National Research
Council postdoctoral research associateship at the National Institute of
Standards and Technology. T.M.T. acknowledges support of the Welch Foundation
(F-1696) and the National Science Foundation (CBET 1065357). J.R.E.
acknowledges financial support of the National Science Foundation (CBET
0828979). The Biowulf Cluster at the National Institutes of Health provided
computational resources for this paper.
References and Notes
## References
* Bhushan et al. (1995) B. Bhushan, J. N. Israelachvili, and U. Landman, Nature 374, 607 (1995).
* Mittal et al. (2006) J. Mittal, J. R. Errington, and T. M. Truskett, Phys. Rev. Lett. 96, 177804 (2006).
* Mittal et al. (2007a) J. Mittal, J. Errington, and T. Truskett, J. Phys. Chem. B 111, 10054 (2007a).
* Mittal et al. (2007b) J. Mittal, J. R. Errington, and T. M. Truskett, J. Chem. Phys. 126, 244708 (2007b).
* Mittal et al. (2007c) J. Mittal, V. K. Shen, J. R. Errington, and T. M. Truskett, J. Chem. Phys. 127, 154513 (2007c).
* Goel et al. (2008) G. Goel, W. P. Krekelberg, J. R. Errington, and T. M. Truskett, Phys. Rev. Lett. 100, 106001 (2008).
* Mittal et al. (2008) J. Mittal, T. M. Truskett, J. R. Errington, and G. Hummer, Phys. Rev. Lett. 100, 145901 (2008).
* Goel et al. (2009) G. Goel, W. P. Krekelberg, M. J. Pond, J. Mittal, V. K. Shen, J. R. Errington, and T. M. Truskett, J. Stat. Mech. 2009, P04006 (2009).
* Mittal (2009) J. Mittal, J. Phys Chem. B 113, 13800 (2009).
* Chopra et al. (2010) R. Chopra, T. M. Truskett, and J. R. Errington, Phys. Rev. E 82, 041201 (2010).
* Grzelak et al. (2010) E. M. Grzelak, V. K. Shen, and J. R. Errington, Langmuir 26, 8274 (2010).
* Grzelak and Errington (2010) E. M. Grzelak and J. R. Errington, Langmuir 26, 13297 (2010).
* Knudsen (1909) M. Knudsen, Ann. Phys. (Leipzig) 28, 75 (1909).
* Arya et al. (2003) G. Arya, H.-C. Chang, and E. J. Maginn, Phys. Rev. Lett. 91, 026102 (2003).
* Malek and Coppens (2001) K. Malek and M. Coppens, Phys. Rev. Lett. 87, 125505 (2001).
* Malek and Coppens (2002) K. Malek and M. Coppens, Colloid Surface A 206, 335 (2002).
* Malek and Coppens (2003) K. Malek and M.-O. Coppens, J. Chem. Phys. 119, 2801 (2003).
* Edmond et al. (2010) K. V. Edmond, C. R. Nugent, and E. R. Weeks, Eur. Phys. J.-Spec. Top. 189, 83 (2010).
* Scheidler et al. (2002) P. Scheidler, W. Kob, and K. Binder, Europhys. Lett. 59, 701 (2002).
* Scheidler et al. (2004) P. Scheidler, W. Kob, and K. Binder, J. Phys Chem. B 108, 6673 (2004).
* Errington et al. (2003) J. R. Errington, P. G. Debenedetti, and S. Torquato, J. Chem. Phys. 118, 2256 (2003).
* Hansen and McDonald (2006) J. P. Hansen and I. R. McDonald, _Theory of Simple Liquids_ (Academic, London, 2006), 3rd ed.
* Lastoskie et al. (1993) C. Lastoskie, K. E. Gubbins, and N. Quirke, J. Phys. Chem. 97, 4786 (1993).
* Olivier et al. (1994) J. Olivier, W. Conklin, and M. Szombathely, Stud. Surf. Sci. Catal. 87, 81 (1994).
* Lastoskie et al. (1997) C. Lastoskie, N. Quirke, and K. Gubbins, in _Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces_ , edited by W. Rudziński, W. Steele, and G. Zgrablich (Elsevier, 1997), vol. 104 of _Studies in Surface Science and Catalysis_ , pp. 745 – 775.
* Rouquerol and Rouquerol (1999) F. Rouquerol and J. Rouquerol, _Adsorption of Powders and Porous Solids_ (Academic, London, 1999).
* Klobes et al. (2006) P. Klobes, K. Meyer, and R. G. Munro, _Porosity and specific surface area measurements for solid materials._ (Gaithersburg, Md.: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology. http://purl.fdlp.gov/GPO/gpo1230, 2006).
* Rapaport (2004) D. C. Rapaport, _The Art of Molecular Dynamic Simulation_ (Cambridge University Press, Cambridge, 2004), 2nd ed.
* Tehver et al. (1998) R. Tehver, F. Toigo, J. Koplik, and J. R. Banavar, Phys. Rev. E 57, R17 (1998).
* Maxwell (1867) J. C. Maxwell, Philos. Trans. R. Soc. London, Ser. A 170, 231 (1867).
* Subramanian and Davis (1979) G. Subramanian and H. T. Davis, Mol. Phys. 38, 1061 (1979).
* Happel and Brenner (1983) J. Happel and H. Brenner, _Low Reynolds Number Hydrodynamics_ (Kluwer, Dordrecht, 1983).
* (33) See supplementary material at [URL will be inserted by AIP] for additional data sets.
* Sofos et al. (2010) F. Sofos, T. Karakasidis, and A. Liakopoulos, Int. J. Heat Mass Tran. 53, 3839 (2010).
Figure Captions
(a)
(b)
Figure 1: Schematics of the geometries of the confined fluid system, as
described in the text. (a) Featureless walls and (b) physically rough walls.
Figure 2: Self diffusivity $D$ versus total density $\rho=N/(AH)$ for the
hard-sphere fluid confined between smooth walls in slit-pore geometry. Figure
3: Self diffusivity $D$ versus total density $\rho$ for the hard-sphere fluid
confined between thermal walls, for various wall separations $H$. Figure 4:
Fraction of collisions in the system involving the wall surface $f_{w}$ for
the hard-sphere fluid confined to a slit-pore geometry. Symbols have the same
meaning as in Fig. 3
(a)
(b)
Figure 5: Ratio of the self-diffusivity between thermal walls $D^{\rm{TW}}$ to
its value when confined between smooth walls $D^{\rm{SW}}$ at the same $\rho$
and $H$ versus (a) $\rho$ and (b) $H^{-1}$. Symbols in (a) are the same as
those in Fig. 3. In (b), all data is for $\rho\sigma^{3}>0.2$, and dashed line
is a fit to the form $D^{\rm{TW}}/D^{\rm{SW}}=1.0-1.44\sigma/H$. Figure 6:
Ratio of predicted to observed self diffusivity (see text) for the thermal
wall system. Dotted blue and red dashed lines provide $5\%$ and $25\%$ error
bounds, respectively. Symbols are the same as those in Fig. 3.
(a)
(b)
Figure 7: Self diffusivity $D^{\rm{RW}}$ versus density $\rho$ for the hard-
sphere fluid confined between rotational walls at (a) $\theta=\pi/2$ and
various wall separations $H$ and (b) $H/\sigma=7$ and various values of
$\theta$.
(a)
(b)
(c)
Figure 8: Ratio of self diffusivity between rotational walls $D^{\rm{RW}}$ to
that between smooth walls $D^{\rm{SW}}$ at the same $\rho$ and $H$ versus
$\rho$ at (a) $\theta=\pi/2$ and various $H$ and (b) $H/\sigma=7$ and various
$\theta$, and versus (c) $H^{-1}$ at various $\theta$. Symbols in (a) and
(b-c) are the same as those in Figs. 7a and 7b, respectively. In (c), all data
points are for $\rho\sigma^{3}>0.2$; lines are linear fits of the form
$D^{\rm{RW}}/D^{\rm{SW}}=1-C(H/\sigma)^{-1}$; the inset to (c) shows the value
of $C$ as a function of $\theta$ from linear fits in main panel of (c). Figure
9: Ratio of predicted to observed self diffusivity (discussed in text) versus
the observed self diffusivity for the rotational wall system with the value of
$\theta$ denoted in each figure. Dotted blue and red dashed lines provide
$10\%$ and $25\%$ error bounds, respectively. Symbols have the same meaning as
in Fig. 7a.
(a)
(b)
(c)
(d)
Figure 10: Self diffusivity in (a,c) smooth $x$ and (b,d) rough $y$ directions
versus density for the hard-sphere fluid confined between physically rough
walls (see Fig. 1b) with $\lambda/\sigma=3.0$ and (a,b) $a_{w}/\sigma=1.0$ and
various $H$ and (c,d) $H/\sigma=7.5$ and various $a_{w}$.
(a)
(b)
Figure 11: Fraction of particle collisions that involve the wall $f_{w}$ for
the hard-sphere fluid confined between physically rough walls versus density.
(a) $a_{w}/\sigma=1.0$ at various $H$ and (b) $H/\sigma=7.5$ and various
$a_{w}$. Symbols have the same meaning as in Fig. 10.
(a)
(b)
(c)
(d)
Figure 12: Ratio of self diffusivity in rough direction $D_{y}$ and smooth
direction $D_{x}$ with $\lambda/\sigma=3.0$ versus (a,b) $\rho$ and (c)
$H^{-1}$ for the hard-sphere fluid confined between physically rough walls.
Symbols in (a-c) are the same as in Fig. 10. In (c) all data points are for
$\rho\sigma^{3}>0.2$ and lines are fits to the form
$D_{y}/D_{x}=1-C(H/\sigma)^{-1}$. Panel (d) displays the values of $C$ versus
$a_{w}$ at all the wavelengths studied.
(a)
(b)
(c)
(d)
Figure 13: Ratio of predicted $D^{\mathrm{pred}}_{y}$ (see text) to observed
value of $D_{y}$ plotted versus $D_{y}$ for the hard-sphere fluid confined
between physically rough walls with $\lambda/\sigma=3.0$ and (a,b)
$a_{w}/\sigma=1$ and various $H$ and (c,d) $H/\sigma=7.5$ and various $a_{w}$.
Symbols in (a,b) and (c,d) correspond to those in Figs. 10a and 10c,
respectively. In (a,c) we use values of $C$ obtained from the fits in Fig.
12c, while in (b,d) we use $C=2.12a_{w}/\sigma$ (see Fig. 12d). The dotted
green and dashed red lines correspond to $10\%$ and $25\%$ error bounds,
respectively.
### Rotational walls
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 14: Properties for rotational walls. (a-d) Self-diffusion versus
density. (e-h) Ratio of the self-diffusion between rotational walls and
between smooth walls at some $H$ and $\rho$ versus density. Values of $\theta$
are indicted above each figure.
### Physically rough walls
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m) (n) (o) (p)
(q)
Figure 15: Self-diffusion for physically rough system with
$\lambda/\sigma=1.5$. (a-d) Self-diffusion in $x$ direction $D_{x}$, (e-h)
self-diffusion in $y$ direction $D_{y}$, (i-l) $D_{y}/D_{x}$, and (m-p)
fraction of wall collisions $f_{w}$ vs density. (q) $D_{y}/D_{x}$ vs $H^{-1}$
for points with $\rho\sigma^{3}>0.2$.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m) (n) (o) (p)
(q)
Figure 16: Same as Fig. 15 with $\lambda/\sigma=2.0$.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m) (n) (o) (p)
(q)
Figure 17: Same as Fig. 15 with $\lambda/\sigma=3.0$.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m) (n) (o) (p)
(q)
Figure 18: Same as Fig. 15 with $\lambda/\sigma=6.0$.
(a)
(b)
Figure 19: Ratio of predicted value of $D_{y}$ from
$D_{y}/D_{x}=1-C[H/\sigma]^{-1}$ to actual value of $D_{y}$ versus $D_{y}$ for
states with $\rho\sigma^{3}>0.2$. In (a), values of $C$ used in prediction are
taken from the full linear fits shown in subfigure (q) of the above figures.
In (b), the value of $C$ is taken from the linear fit $C=2.12a_{w}/\sigma$
(see main text). The dotted green and dashed red lines correspond to $10\%$
and $25\%$ error bounds, respectively.
|
arxiv-papers
| 2011-08-30T17:58:20 |
2024-09-04T02:49:21.833616
|
{
"license": "Public Domain",
"authors": "William P. Krekelberg, Vincent K. Shen, Jeffrey R. Errington, and\n Thomas M. Truskett",
"submitter": "William Krekelberg",
"url": "https://arxiv.org/abs/1108.6024"
}
|
1108.6094
|
# Dimension Reduction Using Rule Ensemble Machine Learning Methods: A
Numerical Study of Three Ensemble Methods
Orianna DeMasi111ODemasi@lbl.gov, Juan Meza222JCMeza@lbl.gov, David H.
Bailey333DHBailey@lbl.gov
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
###### Abstract
Ensemble methods for supervised machine learning have become popular due to
their ability to accurately predict class labels with groups of simple,
lightweight “base learners.” While ensembles offer computationally efficient
models that have good predictive capability they tend to be large and offer
little insight into the patterns or structure in a dataset. We consider an
ensemble technique that returns a model of ranked rules. The model accurately
predicts class labels and has the advantage of indicating which parameter
constraints are most useful for predicting those labels. An example of the
rule ensemble method successfully ranking rules and selecting attributes is
given with a dataset containing images of potential supernovas where the
number of necessary features is reduced from 39 to 21. We also compare the
rule ensemble method on a set of multi-class problems with boosting and
bagging, which are two well known ensemble techniques that use decision trees
as base learners, but do not have a rule ranking scheme. ††footnotetext: This
research was supported in part by the Director, Office of Computational and
Technology Research, Division of Mathematical, Information, and Computational
Sciences of the U.S. Department of Energy, under contract number DE-
AC02-05CH11231
## 1 Introduction
Machine learning algorithms are popular tools for classifying observations.
These algorithms can attain high classification accuracy for datasets from a
wide variety of applications and with complex behavior. In addition, through
automated parameter tuning, it is possible to grow powerful models that can
successfully predict class affiliations of future observations. A
disadvantage, however, is that models can become overly complicated and, as a
result, hard to interpret and expensive to evaluate for large datasets.
Ideally we would like to generate models that are quick to build, cheap to
evaluate, and that give users insight into the data, similar to how the size
of coefficients in a linear regression model can be used to understand
attribute-response relationships and dependencies.
Ensemble methods are a class of machine learning algorithms that develop
simple and fast algorithms by combining many elementary models, called base
learners, into a larger model. The larger model captures more behavior than
each base learner captures by itself and so collectively the base learners can
model the population and accurately predict class labels [14]. Classical
decision tree ensemble methods, such as bagging and boosting, are well known
and have been tested and refined on many datasets [1, 8, 22]. In one such
study, Banfield et al. [1] studied the accuracy of boosting and bagging on a
variety of public datasets and found that in general neither bagging nor
boosting was a statistically significantly stronger method.
In this paper, we modify, extend, and test an implementation [21] of the rule
ensemble method proposed by Friedman and Popescu [16] for binary
classification with bagging and with boosting. The Friedman and Popescu rule
ensemble method is attractive, as it combines the rule weighting or variable
importance that regression provides with the quick decision tree methods and
collective decision making of many simple base learners. The method builds
rules, that take the form of products of indicator functions defined on
hypercubes in parameter space. The rules are fit by growing decision trees, as
each inner node of a tree takes the desired form of a rule. The method then
performs a penalized regression to combine the rules into a sparse model. The
entire method resembles a linear regression model, but with different terms.
Many ensemble methods provide little insight into what variables are important
to the behavior of the system, but by combining the rules with regression, the
rule ensemble method prunes rules of little utility and ranks remaining rules
in order of importance.
We also modified the rule ensemble method to use various coefficient solving
methods on a set of binary and multi-class problems. Previous implementations
of this algorithm are either currently unavailable [11] or have not been fully
tested on a wide set of problems [7]. We extended the rule ensemble method to
multiple class classification problems with one versus all classification [24]
and tested it on classical machine learning datasets from the UC Irvine
machine learning repository [3]. These datasets were chosen because they have
been used to test previous tree ensembles [1, 8, 18, 22, 29] and countless
other machine learning algorithms. Finally, we look at different methods that
can be used to solve for the coefficients and show how one can use the rule
ensemble method to reduce the dimension of a problem. We give an example of
identifying important attributes in a large scientific dataset by applying our
techniques to a set of images of potential supernova [4].
### 1.1 Overview of Rule Ensemble Method
Suppose we are given a set of data points
$\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{N}$, where $\mathbf{x}_{i}$ denotes the
$i$th observation, with label $y_{i}$. Each of the observations,
$\mathbf{x}\in\mathbb{R}^{\mathcal{K}}$, has $\mathcal{K}$ attributes or
feature values that we measure for each observation. The matrix $\mathbf{X}$
will denote the entire set of all $\mathbf{x}_{i}$’s. The $j$th feature of the
$i$th observation is the scalar $x_{ij}$. Our goal then is to be able to
predict what class $y$ a future unlabeled observation $\mathbf{x}$ belongs to.
The method below focuses specifically on the binary decision problem where $y$
can be one of only two classes $\\{-1,+1\\}$. To classify observations we seek
to construct a function $F(\mathbf{x})$ that maps an observation $\mathbf{x}$
to an output variable $\hat{y}=F(\mathbf{x})$ that predicts the true label
$y$.
Define the risk of using any function that maps observations to labels as
$R(F)=E_{\mathbf{x},y}L(y,F(\mathbf{x})),$ (1)
where $E_{\mathbf{x},y}$ is the expectation operator. $L(y,\hat{y})$ is a
chosen loss function that defines the cost of predicting a label $\hat{y}$ for
an observation when the true label is $y$. While various loss functions have
been developed, in practice we will use the ramp loss function as it is
particularly well suited to the binary classification problem we consider [14,
16]. Within this framework we seek to find a function, $F^{*}(\mathbf{x})$,
that minimizes the risk over all such functions
$F^{*}(\mathbf{x})=\underset{F}{\operatorname{argmin}}\
E_{\mathbf{x},y}L(y,F(\mathbf{x})).$
The optimal $F^{*}(\mathbf{x})$ is defined on the entire population. However,
we only have a training sample of observed data
$\mathcal{S}=\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{N}$ so we will construct an
approximation $\hat{F}(\mathbf{x})$ to $F^{*}(\mathbf{x})$ that minimizes the
expected loss on this training set. We assume that the model
$\hat{F}(\mathbf{x})$ has the form of a linear combination of $K$ base
learners $\\{f_{k}(\mathbf{x})\\}_{k=1}^{K}$:
$\hat{F}(\mathbf{x};\mathbf{a})=a_{0}+\sum_{k=1}^{K}a_{k}f_{k}(\mathbf{x}).$
(2)
The next step is to find coefficients
$\mathbf{a}=\\{a_{1},a_{2},\ldots,a_{K}\\}$ that minimize the risk (1). Like
$F^{*}(\mathbf{x})$, the risk is defined over the entire population, so we
will use the approximation $\mathbf{a}^{*}$ that minimizes the risk over the
given sample set of observations $\mathcal{S}$. In particular, we take
$\mathbf{a}^{*}$ to be the solution of
$\displaystyle\mathbf{a}^{*}$
$\displaystyle=\underset{\\{\mathbf{a}\\}}{\operatorname{argmin}}\
E_{\mathcal{S}}L(y,F(\mathbf{x};\mathbf{a})),$ (5)
$\displaystyle=\underset{\\{\mathbf{a}\\}}{\operatorname{argmin}}\frac{1}{N}\sum_{i=1}^{N}L(y_{i},F(\mathbf{x}_{i};\mathbf{a})),$
$\displaystyle=\underset{\\{\mathbf{a}\\}}{\operatorname{argmin}}\frac{1}{N}\sum_{i=1}^{N}L\left(y_{i},a_{0}+\sum_{k=1}^{K}a_{k}f_{k}(\mathbf{x}_{i})\right).$
If the loss function, $L$, is taken to be the mean squared error then this is
simply a linear regression problem.
In many cases, a solution to equation (5) is not be the best for constructing
a sparse interpretable model or a predictive model that is not overfit to the
training data. Instead, one would like to have a solution that has as few
components as possible. To achieve a sparse solution, a penalty term can be
included that prevents less influential terms from entering the model. Here,
we use the $L^{1}$ (lasso [26]) penalty and the approximation
$\hat{\mathbf{a}}$, which is the solution to the penalized problem
$\mathbf{\hat{a}}=\arg\min_{\\{\mathbf{a}\\}}\sum_{i=1}^{N}L\left(y_{i},a_{0}+\sum_{k=1}^{K}a_{k}f_{k}(\mathbf{x}_{i})\right)+\lambda\sum_{k=1}^{K}|a_{k}|.$
(6)
The impact of the penalty is controlled by the parameter $\lambda\geq 0$. This
penalized problem has received a great deal of attention [9, 13, 17] and
enables both estimation of the coefficients as well as coefficient selection.
This section provided a brief introduction to the methods used in this study
and that were developed by Friedman and Popescu [14, 15, 16]. Other papers
provide more details and and justification of the rule ensemble method [14,
16] as well as the method that is used to assemble the rules in the latter
part of the algorithm [15]. Additional sources also provide more details for
the other algorithms that we employed to compute the coefficients [13, 17,
28].
In section 2, we will discuss how to build base learners $f_{k}$. Section 3
will provide more details on the regression method used to solve equation (6).
Sections 5-6 will present computational results comparing the rule ensemble
method with other ensemble methods.
## 2 Base Learners
The base learners $f_{k}$ in equation (LABEL:eq:Fhat) can be of many different
forms. Decision trees, which have been used alone as classification models,
have been used as base learners in ensemble methods such as bagging, random
forests, and boosting. Decision trees are a natural choice to use for a
learner, as many small trees (meaning each tree has few leaves) can be built
quickly and then combined into a larger model. The bagging method grows many
trees, then combines them with equal weights [5]. Boosting is more
sophisticated as it tries to build the rules in an intelligent manner, but it
still gives each tree an equal weight in the ensemble [10].
### 2.1 Using Rules as Base Learners
In the rule ensemble method, simple rules denoted by $r_{k}$ are used as the
base learners and take the form
$r_{k}(\mathbf{x}_{i})=\prod_{j}I(x_{ij}\in p_{kj}),$ (7)
where $I(x_{ij}\in p_{kj})$ is an indicator function. The indicator function
evaluates to $1$ if the observed attribute value $\mathbf{x}_{ij}$ is in the
parameter space defined by $p_{kj}$, and $0$ if the observation is not in that
space. Each $p_{kj}$ is a constraint that the $k$th rule assigns to the $j$th
attribute. For convenience we will denote $\mathbf{p}_{k}=(p_{k1},\dots)$ to
be the vector of parameter constraints that an observation must meet to have
the $k$th rule to evaluate to $1$. Note that a given rule can have multiple
constraints on a single attribute, as well as a different number of
constraints (indicator functions) than other rules. To emphasize that each
rule is defined by a set of parameters we can write
$r_{k}(\mathbf{x}_{i})=r_{k}(\mathbf{x}_{i};\mathbf{p}_{k})$.
To fit a model we need to generate rules by computing parameter sets
$\\{\mathbf{p}_{k}\\}_{k=1}^{K}$. In this study, we will use decision trees to
generate rules, where each internal and terminal node (not the root node) of a
decision tree takes the form of a simple rule defined by (7). Having
$r_{k}(\mathbf{x}_{i};\mathbf{p}_{k})=1$ means that the $k$th rule is obeyed
by the $i$th observation and that it was sorted into the $k$th node of the
decision tree that generated the rule.
### 2.2 Tree Construction - Rule Generation
Decision trees are built using the CART (Classification and Regression Trees)
algorithm [6], which is summarized Table 1 and outlined below. We let
$\mathcal{T}_{m}=\\{r^{m}_{j}\\}_{j=1}^{2(t_{m}-1)}$
denote the set of rules contained in the $m$th tree which has $t_{m}$ terminal
nodes. Let
$T_{m}(\mathbf{x}_{i})=\displaystyle\sum_{j=1}^{2(t_{m}-1)}r^{m}_{j}(\mathbf{x}_{i},p^{m}_{j}))$
denote the prediction that the $m$th tree makes for observation
$\mathbf{x}_{i}$; it is the evaluation of the rules in $\mathcal{T}_{m}$ on
$\mathbf{x}_{i}$.
Each tree is built on a random subset of observations
$S_{m}(\eta)\subset\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{N}$, as training on the
entire dataset can be expensive as well as overfit the tree. $\eta$ is a
parameter that controls the diversity of the rules by defining the number of
observations chosen to be in the subset $S_{m}(\eta)$. As subset size $\eta$
decreases, diversity increases with potentially less global behavior getting
extracted. Diversity between the trees can also be increased by varying the
final size of each tree. Clearly larger trees include more precise rules
defining terminal nodes and thus are inclined to overtrain, but confining the
size of a tree too strictly can prevent it from capturing more subtle behavior
within the dataset. To avoid under or overfitting, we grow each tree until it
has $t_{m}$ terminal nodes, where $t_{m}$ is drawn from an exponential
distribution. The distribution has mean $\bar{L}$, which does have to be
selected a priori. The size of a tree is determined by growing each branch
until no further nodes can be split because one of the following termination
conditions has been met:
The number of observations in a terminal node is less than some selected
cutoff,
The impurity of a node is less than a selected cut off,
The total number of nodes in the tree is greater than $t_{m}$.
The splitting attribute and value is chosen as the split that minimizes the
sum of the impurities (variance of the node) of the two child nodes if that
split were taken. For each split only a random sample of attributes are
considered in order to both increase the diversity of learners and decrease
training time for huge datasets.
### 2.3 Gradient Boosting
To avoid simply regrowing overlapping rules, with no further predictive
capability, we use gradient boosting to intelligently generate diverse rules.
With gradient boosting, each tree is trained on the pseudo residuals
$\mathbf{\rho}_{m}$ of the risk function evaluated on the test set rather than
training directly on the data [18]. The $i$th element of the pseudo residual
vector in the $m$th iteration is given by
$\mathbf{\rho}_{mi}=-\bigg{[}\frac{\partial
L(y_{i},F(\mathbf{x}_{i}))}{\partial
F(\mathbf{x}_{i})}\bigg{]}_{F(\mathbf{x}_{i})=F_{m-1}(\mathbf{x}_{i})}$ (8)
for all $\mathbf{x}_{i}\in S(\eta)_{m}$. Each $\mathbf{\rho}_{m}$ is a vector
with as many entries as there are observations in the subsample $S_{m}(\eta)$
on which it is evaluated. $F_{m}(\mathbf{x})$ is the memory function at the
$m$th iteration. It gives a label prediction based on all the previous
learners (trees) that were built. Note that $F_{m}(\mathbf{x})$ is an
intermediate model of trees that is used in rule generation, while
$F(\mathbf{x})$ is the final prediction model that has rules as linear terms.
Training on the pseudo residuals allows one to account for what the previous
trees were unable to capture. This method is similar to the method of
regressing on residuals in multidimensional linear regression. Using pseudo
residuals also provides another termination condition. If the pseudo residuals
shrink below a chosen value, enough behavior has been captured and no further
rules are generated. A shrinkage parameter, $0\leq\nu\leq 1$, controls the
dependency of the prediction on the previously built learners. Using $\nu=0$
results in no dependence on past calculations, so that the next rule is built
directly on the data labels and have had no part of the labeled value
“accounted for” by dependence on previous calculations.
Rule Generation Algorithm
---
Input: data $\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{N}$ where
$\mathbf{x}_{i}\in\mathbb{R}^{k}$ and $y_{i}\in\mathbb{R}$
$F_{0}(\mathbf{x}_{i})=\arg\displaystyle\min_{c}\sum_{i=1}^{N}L(y_{i},c);\text{
}c\in\mathbb{R}$
For $m=1\dots M$
select random subset $S_{m}(\eta)\subset\\{\mathbf{x}_{i}\\}_{i=1}^{N}$
select number of terminal nodes $t_{m}\sim\exp(\bar{L})$
calculate pseudo residuals $\mathbf{\rho}_{m}$ with (8)
build tree $\mathcal{T}_{m}$ on $\\{y_{i},\rho_{m,i}\\}$ for all $i\in
S_{m}(\eta)$
update $F_{m}(\mathbf{x}_{i})=F_{m-i}(\mathbf{x}_{i})+\nu T(\mathbf{x}_{i})$
End if $|\rho_{m}|_{\infty}$ small enough
total rules $\geq$ max
Return: rules = $\\{\text{internal nodes of }\mathcal{T}_{m}\\}_{m=1}^{M}$
Figure 1: Outline of how to generate rules
## 3 Weighting Rules
To combine the rules into a linear model, we need to approximate the
coefficients $\hat{\mathbf{a}}$ defined in equation (6). Here we implement a
method that approximates $\hat{\mathbf{a}}$ with an accelerated gradient
descent method developed by Friedman and Popescu [15] and summarized in Figure
2. We will refer to this method as Pathbuild, as it does not solve (6)
explicitly, but rather constructs $\hat{\mathbf{a}}$ by starting with a null
solution and then incrementing along a constrained gradient descent path,
distinguished by a parameter $\tau$. Alternative algorithms for approximating
$\hat{\mathbf{a}}$ will be discussed and compared later.
We would like find a value for the lasso penalty that yields the sparsest
solution to (6) while maintaining a model with high accuracy. We initialize
the coefficients to $0$ and find the constant intercept $a_{0}$ by the value
that minimizes
$a_{0}=\arg\min_{\alpha}\sum_{i=1}^{N}L(y_{i},\alpha).$
This may be better understood by considering that $\alpha$ will be the mean of
$(y_{1},\dots,y_{N})$ when the loss is mean squared error. We approximate
$\hat{\mathbf{a}}$ iteratively and calculate the $l+1$st iteration,
$\mathbf{a}^{l+1}$, by taking
$g_{k}(\mathbf{X};\mathbf{a})=\displaystyle\frac{\partial}{\partial
a_{k}}\frac{1}{N}\sum_{i=1}^{N}L(y_{i},F(\mathbf{x}_{i};\mathbf{a})),$
$k^{*}=\\{k:g_{k}(\mathbf{X};\mathbf{a}^{\ell})=||g_{k}(\mathbf{X};\mathbf{a}^{\ell})||_{\infty}\\}.$
We update the coefficients by
$a_{k}^{\ell+1}=\begin{cases}a_{k}^{\ell},&\mathrm{if}\,k=1..K,k\notin
k^{*}\\\ a_{k}^{\ell}+\delta
g_{k}(\mathbf{X};\mathbf{a}^{\ell})&\mathrm{if}\,k\in k^{*},\\\ \end{cases}$
(9)
where $g_{k}(\mathbf{X};\mathbf{a}^{\ell})$ is the gradient of $F(\mathbf{x})$
calculated with the $\ell$th iteration and evaluated on the entire dataset.
The scaling parameter $\delta>0$ can be set constant or chosen at each step in
a clever manner. Note that in equation (9) only a single component of the
coefficient vector is updated at any iteration and thus only a single rule is
able to enter the model at an iteration. The method only progresses in the
direction of rules which have a large effect on the predictive capability and
avoids steps that are of trivial effect. This condition may be relaxed by
incrementing all of the components of $\mathbf{a}$ that have a sufficiently
large gradient
$k^{*}=\\{k:g_{k}(\mathbf{x};\mathbf{a}^{\ell})\geq\tau||g_{k}(\mathbf{x};\mathbf{a}^{\ell})||_{\infty}\\}.$
The parameter $\tau\in[0,1]$ controls how large a component of the gradient
must be relative to the largest component in order for a coefficient to be
updated. Computing the gradient is expensive, but reorganizations and
intelligent approximations to accelerate the computation are presented for
three different loss functions in the appendix [15]. The tricks used for this
“fast” method are most effective for ramp loss and make Pathbuild a
particularly attractive method.
In sections 6.2-6.4 we will compare Pathbuild with three different algorithms
that can be used to solve for the coefficients. Each algorithm uses a slightly
different formulation of the problem defined in equation (6) and a different
technique to encourage a sparse solution that also has little risk. The three
algorithms also use mean squared error to define loss rather than the ramp
loss function that we use in Pathbuild.
Pathbuild: Gradient Regularized Descent Algorithm
---
set constant $a_{0}=\displaystyle\min_{\alpha}\sum_{i=1}^{N}L(y_{i},\alpha)$
$a^{0}_{k}=0,\text{ }k=1,\dots,K$
For $\ell=1,\dots,$ max iterations
$g_{k}(\mathbf{x};\mathbf{a}^{\ell})=\displaystyle\frac{\partial}{\partial
a_{k}}\frac{1}{N}\sum_{i=1}^{N}L(y_{i},F(\mathbf{x}_{i};\mathbf{a}^{\ell}))$
$a_{k}^{\ell+1}=\begin{cases}a_{k}^{\ell},&\mathrm{if}\,g_{k}(\mathbf{x};\mathbf{a}^{\ell})\geq\displaystyle|g_{k}^{\ell}|_{\infty}\\\
a_{k}^{\ell}+\delta
g_{k}(\mathbf{x};\mathbf{a}^{\ell})&\mathrm{otherwise}\end{cases}$
Stop if risk increased
$l>$ max iterations
gradient $<$ tolerance
Figure 2: Outline of Pathbuild method
## 4 Datasets and Methods for Experiments
To test the behavior of the rule ensemble method on a binary classification
problem, we used a dataset of images taken by a telescope [4, 20, 23], the
goal being to identify potential supernovas. The initial data had three images
for each observation. Those images were processed to yield 39 statistics for
each observation that described the distribution and color of individual
pixels within the original three images. These statistics became the
attributes for the dataset and the observations were labeled with +1, -1 if
they were or were not, respectively, an image of a supernova-like object. The
dataset contains a total of 5,000 positive and 19,988 negative observations.
To test how the rule ensemble works on the binary classification problem, we
use a procedure that first randomly selects 2,500 positive observations and
2,500 negative observations for a training set, and then uses the remaining
data for the testing set. This selection process is repeated 10 times for
cross-validation. False positive and false negative error rates were used to
assess the accuracy of the methods in addition to the overall error rate. The
false positive rate is the ratio of observations misclassified as positive to
the total number of negative observations in the test set, while the false
negative rate is the ratio of observations misclassified as negative to the
number of positive observations in the test set. The overall error rate is the
ratio of observations misclassified to the total number of observations in the
test set. The experiments show the effect of the rule complexity (tree depth),
number of rules available (tree size), and $\tau$ thresholding in Pathbuild on
the accuracy of the method. We also consider the effect of substituting
different coefficient solvers in place of Pathbuild.
To assess the overall utility of the rule ensemble we extend our numerical
experiments to multi-class problems, which are described in section 5. We
compare the rule ensemble with classical bagging and boosting methods by
testing all three algorithms on 10 datasets from the UC Irvine Machine
Learning Data Repository [3] with five 2-fold cross-validation tests. A 2-fold
cross-validation test is similar to the method described above except that the
dataset is split into equally sized subsets with the proportion of
observations in each class the same in both subsets. Then one set is used for
training and the other for testing, and then the sets are switched and
retrained and retested. The datasets are briefly described in Table 1. The UC
Irvine sets are chosen since they have been used in many machine learning
studies [8, 22] and are used by Banfield et al. [1] to compare bagging with
boosting. The UC Irvine sets are taken from a wide variety of applications, so
they also present a good breadth of data to test the versatility of methods.
Set | Name | Attributes | Observations | Classes
---|---|---|---|---
1 | breast-w | 9 | 699 | 2
2 | glass | 9 | 214 | 7
3 | ion | 34 | 351 | 2
4 | iris | 4 | 150 | 3
5 | pendigits | 16 | 10992 | 10
6 | phoneme | 5 | 5404 | 2
7 | pima | 8 | 768 | 2
8 | sonar | 60 | 208 | 2
9 | vehicle | 18 | 846 | 4
10 | waveform | 21 | 5000 | 3
Table 1: Description of UC Irvine datasets used to compare ensemble methods on
multi-class problems.
Experiments using the rule ensemble method were run using Matlab™7.10 on a
MacBook Pro with a 2.66 GHz Intel Core i7 processor.
## 5 Multiple Class Classification Results
The rule ensemble method is designed for binary classification problems, but
many datasets contain multiple classes that one needs to identify. To be
applicable to classification in general, we need to extend the rule ensemble
to many class problems. Decision trees easily extend to multiple classes but
the regression performed to assemble the rules in the rule ensemble prevent
the rule ensemble from being extended to classification problems where the
classes are not ordered. To identify multiple classes with the rule ensemble
method we use the one-versus-all (OVA) classification technique that has been
used for successfully extending many binary classification algorithms into
multi-class algorithms [19, 25]. Other methods for extending binary
classification algorithms to multiple class problems exist, such as all-
versus-all classification. However, these methods require a large number of
models to be built and are thus more expensive than OVA and frequently provide
no more utility than the OVA classification method [24].
For a problem with $J$ classes, OVA classification performs $J$ binary tests,
where the $j$th test checks if an observation is a member of the $j$th class
or not the $j$th. Each observation gets a vector label prediction
$\hat{\mathbf{y}}\in\mathbb{R}^{J}$, where each entry $\hat{y}_{j}$ is from
the binary test classifying the $j$th class versus any other class. The
prediction $\hat{\mathbf{y}}$ is a vector of -1’s with a single positive
entry. The index of the positive entry is the class that the observation is
predicted to be from.
To extend the rule ensemble method we perform $J$ binary tests and each test
returns a real valued prediction $F_{j}$. In the binary problem the label
$\hat{y}_{j}$ is predicted to be the sign of the real value returned. However,
in this setting it is possible that one of the binary models will misclassify
the observation and result in $F_{j}$ being positive for more than one value
of $j$. If we just took the sign of each $F_{j}$ then we would have a vector
$\hat{\mathbf{y}}$ with multiple positive entries, indicating the observation
was in multiple classes. In the event that $F_{j}$ is positive for more than
one value of $j$, we take the prediction to be the class that has the most
definitive prediction, i.e. the class $j^{*}$ where $F_{j^{*}}$ is greater
than any other class label prediction. Choosing the largest label prediction
is sensible, since the more confident the algorithm is that the observation is
in a certain class, the closer to $1$ the label prediction will be. The closer
to $0$ a class prediction is, the less certain the algorithm is of the
observation’s class affinity.
Here we compare the rule ensemble method, using Pathbuild, with results from
bagging and boosting tree ensemble methods. To compare we employ 10 datasets
from the UC Irvine data repository [3] and the testing method parameters
previously used to compare various ensemble methods [1]. Bagging uses 1000
trees, boosting uses 50 and both employ random forests for growing trees in
five 2-fold cross validations. Tree ensemble labels can be estimated by a
voting procedure, the prediction is the class that most of the trees predict
the observation to be part of, and an averaging procedure, the label is the
average of the the predictions made by all the trees. Results for both methods
are presented. Minimal tuning was used to run the rule ensemble method on
different datasets.
### 5.1 Results Using OVA Classification on Vehicle Dataset
Figure 3 compares using the rule ensemble and bagging on the vehicle dataset.
Bagging here is used in an OVA classification scheme rather than in its
standard, direct multiple classification method. The error at predicting any
given label in the set is shown. As can be seen in Figure 3, the rule ensemble
beats bagging for the majority of the classes. Figure 3 also shows the varying
level of success that the ensemble techniques had at predicting each class.
Some classes are easier to identify than others (e.g. “opel” is easier to
distinguish than van). Different ensembles were better suited to one class
versus another, and which ensemble was better for a class was not consistent
for all classes in a dataset.
Figure 3: Binary tests on each class of vehicle data. Figure shows accuracy
when using bagging in an OVA classification method rather than with multi-
class decision trees.
### 5.2 Results Using OVA Classification on All Datasets
The results of the multiple class tests are given in Table 2. The rule
ensemble is much stronger than the tree ensembles if averaging of each tree’s
label prediction is used for classification. However, if the trees vote on
which class label is best, then the rule ensemble is better on some datasets
but not others. Voting clearly was better at label prediction than averaging
base learner predictions, but neither boosting nor bagging provided a
universal win over the rule ensemble, as can be seen in Figure 4. What is not
apparent in Table 2 is that the rule ensemble was a much better predictor for
binary labels than the tree ensembles. This result is apparent in Figure 3
where nearly every individual class is better predicted by the rule ensemble
method. Figure 5 shows the accuracy of the rule ensemble method with different
coefficient solvers. Some datasets are easier to classify (larger percent of
data correctly classified) while others, such as the #2 dataset glass, were
more difficult to classify for all the methods.
| | Rule Ensemble | Bagging | Boosting
---|---|---|---|---
Name | # Classes | Pathbuild | Fpc | SPGL1 | Voting | Average | Voting | Average
breast-w | 2 | 4.34 | 4.60 | 4.75 | 3.51 | 5.97 | 3.34 | 10.93
glass | 7 | 37.99 | 33.47 | 35.26 | 26.92 | 39.16 | 29.83 | 54.72
ion | 2 | 9.97 | 10.43 | 9.23 | 7.01 | 13.44 | 7.01 | 24.49
iris | 3 | 4.80 | 4.27 | 5.33 | 4.93 | 5.51 | 5.73 | 5.95
pendigits | 10 | 6.94 | 5.65 | 6.10 | 1.23 | 7.05 | 0.87 | 25.68
phoneme | 2 | 14.97 | 14.33 | 14.16 | 12.06 | 16.97 | 10.81 | 26.61
pima | 2 | 24.45 | 25.76 | 24.56 | 23.65 | 30.13 | 26.22 | 38.78
sonar | 2 | 22.76 | 21.14 | 20.67 | 23.82 | 33.62 | 23.74 | 39.70
vehicle | 4 | 28.35 | 26.69 | 27.63 | 26.24 | 34.05 | 25.18 | 46.36
waveform | 3 | 15.50 | 15.79 | 16.03 | 15.67 | 26.30 | 16.61 | 35.26
Number of wins | 1 | 1 | 1 | 3 | 0 | 5 | 0
Table 2: Error rate of the rule ensemble method compared with that of bagging
and boosting. Error rate is given as the percent of observations in the test
set that were misclassified.
Figure 4: Comparison of the error rate from a model that was generated with
the rule ensemble method with the error rates from models that were generated
with boosting and bagging ensemble methods. Results are summarized in Table 2.
Figure 5: Comparison of the error rate on 10 different datasets from models
that were built with the rule ensemble method using different solvers are used
to find the coefficients $\mathbf{a}$.
## 6 Binary Classification Results
### 6.1 Rule Ensemble with Pathbuild
Our implementation of the algorithm Pathbuild for approximating the rule
coefficients in the rule ensemble method is described in Figure 2. The
coefficients are found by solving equation (5) with a constrained gradient
descent method. In this method, each iteration only advances in directions
where the components of the gradient have magnitude greater than some fraction
$\tau\in[0,1]$ of the absolute value of the largest gradient component. Note
that the set of directions we advance in,
$\\{k:|g_{k}(\mathbf{X};\mathbf{a}^{\ell})|\geq\tau*||g_{k}(\mathbf{X};\mathbf{a}^{\ell})||_{\infty}\\},$
can change at every iteration. By not advancing in directions that have little
change in the risk function, the expense of updating coefficients for
variables of little importance is avoided. Not updating rules of little
importance prevents the coefficient value for that rule from “stepping” off
zero, so that variable is effectively kept out of the model, allowing for a
simpler model. Lower values of $\tau$ should include more rules in the model.
The most inclusive model is when $\tau=0$, which is equivalent to using a
basic gradient descent method to get a standard regression. Larger values of
$\tau$ decrease the total number of rules used in the model. The most
constrained model occurs when $\tau=1$.
Effect of Number of Rules and Tree Size
In Figure 6 we see how the size of the trees and the number of rules used for
the model affect the accuracy of the model. The decision trees are used to
generate rules. Larger decision trees yield more complex rules than small
trees because large trees have nodes that are deeper. Nodes deep in a tree
capture subtle interactions within the training data since they depend on more
splits and are more complex than nodes that are closer to the root node.
Figure 6 shows that ensembles built with larger trees have higher error rates
than ensembles that use smaller trees. The increase in error rate when larger
trees are built shows that when the model uses more complex rules, the model
overfits the training data. However, the size of the trees does not have a
strong effect on the how large of an error rate the rule ensemble has.
Further, the accuracy of the rule ensemble is highly variable and the variance
increases when larger trees are built. Ensembles built with trees that average
40 leaves had 4-7% error, which is a large range when one considers that the
mean classification error is only about 5.5%. This error is larger than and
has more variance than the error when trees with an average of 5 leaves are
built, which is 3-4.2% error. It is not clear why there is so much variance in
the error rate in general. One should recall that the average number of
terminal nodes in the decision trees are exponentially distributed, only the
mean of the distribution is changed, so there is a variety of sizes of trees
in each ensemble and complexity between rules in each ensemble. Because there
is a variety of tree sizes there is some stability in the error rate as the
mean size of the trees is changed.
The bottom of Figure 6 also shows that using more rules can decrease the mean
error rate of the rule ensemble method as well as the variance in the error
rate. Increasing the number of rules built from 100 to 600 allowed the
ensemble to capture more behavior and, as a result, nearly halved the error
rate of the method. However, the error rate only decreases down to a certain
point, after which adding more rules does not improve the model. For our data
set, the error decreases to under 5.0% when 600 rules are built, but does not
continue to decrease substantially when more than 600 rules are used. We also
see that the error rates between ensembles that are built on more rules have
less variance than the error rates from ensembles that are built out of fewer
rules. This result is reasonable, as having more rules gives the ensemble a
better chance of finding good rules that successfully separate the data into
classes.
Figure 6: The figure at the top shows that growing large trees (complex rules)
increases the error rate. The bottom figure was made by growing trees with an
average of 50 terminal nodes and shows that ensembles that have more rules
have lower error rates. Tests were run with 500 maximum rules in each model.
The $\tau$ tolerance was 0.5. Asterisks indicate the mean error rate from
multiple tests.
In the initial tree building phase, a subsample of data is selected and a tree
is grown on each random subsample. Our initial experiments took subsamples of
2,500 observations ($25\%$ of the total number of observations in the training
set). When we decreased the subsample size to 500 observations ($10\%$ of
training set), error rates did not significantly change even for a variety of
tree sizes that had between 5 and 80 terminal nodes. The lack of significant
difference indicates that 500 observations give us a large enough sample to
catch the same amount of behavior that is captured when larger subsamples of
data are used to build each tree.
Effect of Using Rules Versus Linear Terms
In Figure 7 we see the effect of allowing the model to have linear
dependencies on individual features. If only linear terms are used, then the
model is a standard multiple linear regression. Allowing the model to be built
with both linear terms and the rules generated by the trees yields a mixed
model. Using rules for the regression terms provides a clear advantage over
the standard regression model by reducing the error rate from nearly 30% error
to less than 5%. The linear regression is also more biased in its error than
the rule ensemble. This bias can be seen by the false negative rate being
close to zero; this means nearly all the error is caused by mislabeling
observations with negative labels. We would not expect a linear regression to
capture any of the complex nonlinear behavior in the dataset, and the error
rates show that such an conjecture is correct – rules are needed to get
significant predictive capability.
Figure 7: Using rules in an ensemble was six times more accurate than only
using linear terms, a classical multiple linear regression of the labels on
the attribute variables. Linear regression was not reliable for predicting the
labels, but using the rule ensemble allowed for only 5% error in prediction.
This experiment was run using Pathbuild to solve for coefficients.
Effect of Using the $\tau$ Threshold as Penalty
The variable $\tau$ controls how many directions are updated at each iteration
of Pathbuild in the thresholded gradient descent method. The results of
increasing $\tau$ are shown in Figure 8. The model becomes less accurate and
the variance of the error rate increases, as $\tau$ increases. An increase in
$\tau$ causes a higher threshold that results in fewer terms being included in
each iteration of the coefficient finding method and a ensemble model that is
less accurate. It is interesting to note that within a certain range,
decreasing $\tau$ further does not offer much increase in the predictive
capability of the model. In this example, we see that when $\tau$ is between 0
and 0.3 there isn’t a large increase in error rate. This indicates that using
a weaker threshold of $\tau=0.3$ or even $\tau=0.4$ will not significantly
compromise the accuracy of our model. This is a good result, as using a larger
threshold decreases the computational expense of each iteration of the
gradient descent method. The result that $\tau=0.3$ produces similar error
rates to using $\tau=0$ means that we can get the same accuracy with less
computation.
Figure 8: Error rate increases as $\tau$ increases and restricts the number of
coordinates the algorithm advances in at each iteration. This experiment was
run with each tree having an average of 20 terminal nodes and 600 maximum
rules.
### 6.2 Rule Ensemble with Glmnet
In this experiment we use the Glmnet package [13], which returns
approximations to solutions of elastic-net regularized general linear models,
to solve for the coefficients $\mathbf{a}$ within the rule ensemble method.
Glmnet approximates a solution to the least squared error regression subject
to an elastic net penalty, which is
$\underset{\mathbf{a}\in\mathbb{R}^{n}}{\min}\frac{1}{N}||\mathbf{X}\mathbf{a}-\mathbf{y}||_{2}+\lambda
P_{\alpha}(\mathbf{a}),$ (10)
with a coordinate-wise gradient descent method [13]. The elastic net is
defined as
$P_{\alpha}(x)=\alpha||\mathbf{a}||_{1}+(1-\alpha)||\mathbf{a}||_{2}^{2}$
for $\alpha\in[0,1]$. When $\alpha=0$ the problem is referred to as ridge
regression, and when we set $\alpha=1$ we get the same problem as in equation
(6). The coordinate-wise gradient descent method starts with the null
solution, similar to Pathbuild, then cycles over the coefficients and uses
partial residuals and a soft-thresholding operator to update each coefficient
one by one [12]. Glmnet has some modifications that also allow some parameters
to be associated with and updated at the same time as neighboring parameters.
The null solution corresponds to solving equation (10) with $\lambda=\infty$.
As the coefficients are updated, $\lambda$ is decreased exponentially until
the lower bound $\lambda_{min}$, the desired and pre-specified penalty weight,
is met. Glmnet calculates a set of coefficients along each increment of the
path $\lambda=\infty$ to $\lambda=\lambda_{min}$ and uses the previous
solution as a “warm start” to approximate the next solution. Note that
$\lambda_{min}$ should be small enough to prevent the penalty from being so
large that it causes the vector to be overly sparse. However, $\lambda_{min}$
should also be positive and large enough to ensure a sparse solution that is
robust to the training data. A robust solution includes terms for interactions
that are inherent to the application generating the data, not interactions
that are only figments the subset selected for training. It is not clear how
to pick the penalty weight $\lambda$ to maintain sparsity of the solution and
prevent overfitting while also capturing enough characteristics of the
dataset.
Figure 9: Error rate decreases when Glmnet is used to solve for the
coefficients and the constraint parameter $\lambda$ decreases.
Here we use the rules generated in the previous experiment with Glmnet and
build models using the coefficients that are generated at each step of the
path $\lambda\in[\lambda_{min},\infty]$. Figure 9 shows how the accuracy of
the method changes as the weight of the penalty used to find the coefficients
changes. The solution with Glmnet when $\lambda$ is small results in slightly
less error than the solution with Pathbuild when $\tau$ is small. The variance
in the error rates from solutions found with Pathbuild is less than the
variance of error rates from solutions found with Glmnet. Both solutions yield
false positive rates that are more than twice as large as the false negative
rates; this is probably a result of the ratio of positive to negative
observations in the test set is small. The error rate slowly decreases as
$\lambda$ decreases, but then the error rate stabilizes when $\lambda$ is very
small, $<0.01$. It is interesting that the variance in error rates of the
solutions is relatively constant as $\lambda$ changes.
### 6.3 Rule Ensemble with Spgl1
In this experiment, we used the Spgl1 (sparse projected-gradient $l$1)
Matlab™package [27] to solve for the coefficients $\mathbf{a}$ in
$\min||\mathbf{X}\mathbf{a}-\mathbf{y}||_{2}\text{ subject to
}||\mathbf{a}||_{1}<=\sigma.$ (11)
At each iteration of the algorithm, a convex optimization problem is
constructed, whose solution yields derivative information that can be used by
a Newton-based root-finding algorithm [28]. Each iteration of the Spgl1 method
has an outer/inner iteration structure, where each outer iteration first
computes an approximation to $\sigma$. The inner iteration then uses a
spectral gradient-projection method to approximately minimize a least-squares
problem with an explicit one-norm constraint specified by $\sigma$. Some
advantages of the Spgl1 method are that only matrix-vector operations are
required and numerical experience has shown that it scales well to large
problems.
The results using Spgl1 are shown in Figure 10. The accuracy of the Spgl1
solution increases when $\sigma$ increases. The error rates are similar to
those found by Pathbuild and Glmnet, but slightly higher than Glmnet even when
$\sigma$ is large.
Figure 10: Error rate decreases when Spgl1 is used to solve for the
coefficients and the constraint parameter $\sigma$ increases.
### 6.4 Rule Ensemble with Fpc
In this experiment, we used a fixed point continuation method (Fpc) [17] that
approximates the solution $\mathbf{a}$ in
$\underset{\mathbf{a}\in\mathbb{R}^{n}}{\min}\text{
}||\mathbf{a}||_{1}+\frac{\mu}{2}*||\mathbf{Xa}-\mathbf{y}||_{2}^{2}.$ (12)
This problem formulation seeks to minimize the weighted sum of the norm of the
coefficients and the error of the solution, the left and right terms
respectively. The sparsity of $\mathbf{a}$ is controlled by the size of the
weighting parameter $\mu$. Increasing $\mu$ places more importance on
minimizing the error, and reduces the ratio of the penalty to the error. The
reduction of penalty importance allows more coefficients to become non-zero
(the $\ell_{1}$ norm of the coefficients to increase) and thus find a closer
fit to the problem. Equation (12) is simply a reformulation of problem (6)
with the lasso penalty, and is referred to as a basis pursuit problem in
signal processing. The relation of the two problems can clearly be seen if,
for any $\lambda$ value, $\mu$ is chosen to be
$\mu=\frac{2}{N\lambda}$
and equation (12) is multiplied by $\lambda$. Fpc was developed for
compressing signals by extracting the central components of the signal.
Fpc exploits the properties of the $l_{2}$ norm and declares three equivalent
conditions for reaching an optimal solution. Fpc uses the reformulations of
the optimality conditions to declare a shrinkage operator $s_{\nu}$, where
$\nu$ is a shrinkage parameter that has both an effect on the speed of
convergence and how many non-zero entries $\mathbf{a}^{*}$ has. The operator
$s_{\nu}$ acts on a supplied initial value $\mathbf{a}^{0}$ (which we chose to
be the null solution) and finds our solution $\mathbf{a}^{*}$ through a fixed
point iteration
$\mathbf{a}^{*}=s_{\nu}(\mathbf{a}^{*}).$
The given condition for the threshold of $s_{\nu}$ is
$\text{if }\nu-|\mathbf{y}|>0\text{ then }s_{\nu}(\mathbf{y})\rightarrow 0.$
Fpc forms a path of solutions that starts with $\mu$ initialized to
$\mu=\frac{\eta}{||X^{t}\mathbf{y}||_{\infty}}$ (where $\eta$ is a ratio of
possible optimal square error at the next step to the square error at the
current step). The parameter $\mu$ is altered at each step, which forces the
shrinkage parameter to expand and contract but the upper bound for $\mu$ is
supplied by the user. All results presented here use Fpc with projected
gradient steps and optionally using a variant of Barzilai-Borwein steps [2].
The results of solutions generated by Fpc are shown in Figure 11. They are
roughly as accurate as the solutions generated with the previous solvers. Fpc
also has an explicit display of the thresholding as seen in Figure 13; the
norm of the coefficients increases dramatically then asymptotically approaches
a certain value. The asymptotic behavior is caused by the threshold
constricting the coefficients and essentially preventing another coefficient
from stepping off of zero. The thresholding is also seen in the error rate
decreases as the weight on the mean squared error is increased, but stabilizes
once the training set is reasonably fit. The value of $\mu$ where the error
stabilizes is the value needed to build the model, but unfortunately it is not
clear how to choose this value of $\mu$ a priori. The need for a selection of
the penalty parameter is one of the difficulties that Fpc, Spgl1, and Glmnet
have. Pathbuild shares a similar problem with the need to selection the
gradient descent constriction parameter $\tau$.
Figure 11: Error rate decreases when Fpc is used and the weight on the risk or
mean squared error is increased ($\mu$ increased). Figure 12: There is little
fluctuation in the overall error rate when Fpc is used on rules that were
built with different size trees. Only the mean of cross validation tests is
plotted here for simplicity. Little fluctuation implies that simpler rules,
which come from smaller trees, can be used to build a model without
sacrificing predictive capability. Figure 13: Sparsity of solution is
indicated by the number of coefficients not equal to zero. As $\mu$ is
increased, the solution becomes less penalized and more coefficients step off
zero and allow more terms to be included in the model. The sparsity of the
solution stops decreasing when $\mu$ is large and the penalty is relatively
small compared to the emphasis on minimizing the risk or second term in
equation (12). Here 78% of the coefficients are trivial when $\mu=0.19$.
### 6.5 Identifying Important Attributes Via Rule Importance
Figure 11 shows that the rule ensemble method is quite successful at correctly
classifying observations when all of the attributes are used to generate rules
and build the model. Attributes have variable importance in the model and we
suspect that not all of the 39 attributes in the full dataset are needed to
model and correctly predict class labels. We want to use the rule ensemble
method to select only the attributes that are important and save the expense
of considering the other less important variables.
The importance of a rule is indicated by the magnitude of the coefficient for
that rule. The larger a coefficient is in magnitude, the more important the
corresponding rule is, since that rule will have a larger contribution to the
model. To sift out the most important attributes, we look at which rules Fpc
considered important at different values of $\mu$. Rules are ordered by the
magnitude of their corresponding coefficient and the rules corresponding to
the 20 largest (in magnitude) coefficients are selected. An example of
ordering the rules is in Table 3 where the 5 most important rules from one
test are ordered. This process is continued for 5 different repetitions of
training and testing, which yields 5 sets of 20 most important rules. The sets
of rules are decomposed into sets of attributes that are used to make up the
rules in each set. Then we let the 5 repetitions vote on which attributes are
influential and keep only attributes that are in the set of important
attributes for at least 3 out of the 5 repetitions. Figure 14 shows how many
votes the highest ranking rules get and indicates that certain rules are
important in all solutions while others are considered important in only some
solutions. This set of attributes forms a smaller subset of the 39 attributes
available in the initial dataset. The subset of rules only contains attributes
that are used in at least one of the 20 most important rules in at least 3 of
the 5 repetitions.
The importance of a rule is indicated by the magnitude of the coefficient for
that rule. The larger a coefficient is in magnitude, the more important the
corresponding rule is, as that rule will have a larger contribution to the
model. To sift out the most important attributes, we look at which rules Fpc
considered important at different values of $\mu$. Rules are ordered by the
magnitude of their corresponding coefficient and if a rule is one of the top
20 most important in a solution generated with a certain $\mu$ (13 values of
$\mu$ we considered), then that rule receives a vote. An example of ordering
the rules is in Table 3 where the 5 most important rules from one test with a
given $\mu$ are ordered. Figure 14 shows for how many values of $\mu$ each
rule was considered to be in the top 20 most important; this indicates that
certain rules are important in solutions with all values of $\mu$ tried while
others are considered important only when certain $\mu$ are used. This process
is continued for 5 different cross-validation sets, which yields 5 sets of
rules that were in the top 20 most important rules for at least one value of
$\mu$. The sets of rules are decomposed into sets of the attributes that were
used to make up the rules in each set. Then we let the 5 repetitions vote on
which attributes are needed to make the most influential rules and keep only
the attributes that are in the set of important attributes for at least 3 out
of the 5 repetitions. This set of attributes forms a smaller subset of the
total attributes available in the initial dataset; it is the subset attributes
that are used in at least one of the most important rules in at least 3 of the
5 repetitions.
For the supernova dataset, the smaller subset of attributes included only 21
of the 39 original attributes. Tests were repeated using the same cross-
validation sets and method parameters as were used in Figure 11, but using
only the smaller subset of 21 attributes to train on rather than all 39
attributes. Figure 15 compares the error rate of the method when 21 attributes
were used with the error rate of the method when all 39 attributes were used.
The results show that the accuracy of the method improves when we reduce the
number of attributes used in the model. The method successfully ranks rules
and identifies more important attributes. The method loses accuracy when the
less important features are included; in essence, the extra attributes act as
noise. After the method identifies these attributes as less important and we
remove them, the method is able to return an even more accurate model and the
insight of which attributes are not adding predictive capability to the model.
Garnering better accuracy with fewer attributes may allow the extra attributes
to be excluded from the data collection, which will save time in collecting
data, save space in storing data, and allow an overall better analysis.
Figure 14: This histogram shows how many times a rule was one of the top 20 most important rules in a solution. A solution was generated at each of 13 different values of $\mu$, as shown in Figure 11. Rules that received 13 votes were one of 20 most influential rules for every value of $\mu$ tried. Only rules that were in the top 20 most influential for at least one solution are shown. The attributes that were used in the rules shown here were used to find a smaller subset of attributes to train on for the results in Figure 15. Rule $r_{k}$ | $|a_{k}|$
---|---
$x_{2}\geq$ -0.315 & $x_{18}\geq$ 0.047 | 0.1045
$x_{29}<$-0.251 | 0.0725
$x_{23}\geq$ -0.606 | 0.0317
$x_{1}<$ -0.324 | 0.0274
$x_{12}\geq$ 0.260 | 0.0193
Table 3: Example of ordering rules by importance. These are the five rules
with greatest importance in the first model as selected by Fpc with
$\mu=0.25$. Figure 15: Comparison of overall error rate when fewer attributes
are used. The preliminary tests used all 39 attributes in the dataset. The
subsequent tests used only the subset of 21 attributes that were used to
construct the most important rules in the preliminary tests. Using the
restricted set of attributes gives a lower error rate indicating that the rule
ensemble method successfully identified which attributes are important in the
dataset.
## 7 Conclusions
Figure 16: Length of bars indicate the magnitude of coefficients as calculated
by different solvers. Only the coefficients with the 10 largest magnitudes
from each solver are displayed. Coefficients plotted come from solutions that
yielded similar error rates: $\tau=0.4$ , $\mu=.11$, $\sigma=8.5$,
$\lambda=0.014$.
We compared several variations of a rule ensemble method with some well-known
tree ensemble methods, namely boosting and bagging, on a variety of multi-
class problems. We extended the rule ensemble to work on multi-class problem
by using the OVA technique and found that with this extension the rule
ensemble method performed comparably to the tree methods on a set of 10
classical datasets. This result highlights the power of the rule ensemble
method, as we had expected the tree ensemble methods to do better on multi-
class problems. Tree ensembles can use multi-class decision trees, which
provide what one would think is a more natural extension to multi-class
problems than using the OVA method. However, the rule ensemble method returned
comparable rates of accuracy on most datasets and even performed better on
some of the datasets. The discrepancy between the tree ensembles with voting
and the rule ensemble was larger on problems that had a relatively large
number of labels, such as the pendigits dataset, which had the most labels out
of all the datasets, than on datasets with fewer labels. To improve the
accuracy of the rule ensemble on problems with many classes, we would like to
try using multi-class decision trees to build the rules and then relabel the
nodes for each binary problem. This technique might yield better rules as it
would allow for differentiation between the classes in the rule building
phase. Better rules would then allow for a clearer separation of binary labels
in the regression phase. This technique would also make the training phase
more efficient as it would only require one set of rules to be constructed
rather the as many sets of rules as there are classes.
We also looked at using 4 different methods to find coefficients to assemble
the rules. All 4 methods present the challenge of needing to select a
constraint parameter that controls the sparsity/accuracy trade-off of the
solution that they return. If each parameter is chosen correctly then the
methods are capable of producing coefficients that allow for similar accuracy
in the model. The different approaches that the methods take for finding the
coefficients do result in slightly different rankings of the rules. The
difference in coefficients that each method considers important is shown in
Figure 16. Ideally all solvers would select the same terms to be the most
significant and would order the terms by importance the same way. Figure 16
shows that some rules that one method considers important are not considered
to be important to another method. Fpc and Spgl1 order coefficients similarly,
which is indicated by Spgl1 giving a significant magnitude to coefficients
that Fpc also gives a significant magnitude to. Glmnet’s and Pathbuild’s
ordering share less similarity with Fpc and Spgl1 as indicated by coefficients
such as 9 and 18 that Glmnet and Pathbuild give a significant magnitude to,
but both Fpc and Spgl1 give trivial values to. The difference in methods is
also reflected in the sparsity of the solutions that they return. To achieve
similar accuracy (taken here at 96% accuracy) Pathbuild returns a solution
with 40-50% of the coefficients non-zero while the other methods return much
sparser solutions that have only 12-19% of the coefficients non-zero. In
general, Spgl1 returned the sparsest solutions and Pathbuild returned the
least sparse solutions for models with similar error rates.
As a final step, we showed the utility of the rule ensemble method for
identifying important attributes in a dataset containing images of potential
supernovas. The rule ensemble method has the benefit over tree methods of
providing insight into a dataset by returning weighted rules. Rules with large
weights have a larger effect on the model and thus can be thought of as more
important than other rules. We used the importance of such rules to alert us
to the more significant features in the dataset by looking at which features
the important rules are defined on. This technique allowed us to select 21
attributes out of the 39 available and reduce the error rate of the model by
building models only on the reduced set of attributes. Traditional algorithms
that use ensembles of decision trees, such as boosting and bagging, aren’t
able to provide this insight into the importance of certain variables of a
dataset because they do not rank or weight of rules.
The rule ensemble method has the advantage over some other methods by being
able to identify relationships and hierarchies between variables to a certain
extent when building the decision trees. The rules in the decision trees get
more complex the deeper the tree is grown and also are able to have limited
support in the parameter space, so they only affect certain observations that
fall in that space. By including more variables, complex rules can be seen as
resembling discrete correlations, and the post-processing of the rules allows
for overly simplified correlations (that precede more complex rules in depth)
to be removed from the model. The post-processing also allows for overly
complex rules to be pruned from the model. Thus some variable interactions can
be captured by the rule ensemble method without any a priori assumption that
they exist, as is needed in standard regression models, and excessive
computation is not spent considering correlations that do not exist.
We do not compare the computational efficiency of the rule ensemble method
with tree ensemble methods here, since it is currently written in Matlab™,
while the tree ensemble methods used are written in C. However, we do not
expect that the rule ensemble method will reduce the amount of time necessary
for the training portion of the algorithm to run because it must perform the
coefficient solving method in addition to the tree growing. If the rule
ensemble method is able to prune a substantial number of repetitive or
unnecessary rules, then it is likely to run substantially more quickly than
the tree methods. Comparing the time efficiency of the rule ensemble with
other tree methods and other machine learning techniques will be part of
future work. We do not present the computational efficiency of the coefficient
solving methods used in the rule ensemble method for the same reason. Each
solver is written in a different programming language, and each will have to
be implemented in the same language and level of optimization before a
meaningful study can be performed.
## Acknowledgements
We would like to thank Sean Peisert and Peter Nugent for their valuable
comments and suggestions.
Here we discuss the gradient method Pathbuild, which is described in section
3, in greater detail. Simplifications of the gradient method are presented and
considered as the “fast method”.
## Appendix A Derivation of the Negative Gradient of Risk
The negative gradient $\mathbf{g}\in\mathbb{R}^{K}$ of the loss on the
observations is found by taking partial derivatives of the sum of the loss on
each observation with respect to each coefficient. The components of the
negative gradient are given by
$g_{k}=-\frac{\partial}{\partial{a_{k}}}\frac{1}{N}\sum_{i=1}^{N}L(y_{i},F(\mathbf{x}_{i})),$
(13)
where $k=1,\dots,K$. Note that $g_{0}=0$ as $a_{0}$ is the constant intercept
that minimizes the risk when $F(\mathbf{x}_{i})=a_{0}$ and all the other
coefficients have not moved off their initial zero value.
$\\{g_{k},\,k=1..K\\}$ are the non trivial components of the gradients.
$\frac{\partial{}}{\partial{}a_{k}}L(y_{i},F(\mathbf{x}_{i}))=\frac{\partial{}L(y_{i},F(\mathbf{x}_{i}))}{\partial{}F(\mathbf{x}_{i})}\frac{\partial{}F(\mathbf{x}_{i})}{\partial{}a_{k}}.$
(14)
Note that the second term is easily computed from the linear form of
$F(\mathbf{x}_{i})$ and is given by
$\frac{\partial{}F(\mathbf{x}_{i})}{\partial{}a_{k}}=x_{ik}.$ (15)
### A.1 Negative gradient squared error ramp loss is used
The previous discussion has been generalized for the use of any loss function
$L(\cdot)$. Now consider the case when the loss function is given by
$\begin{array}[]{c}L(y_{i},F(\mathbf{x}_{i}))=[y_{i}-H(F(\mathbf{x}_{i}))]^{2}\\\
H(F(\mathbf{x}_{i}))=\max[-1,\min(1,F(\mathbf{x}_{i}))],\end{array}$ (16)
which is the squared error ramp loss for the $i$-th observation. We want to
find the derivative with respect to $\mathbf{a}$ for this loss function. Begin
by taking a partial derivative with respect to $F$
$\frac{\partial{}}{\partial{}F}L(y_{i},F(\mathbf{x}_{i}))=-2(y_{i}-F(\mathbf{x}_{i}))\,I(|F(\mathbf{x}_{i})|<1).$
(17)
Substitute (15) and (DrampDF) into (14) to get the derivative for the squared
error ramp loss
$\frac{\partial{}}{\partial{}a_{k}}L(y_{i},F(\mathbf{x}_{i}))=-2(y_{i}-F(\mathbf{x}_{i}))\,x_{ik}\,I(|F(\mathbf{x}_{i})|<1).$
(18)
Using the form of $F(\mathbf{x}_{i})$ in the partial derivative (18) and then
substituting into (13), we get the gradient for the risk using the squared
error ramp loss function
$\begin{array}[]{r
l}g_{k}&=\displaystyle\frac{2}{N}\sum_{i=1}^{N}\,\left(y_{i}-a_{0}-\sum_{j=1}^{K}\,a_{j}\,x_{ij}\right)x_{ik}\,I(|F(\mathbf{x}_{i})|<1).\\\
\\\ \end{array}$ (19)
Rearranging, switching the order of summation, and evaluating at the $\ell$-th
step $\mathbf{a}^{\ell}$ in the approximation of $\mathbf{a}$ we can write the
gradient at the $\ell$-th step as
$\begin{array}[]{r
l}g_{k}^{\ell}=&\displaystyle\frac{2}{N}\sum_{i=1}^{N}\,y_{i}\,x_{ik}\,I(|F^{\ell}(\mathbf{x}_{i})|<1)\\\
\\\ &\displaystyle-
a_{0}\frac{2}{N}\sum_{i=1}^{N}\,x_{ik}\,I(|F^{\ell}(\mathbf{x}_{i})|<1)\\\ \\\
&\displaystyle-\sum_{j=1}^{K}\,a_{j}^{\ell}\left[\frac{2}{N}\sum_{i=1}^{N}\,x_{ij}\,x_{ik}\,I(|F^{\ell}(\mathbf{x}_{i})|<1)\right].\end{array}$
(20)
### A.2 Negative gradient with auxiliary functions $v$, $u$
We need to keep track of the dependencies and update properly at each
iteration. The goal of the method is to update the coefficients $\mathbf{a}$.
We take a step with respect to $\mathbf{a}$ and then update everything, so let
$\mathbf{a}$ act as the independent variable. Recall that $i$ is the index
over the observations so $x_{i}$ is the attribute values for the $i$-th
observation and $F_{i}$ is the predicted value for that observation. This
leaves us with
$\begin{array}[]{l}F^{\ell}(\mathbf{x}_{i})=a_{0}+\displaystyle\sum_{k=1}^{K}a^{\ell}_{k}\,x_{ik}=F_{i}(a^{\ell}).\end{array}$
Defining the indicators $v_{i}^{\ell}$
$\begin{array}[]{l}v^{\ell}_{i}=v_{i}(a^{\ell})=I(|F_{i}(a^{\ell})|<1),\\\ \\\
\end{array}$
we can define a new function by
$u^{\ell}(p,q)=u(v^{\ell};p,q)=\frac{2}{N}\sum_{i=1}^{N}\,p_{i}\,q_{i}\,v_{i}^{\ell}$
(21)
where $p$ and $q$ are scalars and
$v_{i}^{\ell}=I(|F^{\ell}(\mathbf{x}_{i})|<1)$. Using the two functions $v,u$
the negative gradient at the $\ell$-th step (20) can be written in a simpler
form
$g_{k}^{\ell}=u^{\ell}(v^{\ell};y,x_{k})-a_{0}\,u^{\ell}(v^{\ell};1,x_{k})-\sum_{j=1}^{K}\,a_{j}^{\ell}\,u^{\ell}(v^{\ell};x_{j},x_{k}).$
(22)
## Appendix B Fast Algorithm
To “step” we move proportional to the largest component of the negative
gradient (13). Let $g^{\ell}_{j^{*}}$ be the largest absolute component of the
gradient
$j^{*}=\arg\max_{1\leq j\leq K}|g_{j}^{\ell}|$
at the $\ell$-th step. Then call the length of the next step
$\delta_{*}=\Delta_{\nu}\,g_{j^{*}}^{\ell}$ and update the coefficients with
$a_{j^{*}}^{\ell+1}=a_{j^{*}}^{\ell}+\delta_{*}$. The coefficients at the
$(\ell+1)$-th step are
$\mathbf{a}^{\ell+1}=\begin{cases}a_{j}^{\ell},&\mathrm{if}\,j=1..K,j\neq
j^{*}\\\ a_{j^{*}}^{\ell}+\delta_{*}&\mathrm{if}\,j=j^{*}.\\\ \end{cases}$
(23)
After a step the gradient must be recomputed before another step can be taken.
Rather than fully recomputing an update can be applied only to the components
of the gradient that are affected by the step. There are two cases of how the
update to the gradient can be made. One update occurs when the step in the
coefficients has caused indicator functions to change; this update requires
more work and is expensive. The other update is cheap and is given as follows.
### B.1 Case when indicators do not change
The step size $\Delta_{\nu}$ should be small; in practice it is taken to be
0.01. The idea is that with a small stepsize $|F(\mathbf{x}_{i})|$ will not
exceed 1 “often.” On the steps where this is true the indicators do not change
so $v^{\ell},u(v^{\ell};y,x_{k}),u(v^{\ell};x_{j},x_{k})$ do not change and
the negative gradient at the $(\ell+1)$-th step is found by substituting (23)
into (22)
$\begin{array}[]{r c l}g_{k}^{\ell+1}=&\displaystyle
u(v^{\ell};y,x_{k})-a_{0}\,u^{\ell}(v^{\ell};1,x_{k})-\sum_{j=1}^{K}\,a_{j}^{\ell}\,u(v^{\ell};x_{j},x_{k})&-\delta_{*}\,u(v^{\ell};x_{j^{*}},x_{k})\\\
=&\displaystyle
g_{k}^{\ell}&-\delta_{*}\,u(v^{\ell};x_{j^{*}},x_{k}).\end{array}$ (24)
### B.2 Case when indicators change - adjustments
If the assumption fails and the indicators change on a step, then
$v^{\ell}\neq v^{\ell+1}$ $u^{\ell}(v^{\ell})\neq u^{\ell+1}(v^{\ell})$ and
(24) does not hold. To find $g^{\ell+1}$, consider the cases of how $v$ can
change and and define the variable
$z_{i}^{\ell}=\begin{cases}-1,&\mathrm{if}\;v_{i}^{\ell}=1\;\text{and}\;v_{i}^{\ell+1}=0\\\
\;\;\,0,&\mathrm{if}\;v_{i}^{\ell}=v_{i}^{\ell+1}\\\
+1,&\mathrm{if}\;v_{i}^{\ell}=0\;\text{and}\;v_{i}^{\ell+1}=1.\end{cases}$
(25)
$z_{n}$ can be thought of adding in observations where the indicators have
turned on and subtracting observations where indicators have turned off. Using
$z_{n}$, $u$ can be adjusted
$\begin{array}[]{r l
l}u(v^{\ell+1};y,x_{k})=&u(v^{\ell};y,x_{k})&+\displaystyle\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}\,y_{n}\,x_{nk}\\\
u(v^{\ell+1};x_{j},x_{k})=&u(v^{\ell};x_{j},x_{k})&+\displaystyle\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}\,x_{nj}\,x_{nk}\\\
u(v^{\ell+1};1,x_{k})=&u^{\ell}(v^{\ell};1,x_{k})&+\displaystyle\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}\,x_{nk}\\\ \end{array}$ (26)
and used with (23) and (19) to find the $\ell$-th update of the negative
gradient
$\begin{array}[]{r c c l}g_{k}^{\ell+1}=&\displaystyle
u(v^{\ell+1};y,x_{k})&-a_{0}\,u(v^{\ell+1};1,x_{k})&-\displaystyle\sum_{j=1}^{K}\,a_{j}^{\ell+1}\,u(v^{\ell+1};x_{j},x_{k})\\\
=&\displaystyle
u(v^{\ell};y,x_{k})&-a_{0}\,u(v^{\ell};1,x_{k})&-\displaystyle\sum_{j=1}^{K}\,a_{j}^{\ell}\,u(v^{\ell+1};x_{j},x_{k})\\\
&&+\displaystyle\frac{2}{N}\sum_{z_{n}\neq 0}z_{n}y_{n}x_{nk}&\displaystyle-
a_{0}\,\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}x_{nk}-\delta_{*}\,u(v^{\ell+1};x_{j^{*}},x_{k})\end{array}$ (27)
With a little more rearrangement the update to the gradient as
$\begin{array}[]{r l l}g_{k}^{\ell+1}=&g_{k}^{\ell}&\text{adjust for obs. with
changed $I(|F_{i}|<1)$}\\\ \\\ &+\displaystyle\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}y_{n}x_{nk}&\text{update $u^{l}(y,x_{k})$ from indicator change}\\\
&-\displaystyle a_{0}\,\frac{2}{N}\sum_{z_{n}\neq 0}z_{n}x_{nk}&\text{update
$u(v^{\ell+1};1,x_{k})$ from indicator change}\\\
&-\displaystyle\delta_{*}u(v^{\ell}x_{j^{*}},x_{k})&\text{step in $j^{*}$
direction}\\\ \\\ &-\displaystyle\delta_{*}\frac{2}{N}\sum_{z_{n}\neq
0}z_{n}x_{nj^{*}}x_{nk}&\text{terms not included in update due to old
}v^{\ell}\\\
&-\displaystyle\frac{2}{N}\sum_{j=1}^{K}\,a_{j}^{\ell}\sum_{z_{n}\neq
0}z_{n}x_{nj^{*}}x_{nk}&\text{adjust for observations with changed
$I(|F_{i}|<1).$}\end{array}$ (28)
## References
* [1] R.E. Banfield, L.O. Hall, K.W. Bowyer, and D. Bhadoria. A comparison of ensemble creation techniques. Lecture Notes in Computer Science, 3077:223–32, 2004.
* [2] J. Barzilai and J. Borwein. Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8:141–8, 1988.
* [3] C.L. Blake and C.J. Merz. UCI Repository of Machine Learning Databases. UC Irvine, 1998. http://www.ics.uci.edu/?mlearn/MLRepository.html.
* [4] JS Bloom, JW Richards, PE Nugent, RM Quimby, MM Kasliwal, DL Starr, D. Poznanski, EO Ofek, SB Cenko, NR Butler, et al. Automating discovery and classification of transients and variable stars in the synoptic survey era. Arxiv preprint arXiv:1106.5491, 2011. http://arxiv.org/abs/1106.5491.
* [5] L. Breiman. Bagging predictors. Machine learning, 24(2):123–40, 1996.
* [6] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classification and regression trees. Chapman & Hall/CRC, 1998.
* [7] R. Brun and F. Rademakers. Root - an object oriented data analysis framework. Nucl. Inst. & Meth. in Phys. Res., pages 81–6, 1997. http://root.cern.ch/.
* [8] T.G. Dietterich. An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization. Machine Learning, 4(2):139–57, 2000.
* [9] D. Donoho, I. Johnstone, G. Kerkuacharian, and D. Picard. Wavelet shrinkage; asymptotia? (with discussion). Journal Royal Statistical Society, 57(2):201–37, 1995.
* [10] Y. Freund and R.E. Schapire. Experiments with a new boosting algorithm. In Machine Learning- International Workshop then Conference, pages 148–56, 1996.
* [11] J.H. Friedman. RuleFit with R. Technical report, Department of Statistics Stanford University, 2005. http://www-stat.stanford.edu/~jhf/R-RuleFit.html.
* [12] J.H. Friedman, T. Hastie, H. Höfling, and R. Tibshirani. Pathwise coordinate optimization. Annals ofApplied Statistics, 1(2):302–32, 2007.
* [13] J.H. Friedman, T. Hastie, and R. Tibshirani. Regularization paths for generalized linear models via coordinate descent. Technical report, Department of Statistic Stanford University, 2008. http://www-stat.stanford.edu/~jhf/ftp/glmnet.pdf.
* [14] J.H. Friedman and B.E. Popescu. Importance sampled learning ensembles. Technical report, Department of Statistics Stanford University, 2003. http://www-stat.stanford.edu/~jhf/ftp/isle.pdf.
* [15] J.H. Friedman and B.E. Popescu. Gradient directed regularization. Technical report, Department of Statistics Stanford University, 2004. http://www-stat.stanford.edu/~jhf/ftp/pathlite.pdf.
* [16] J.H. Friedman and B.E. Popescu. Predictive learning via rule ensembles. Annals ofApplied Statistics, 2(3):916–54, 2008.
* [17] E. Hale, W. Yin, and Y. Zhang. A fixed-point continuation method for $l_{1}$-regularized minimization with applications to compressed sensing. Technical report, Rice University CAAM, 2007. http://www.caam.rice.edu/~zhang/reports/tr0707.pdf.
* [18] T. Hastie, R. Tibshirani, and J.H. Friedman. Elements of Statistical Learning. Springer, 2001.
* [19] C.W. Hsu and C.J. Lin. A comparison of methods for multiclass support vector machines. Neural Networks, IEEE Transactions on, 13(2):415–25, 2002.
* [20] N.M. Law and S.R. Kulkarni et al. The Palomar Transient Factory: System Overview, Performance, and First Results. Publications of the Astronomical Society of the Pacific, 121:1395–408, 2009. http://adsabs.harvard.edu/abs/2009PASP..121.1395L.
* [21] J. Meza and M. Woods. A numerical comparison of rule ensemble methods and support vector machines. Technical report, Lawrence Berkeley National Laboratory, 2009.
* [22] J. Quinlan. Bagging, boosting, and C4.5. Proceedings Thirteenth American Association for Artificial Intelligence National Conference on Artificial Intelligence, pages 725–30, 1996\.
* [23] A. Rau and S.R. Kulkarni et al. Exploring the Optical Transient Sky with the Palomar Transient Factory. Publications of the Astronomical Society of the Pacific, 121:1334–51, 2009. http://adsabs.harvard.edu/abs/2009PASP..121.1334R.
* [24] R. Rifkin and A. Klautau. In defense of one-vs-all classification. The Journal of Machine Learning Research, 5:101–41, 2004.
* [25] A.C. Tan, D. Gilbert, and Y. Deville. Multi-class protein fold classification using a new ensemble machine learning approach. Genome informatics series, pages 206–17, 2003.
* [26] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, 58(1):267–88, 1996.
* [27] E. van den Berg and M. P. Friedlander. SPGL1: A solver for large-scale sparse reconstruction, 2007. http://www.cs.ubc.ca/labs/scl/spgl1.
* [28] E. van den Berg and M. P. Friedlander. Probing the pareto frontier for basis pursuit solutions. SIAM Journal on Scientific Computing, 31(2):890–912, 2008. http://link.aip.org/link/?SCE/31/890.
* [29] I.H. Witten and E. Frank. Data Mining: Practical machine learning tools and techniques. Morgan Kaufmann Pub, 2005.
|
arxiv-papers
| 2011-08-30T22:36:15 |
2024-09-04T02:49:21.841808
|
{
"license": "Public Domain",
"authors": "Orianna DeMasi, Juan Meza, David H. Bailey",
"submitter": "David Bailey",
"url": "https://arxiv.org/abs/1108.6094"
}
|
1108.6229
|
Oriented unicyclic graphs with extremal skew energy
Yaoping Hou, Xiaoling Shen and Chongyan Zhang
Department of Mathematics, Hunan Normal University
Changsha, Hunan 410081, China
email: yphou@hunnu.edu.cn
Abstract
Let $\overrightarrow{G}$ be an oriented graph of order $n$ and
$\lambda_{1},\lambda_{2},\cdots,\lambda_{n}$ denote all the eigenvalues of the
skew-adjacency matrix of $\overrightarrow{G}.$ The skew energy
$\displaystyle{\cal E}_{s}(\overrightarrow{G})=\sum_{i=1}^{n}|\lambda_{i}|.$
In this paper, the oriented unicyclic graphs with minimal and maximal skew
energy are determined.
_Keywords: Unicyclic graph; oriented graph; skew-adjacency matrix, skew
energy_
AMS Classification 05C50, 15A03
## 1 Introduction
An important quantum–chemical characteristic of a conjugated molecule is its
total $\pi$-electron energy. The energy of a graph has closed links to
chemistry. Since the concept of the energy of simple undirected graphs was
introduced by Gutman in [5], there have been lots of research papers on this
topic. For the extremal energy of unicyclic graph, Hou [9] showed that
$S^{3}_{n}$ is the graph with minimal energy in all unicyclic graphs; In [7],
Huo and Li showed that $P^{6}_{n}$ is the graph with maximal energy in all
unicyclic graphs. More results on the energy of unicyclic graphs see [6, 8,
10, 12, 15, 16].
There are various generalizations of energy to graph matrices [13], such as
Laplacian energy, incidence energy and distance energy. Let $G$ be a simple
undirected finite graph of order $n$ and $\overrightarrow{G}$ be an
orientation of $G,$ which assigns to each edge of $G$ a direction so that
$\overrightarrow{G}$ becomes a directed graph. All digraphs in this paper are
oriented graphs of some graphs. Let $\overrightarrow{G}$ be an oriented graph
of order $n$. The skew adjacency matrix $S(\overrightarrow{G})=(s_{i,j})$ is a
real skew symmetric matrix, where $s_{i,j}=1$ and $s_{j,i}=-1$ if
$i\rightarrow j$ is an arc of $\overrightarrow{G},$ otherwise
$s_{i,j}=s_{j,i}=0.$ The skew spectrum $Sp(\overrightarrow{G})$ of
$\overrightarrow{G}$ is defined as the spectrum of $S(\overrightarrow{G}).$
Note that $Sp(\overrightarrow{G})$ consists of only purely imaginary
eigenvalues because $S(\overrightarrow{G})$ is real skew symmetric.
Recently, the skew-energy of an oriented graph $\overrightarrow{G}$ was
defined as the energy of matrix $S(\overrightarrow{G})$ in [2], that is,
${\cal E}_{s}(\overrightarrow{G})=\sum_{\lambda\in
Sp(\overrightarrow{G})}|\lambda|.$
There are situations when chemists use digraphs rather than graphs. One such
situation is when vertices represent distinct chemical species and arcs
represent the direction in which a particular reaction takes place between the
two corresponding species. It is possible that the skew energy has similar
applications as energy in chemistry. For a graph $G$, there are any
orientations on it, it is also interesting to find what orientation has
extremal energy among all orientations of a given graph.
An unicyclic graph is the connected graph with the same number of vertices and
edges. In this paper, we are interested in studying the orientations of
unicyclic graphs with extremal skew energy. Let $G(n,\ell)$ be the set of all
connected unicyclic graphs on $n$ vertices with girth $\ell$. Denote, as
usual, the $n$-vertex path and cycle by $P_{n}$ and $C_{n}$, respectively. Let
$P^{\ell}_{n}$ be the unicyclic graph obtained by connecting a vertex of
$C_{\ell}$ with a terminal vertex of $P_{n-\ell}$, $S_{n}^{\ell}$ be the graph
obtained by connecting $n-\ell$ pendant vertices to a vertex of $C_{\ell}$
(see Fig. $1$).
The rest of the paper is organized as follows. In section $2$, a new integral
formal for ${\cal E}_{s}(\overrightarrow{G})$ is obtained and the oriented
unicyclic graph with minimal skew energy is determined. In section $3$, the
oriented unicyclic graph with maximal skew energy is determined.
## 2 Oriented unicyclic graph with minimal skew energy
Let $G$ be a graph. A _linear subgraph_ $L$ of $G$ is a disjoint union of some
edges and some cycles in $G$ ([3]). A _$k$ -matching_ $M$ in $G$ is a disjoint
union of $k$-edges. If $2k$ is the order of $G$, then a $k$-matching of $G$ is
called a _perfect matching_ of $G$. The number of $k$-matching is denoted by
$m({G},k)$.
If $C$ be any undirected even cycle of ${G},$ we say $C$ is _evenly oriented_
relative to the orientation $\overrightarrow{G}$ if it has an even number of
edges oriented in clockwise direction. Otherwise $C$ is _oddly oriented_.
We call a linear subgraph $L$ of $G$ _evenly linear_ if $L$ contains no odd
cycle and denote by $\mathcal{E}\mathcal{L}_{i}(G)$ (or
$\mathcal{E}\mathcal{L}_{i}$ for short) the set of all evenly linear subgraphs
of $G$ with $i$ vertices. For a linear subgraph
$L\in\mathcal{E}\mathcal{L}_{i}$ denote by $p_{e}(L)$(resp., $p_{o}(L)$ ) the
number of evenly (resp., oddly) oriented cycles in $L$ relative to
$\overrightarrow{G}$. Denote the characteristic polynomial of
$S(\overrightarrow{G})$ by
$P_{s}(\overrightarrow{G};x)=det(xI-S(\overrightarrow{G}))=\sum^{n}_{i=0}b_{i}x^{n-i}.$
(1)
Then (i) $b_{0}=1$, (ii) $b_{2}$ is the number of edges of $G,$ (iii) all
$b_{i}\geq 0$ and (iv) $b_{i}=0$ for all odd $i$ since the determinant of
every real skew symmetric matrix is nonnegative and is 0 if its order is odd.
We have
###### Lemma 2.1
([11]) Let $\overrightarrow{G}$ be an orientation of a graph $G$. Then
$b_{i}(\overrightarrow{G})=\sum_{L\in\mathcal{E}\mathcal{L}_{i}}(-2)^{p_{e}(L)}2^{p_{o}(L)},$
(2)
where $p_{e}(L)$ is the number of evenly oriented cycles of $L$ and $p_{o}(S)$
is the number of oddly oriented cycles of $L$ relative to
$\overrightarrow{G}$, respectively.
From Lemma 2.1, we can get the following statement.
###### Corollary 2.2
Let $G$ be a unicyclic graph with unique cycle $C$ of length $\ell$ and
$\overrightarrow{G}$ be an orientation of $G.$ Then
$b_{2k}(\overrightarrow{G})=\left\\{\begin{array}[]{cc}m({G},k),&C\ is\ an\
odd\ cycle;\\\ m({G},k)-2m({G}-C,k-\frac{\ell}{2}),&C\ is\ evenly\
oriented;\\\ m({G},k)+2m({G}-C,k-\frac{\ell}{2}),&C\ \ is\ oddly\
oriented.\end{array}\right.$
###### Lemma 2.3
([11]) Let $e=uv$ be an edge of $G$ that is on no any even cycle of $G.$ Then
$P_{s}(\overrightarrow{G};x)=P_{s}(\overrightarrow{G}-e;x)+P_{s}(\overrightarrow{G}-u-v;x).$
By Lemma 2.3, we obtain that
###### Corollary 2.4
Let $e=uv$ be an edge of $G$ that is on no even cycle of $G.$ Then
$b_{2k}(\overrightarrow{G})=b_{2k}(\overrightarrow{G}-e)+b_{2k-2}(\overrightarrow{G}-u-v).$
(3)
Furthermore, if $e=uv$ is a pendant edge with the pendant vertex $v.$ Then
$b_{2k}(\overrightarrow{G})=b_{2k}(\overrightarrow{G}-v)+b_{2k-2}(\overrightarrow{G}-u-v).$
(4)
For any orientation of a graph which does not contain any even cycle (in
particular, a tree, a unicyclic non-bipartite graph) by Theorem 2.9, we have
$b_{2k}(\overrightarrow{G})=m({G},k)$.
For the $k$-matching number of a graph $G$, the following result is well-
known.
###### Lemma 2.5
Let $e=uv$ be an edge of $G$. Then
(i) $m({G},k)=m({G}-e,k)+m({G}-u-v,k-1).$
(ii) If $G$ is a forest, then $m({G},k)\leq m(P_{n},k)$, $k\geq 1$.
(iii) If ${H}$ is a subgraph of ${G}$, then $m({H},k)\leq m({G},k)$, $k\geq
1$. Moreover, if ${H}$ is a proper subgraph of ${G}$, then the inequality is
strict.
From [2], an integral formula for skew energy was given:
${\cal
E}_{s}(\overrightarrow{G})=\frac{1}{\pi}\int^{\infty}_{-\infty}(n+x\frac{P^{\prime}_{s}(\overrightarrow{G},-x)}{P_{s}(\overrightarrow{G},-x)})dx,$
(5)
where $P^{\prime}_{s}(\overrightarrow{G},-x)$ is the derivative of
$P_{s}(\overrightarrow{G},-x).$
###### Theorem 2.6
Let $\overrightarrow{G}$ be an orientation of a graph $G$. Then
${\cal
E}_{s}(\overrightarrow{G})=\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{1}{t^{2}}ln(1+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}b_{2k}t^{2k})dt.$
Proof. From equality (5), we have
$\displaystyle{\cal E}_{s}(\overrightarrow{G})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}(n+x\frac{P^{\prime}_{s}(\overrightarrow{G},-x)}{P_{s}(\overrightarrow{G},-x)})dx=\frac{1}{\pi}\int^{\infty}_{-\infty}(n-\mu\frac{P^{\prime}_{s}(\mu)}{P_{s}(\mu)})d\mu$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{0}(n-\frac{1}{t}\frac{P^{\prime}_{s}(\frac{1}{t})}{P_{s}(\frac{1}{t})})(-\frac{1}{t^{2}})dt+\frac{1}{\pi}\int^{0}_{-\infty}(n-\frac{1}{t}\frac{P^{\prime}_{s}(\frac{1}{t})}{P_{s}(\frac{1}{t})})(-\frac{1}{t^{2}})dt$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}(n-\frac{1}{t}\frac{P^{\prime}_{s}(\frac{1}{t})}{P_{s}(\frac{1}{t})})\frac{1}{t^{2}}dt=\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{1}{t}d[ln(t^{n}P_{s}(\frac{1}{t})]$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{1}{t^{2}}ln(t^{n}P_{s}(\frac{1}{t}))dt.=\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{1}{t^{2}}ln(1+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}b_{2k}t^{2k})dt.\;\;\;\Box$
###### Example 2.7
Let $\overrightarrow{C_{4}}^{-}$ be an orientation of $C_{4}$ such that all
edges have the same direction. Then by Lemma 2.1, the characteristic
polynomial of $S(\overrightarrow{C_{4}}^{-})$ is $x^{4}+4x^{2}$. By Lemma 2.6,
$\displaystyle{\cal E}_{s}(\overrightarrow{C_{4}}^{-},x))$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{1}{t^{2}}ln(1+4t^{2})dt$
$\displaystyle=$
$\displaystyle\frac{8}{\pi}\int^{\infty}_{0}\frac{1}{1+4t^{2}}dt=4.$
From Theorem 2.6, ${\cal E}_{s}(\overrightarrow{G})$ is an increasing function
of $b_{2k}(\overrightarrow{G})$, $k=0$, $1$, $\cdots$,
$\lfloor\frac{n}{2}\rfloor$. Consequently, if $\overrightarrow{G_{1}}$ and
$\overrightarrow{G_{2}}$ are oriented graphs of $G_{1}$ and $G_{2},$
respectively, for which
$b_{2k}(\overrightarrow{G_{1}})\geq b_{2k}(\overrightarrow{G_{2}})$ (6)
for all $\lfloor{\frac{n}{2}}\rfloor\geq k\geq 0$, then
${\cal E}_{s}(\overrightarrow{G_{1}})\geq{\cal
E}_{s}(\overrightarrow{G_{2}}).$ (7)
Equality in (7) is attained only if (6) is an equality for all
$\lfloor{\frac{n}{2}}\rfloor\geq k\geq 0$.
If relations (6) hold for all $k$, then we write $G_{1}\succeq G_{2}$ or
$G_{2}\preceq G_{1}$. If $G_{1}\succeq G_{2}$, but not $G_{2}\succeq G_{1}$,
then we write $G_{1}\succ G_{2}$.
Let $\overrightarrow{G}$ be an orientation of a graph $G$. Let $W$ be a subset
of $V(G)$ and $\overline{W}=V(G)\setminus W$. The orientation
$\overrightarrow{G^{\prime}}$ of $G$ obtained from $\overrightarrow{G}$ by
reversing the direction of all arcs between $\overline{W}$ and $W$ is said to
be obtained from $\overrightarrow{G}$ by switching with respect to $W$.
Moreover, two orientations $\overrightarrow{G}$ and
$\overrightarrow{G^{\prime}}$ of a graph $G$ are said to be switching-
equivalent if $\overrightarrow{G^{\prime}}$ can be obtained from
$\overrightarrow{G}$ by a sequences of switching. If two orientations
$\overrightarrow{G}$ and $\overrightarrow{G}^{\prime}$ of a graph $G$ are
switching-equivalent then their skew-adjacency matrices are similar by a
diagonal matrix whose $(i,i)$-entry is $-1$ when $i\in W$ and $1$ when
$i\not\in W$ by Lemma 3.1 in [2] and hence they have the same skew spectrum.
Thus,
###### Lemma 2.8
Let $\overrightarrow{G}$ and $\overrightarrow{G}^{\prime}$ be two orientations
of a graph $G.$ If $\overrightarrow{G}$ and $\overrightarrow{G}^{\prime}$ are
switching-equivalent, then ${\cal E}_{s}(\overrightarrow{G})={\cal
E}_{s}(\overrightarrow{G}^{\prime})$.
By Lemma 2.8 and switching-equivalence, there are only two different
orientations on a unicyclic graph $G.$ All edges on the unique cycle $C$ have
the same direction or just one edge on the cycle has the opposite direction to
the directions of other edges on the cycle regardless how the edges not on the
cycle $C$ are oriented. Denote by ${\overrightarrow{G}}^{-}$
(${\overrightarrow{G}}^{+},$ resp.) the orientation of $G$ in first (second,
resp.) case above.
###### Lemma 2.9
For any unicyclic graph $G$,
${\overrightarrow{G}}^{+}\succeq{\overrightarrow{G}}^{-}.$
Proof. By Corollary (2.2), if the girth $\ell$ of $G$ is odd, then
$b_{2k}({{\overrightarrow{G}}^{+}})=b_{2k}({{\overrightarrow{G}}^{-}})=m(G,k)\mbox{
for all $0\leq k\leq\lfloor\frac{n}{2}\rfloor$.}$
If the girth $\ell$ of $G$ is even, then
$b_{2k}({{\overrightarrow{G}}^{+}})=m(G,k)+2m(G-C_{\ell},k-\lfloor\frac{\ell}{2}\rfloor),$
$b_{2k}({{\overrightarrow{G}}^{-}})=m(G,k)-2m({G}-C_{\ell},k-\lfloor\frac{\ell}{2}\rfloor)$
for all nonnegative integer $\lfloor\frac{n}{2}\rfloor\geq
k\geq\lfloor\frac{\ell}{2}\rfloor,$ and
$b_{0}({{\overrightarrow{G}}^{+}})=b_{0}({{\overrightarrow{G}}^{-}})=1,\;\;b_{2k}({{\overrightarrow{G}}^{+}})=b_{2k}({{\overrightarrow{G}}^{-}})=m(G,k)$
for all $0<k<\lfloor\frac{\ell}{2}\rfloor$. Thus the result follows
immediately. $\Box$
###### Lemma 2.10
Let ${\overrightarrow{G}}$ be an orientation of a unicyclic graph $G\in
G(n,\ell)$, $G\neq S_{n}^{\ell}$. If unique cycle $C_{\ell}$ in
${\overrightarrow{G}}$ and $\overrightarrow{S_{n}^{\ell}}$ is the same
orientation, then ${\overrightarrow{G}}\succ\overrightarrow{S_{n}^{\ell}}$.
Proof. We prove the statement by induction on $n$. Since $G\neq S_{n}^{\ell}$,
$n\geq\ell+2$. For $n=\ell+2$, $G$ is one of the two graphs in Fig. $2$.
By Corollary 4,
$b_{2k}(\overrightarrow{P_{\ell+2}^{\ell}})=b_{2k}(\overrightarrow{P_{\ell+1}^{\ell}})+b_{2k-2}(\overrightarrow{C_{\ell}}),$
$b_{2k}({\overrightarrow{G}_{1}})=b_{2k}(\overrightarrow{P_{\ell+1}^{\ell}})+b_{2k-2}(\overrightarrow{T}),$
where $T$ is a graph obtained by connecting one of the vertices in
$P_{\ell-1}$ to a pendant vertex.
$b_{2k}(\overrightarrow{S_{\ell+2}^{\ell}})=b_{2k}(\overrightarrow{P_{\ell+1}^{\ell}})+b_{2k-2}(\overrightarrow{P_{\ell-1}}).$
By Corollary 4, if $\ell$ is odd or $\ell$ is even but $k\leq\frac{\ell}{2}$
then
$b_{2k-2}(\overrightarrow{C_{\ell}})=m({C_{\ell}},k-1);$
for $k=\frac{\ell}{2}+1.$ If $C$ is oddly oriented, then
$b_{l}(\overrightarrow{C_{\ell}})=m({C_{\ell}},\frac{\ell}{2})+2=4;$
If $C$ is evenly oriented, then
$b_{\ell}(\overrightarrow{C_{\ell}})=m({C_{\ell}},\frac{\ell}{2})-2=0,\;\;b_{\ell}(\overrightarrow{P_{\ell-1}})=0.$
Since both $T$ and $P_{\ell-1}$ are trees,
$b_{2k-2}(\overrightarrow{T})=m({T},k-1),$
$b_{2k-2}(\overrightarrow{P_{\ell-1}})=m({P_{\ell-1}},k-1).$
Since $P_{\ell-1}$ is a proper subgraph of both $C_{\ell}$ and $T$, by Lemma
2.5,
$m(\overrightarrow{P_{\ell-1}},k-1)<m({T},k-1),m({P_{\ell-1}},k-1)<m({C_{\ell}},k-1).$
The result holds immediately for $n=\ell+2$.
Suppose that ${\overrightarrow{G}}\succ\overrightarrow{S_{n^{\prime}}^{\ell}}$
for all $n^{\prime}<n$. Since ${\overrightarrow{G}}$ is a unicyclic digraph,
there is at least a pendant edge $uv$ with pendant vertex $v$ in
${\overrightarrow{G}}$, by equality (4), we get
$\displaystyle b_{2k}({\overrightarrow{G}})$ $\displaystyle=$ $\displaystyle
b_{2k}({\overrightarrow{G}}-v)+b_{2k-2}({\overrightarrow{G}}-u-v),$
$\displaystyle b_{2k}(\overrightarrow{S_{n}^{\ell}})$ $\displaystyle=$
$\displaystyle
b_{2k}(\overrightarrow{S_{n-1}^{\ell}})+b_{2k-2}(\overrightarrow{P_{\ell-1}}).$
By induction assumption, it suffices to prove that
$b_{2k-2}({\overrightarrow{G}}-u-v)\geq b_{2k-2}(\overrightarrow{P_{\ell-1}})$
for all $0\leq k\leq\lfloor\frac{n-\ell}{2}\rfloor$. For
$k>\lfloor\frac{\ell-1}{2}\rfloor$, we have
$b_{2k-2}({\overrightarrow{G}}-u-v)\geq
b_{2k-2}(\overrightarrow{P_{\ell-1}})=0$. For
$\lfloor\frac{\ell-1}{2}\rfloor\geq k\geq 0$,
$b_{2k-2}({\overrightarrow{G}}-u-v)=m({\overrightarrow{G}}-u-v,k-1)\geq
m(\overrightarrow{P_{\ell-1}},k-1)=b_{2k-2}(\overrightarrow{P_{\ell-1}})$
since ${P_{\ell-1}}$ is a subgraph of ${G}-u-v$. $\Box$
###### Lemma 2.11
Let $n\geq\ell\geq 6$ or $n>\ell=5$, then
${\overrightarrow{S_{n}^{4}}}^{-}\prec{\overrightarrow{S_{n}^{4}}}^{+}\prec{\overrightarrow{S_{n}^{\ell}}}^{-}\preceq{\overrightarrow{S_{n}^{\ell}}}^{+}$.
Proof. By Lemma 2.1, the characteristic polynomial of
$\overrightarrow{S_{n}}^{4}$ is:
$\displaystyle P_{s}({\overrightarrow{S_{n}^{4}}}^{+},x)$ $\displaystyle=$
$\displaystyle x^{n-4}(x^{4}+nx^{2}+2n-4);$ $\displaystyle
P_{s}({\overrightarrow{S_{n}^{4}}}^{-},x)$ $\displaystyle=$ $\displaystyle
x^{n-4}(x^{4}+nx^{2}+2n-8);$
Obviously,
${\overrightarrow{S_{n}^{4}}}^{-}\prec{\overrightarrow{S_{n}^{4}}}^{+}$.
So it suffices to prove that $b_{4}(\overrightarrow{S_{n}^{\ell}})>2n-4$. By
equality (2),
$\displaystyle b_{4}({\overrightarrow{S_{n}^{\ell}}}^{+})$ $\displaystyle=$
$\displaystyle b_{4}({\overrightarrow{S_{n}^{\ell}}}^{-})$ $\displaystyle=$
$\displaystyle m(P_{\ell-1},2)+(n-\ell)m(P_{\ell-1},1)+2m(P_{\ell-2},1)$
$\displaystyle=$
$\displaystyle\frac{(\ell-3)(\ell-4)}{2}+(n-\ell)(\ell-2)+2(\ell-3)$
$\displaystyle=$ $\displaystyle\frac{2n\ell-\ell^{2}+\ell-4n}{2}.$
For $n>\ell=5$, $b_{4}(\overrightarrow{S_{n}^{\ell}})=3n-10\geq 2n-4$; For
$n\geq\ell=6$, $b_{4}(\overrightarrow{S_{n}^{\ell}})=4n-15>2n-4$; for
$n\geq\ell=7$, $b_{4}(\overrightarrow{S_{n}^{\ell}})=5n-21>2n-4$; for
$\ell\geq 8$,
$b_{4}(\overrightarrow{S_{n}^{\ell}})\geq\frac{n\ell+\ell-4n}{2}\geq
2n+4>2n-4.$
By Lemma 2.9, the proof is completed. $\Box$
For $n=\ell=5$,
$b_{4}({\overrightarrow{S_{n}^{\ell}}}^{+})=\frac{2n{\ell}-{\ell}^{2}+\ell-4n}{2}=5$,
$b_{4}({\overrightarrow{S_{n}^{4}}}^{+})=10-4=6$,
$b_{4}({\overrightarrow{S_{n}^{4}}}^{-})=10-8=2$. It gives
${\overrightarrow{S_{5}^{4}}}^{-}\prec{\overrightarrow{S_{5}^{5}}}^{+}\prec{\overrightarrow{S_{5}^{4}}}^{+}$.
###### Lemma 2.12
Let $\overrightarrow{S_{n}^{3}}$ be any orientation of unicyclic graph
$S_{n}^{3}$. Then $\overrightarrow{S_{n}^{3}}\prec\overrightarrow{S_{n}^{4}}$
for $n\geq 6$;
$\overrightarrow{S_{n}^{3}}={\overrightarrow{S_{n}^{4}}}^{-}\prec{\overrightarrow{S_{n}^{4}}}^{+}$
for $n=5$;
${\overrightarrow{S_{n}^{4}}}^{-}\prec\overrightarrow{S_{n}^{3}}\prec{\overrightarrow{S_{n}^{4}}}^{+}$
for $n=4$.
Proof. By Lemma 2.1, the characteristic polynomial of
$\overrightarrow{S_{n}^{3}}$ is
$P_{s}(\overrightarrow{S_{n}^{3}},x)=x^{n-4}(x^{4}+nx^{2}+n-3).$
Since $n-3<2n-8$, the result holds by Lemmas 2.11 and 2.9.
For $n=5$, it is easy to get that for any orientation ${\cal
E}_{s}(\overrightarrow{S_{5}^{3}})=2\sqrt{5+2\sqrt{2}}$, ${\cal
E}_{s}({\overrightarrow{S_{5}^{4}}}^{-})=2\sqrt{5+2\sqrt{2}}<2(\sqrt{2}+\sqrt{3})$,
${\cal E}_{s}({\overrightarrow{S_{5}^{4}}}^{+})=2(\sqrt{2}+\sqrt{3})$. Then
$\overrightarrow{S_{n}^{3}}={\overrightarrow{S_{5}^{4}}}^{-}\prec{\overrightarrow{S_{5}^{4}}}^{+}$.
For $n=4$, ${\cal E}_{s}(\overrightarrow{S_{5}^{3}})=2\sqrt{6}$, ${\cal
E}_{s}({\overrightarrow{C_{4}}}^{+})=4\sqrt{2}$, ${\cal
E}_{s}({\overrightarrow{C_{4}}}^{-})=4$. Thus
${\overrightarrow{S_{5}^{4}}}^{-}\prec\overrightarrow{S_{n}^{3}}\prec{\overrightarrow{S_{5}^{4}}}^{+}$.
$\Box$
From Lemmas 2.10, 2.11 and 2.12, we obtain one of the main result of this
paper.
###### Theorem 2.13
Among all orientations of unicyclic graphs on $n$ vertices,
$\overrightarrow{S_{n}^{3}}$ has the minimal skew energy and
${\overrightarrow{S_{n}^{4}}}^{-}$ has the second minimal skew energy for
$n\geq 6$; both ${\overrightarrow{S_{5}^{3}}}$ and
${\overrightarrow{S_{5}^{4}}}^{-}$ have the minimal skew energy,
${\overrightarrow{S_{5}^{4}}}^{+}$ has the second minimal skew energy for
$n=5$; ${\overrightarrow{C^{4}}}^{-}$ has the minimal skew energy,
$\overrightarrow{S_{4}^{3}}$ has the second minimal skew energy for $n=4$.
## 3 Oriented unicyclic graph with maximal skew energy
By Lemma 2.9, we only need to consider ${{\overrightarrow{G}}^{+}}$ for
considering of maximum skew energy.
###### Lemma 3.1
Let ${\overrightarrow{G}}$ be an orientation of unicyclic graph $G\in
G(n,\ell)$ and $G\neq P_{n}^{\ell}$. Then
${{\overrightarrow{G}}^{+}}\prec{\overrightarrow{P_{n}^{\ell}}}^{+}$.
Proof. We prove the statement by induction on $n$. For $n=\ell+2$, there are
only two cases for $G\neq P_{n}^{l}$: one is $S_{\ell+2}^{l}$, the other is
the graph $G_{1}$ in Fig. $2$. By the proof in Lemma 2.10, we only need to
prove that $b_{2k-2}(\overrightarrow{C_{l}})\geq m({T},k-1)$. By Lemma 2.5,
$\displaystyle m({\overrightarrow{C_{\ell}}}^{+},k-1)$ $\displaystyle=$
$\displaystyle m({C_{\ell}},k-1)$ $\displaystyle=$ $\displaystyle
m({P_{\ell}},k-1)+m({P_{\ell-2}},k-2),$ $\displaystyle m({T},k-1)$
$\displaystyle=$ $\displaystyle
m({P_{\ell-1}},k-1)+m({P_{s}}\bigcup{P_{t}},k-2)$ $\displaystyle\leq$
$\displaystyle m({P_{\ell-1}},k-1)+m({P_{\ell-2}},k-2)$ $\displaystyle<$
$\displaystyle m({P_{\ell}},k-1)+m({P_{\ell-2}},k-2)$ $\displaystyle=$
$\displaystyle m({{C_{\ell}}},k-1)$ $\displaystyle\leq$ $\displaystyle
b_{2k-2}({\overrightarrow{C_{\ell}}}^{+}),\;\;\;\;\;\;\;\mbox{(}by\ Corollary\
\ \ref{bre})$
where $s+t=l-2$.
Suppose
${{\overrightarrow{G}}^{+}}\prec{\overrightarrow{P_{n^{\prime}}^{\ell}}}^{+}$
for all $n^{\prime}<n$. Then there is a pendant edge, say $uv$, with pendant
vertex $v$. By Lemma 2.3,
$\displaystyle b_{2k}({{\overrightarrow{G}}^{+}})$ $\displaystyle=$
$\displaystyle
b_{2k}({{\overrightarrow{G}}^{+}}-v)+b_{2k-2}({{\overrightarrow{G}}^{+}}-v-u);$
$\displaystyle b_{2k}({\overrightarrow{P_{n}^{\ell}}}^{+})$ $\displaystyle=$
$\displaystyle
b_{2k}({\overrightarrow{P_{n-1}^{\ell}}}^{+})+b_{2k-2}({\overrightarrow{P_{n-2}^{\ell}}}^{+}).$
By induction hypothesis, it suffices to prove that
$b_{2k-2}({{\overrightarrow{G}}^{+}}-v-u)\leq
b_{2k-2}({\overrightarrow{P_{n-2}^{\ell}}}^{+})$. If
${{\overrightarrow{G}}^{+}}-v-u$ contains a cycle, then by induction
hypothesis the inequality holds. Suppose that ${{\overrightarrow{G}}^{+}}-v-u$
is a forest, then by Lemmas 2.5 and 2.9,
$\displaystyle b_{2k-2}({{\overrightarrow{G}}^{+}}-v-u)$ $\displaystyle=$
$\displaystyle m({{G}}-v-u,k-1)$ $\displaystyle\leq$ $\displaystyle
m({P_{n-2}},k-1)\leq m({{P_{n-2}^{\ell}}},k-1)$ $\displaystyle\leq$
$\displaystyle b_{2k-2}({{P_{n-2}^{\ell}}}).\mbox{ \hskip 28.45274pt $\Box$ }$
###### Lemma 3.2
For $3\leq\ell\leq n$, $\ell\neq 4$,
${\overrightarrow{P_{n}^{\ell}}}\prec{\overrightarrow{P_{n}^{4}}}^{+}$.
Proof. We proceed the proof by induction on $n$. For $n=\ell=5$,
$b_{4}({\overrightarrow{P_{5}^{5}}}^{+})=b_{4}({\overrightarrow{C_{5}}}^{+})=5$,
$b_{4}({\overrightarrow{P_{5}^{4}}}^{+})=b_{4}({\overrightarrow{S_{5}^{4}}}^{+})=6$,
thus
$b_{4}({\overrightarrow{P_{5}^{5}}}^{+})<b_{4}({\overrightarrow{P_{5}^{4}}}^{+})$.
Suppose that
${\overrightarrow{P_{n^{\prime}}^{\ell}}}^{+}\prec{\overrightarrow{P_{n^{\prime}}^{4}}}^{+}$
for all $n^{\prime}<n$, $5\leq\ell$. By Lemma 2.1, we have
$\displaystyle b_{2k}({\overrightarrow{P_{n}^{\ell}}}^{+})$ $\displaystyle=$
$\displaystyle
b_{2k}({\overrightarrow{P_{n-1}^{\ell}}}^{+})+b_{2k-2}({\overrightarrow{P_{n-2}^{\ell}}}^{+}),$
$\displaystyle b_{2k}({\overrightarrow{P_{n}^{4}}}^{+})$ $\displaystyle=$
$\displaystyle
b_{2k}({\overrightarrow{P_{n-1}^{4}}}^{+})+b_{2k-2}({\overrightarrow{P_{n-2}^{4}}}^{+}).$
By induction hypothesis, the result follows.
Similarly, we can prove that for $\ell=3$ the inequality holds too. $\Box$
By Lemmas 3.1 and 3.2, we have
###### Theorem 3.3
Among all orientations of unicyclic graph, ${\cal
E}_{s}({\overrightarrow{P_{n}^{4}}}^{+})$ is the unique oriented graph (under
switching-equivalent) with maximal skew energy.
Acknowledgments. This project is supported by National Natural Science
Foundation of China
## References
* [1]
* [2] C. Adiga, R. Balakrishnan and Wasin So, The skew energy of a digraph, Linear Algebra and its Application 432 (2010) 1825-1835.
* [3] D. M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1979.
* [4] M. Dehmer, F. Emmert-Streib, Analysis of complex networks: from biology to linguistics. WILEY-VCH Verlag GmbH Co. hGaA, Weinheim, 2009.
* [5] I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungszentrum Graz 103 (1978) 1–22.
* [6] I. Gutman, Y. Hou, Bipartite unicyclic graphs with greatest energy, MATCH Commun. Math. Comput. Chem. 43 (2001) 17–28.
* [7] B. Huo, X. Li, Complete solution to a conjecture on the maximal energy of unicyclic graphs. European journal of combinatorics. 32 (2011) 662–673.
* [8] H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Lin. Algebra Appl. 426 (2007) 478–489.
* [9] Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163–168.
* [10] Y. Hou, I. Gutman, C. W. Woo, Unicyclic graphs with maximal energy, Lin. Algebra Appl. 356 (2002) 27–36.
* [11] Y. Hou, T. Lei, Characteristic polynomials of skew-adjacency matrices of oriented graphs, The Electronic Journal of Combinatorics 18(2011) # P156.
* [12] X. Li, J. Zhang, B. Zhou, On unicyclic conjugated molecules with minimal energies, J. Math. Chem. 42 (2007) 729–740.
* [13] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 320,(2007) 1472–1475.
* [14] I. Pena, J. Rada, Energy of digraphs, Linear and Multilinear Algebra 56(2008) 565-579.
* [15] W. H. Wang, A. Chang, L. Z. Zhang, D. Q. Lu, Unicyclic H$\ddot{u}$ckel molecular graphs with minimal energy, J. Math. Chem. 39 (2006), 231–241.
* [16] W. H. Wang, A. Chang, D. Q. Lu, Unicyclic graphs possessing Kekul structures with minimal energy, J. Math. Chem. 42 (2007) 311–320.
|
arxiv-papers
| 2011-08-31T13:51:02 |
2024-09-04T02:49:21.851247
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hou Yaoping, Shen Xiaoling and Zhang Chongyan",
"submitter": "Hou Yaoping",
"url": "https://arxiv.org/abs/1108.6229"
}
|
1109.0086
|
arxiv-papers
| 2011-09-01T04:14:34 |
2024-09-04T02:49:21.865476
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qiang Zeng and Hai Zhuge",
"submitter": "Qiang Zeng",
"url": "https://arxiv.org/abs/1109.0086"
}
|
|
1109.0217
|
# Vessel Segmentation in Medical Imaging Using a Tight-Frame Based Algorithm
Xiaohao Cai, Raymond Chan, Serena Morigi, Fiorella Sgallari Department of
Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
(xhcai@math.cuhk.edu.hk and rchan@math.cuhk.edu.hk).Department of Mathematics-
CIRAM, University of Bologna, Bologna, Italy (morigi@dm.unibo.it and
sgallari@dm.unibo.it).
###### Abstract
Tight-frame, a generalization of orthogonal wavelets, has been used
successfully in various problems in image processing, including inpainting,
impulse noise removal, super-resolution image restoration, etc. Segmentation
is the process of identifying object outlines within images. There are quite a
few efficient algorithms for segmentation that depend on the variational
approach and the partial differential equation (PDE) modeling. In this paper,
we propose to apply the tight-frame approach to automatically identify tube-
like structures such as blood vessels in Magnetic Resonance Angiography (MRA)
images. Our method iteratively refines a region that encloses the possible
boundary or surface of the vessels. In each iteration, we apply the tight-
frame algorithm to denoise and smooth the possible boundary and sharpen the
region. We prove the convergence of our algorithm. Numerical experiments on
real 2D/3D MRA images demonstrate that our method is very efficient with
convergence usually within a few iterations, and it outperforms existing PDE
and variational methods as it can extract more tubular objects and fine
details in the images.
###### Index Terms:
Tight-frame, thresholding, image segmentation, wavelet transform.
## I Introduction
Segmentation problem of branching tubular objects in 2D and 3D images arises
in many applications, for examples, extracting roads in aerial photography,
and anatomical surfaces of blood vessels in medical images. In this paper, we
are concerned with identifying tube-like structures in Magnetic Resonance
Angiography (MRA) images. Because of the necessity to obtain as much fine
details as possible in real time, automatic, robust and efficient methods are
needed. However, due to the low contrast, intensity inhomogeneity, partial
occlusions and intersections, and the presence of noise and possible blur in
the images, this segmentation problem is very challenging.
There are many different approaches for vessel segmentation, see for example
[16, 20, 23, 26, 31, 34, 42] and the extended reviews [3, 30]. In the
following, we concentrate on two approaches that are related to our approach:
the active contour approach and the partial differential equation (PDE)
approach. For the vessel segmentation algorithms that are based on deformable
models [35], because the explicit deformable model representation is usually
impractical, level set techniques to evolve a deformable model have been
introduced, and they provide implicit representation of the deformable model.
However, the level set segmentation approach is computationally more expensive
as it needs to cover the entire domain of interest, which is generally one
dimension higher than the original one. Interested readers can refer to recent
literature on the level set segmentation strategy for tubular structures [26,
29, 40].
Based on the curve evolution techniques, Mumford-Shah functional and level
sets, a new model for active contours to detect objects in a given image was
proposed in [16]. Unlike the classical active contour models, this model does
not depend on the gradient of the image. In [38], a generalization of the
active contour without edges model was proposed for object detection using
logic operations. However, both the active contour model [16] and the logic
framework [38] are not suitable for detecting tubular structures with low
contrast and intensity inhomogeneity, see numerical results in Section IV.
In [23], a geometric deformable model for the segmentation of tubular-like
structures was proposed. The model is characterized mainly by two components
by using a suitable diffusion tensor: the mean curvature flow and the
directionality of the tubular structures. The major advantage of this
technique is the ability to segment twisted, convoluted and occluded
structures without user interactions; and it can follow the branching of
different layers, from thinner to larger structures. The dependence on the
grid resolution chosen to solve the discretized PDE model is still an open
problem. The authors in [23] have also applied a variant of the proposed PDE
model to the challenging problem of composed segmentation in [24].
Besides the methods above, there were some initial work by using wavelets or
tight-frames to do texture classification and segmentation [1, 43]. The tight-
frame approach is a versatile and effective tool for many different
applications in image processing, see [4, 7, 12, 13, 39, 41, 44]. There are
many kinds of tight-frame systems, such as those from framelets [36],
contourlets [19] and curvelets [10, 11], etc. Recently, the authors in [20]
proposed to combine the tight-frame image restoration model of [8] and the
total variation based segmentation model of [2, 15, 16] to do segmentation.
Their approach results in a minimization problem. In this paper, we also
derive a segmentation algorithm that uses the tight-frames. However our method
is not based on minimizing any variational model and hence it is different
from the method in [20]. In fact, our algorithm iteratively updates the set of
possible boundary pixels to change the given image into a binary image. Like
the method in [23], our method also has the ability to segment twisted,
convoluted and occluded structures. We will show that our algorithm is
convergent. In fact, numerical results show that it converges within 10
iterations for 2D as well as 3D Magnetic Resonance Angiography (MRA) images.
We will see that our method can extract much more details from the given image
than the method in [16], [20] and [23]. We remark that a preliminary version
of our segmentation algorithm has been given in our proceeding paper [9]. The
main contributions in this paper are: (1) a simpler strategy to initialize and
refine the regions enclosing the possible boundary of the vessels, (2) the new
strategy leads to a simple proof of convergence and easier choice of
parameters, (3) a different tight-frame with better directionally selective
property is used to obtain more details in the image, (4) a new 3D result with
higher noise and more complicated vessel is added.
The rest of the paper is organized as follows. In Section II, we recall some
basic facts about tight-frame and tight-frame algorithms. Our segmentation
algorithm is given in Section III. In Section IV we test our algorithm on
various real 2D and 3D MRA images and compare it with those representative
algorithms from different approaches: PDE-based [23], tight-frame [20] and
active contour models [16]. Conclusions are given in Section V.
## II Tight-Frame Algorithm
In this section, we briefly introduce the tight-frame algorithm which are
based on tight-frame transforms. All tight-frame transforms ${\cal A}$ have a
very important property, the “perfect reconstruction property”: ${\cal
A}^{T}{\cal A}={\cal I}$, the identity transform, see [36]. Unlike the
wavelets, in general, ${\cal A}{\cal A}^{T}\not={\cal I}$. For theories of
framelets and tight-frame transforms, we refer the readers to [17] for more
details. In order to apply the tight-frame algorithm, one only needs to know
the filters corresponding to the framelets in the tight-frame. In the
followings, we give two examples of tight-frames: the piecewise linear
B-spline tight-frame [18] and the dual-tree complex wavelet tight-frame [37].
The filters in the piecewise linear B-spline tight-frame are:
$h_{0}=\frac{1}{4}[1,\ 2,\ 1],\quad h_{1}=\frac{\sqrt{2}}{4}[1,\ 0,\ -1],\quad
h_{2}=\frac{1}{4}[-1,\ 2,\ -1],$ (1)
see [36]. The tight-frame coefficients of any given vector ${\bf v}$
corresponding to filter $h_{i}$ can be obtained by convolving $h_{i}$ with
${\bf v}$. In matrix terms, we can construct, for each filter, its
corresponding filter matrix which is just the Toeplitz matrix with diagonals
given by the filter coefficients, e.g. $H_{0}=\frac{1}{4}{\rm tridiag}[1,\ 2,\
1]$. Then the 1D tight-frame forward transform is given by
${\cal A}=\begin{bmatrix}H_{0}\\\ H_{1}\\\ H_{2}\end{bmatrix}.$ (2)
To apply the tight-frame transform onto ${\bf v}$ is equivalent to computing
${\cal A}{\bf v}$, and $H_{i}{\bf v}$ gives the tight-frame coefficients
corresponding to the filter $h_{i}$, $i=1,2,3$.
The $d$-dimensional piecewise linear B-spline tight-frame is constructed by
tensor products from the 1D tight-frame above, see [21]. For example, in 2D,
there are nine filters given by $h_{ij}\equiv h_{i}^{T}\otimes h_{j}$ for
$i,j=1,2,3$, where $h_{i}$ is given in (1). For any 2D image $f$, the tight-
frame coefficients with respect to $h_{ij}$ are obtained by convolving
$h_{ij}$ with $f$. The corresponding forward transform ${\cal A}$ will be a
stack of nine block-Toeplitz-Toeplitz-block matrices (cf. (2)). The tight-
frame coefficients are given by the matrix-vector product ${\cal A}{\bf f}$,
where ${\bf f}=\textrm{vec}(f)$ denotes the vector obtained by concatenating
the columns of $f$.
Dual-tree complex wavelet transform (D$\mathbb{C}$WT) was firstly introduced
by Kingsbury [32, 33]. Apart from having the usual perfect reconstruction
property, shift-invariance property and linear complexity, it also has the
nice directionally selective property at $\pm 15^{\circ},\pm 45^{\circ},\pm
75^{\circ}$. The idea is to use two different sets of filters: one gives the
real part of the transform and the other gives the imaginary part. Let
{$g_{0},g_{1}$} and {${h}_{0},{h}_{1}$} denote the two different sets of
orthonormal filters, where $g_{0},{h}_{0}$ are the low pass filters and
$g_{1},{h}_{1}$ are the high pass filters. Let the square matrix
$A_{g_{i}{h}_{j}}$ denote the 2D separable wavelet transform implemented using
$g_{i}$ along the rows and ${h}_{j}$ along the columns, and define
$A_{{h}_{i}{h}_{j}},A_{g_{i}g_{j}},A_{{h}_{i}g_{j}}$ similarly, where
$i,j=1,2$. Then the forward transform for the 2-dimensional D$\mathbb{C}$WT is
represented by
${\cal A}{\bf f}:=\frac{1}{\sqrt{8}}\begin{bmatrix}I&-I&0&0\\\ I&I&0&0\\\
0&0&I&I\\\ 0&0&I&-I\end{bmatrix}\begin{bmatrix}A_{g_{i}g_{j}}\\\
A_{{h}_{i}{h}_{j}}\\\ A_{{h}_{i}g_{j}}\\\ A_{g_{i}{h}_{j}}\\\
\end{bmatrix}{\bf f},\quad i,j=1,2.$
We see that the 2-dimensional D$\mathbb{C}$WT requires four different wavelet
transforms in parallel. We refer the readers to [14, 17, 28, 37] and the
references therein for more details and for the implementation of
3-dimensional D$\mathbb{C}$WT. In practice, the D$\mathbb{C}$WT are
implemented by using different sets of filters for the first level and the
remaining levels, see [37]. The filters and the Matlab code for
D$\mathbb{C}$WT can be obtained from [14].
The tight-frame algorithms, as given in [4, 5, 6, 7, 13], are of the following
generic form:
$\displaystyle{{\bf f}}^{(i+{\frac{1}{2}})}$ $\displaystyle=$
$\displaystyle\mathcal{U}({{\bf f}}^{(i)}),$ (3) $\displaystyle{{\bf
f}}^{(i+1)}$ $\displaystyle=$ $\displaystyle{\cal
A}^{T}\mathcal{T}_{\lambda}({\cal A}{{\bf f}}^{(i+{\frac{1}{2}})}),\quad
i=1,2,\ldots.$ (4)
Here ${\bf f}^{(i)}$ is an approximate solution at the $i$-th iteration,
$\mathcal{U}$ is a problem-dependent operator, and
$\mathcal{T}_{\lambda}(\cdot)$ is the soft-thresholding operator defined as
follows. Given vectors ${\bf v}=[v_{1},\cdots,v_{n}]^{T}$ and
${\bf\lambda}=[\lambda_{1},\cdots,\lambda_{n}]^{T}$,
$\mathcal{T}_{\lambda}({\bf
v})\equiv[t_{\lambda_{1}}(v_{1}),\cdots,t_{\lambda_{n}}(v_{n})]^{T}$, where
$t_{\lambda_{k}}(v_{k})\equiv\left\\{\begin{array}[]{lcl}\textrm{sgn}(v_{k})(|v_{k}|-\lambda_{k}),&&\text{if}\
|v_{k}|>\lambda_{k},\\\ 0,&&\text{if}\
|v_{k}|\leq\lambda_{k}.\end{array}\right.$ (5)
For how to choose $\lambda_{k}$, see [22].
We remark that (4) performs a tight-frame denoising and smoothing on the image
while (3) performs a data-fitting according to the specific problem at hand.
For high-resolution image reconstruction problem [13], astronomical infra-red
imaging [5], impulse noise removal [6], and inpainting problem [4], the tight-
frame algorithm (3)–(4) has been shown to be convergent to a functional with
the regularization term being the 1-norm of the tight-frame coefficients, see
[7].
## III Tight-Frame Based Algorithm for Segmentation
The technology of Magnetic Resonance Angiography (MRA) imaging is based on
detection of signals from flowing blood and suppression of signals from other
static tissues, so that the blood vessels appear as high intensity regions in
the image, see Fig. 1(a). The structures to be segmented are vessels of
variable diameters which are close to each other. In general in medical
images, speckle noise and weak edges make it difficult to identify the
structures in the image. Fortunately, the MRA images contain some properties
that can be exploited to derive a good segmentation algorithm. From Fig. 1(a),
we see that the pixels near the boundary of the vessels are not exactly of one
value, but they are in some range, whereas the values of the pixels in other
parts are far from this range. Thus the main idea of our algorithm is to
approximate this range accurately. We will obtain the range iteratively by a
tight-frame algorithm. The main steps are as follows. Suppose in the beginning
of the $i$th iteration, we are given an approximate image $f^{(i)}$ and a set
$\Lambda^{(i)}$ that contains all possible boundary pixels. Then we (i) use
$\Lambda^{(i)}$ to estimate an appropriate range $[\alpha_{i},\beta_{i}]$ that
contains the pixel values of possible boundary pixels; (ii) use the range to
separate the image into three parts—those below the range (background pixels),
inside the range (possible boundary pixels), and above the range (pixels in
the vessels); (iii) denoise and smooth the inside part by the tight-frame
algorithm to get a new image $f^{(i+1)}$. We stop when the image becomes
binary and this happens within 10 iterations for the real MRA images we
tested, see Table I in Section IV. In the followings, we elaborate each of the
steps. Without loss of generality, we assume that the given image $f$ has
dynamic range in [0, 1].
Initialization. To start the algorithm at $i=0$, we need to define $f^{(0)}$,
the initial guess, and $\Lambda^{(0)}$, the initial set of possible boundary
pixels. Naturally, we set $f^{(0)}=f$, the given image. For $\Lambda^{(0)}$,
since we do not have any knowledge of where the boundary pixels are at the
beginning, we identify them by using the gradient of $f$, i.e. we locate them
as pixels where the gradient is bigger than a threshold $\epsilon$. More
precisely, let $\Omega$ be the index set of all the pixels in the image, then
we define
$\Lambda^{(0)}\equiv\\{j\in\Omega\ |\ \|[\nabla f]_{j}\|_{1}\geq\epsilon\\}.$
(6)
Here $[\nabla f]_{j}$ is the discrete gradient of $f$ at the $j$th pixel. Once
$f^{(0)}$ and $\Lambda^{(0)}$ are defined, we can start the iteration.
Step (i): computing the range $[\alpha_{i}$, $\beta_{i}]$. Given
$\Lambda^{(i)}$, we first compute the mean pixel value on $\Lambda^{(i)}$:
$\mu^{(i)}=\frac{1}{|\Lambda^{(i)}|}\sum_{j\in\Lambda^{(i)}}f^{(i)}_{j},$ (7)
where $|\cdot|$ denotes the cardinality of the set and $f^{(i)}_{j}$ is the
pixel value of pixel $j$ in image $f^{(i)}$. Then we compute the mean pixel
values of the two sets separated by $\mu^{(i)}$:
$\mu^{(i)}_{-}=\frac{1}{|\\{j\in\Lambda^{(i)}:f^{(i)}_{j}\leq\mu^{(i)}\\}|}\sum_{\\{j\in\Lambda^{(i)}:f^{(i)}_{j}\leq\mu^{(i)}\\}}f^{(i)}_{j},$
(8)
and
$\mu^{(i)}_{+}=\frac{1}{|\\{j\in\Lambda^{(i)}:f^{(i)}_{j}\geq\mu^{(i)}\\}|}\sum_{\\{j\in\Lambda^{(i)}:f^{(i)}_{j}\geq\mu^{(i)}\\}}f^{(i)}_{j}.$
(9)
While $\mu^{(i)}$ reflects the mean energy of the set of possible boundary
pixels, $\mu^{(i)}_{-}$ and $\mu^{(i)}_{+}$ reflect the mean energies of the
pixels on the boundary closer to the background and closer to the vessels
respectively. We define
$\alpha_{i}\equiv\max\left\\{\frac{\mu^{(i)}+\mu^{(i)}_{-}}{2},0\right\\},\quad\beta_{i}\equiv\min\left\\{\frac{\mu^{(i)}+\mu^{(i)}_{+}}{2},1\right\\}.$
(10)
Step (ii): thresholding the image into three parts. Using the range
$[\alpha_{i},\beta_{i}]\subseteq[0,1]$, we can separate the image $f^{(i)}$
into three parts—those below, inside, and above the range, see Fig. 1(b) for
$i=0$. Since our aim is to create a binary image, we threshold those pixel
values that are smaller than $\alpha_{i}$ to 0, those larger than $\beta_{i}$
to 1, and those in between, we stretch them between 0 and 1 using a simple
linear contrast stretch, see [27]. If there are no pixels in between
$\alpha_{i}$ and $\beta_{i}$, then the threshold image is binary and the
algorithm stops. More precisely, let
$\displaystyle M_{i}={\rm max}\\{f^{(i)}_{j}\ |\ \alpha_{i}\leq
f^{(i)}_{j}\leq\beta_{i},j\in{\Lambda^{(i)}}\\},\quad m_{i}={\rm
min}\\{f^{(i)}_{j}\ |\ \alpha_{i}\leq
f^{(i)}_{j}\leq\beta_{i},j\in{\Lambda^{(i)}}\\},$ (11)
then we define
$f^{(i+{\frac{1}{2}})}_{j}=\left\\{\begin{array}[]{lclll}0,&&{\rm if}\
f^{(i)}_{j}\leq\alpha_{i},\\\
\frac{f^{(i)}_{j}-m_{i}}{M_{i}-m_{i}},&&\alpha_{i}\leq
f^{(i)}_{j}\leq\beta_{i},&&\quad{\rm for\ all}\ j\in{\Omega}.\\\ 1,&&{\rm if}\
\beta_{i}\leq f^{(i)}_{j},\end{array}\right.$ (12)
Fig. 1(c) shows the threshold and stretched image from Fig. 1(b), where the
yellow pixels are pixels we have classified not on the boundary, i.e.
$f^{(i+{\frac{1}{2}})}_{j}=0$ (signifying pixel $j$ is in the background) or
$f^{(i+{\frac{1}{2}})}_{j}=1$ (signifying pixel $j$ is inside the vessel). The
remaining pixels are remained to be classified and we denote the set by:
$\Lambda^{(i+1)}=\\{j\ |\ 0<f^{(i+{\frac{1}{2}})}_{j}<1,j\in{\Omega}\\}.$ (13)
Note that those pixels with values $m_{i}$ and $M_{i}$ are mapped to 0 and 1
respectively and hence will not be in $\Lambda^{(i+1)}$. Next we denoise and
smooth $f^{(i+{\frac{1}{2}})}$ on $\Lambda^{(i+1)}$.
| |
---|---|---
(a) | (b) | (c)
Figure 1: (a) Given MRA image. (b) Three parts of the given image
(green–below, red–in between, and yellow–above). (c) Threshold and stretched
image by (12) (yellow pixels are with value 0 or 1).
Step (iii): tight-frame iteration. To denoise and smooth the image
$f^{(i+{\frac{1}{2}})}$ on $\Lambda^{(i+1)}$, we apply the tight-frame
iteration (4) on $\Lambda^{(i+1)}$. More precisely, if
$j\not\in\Lambda^{(i+1)}$, then we set
$f^{(i+1)}_{j}=f^{(i+{\frac{1}{2}})}_{j}$; otherwise, we use (4) to get
$f^{(i+1)}_{j}$. To write it out clearly, let ${\bf
f}^{(i+{\frac{1}{2}})}={\rm vec}(f^{(i+{\frac{1}{2}})})$, and $P^{(i+1)}$ be
the diagonal matrix where the diagonal entry is 1 if the corresponding index
is in $\Lambda^{(i+1)}$, and 0 otherwise. Then
${\bf f}^{(i+1)}\equiv(I-P^{(i+1)}){\bf f}^{(i+{\frac{1}{2}})}+P^{(i+1)}{\cal
A}^{T}\mathcal{T}_{\lambda}({\cal A}{\bf f}^{(i+{\frac{1}{2}})}).$ (14)
By reordering the entries of the vector ${\bf f}^{(i+1)}$ into columns, we
obtain the image $f^{(i+1)}$. Note that the effect of (14) is to denoise and
smooth the image on $\Lambda^{(i+1)}$, see [7]. Since the pixel values of all
pixels outside $\Lambda^{(i+1)}$ are either $0$ or $1$, the cost of the tight-
frame transform in (14), such as the computation of ${\cal A}{\bf
f}^{(i+{\frac{1}{2}})}$, can be reduced significantly by taking advantage of
the fact that all computations can be done only on pixels around
$\Lambda^{(i+1)}$.
Stopping criterion. We stop the iteration when all the pixels of
$f^{(i+{\frac{1}{2}})}$ are either of value $0$ or $1$, or equivalently when
$\Lambda^{(i)}=\emptyset$. For the binary image $f^{(i+{\frac{1}{2}})}$, all
the pixels with value 0 are considered as background pixels and pixels with
value 1 constitute the tubular structures.
Below we give the full algorithm and show that it always converges to a binary
image.
Algorithm 1: Tight-frame algorithm for segmentation
1. | Input: given image $f$.
---|---
2. | Set $f^{(0)}=f$ and $\Lambda^{(0)}$ by (6)
3. | Do $i=0,1,\ldots,$ until stop
| (a) Compute $[\alpha_{i},\beta_{i}]$ by $(\ref{int_i})$.
| (b) Compute $f^{(i+{\frac{1}{2}})}$ by (12).
| (c) Stop if $f^{(i+{\frac{1}{2}})}$ is a binary image.
| (d) Compute $\Lambda^{(i+1)}$ by (13).
| (e) Update $f^{(i+{\frac{1}{2}})}$ to $f^{(i+1)}$ by (14).
4. | Output: binary image $f^{(i+{\frac{1}{2}})}$.
###### Theorem 1.
Our tight-frame algorithm will converge to a binary image.
###### Proof.
From (13), it suffices to prove that $|\Lambda^{(i)}|=0$ at some finite step
$i>0$. By (11), if $f^{(i+{\frac{1}{2}})}$ is not yet a binary image, then
there will be at least one $j\in\Lambda^{(i)}$ such that $f^{(i)}_{j}=M_{i}$.
By (12), $f^{(i+{\frac{1}{2}})}_{j}$ will be set to 1 and hence by (13),
$j\not\in\Lambda^{(i+1)}$. Hence $|\Lambda^{(i+1)}|<|\Lambda^{(i)}|$. Since
$|\Lambda^{(0)}|$ is finite, there must exist some $i$ such that
$|\Lambda^{(i)}|=0$. ∎
We emphasize that the algorithm actually converges within 10 iterations for
the 2D and 3D real images we have tested.
Finally, let us estimate the computation cost of our method for a given image
with $n$ pixels. Since the costs of computing $\mu^{(i)}$, $\mu^{(i)}_{-}$,
$\mu^{(i)}_{+}$, and $[\alpha_{i},\beta_{i}]$ are all of $O(n)$ operations,
see (7)–(10); and the cost of a tight-frame transform is also linear with
respect to $n$ (see e.g. (2) where $H_{i}$ are all tri-diagonal matrices), we
see that the complexity of our algorithm is $O(n)$ per iteration. In fact, one
can speed up the computation tremendously as all computations can be done only
on pixels around $\Lambda^{(i)}$ and there is no need to carry out the
computations in $\Omega$—though in the numerical tests, we did not optimize
the code and we just carried out the computations in $\Omega$. According to
Table I, after just 3 iterations, the set $\Lambda^{(i)}$ contains only a
hundred pixels in the 2D case and just two thousand pixels in the 3D case.
## IV Numerical Examples
In this section, we try our tight-frame segmentation algorithm (Algorithm 1)
on 2D and 3D real MRA images tested in [23] and [24]. Our algorithm is written
in Matlab. The thresholding parameters $\lambda_{k}$ used in (5) are all
chosen to be $\lambda_{k}\equiv 0.1$; and we choose $\epsilon=0.003$ and 0.06
for 2D and respectively 3D images in (6). To display the boundary and the
surface in the binary images visually, we use the Matlab commands “contour”
and “isosurface” for 2D and 3D images respectively. The tight-frame we used is
the D$\mathbb{C}$WT downloaded from [14] where all parameters in the code are
chosen to be the default values. In particular, the number of wavelet levels
used in the code is 4. The 2D images are tested in a MacBook with 2.4 GHz
processor and 4GB RAM, while the 3D images, because of their sizes, are tested
on a node with 120GB RAM in a PC-cluster. We compare our method with those
representative algorithms from different approaches: PDE-based [23], tight-
frame [20] and active contour models [16], where the programs are provided by
the authors.
Example 1. The test image is a $182\times 182$ MRA image of a carotid vascular
system, see Fig. 2(a). The blood vessels contain regions with high and low
intensities, including some very thin vessels in the middle with intensities
as low as the intensity of the background. Intersections of partial structures
even increase the difficulty of the segmentation. The segmentation results by
different methods are given in Fig. 2 where the given image is overlaid so
that we can compare the accuracy of the methods. Clearly, the results of Figs.
2(b) and (c) are not satisfactory since the vessels obtained in Fig. 2(b) are
disconnected and some vessels in Fig. 2(c) are not detected. By comparing the
parts inside the rectangles in Fig. 2(d) with those in Fig. 2(e), we see that
our method can extract clearer boundary than the method in [23], especially
for handling the artifacts near the boundary. We also see that the noise in
the image has been removed because of the denoising property of formula (4) as
explained in Section II. Our method converges in 5 iterations and requires
0.64 seconds. The first column of Table I gives $|\Lambda^{(i)}|$ at each
iteration.
| | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
Figure 2: Carotid vascular system segmentation. (a) Given image. (b), (c) and (d) Results by the methods in [16], [20] and [23] respectively. (e) Result of our method. TABLE I: Cardinality of $\Lambda^{(i)}$ at each iteration of the four examples | Example 1 | Example 2 | Example 3 | Example 4
---|---|---|---|---
$|\Lambda^{(i)}|$ | $|\Omega|=11284$ | $|\Omega|=66049$ | $|\Omega|=8120601$ | $|\Omega|=6000000$
$i=0$ | 1721 | 9444 | 137330 | 152898
$i=1$ | 354 | 1943 | 32760 | 32064
$i=2$ | 82 | 464 | 8795 | 8565
$i=3$ | 26 | 133 | 2475 | 2391
$i=4$ | 4 | 32 | 689 | 650
$i=5$ | 0 | 8 | 189 | 187
$i=6$ | - | 0 | 56 | 59
$i=7$ | - | - | 15 | 12
$i=8$ | - | - | 3 | 2
$i=9$ | - | - | 0 | 0
Example 2. The test image is a $256\times 256$ MRA image of a kidney vascular
system as shown in Fig. 3(a). This example shows the ability of our method to
reconstruct structures which present small occlusions along the coherence
direction. Our method converges in 6 iterations with 0.78 seconds; and the
second column in Table I gives $|\Lambda^{(i)}|$ at each iteration. The
results by the method in [16] (Fig. 3(b)) and by the method in [20] (Fig.
3(c)) are not good since they can not recover the small occlusions along the
coherence direction, while this can be done by our method and the method in
[23], see Figs. 3(d) and (e). Furthermore, our method is better than the
method in [23] by comparing the rectangular parts of Fig. 3(d) with those in
Fig. 3(e), since our method can detect smoother edges; see Figs. 3(f)–(k)
which are the results of zooming in the rectangular parts of Figs. 3(d) and
(e) respectively. The results also show that our method is very effective in
removing artifacts.
| |
---|---|---
(a) | (b) | (c)
|
---|---
(d) | (e)
| |
---|---|---
(f) | (g) | (h)
| |
(i) | (j) | (k)
Figure 3: Kidney vascular system segmentation. (a) Given image. (b), (c) and
(d) Results by the methods in [16], [20] and [23] respectively. (e) Results by
our method. (f)–(k) are the zoomed-in parts of (d) and (e).
Example 3. This is a 3D example where we extracted a volumetric data set of
size $201\times 201\times 201$ from a $436\times 436\times 540$ CTA (Computed
Tomographic Angiography) image of the kidney vasculature system, see Fig.
4(a). Because of different curvatures, diameters, bifurcations, and weak
surfaces impaired by noise, it is hard to detect the tips of the thin vessels.
Figs. 4(b) and (c) give the results by using the method in [24] and our
proposed method respectively. The figure shows that our method can give much
more details. A visual comparison of the given image with our result shows
that almost all the vessels are correctly segmented. We note that our method
converges in 9 iterations, see the third column in Table I.
---
(a)
|
---|---
(b) | (c)
Figure 4: Segmentation of the kidney volume data set. (a) Given CTA image. (b)
Result by the method in [24]. (c) Result by our method.
Example 4. Our second 3D example is an MRA data set of a brain aneurysms
(vessel wall dilatations). The $120\times 250\times 200$ volumetric data set
has been extracted from a $120\times 448\times 540$ MRA image of the brain-
neck vasculature system, see Fig. 5(a). As in Example 3, the different
curvatures, diameters and bifurcations of the vessels make it a difficult
problem. In addition, the high noise makes the thin vessels hard to see even
by the naked eyes. For this topological complex tubular structures, Figs. 5(b)
and (c) give the results by the method in [23] and our method respectively.
Obviously, our method can segment many more thin vessels, especially those
corrupted by the high noise. The results reflect that our new method is very
effective in handling the noise spread on the surface of the vessels. For this
complicated example, our method converges in 9 iterations only, see the fourth
column of Table I. Our segmentation technique can be used to compute vessel
radii and other clinically useful measurements in case of aneurysms. In Fig.
5(c), one may argue that there are some small isolated points in the image and
the surface of the vessels is not smooth. This can easily be remedied by
smoothing our final binary image by the tight-frame formula (4) one time
before we show the image, since (4) has the denoising property as we explained
in Section II. See Fig. 5(d) for the denoised-and-smoothed image after one
iteration of (4).
---
(a)
|
---|---
(b) | (c)
---
(d)
Figure 5: Segmentation of the brain volume data set. (a) Given MRA image. (b)
Result by the method in [23]. (c) Result by our method. (d) Result of our
method after smoothing by (4) once.
## V Conclusions and Future Work
In this paper, we introduced a new and efficient segmentation method based on
the tight-frame approach. The numerical results demonstrate the ability of our
method in segmenting tubular structures. The method can be implemented fast
and give very accurate, smooth boundaries or surfaces. In addition, since the
pixel values of more and more pixels will be set to either 0 or 1 during the
iterations, by taking advantage of this, one can construct a sparse data
structure to accelerate the method. Moreover, one can use different tight-
frame systems such as those from contourlets, curvelets or steerable-wavelet
[19, 11, 25] to capture more directions along the boundary. Though we have
proved that our algorithm will always converge to a binary image, it will be
interesting to see what functional the binary image is minimizing. The
framework for proving convergence for tight-frame algorithms, as developed in
[7], may be useful here. These are the directions we will explore in the
future.
### Acknowledgments.
This work was supported by HKRGC Grant CUHK 400510 and CUHK Direct Allocation
Grant 2060408.
## References
* [1] S. Arivazhagan and L. Ganesan. Texture segmentation using wavelet transform. Pattern Recognition Letters, 24, 3197–3203, 2003.
* [2] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran, and S. Osher. Fast global minimization of the active contour/snake model. J. Math. Imaging Vision, 28(2), 151–167, 2007.
* [3] D. Cremers, M. Rousson, and R. Deriche. A review of statistical approaches to level set segmentation: Integrating color, texture, motion, and shape Int. J. Comput. Vis., 72(2), 195–215, 2007.
* [4] J.F. Cai, R.H. Chan, L.X. Shen, and Z.W. Shen. Simultaneously inpainting in image and transformed domains. Numer. Math., 112(4), 509–533, 2009.
* [5] J.F. Cai, R.H. Chan, L.X. Shen, and Z.W. Shen. Restoration of chopped and nodded images by framelets. SIAM J. Sci. Comput., 30, 1205–1227, 2008.
* [6] J.F. Cai, R.H. Chan, L.X. Shen, and Z.W. Shen. Convergence analysis of tight framelet approach for missing data recovery. Adv. Comput. Math., 31, 87–113, 2009.
* [7] J.F. Cai, R.H. Chan, and Z.W. Shen. A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal., 24, 131–149, 2008.
* [8] J.F. Cai, S. Osher, and Z.W. Shen. Split Bregman methods and frame based image restoration. Multiscale Modeling and Simulation, 8(2), 337–369, 2009.
* [9] X.H. Cai, R.H. Chan, S. Morigi, and F. Sgallari. Framelet-Based Algorithm for Segmentation of Tubular Structures. SSVM 2011, LNCS6667. Springer, 2011\.
* [10] E. Candes and D. Donoho. New tight frames of curvelets and optimal representations of objects with C2 singularities. Comm. Pure Appl. Math, 56, 219–266, 2004.
* [11] E. Candès, L. Demanet, D. Donoho, and L. Ying. Fast discrete curvelet transforms. Multiscale Modeling and Simulation, 5(3), 861–899, 2006.
* [12] R.H. Chan, S. Setzer, and G. Steidl. Inpainting by flexible Haar-wavelet shrinkage. SIAM J. Imaging Sci., 1(3), 273–293, 2008.
* [13] R.H. Chan, T.F. Chan, L.X. Shen, and Z.W. Shen. Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput., 24(4), 1408–1432, 2003.
* [14] S.H. Cai and K.Y. Li. Matlab Implementation of Wavelet Transforms. http://taco.poly.edu/WaveletSoftware/
* [15] T.F. Chan, S. Esedoglu, and M. Nikolova. Algorithms for finding global minimizers of image segmentation and denoising models. Technical Report, 54, UCLA, 2004.
* [16] T.F. Chan and L.A. Vese. Active contours without edges. IEEE Trans. Image Process., 10(2), 266–77, 2001.
* [17] I. Daubechies. Ten lectures on wavelets. vol. CBMS-NSF Lecture Notes, SIAM, nr., 61, 1992.
* [18] I. Daubechies, B. Han, A. Ron, and Z. Shen. Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal., 14(1), 1–46, 2003.
* [19] M.N. Do and M. Vetterli. The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image Process., 14(12), 2091–2106, 2004.
* [20] B. Dong, A. Chien, and Z.W. Shen. Frame based segmentation for medical images. Technical Report, 22, UCLA, 2010.
* [21] B. Dong and Z.W. Shen. MRA based wavelet frames and applications. IAS Lecture Notes Series, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute, 2010.
* [22] D.L. Donoho. De-noising by soft-thresholding. IEEE Trans. Inform. Theory, 41(3), 613–627, 1995.
* [23] E. Franchini, S. Morigi, and F. Sgallari. Segmentation of 3D tubular structures by a PDE-based anisotropic diffusion model. M. Dæhlen et al. (eds.): MMCS 2008, LNCS5862, pp. 224–241, 2010, Springer-Verlag Berlin Heidelberg, 2010.
* [24] E. Franchini, S. Morigi, and F. Sgallari. Composed segmentation of tubular structures by an anisotropic PDE model. X.-C. Tai et al. (eds.): SSVM 2009, LNCS5567, pp. 75-86, 2009, Springer-Verlag Berlin Heidelberg, 2009.
* [25] W.T. Freeman and E.H. Adelson. The design and use of steerable filters. IEEE Trans. Pattern Analysis and Machine Intelligence, 13(9), 891–906, 1991.
* [26] A. Gooya, H. Liao, et al. A variational method for geometric regularization of vascular segmentation in medical images. IEEE Trans. Image Process., 17(8), 1295–1312, 2008.
* [27] R.C. Gonzales and R.E. Woods. Digital Image Processing, 3rd Ed., PrenticeHall, 2008.
* [28] X.Q. Gao, T.Q. Nguyen, and G. Strang. Theory and lattice structure of complex paraunitary filterbanks with filters of (hermitian-) symmetry/antisymmetry properties. IEEE Trans. Signal Processing, 49(5), 1028–1043, 2001.
* [29] H. Hassan and A.A. Farag. Cerebrovascular segmentation for MRA data using levels set. Proc. CARS, pp. 246–252, 2003.
* [30] C. Kirbas and F. Quek. A review of vessel extraction techniques and algorithms. ACM Computing Surveys, 36, 81–121, 2004.
* [31] K. Krissian, G. Malandain, N. Ayache, R. Vaillant, and Y. Trousset. Model-based detection of tubular structures in 3D images. CVIU, 80,130–171, 2000.
* [32] N. Kingsbury. Image processing with complex wavelets. Philos. Trans. R. Soc. London A, Math. Phys. Sci., 357(1760), 2543–2560, 1999.
* [33] N. Kingsbury. Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. Anal., 10(3), 234 253, 2001.
* [34] L.M. Lorigo, O.D. Faugeras, E.L. Grimson, et al. Curves: Curve evolution for vessel segmentation. Medical Image Analysis, 5, 195–206, 2001.
* [35] T. Mcinerney and D. Terzopoulos. Deformable models in medical image analysis: A survey. Medical Image Analysis, 1(2), 91–108, 1996.
* [36] A. Ron and Z.W. Shen. Affine systems in L2(Rd): the analysis of the analysis operator. J. Funct. Anal., 148, 408–447, 1997.
* [37] I. Selesnick, R.G. Baraniuk, and N.G. Kingsbury. The dual-tree complex wavelet transform. IEEE Signal Process. Mag., 22(6), 123–151, 2005.
* [38] B. Sandberg and T.F. Chan. A logic framework for active contours on multi-channel images. J. Vis. Commun. Image R., 16, 333–358, 2005.
* [39] F. Shi and I.W. Selesnick. Video denoising using oriented complex wavelet transforms. Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP) 2, 949–952, 2004.
* [40] H. Scherl, et al. Semi automatic level set segmentation and stenosis quantification of internal carotid artery in 3D CTA data sets. Medical Image Analysis, 11(1), 21–34, 2007.
* [41] J.-L. Starck, E.J. Candès, and D.L. Donoho. The curvelet transform for image denoising. IEEE Trans. Image Processing, 11(6), 670–684, 2000.
* [42] K. Sum and P. Cheung. Vessel extraction under non-uniform illumination: A level set approach. IEEE Trans. Biomed. Eng., 55(1), 358–360, 2008.
* [43] M. Unser. Texture classification and segmentation using wavelet frames. IEEE Trans. Image Process., 4(11), 1549–1560, 1995.
* [44] Z. Ye and C.-C. Lu. A complex wavelet domain Markov model for image denoising. Proc. IEEE Int. Conf. Image Processing, Barcelona. 3, 365–368, 2003.
|
arxiv-papers
| 2011-08-13T16:24:12 |
2024-09-04T02:49:21.871701
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaohao Cai, Raymond Chan, Serena Morigi, Fiorella Sgallari",
"submitter": "Xiaohao Cai",
"url": "https://arxiv.org/abs/1109.0217"
}
|
1109.0409
|
††institutetext: Centre for Theoretical Physics, Rijksuniversiteit Groningen
# The $\mathbb{Z}_{2}$ Two Higgs Doublet Model and the hierarchy problem
Maikel de Vries
###### Abstract
The hierarchy problem is investigated in the $\mathbb{Z}_{2}$ Two Higgs
Doublet extension of the Standard Model. From the quadratic divergences of the
Higgs field propagators fine-tuning constraints are extracted and solved. This
leads to the mass relation $m_{h^{0}}^{2}<2m_{t}^{2}<m_{H^{0}}^{2}$, where
$h^{0}$ is the analogue of the SM Higgs boson and $H^{0}$ a heavier neutral
scalar. The constraint leads to an admissible condition on the Higgs boson
mass, in contrast to a similar treatment for the Standard Model. Based on this
analysis it is argued that there may not be a little hierarchy problem for the
$\mathbb{Z}_{2}$ model, as it is the low energy counterpart of the MSSM.
## 1 Introduction
The mechanism for the introduction of masses for the Standard Model fermions
and gauge bosons remains experimentally unverified. Although the Higgs
mechanism is the most promising approach, there is still a lot of freedom to
extend the Standard Model. However, the Higgs mechanism suffers from the
hierarchy problem, i.e. the absence of hierarchy in energy scales of physical
theories. At energies of $100$ GeV physics is described by the experimentally
confirmed Standard Model, though beyond these energies only the GUT scale at
$10^{15}$ GeV — where the electroweak and strong couplings meet — and the
Planck scale at $10^{19}$ GeV — where quantum gravity appears — are
conjectured. In summary, beyond the Standard Model there is an unnatural
“energy desert” covering over twelve orders of magnitude.
## 2 Hierarchy and fine-tuning
In the Standard Model the lack of hierarchy translates to a problem called
fine-tuning. The Higgs boson mass receives quadratically divergent corrections
and the one-loop correction equals bib:VeltmanHierarchy
$\delta
m_{h}^{2}=\frac{\Lambda^{2}}{\pi^{2}v^{2}}\left(\frac{3m_{h}^{2}}{16}+\frac{3m_{W}^{2}}{8}+\frac{3m_{Z}^{2}}{16}-\frac{3m_{t}^{2}}{4}\right),$
(1)
where $v\approx 246\textrm{ GeV}$ and $\Lambda$ is the momentum cut-off in the
loop. The physical mass of the Higgs boson, equal to the pole of the
propagator, is given by
$m_{h,\textrm{phys}^{2}}=m_{h}^{2}+\delta m_{h}^{2}.$ (2)
Electroweak theory predicts a physical mass around $100$ GeV, which receives a
contribution of approximately $10^{15}$ GeV, as the GUT scale is inserted for
$\Lambda$. For the physical mass to stay around the electroweak scale the one-
loop contribution must be cancelled by the bare parameter in the Lagrangian.
This implies a fine-tuning of $28$ decimal places in this bare parameter. In
conclusion, the “energy desert” manifests itself as a fine-tuning problem in
the Higgs sector of the Standard Model. Many attempts to solve this issue have
been made, two of them are reviewed in the following sections.
### 2.1 Fine-tuning constraints
The first approach is obtained by demanding the absence of quadratic
divergences in a theory, which leads to constraints on that theory
bib:OslandGrzadkowski ; bib:OslandWu . For the Standard Model there is one
constraint, which reads:
$m_{h}^{2}=4m_{t}^{2}-2m_{W}^{2}-m_{Z}^{2}.$ (3)
With the fermion and gauge boson masses known from experiments
bib:ParticleData , the mass of the Higgs boson is required to equal
$m_{h}=311.6\pm 3.5\textrm{ GeV}$. This value is unrealistic within the known
mass bounds
$114\textrm{ GeV}<m_{h}<185\textrm{ GeV},$ (4)
obtained by the LEP Electroweak Working Group bib:LEPWorkingGroup . Note that
the upper bound from perturbation theory only holds in the framework of the
Standard Model.
#### Exact solution
The fine-tuning constraints originate from one-loop corrections, for an exact
solution of the fine-tuning problem the discussion should be extended to any
number of loops. In bib:EinhornJones it is argued that two scenarios may
appear: the cancellation of quadratic divergences order by order and
cancellation up to a certain order. The first scenario implies an infinite
number of constraints that must be satisfied. At first sight the second
scenario seems to differ, however, it is shown that also here an infinite
number of constraints appear bib:EinhornJones . Satisfying an infinite number
of constraints is highly unfeasible unless there is a symmetry in the theory,
e.g. supersymmetry. In conclusion, an exact solution to the fine-tuning
problem is not obtainable through the method of adopting fine-tuning
constraints.
#### Alleviating fine-tuning
Instead of attaining an exact solution, the fine-tuning problem may be
lessened. Through perturbation theory it is argued in bib:KoldaMurayama that
one-loop corrections dominate the fine-tuning problem up to a certain energy
scale $\Lambda$. Then, if the one-loop corrections vanish, i.e. the fine-
tuning constraints are satisfied, fine-tuning is absent below that energy
scale. Beyond that energy scale the amount of fine-tuning is reduced, and the
problem is alleviated.
Actually, it is enlightening to determine the energy scale below which no
fine-tuning is necessary. That energy scale is denoted as
$\Lambda^{\textrm{NP}}$, the scale at which new physics should appear for the
fine-tuning problem to be nonexistent. In the Standard Model no fine-tuning is
present if the one-loop correction to the Higgs boson mass (1) is roughly less
than the electroweak scale $v/2$, hence it is required that
$\left|\frac{1}{16\pi^{2}v^{2}}\left(\Lambda^{\textrm{NP}}_{\textrm{SM}}\right)^{2}\left(3m_{h}^{2}+6m_{W}^{2}+3m_{Z}^{2}-12m_{t}^{2}\right)\right|\lesssim\frac{v^{2}}{4}.$
(5)
Insertion of the experimental Higgs mass bounds (4) gives an energy scale of
new physics $\Lambda^{\textrm{NP}}_{\textrm{SM}}\approx 750-900\textrm{ GeV}$.
### 2.2 Minimal Supersymmetric Standard Model
The straightforward approach to the fine-tuning problem is removing all
quadratic divergences in a theory. Supersymmetry meets this demand through the
introduction of a symmetry between bosons and fermions. The simplest
supersymmetric extension of the Standard Model is the Minimal Supersymmetric
Standard Model (MSSM). For each of the particles in the Standard Model a
superpartner is introduced in order to cancel its quadratic divergent
corrections. Due to the nature of supersymmetry two scalar doublets are needed
to provide masses for both types of quarks, up and down. The problem of fine-
tuning is absent in the MSSM, although it still suffers from another problem:
the little hierarchy problem.
#### Little hierarchy problem
The little hierarchy problem occurs in the transition region from an effective
low energy theory (SM) and its supersymmetric counterpart (MSSM), for a review
see bib:Martin . Supersymmetry must be broken at some scale
$\Lambda_{\textrm{SUSY}}$, since superpartners have not been found in
experiments yet. Above this energy scale supersymmetry is exact, quadratic
divergences are cancelled and there is no need for fine-tuning. The same holds
at Standard Model energies $\Lambda_{\textrm{SM}}$, since the loop corrections
and the physical Higgs mass are of the same order. However, a problem appears
just below the scale of supersymmetry breaking where loop corrections are of
order $\Lambda_{\textrm{SUSY}}$. This implies a fine-tuning of roughly
$\Lambda_{\textrm{SUSY}}^{2}/\Lambda_{\textrm{SM}}^{2}$.
An estimate of the supersymmetry breaking scale can be obtained from the upper
bound on the mass of the lightest scalar $h^{0}$ in the MSSM. The bound
$m_{h^{0}}<m_{Z}$ is inconsistent with the lower bound on the Higgs mass (4),
however loop corrections from sparticles can raise the theoretical upper bound
on $m_{h^{0}}$ bib:Martin . To raise the upper bound to a realistic value
above the LEP bound, sparticle masses and thereby the supersymmetry breaking
scale must be around $1.5$ TeV. Since
$\Lambda_{\textrm{SUSY}}>\Lambda^{\textrm{NP}}_{\textrm{SM}}$, the transition
from Standard Model to MSSM does not occur without a little fine-tuning.
## 3 $\mathbb{Z}_{2}$ Two Higgs Doublet Model
Two Higgs Doublet Models (2HDMs) have two left-handed scalar doublets and were
first introduced in bib:Lee . These models appear in different types, for
comprehensive overviews see bib:2HDMOverview ; bib:Diaz2HDM . To analyse the
hierarchy problem and related topics as discussed in section 2 a specific
model needs to be chosen. The $\mathbb{Z}_{2}$ Two Higgs Doublet Model is
chosen, the justification for this choice is based on the facts that it:
* •
is the minimal non-trivial extension to the Standard Model Higgs sector.
* •
resembles the Standard Model Higgs sector as CP is conserved and tree-level
FCNCs are absent, beyond tree-level FCNCs are suppressed by the GIM mechanism
bib:GIMMechanism .
* •
equals the MSSM Higgs sector if the Yukawa Lagrangian is type-II and if the
potential satisfies additional constraints compared to the $\mathbb{Z}_{2}$
model discussed below bib:HaberHempfling .
A brief overview of the $\mathbb{Z}_{2}$ Two Higgs Doublet Model will be given
to clarify some aspects relevant for the discussion of the hierarchy problem.
### 3.1 2HDM Lagrangian
The Lagrangian of the $\mathbb{Z}_{2}$ 2HDM is the sum of the massless
Standard Model Lagrangian and the Lagrangian defined as
$\mathcal{L}_{\textrm{2HDM}}=(D_{\mu}\Phi_{1})^{\dagger}D^{\mu}\Phi_{1}+(D_{\mu}\Phi_{2})^{\dagger}D^{\mu}\Phi_{2}-V\left(\Phi_{1},\Phi_{2}\right)+L_{Y}$
(6)
The Lagrangian should be invariant under both CP and a $\mathbb{Z}_{2}$
symmetry, which is introduced between the doublets as $\Phi_{1}\to\Phi_{1}$,
$\Phi_{2}\to-\Phi_{2}$. The kinetic terms are invariant under these
symmetries, however, for the potential this implies a reduction from fourteen
to seven parameters bib:2HDMOverview . With the four gauge invariant
combinations of the scalar doublets defined as
$A\equiv\Phi_{1}^{\dagger}\Phi_{1},\quad
B\equiv\Phi_{2}^{\dagger}\Phi_{2},\quad
C\equiv\textrm{Re}\left(\Phi_{1}^{\dagger}\Phi_{2}\right),\quad
D\equiv\textrm{Im}\left(\Phi_{1}^{\dagger}\Phi_{2}\right),$ (7)
the potential is given by
$V\left(\Phi_{1},\Phi_{2}\right)=-\mu_{1}^{2}A-\mu_{2}^{2}B+\lambda_{1}A^{2}+\lambda_{2}B^{2}+\lambda_{3}C^{2}+\lambda_{4}D^{2}+\lambda_{5}AB.$
(8)
In the most general Yukawa Lagrangian both doublets couple to all fermion
types — up-type quarks, down-type quarks and charged leptons. The
$\mathbb{Z}_{2}$ symmetry restrict these couplings such that only one doublet
may couple to one type of fermion, dependent on how the symmetry is introduced
for the fermions. To resemble the MSSM, Type-II Yukawa interactions are
employed. In the fermionic sector the $\mathbb{Z}_{2}$ symmetry has to be
introduced as $e_{R},d_{R}\to e_{R},d_{R}$ and $u_{R}\to-u_{R}$, which leads
to a Yukawa Lagrangian of the form
$\mathcal{L}_{Y}=-\lambda_{e}\bar{l}_{L}\Phi_{1}e_{R}-\lambda_{d}\bar{q}_{L}\Phi_{1}d_{R}-\lambda_{u}\bar{q}_{L}i\sigma_{2}\Phi_{2}^{\ast}u_{R}+\;\textrm{h.c.}$
(9)
The $\mathbb{Z}_{2}$ symmetry ensures absence of Flavour Changing Neutral
Currents in the Yukawa sector bib:GlashowWeinberg and CP violation in the
potential bib:SantosBarrosoII . The only source for CP violation in this model
resides, as in the Standard Model, in the complex phase of the CKM matrix.
### 3.2 2HDM Phenomenology
In order to obtain the mass states it is necessary to find the minima of the
potential, both doublets then acquire a vacuum expectation value just as in
the Standard Model. There are two possibilities
$v_{1}=\frac{\mu_{1}^{2}}{\lambda_{1}},\;v_{2}=0\quad\textrm{and}\quad
v_{1}^{2}=\frac{\lambda_{2}\mu_{1}^{2}-\lambda_{+}\mu_{2}^{2}}{\lambda_{1}\lambda_{2}-\lambda_{+}^{2}},\;v_{2}^{2}=\frac{\lambda_{1}\mu_{2}^{2}-\lambda_{+}\mu_{1}^{2}}{\lambda_{1}\lambda_{2}-\lambda_{+}^{2}},$
(10)
where $\lambda_{+}\equiv\tfrac{1}{2}\left(\lambda_{3}+\lambda_{5}\right)$. The
choice where either $v_{1}$ or $v_{2}$ is zero leads to the Inert 2HDM (IDM)
bib:GustafssonIDM . After spontaneous symmetry breaking in the potential there
are three massless Goldstone scalars $G^{\pm},G^{0}$ corresponding to the weak
gauge bosons and five massive scalars $H^{\pm},H^{0},h^{0},A^{0}$. The masses
of the massive scalars in terms of the potential parameters read
$\displaystyle m_{H^{+}}^{2}$
$\displaystyle=-\frac{1}{2}\lambda_{3}\left(v_{1}^{2}+v_{2}^{2}\right)$
$\displaystyle m_{H^{0}}^{2}$
$\displaystyle=\lambda_{1}v_{1}^{2}+\lambda_{2}v_{2}^{2}+\sqrt{\left(\lambda_{1}v_{1}^{2}-\lambda_{2}v_{2}^{2}\right)^{2}+4\lambda_{+}^{2}v_{1}^{2}v_{2}^{2}}$
$\displaystyle m_{h^{0}}^{2}$
$\displaystyle=\lambda_{1}v_{1}^{2}+\lambda_{2}v_{2}^{2}-\sqrt{\left(\lambda_{1}v_{1}^{2}-\lambda_{2}v_{2}^{2}\right)^{2}+4\lambda_{+}^{2}v_{1}^{2}v_{2}^{2}}$
$\displaystyle m_{A^{0}}^{2}$
$\displaystyle=\frac{1}{2}\left(\lambda_{4}-\lambda_{3}\right)\left(v_{1}^{2}+v_{2}^{2}\right).$
(11)
The mass states originate from mixing between the doublets, which leads to two
mixing angles $\alpha,\beta$ defined as
$\tan\beta\equiv\frac{v_{2}}{v_{1}},\quad\tan
2\alpha\equiv\frac{2v_{1}v_{2}\lambda_{+}}{\lambda_{1}v_{1}^{2}-\lambda_{2}v_{2}^{2}}.$
(12)
There are two theoretical issues to address. The minimum of the potential must
be stable, which requires all scalar masses to be positive. Furthermore the
potential must be bounded from below, resulting in bib:Ivanov
$\lambda_{1}>0,\quad\lambda_{2}>0,\quad\lambda_{5}>-2\sqrt{\lambda_{1}\lambda_{2}},\quad\lambda_{3}+\lambda_{5}>-2\sqrt{\lambda_{1}\lambda_{2}},\quad\lambda_{4}+\lambda_{5}>-2\sqrt{\lambda_{1}\lambda_{2}}.$
(13)
As the mass states are known — the electroweak gauge mass states remain as in
the Standard Model — the Lagrangian (6) can be be expanded in terms of these
mass states, from which the Feynman rules involving Higgs scalars can be
straightforwardly obtained. They depend on six free parameters — the scalar
masses and mixing angles — and $v=v_{1}+v_{2}\approx 246\textrm{ GeV}$ and the
fermion and gauge boson masses. The Feynman rules for the $\mathbb{Z}_{2}$
2HDM have been presented in bib:SantosBarroso , these have been rederived to
provide a thorough check.
### 3.3 Fine-tuning in the $\mathbb{Z}_{2}$ 2HDM
In this section fine-tuning will be discussed in the context of the
$\mathbb{Z}_{2}$ 2HDM. The fine-tuning constraints can not be satisfied
consistently for the IDM bib:OslandGrzadkowski ; bib:NewtonWu and hence only
the regular model will be treated.
#### Fine-tuning constraints
To obtain the fine-tuning constraints as discussed in section 2.1 for the
$\mathbb{Z}_{2}$ 2HDM, all quadratically divergent corrections to each of the
six propagators are needed. This leads to six constraints, though only two of
those are independent and read
$\displaystyle 24m_{t}^{2}\cos^{2}\beta=\;$ $\displaystyle
6m_{H^{0}}^{2}\left(\cos 2\beta-\cos 2\alpha\right)+6m_{h^{0}}^{2}\left(\cos
2\beta+\cos 2\alpha\right)$ $\displaystyle 24m_{t}^{2}\cos^{2}\beta=\;$
$\displaystyle\sin^{2}2\beta\left(4m_{H^{+}}^{2}+2m_{A^{0}}^{2}+6m_{W}^{2}+3m_{Z}^{2}\right)$
$\displaystyle+m_{H^{0}}^{2}\left(6-6\cos 2\alpha\cos 2\beta+4\sin 2\alpha\sin
2\beta\right)$ $\displaystyle+m_{h^{0}}^{2}\left(6+6\cos 2\alpha\cos
2\beta-4\sin 2\alpha\sin 2\beta\right).$ (14)
The equations (3.3), together with the constraints (13), admit a solution for
$\alpha$ and $\beta$ provided that:
$\displaystyle 0\leq m_{H^{+}}^{2}<\infty,\quad 0\leq m_{A^{0}}^{2}<\infty,$
$\displaystyle 0<m_{h^{0}}^{2}<2m_{t}^{2}<m_{H^{0}}^{2}<\infty.$ (15)
If these relations for the scalar masses hold, a unique solution in the mixing
angles $\alpha$ and $\beta$ can be found numerically. This solution will
satisfy the constraint $\sin 2\beta>0$ and is invariant under shifts over
$\pi$. Hence, solutions are in correspondence with the experimental bounds on
the Higgs mass (4).
#### Little hierarchy problem
There is reason to expect that fine-tuning can be alleviated significantly
compared to the Standard Model, since after the fine-tuning constraints have
been satisfied additional free parameters are available in the
$\mathbb{Z}_{2}$ 2HDM. For this model the calculations are extended to six
different propagators, each of them receiving quadratically divergent
corrections, leading to six equations equivalent to (5). Numerical analysis
reveals that as long as equations (3.3) are satisfied, the scale of new
physics for the $\mathbb{Z}_{2}$ 2HDM can be raised to
$\Lambda_{\textrm{2HDM}}^{\textrm{NP}}\approx 2.5\textrm{ TeV}$ if the mixing
angles $\alpha$ and $\beta$ are chosen in the neighbourhood of the unique
solution.
In the analysis of the little hierarchy problem in section 2.2 a supersymmetry
breaking scale of $\Lambda_{\textrm{SUSY}}\approx 1.5\textrm{ TeV}$ has been
obtained. It was argued that a little fine-tuning is needed in the transition
from the Standard Model to the MSSM. However, this is a rather naive analysis,
since in fact the $\mathbb{Z}_{2}$ 2HDM is the effective low energy theory of
the MSSM. Therefore one should consider the scale of new physics for the
$\mathbb{Z}_{2}$ 2HDM, which results in
$\Lambda_{\textrm{SUSY}}\approx\Lambda_{\textrm{2HDM}}^{\textrm{NP}}$. Now
there is no need for any fine-tuning and the little hierarchy problem is
absent.
## 4 Conclusions
The hierarchy problem has been analysed in terms of fine-tuning constraints,
which arise from demanding the absence of quadratic divergences in a theory at
one-loop order. It has been shown that these cannot be satisfied within
experimental bounds (4) on the Higgs mass for the Standard Model. However, for
the $\mathbb{Z}_{2}$ Two Higgs Doublet Model a solution can be found provided
that $m_{h^{0}}^{2}<m_{t}^{2}<m_{H^{0}}^{2}$.
It has been shown that the method of adopting the fine-tuning constraints is
not able to solve the hierarchy problem to all orders. Nonetheless with this
method the problem of fine-tuning may be alleviated, and accordingly a scale
of new physics may be obtained below which there is no need for fine-tuning.
For the Standard Model that energy scale is around $800$ GeV. In the case of
the $\mathbb{Z}_{2}$ 2HDM the situation is improved and the scale of new
physics is around $2.5$ TeV as long as $m_{h^{0}}^{2}<m_{t}^{2}<m_{H^{0}}^{2}$
and the mixing angles are chosen suitably.
Furthermore it was argued that the little hierarchy problem may be absent,
since the $\mathbb{Z}_{2}$ 2HDM is the low energy effective theory of the
Minimal Supersymmetric Standard Model. For the $\mathbb{Z}_{2}$ 2HDM the scale
of new physics can be lifted to the scale of supersymmetry breaking and there
is no need for any fine-tuning.
###### Acknowledgements.
The author would like to thank M. de Roo and D. Boer for their valuable
feedback and suggestions, which greatly improved this article.
## References
* (1) M. Veltman, The Infrared-Ultraviolet Connection, Acta Physica Polonica B 12 (1981) 437–457.
* (2) B. Grzadkowski and P. Osland, A Natural Two-Higgs-Doublet Model, arXiv:0910.4068v1 [hep-ph] (2009).
* (3) P. Osland and T.T. Wu, Parameters in the electroweak theory I, II and III, Zeitschrift für Physik C 55 (1992) 569–583, 585–591, 593–604.
* (4) K. Nakamura et al., (Particle Data Group), J. Phys. G: Nucl. Part. Phys. 37 (2010) 075021.
* (5) LEP Electroweak Working Group. http://www.cern.ch/LEPEWWG/, July, 2010.
* (6) M. Einhorn and D. Jones, Effective potential and quadratic divergences, Physical Review D 46 (1992) 5206.
* (7) C. Kolda and H. Murayama, The Higgs mass and new physics scales in the minimal standard model, J. High Energy Phys. 07 (2000) 035.
* (8) S. Martin, A Supersymmetry Primer, arXiv:hep-ph/9709356v5 (2008).
* (9) T. Lee, A Theory of Spontaneous T Violation, Physical Review D 8 (1973) 1226–1239.
* (10) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Theory and phenomenology of two-Higgs-doublet models, arXiv:1106.0034v2 [hep-ph] (2011).
* (11) R. Diaz, Phenomenological analysis of the Two Higgs Doublet Model. PhD thesis, Universidad Nacional de Colombia, arXiv:hep-ph/0212237v2, 2003\.
* (12) S. Glashow, J. Iliopoulos and L. Maiani, Weak Interactions with Lepton-Hadron Symmetry, Physical Review D 2 (1970) 1285–1292.
* (13) H. Haber and R. Hempfling, Renormalization-group-improved Higgs sector of the minimal supersymmetric model, Physical Review D 48 (1993) 4280–4309.
* (14) S. Glashow and S. Weinberg, Natural conservation laws for neutral currents, Physical Review D 15 (1977) 1958–1965.
* (15) J. Velhinho, R. Santos and A. Barroso, Tree level vacuum stability in two-Higgs-doublet models, Physics Letters B 322 (1993) 213.
* (16) M. Gustafsson, The Inert Doublet Model and its Phenomenology, arXiv:1106.1719v1 [hep-ph] (2011).
* (17) I. Ivanov, Minkowski space structure of the Higgs potential in the two-Higgs-doublet model, Physical Review D 75 (2007) 035001.
* (18) R. Santos and A. Barroso, Renormalization of two-Higgs-doublet models, Physical Review D 56 (1997) 5366–5385.
* (19) C. Newton and T.T. Wu, Mass relations in the two Higgs doublet model from the absence of quadratic divergences, Zeitschrift für Physik C 62 (1994) 253–263.
|
arxiv-papers
| 2011-09-02T11:03:51 |
2024-09-04T02:49:21.884213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maikel de Vries",
"submitter": "Maikel de Vries",
"url": "https://arxiv.org/abs/1109.0409"
}
|
1109.0470
|
Non Hamiltonian Chaos from Nambu Dynamics of Surfaces111Published in Chaos
Theory: Modeling, Simulation and Applications,
C.H. Skiadas, I. Dimotikalis and C. Skiadas (Eds),
World Scientific Publishing Co, pp. 110-119.
Minos Axenides
Institute of Nuclear Physics, NCSR Demokritos, 15310 Agia Paraskevi, Attiki,
Greece
(E-mail: axenides@inp.demokritos.gr)
###### Abstract
We discuss recent work with E.Floratos (JHEP 1004:036,2010) on Nambu Dynamics
of Intersecting Surfaces underlying Dissipative Chaos in $R^{3}$. We present
our argument for the well studied Lorenz and Rössler strange attractors. We
implement a flow decomposition to their equations of motion. Their volume
preserving part preserves in time a family of two intersecting surfaces, the
so called Nambu Hamiltonians. For dynamical systems with linear dissipative
sector such as the Lorenz system, they are specified in terms of Intersecting
Quadratic Surfaces. For the case of the Rössler system, with nonlinear
dissipative part, they are given in terms of a Helicoid intersected by a
Cylinder. In each case they foliate the entire phase space and get deformed by
Dissipation , the irrotational component to their flow. It is given by the
gradient of a surface in $R^{3}$ specified in terms of a scalar function. All
three intersecting surfaces reproduce completely the dynamics of each strange
attractor.
## 1 Introduction
Dissipative dynamical systems, with a low dimensional phase space, present an
important class of simple non-linear physical systems with intrinsic complex
behavior (homoclinic bifurcations, period doubling, onset of chaos,
turbulence), which generated intense experimental, theoretical and numerical
work in the last few decades [1, 2].
Recently [3] we have reexamined dissipative dynamical systems with a
3-dimensional phase space from the perspective of Nambu-Hamiltonian
Mechanics(NHM) [4, 5]. The latter represents a generalization of Classical
Hamiltonian Mechanics, mostly appropriate for the study of odd-dimensional
phase-space volume preserving flows( Liouville’s theorem). As such and in
order to make it directly applicable to the dynamics of dissipative systems in
$R^{3}$ we must associate it to a volume preserving dynamics sector. We have
done so by introducing a flow decomposition to their equations of motion and
therefore isolate in their flow vector field its rotational(solenoidal) part.
It is manifestly volume preserving, non-dissipative and hence directly
describable in terms of the intersecting surfaces of NHM. Along with the
remaing irrotational component to the flow the decomposition fully recovers
the dissipative dynamics of the system in question, presenting itself as an
equivalent formulation of the odd-dimensional dynamical system.
We have applied the above recipe to the famous examples of Lorenz[6] and
Rössler [7] chaotic attractors which represent the prototype models for the
onset of turbulence.[2].
In sect. 2 we start off with a discussion of flow decompostion for the most
elementary and familiar of all dissipative systems in $R^{2}$ without chaotic
behavior : the dissipative harmonic oscillator (DHO). In this case the
existence of a Hamiltonian formalism identifies for its ”closed”
nondissipative sector the harmonic oscilator (HO) with a well defined
integrable classical and quantum evolution.
The presence of chaotic flows for dissipative systems in $R^{3}$ is argued, by
analogy, to nescessitate intersecting Nambu Surfaces which define
rerspectively ”closed” physical systems with integrable $R^{3}$ periodic
orbits and simultaneously a well defined classical and quantum behaviour.
In sect. 3 we apply the framework for the cases of the Lorenz and Rössler
strange attractors. We isolate their non-dissipative sector parametrized by
two intersecting surfaces: a cylinder and a paraboid for the Lorenz attractor
as well as a helicoid with a cylinder for the Rössler system. They account,
amazingly, for the double scroll topology of the full ” butterfly” Lorenz
attractor. and the single scroll topology for the Rössler case. We end our
presentation with Conclusions and open problems.
## 2 Flow Decomposition in Dissipative Systems: $R^{2}$ versus $R^{3}$
Dissipation is a necessary condition for dynamical systems to exhibit chaos.
Yet it is not suficient. This is enunciated through a powerful No-go theorem .
Indeed the Poincare-Bendixon alows for only fixed points and limits cycles in
two dimensions. Chaotic Flows need space to emerge. At the minimal level they
emerge with dissipative systems in $R^{3}$ . They are typically associated
with Strange Attractors such as the famous ones of Lorenz and Rössler. They
belong to a large class of dynamical systems whose dynamics is governed by
continuous set of 1st order ordinary differental flow equations
$\dot{\vec{x}}\ =\ \vec{v}(\vec{x}(t),t,\lambda)$ (1)
where $\vec{v}$ is a velocity field flow with $\lambda$ some control external
parameter. Their phase space dynamics, depending on whether they exchange
energy with their environment or not, can be either open-dissipative or closed
(conservative-Hamiltonian) . This is reflected on their velocity flow field
being divergenceless or not. Let us see all these issues for the simple case
of the Dissipative Harmonic Oscillator (DHO) whose phase space equations of
motion are well known to be for $x_{1}=q$ and $x_{2}=p$:
$\displaystyle\dot{q}\ $ $\displaystyle=$ $\displaystyle\ p\ +\ \alpha q$
$\displaystyle\dot{p}\ $ $\displaystyle=$ $\displaystyle\ -q+\beta p$ (2)
The velocity flow is given by $\nabla\dot{v}_{tot}=(\alpha+\beta)\neq 0$ for
appropriate values $\alpha\neq-\beta\neq 0$ with a net outflow or inflow of
energy depending on the sign of $\nabla\dot{v}_{tot}$ (inflow $<0$ or outflow
$>0$). Nevertheless we split it into a nondissipative component
$v_{ND}=(p,-q)$ with zero flow $\nabla\dot{v}_{ND}=0$. Its dissipative part is
given by $v_{D}=\alpha q+\beta p$ with its flow being simply the total flow of
the HO $\nabla v_{D}=\alpha+\beta$. The dynamics of the HO in one dimension
can take the form of a 2nd order differential equation
$\ddot{q}+(\alpha+\beta)\dot{q}+(1+\alpha\beta)q=0$ (3)
The damping factor is $\gamma=\alpha+\beta$ and the effective natural
frequancy which is dissipation induced is given by
$\omega_{eff}=1+\alpha\beta$ .
The existence of the Harmonic Oscillator Hamiltonian
$H=\frac{1}{2}(p^{2}+q^{2})$ along with a Dissipation function
$D=\frac{1}{2}(\alpha q^{2}+\beta p^{2})$ reproduce the eqs. of motion of
(2.2) in a more compact form
$\dot{x}^{i}\ =\ \epsilon^{ij}(\partial_{j}H+\partial_{j}D)$ (4)
where $i,j=1,2$ and $\epsilon^{12}=-\epsilon^{21}=1$ as usual. The non-
dissipative sector of the HO is given by
$\displaystyle\dot{q}\ $ $\displaystyle=$ $\displaystyle\ \frac{\partial
H}{\partial p}\ =\ p$ $\displaystyle\dot{p}\ $ $\displaystyle=$
$\displaystyle\ -\frac{\partial H}{\partial q}\ =\ -q$ (5)
Firstly it identifies a circular periodic orbit, the famous HO as the
integrable progenitor of the transient DHO . This is a well defined closed
physical system with a most familiar quantum behavior, the Quantum Harmonic
Oscillator.
Secondly the linear dissipation of the DHO a transient spirals it inwards to a
fixed point of zero energy. The harmonic oscillator localizes in effect such
a dissipative evolution .
Lastly the rate at which the energy of the harmonic oscillator loses its
energy as it spirals in depends on its damping strength. It is the Quality
factor which is easily computed to be
$Q\ =\ \frac{\omega_{eff}}{\gamma}\ =\ \frac{1+\alpha\beta}{\alpha+\beta}$ (6)
All in all Flow decomposition for the transient DHO has three immediate
implications for the DHO: 1\. Existence of Integrable Harmonic oscillator
associated with a periodic orbit 2\. Localization of the DHO evolution and
asymptotic fixed point state 3\. Quality factor for DHO as a measure of its
energy loss rate and in effect of the damping strenth.
We will proceed now to examine whether it is feasible to implement this
methodology for the case of Strange attractors which possess chaotic flows in
$R^{3}$.
Nambu-Hamiltonian mechanics is a specific generalization of classical
Hamiltonian mechanics, where the invariance group of canonical symplectic
transformations of the Hamiltonian evolution equations in 2n dimensional phase
space is extended to the more general volume preserving transformation group
$SDiff({\cal M})$ with an arbitrary phase space manifold M of any dimension
$d=\mbox{dim}({\cal M})$ .
In [3] we work with the case of a three dimensional flat phase space manifold.
Nevertheless our results could be generalized to curved manifolds of any
dimension[5, 8].
Nambu-Hamiltonian mechanics of a particular dynamical system in $R^{3}$ is
defined once two scalar functions $H_{i}\in C^{\infty}(R^{3}),i=1,2$, the
generalized Hamiltonians [4, 5] are provided. The evolution equations are:
$\dot{x}^{i}\ =\ \\{x^{i},H_{1},H_{2}\\}\ \ \ \ \ \ \ \ i=1,2,3$ (7)
where the Nambu 3-bracket, a generalization of Poisson bracket in Hamiltonian
mechanics, is defined as
$\\{f,g,h\\}\ =\ \epsilon^{ijk}\partial^{i}f\partial^{j}g\partial^{k}h\ \ \ \
\ \ i,j,k=1,2,3\ \ \ \ \ \ \forall f,g,h\in C^{\infty}(R^{3})$ (8)
Any local coordinate transformation
$x^{i}\ \rightarrow\ y^{i}\ =\ y^{i}(x)\ \ \ \ \ \ \ \ \ i=1,2,3$ (9)
which preserves the volume of phase space
$\mbox{det}\left(\frac{\partial y^{i}}{\partial x^{j}}\right)\ =\ 1\ \ \ \ \
\forall\ x=(x^{1},x^{2},x^{3})\in R^{3}$ (10)
leaves invariant the 3-bracket and therefor it is a symmetry of Nambu-
Mechanics. Except for the linearity and antisymmetry of the bracket with
respect to all of its arguments it also satisfies an important identity, the
so called ”Fundamental identity”[FI] [3, 8].
The evolution eq.(2.7) has a flow vector field:
$v^{i}(x)\ \ =\ \ \epsilon^{ijk}\partial^{j}H_{1}\partial^{k}H_{2}\ \ \ \ \ \
\ \ \ i,j,k=1,2,3$ (11)
which is volume preserving
$\partial^{i}v^{i}\ =\ 0$ (12)
The reverse is also true. We will name the phase-space volume preserving flows
”Non-dissipative” while the non-conserving ones ”Dissipative”
($\partial^{i}v^{i}(x)>0(<0))$.
The flow equations of a general dissipative system may take the general vector
form
$\dot{\vec{x}}\ \ =\ \ \vec{\nabla}H_{1}\times\vec{\nabla}H_{2}+\vec{\nabla}D$
(13)
We notice that given a pair of functions $H_{1},H_{2}$ such that
$\vec{v}_{ND}\ \ =\ \ \vec{\nabla}H_{1}\times\vec{\nabla}H_{2}$ (14)
any transformation of $H_{1},H_{2}$
$H_{i}\rightarrow H_{i}^{\prime}(H_{1},H_{2})\ \ \ \ \ i=1,2$ (15)
with unit Jacobian
$\mbox{det}\left(\frac{\partial H_{i}^{\prime}}{\partial H_{j}}\right)\ =\ 1$
(16)
gives also
$\vec{v}_{ND}\ =\ \vec{\nabla}H_{1}^{\prime}\times\vec{\nabla}H_{2}^{\prime}$
(17)
In ref.[8] we reduced the evolution equation of the form (2.32) Nambu
Mechanics in Hamiltonian-Poisson form as follows:
$\dot{x}^{i}\ =\ \\{x^{i},H_{1},\\}_{H_{2}}$ (18)
where the induced Poisson bracket
$\\{f,g\\}_{H_{2}}\ =\
\epsilon^{ijk}\partial^{i}f\partial^{j}g\partial^{k}H_{2}$ (19)
satisfies all the required properties like linearity, antisymmetry and the
Jacobi identity.
In the following sections we are going to present a detailed investigation of
the Lorenz and Rössler attractors from the point of view of Dissipative Nambu-
Hamiltonian Dynamics.
## 3 The Lorenz Attractor from Dissipative Dynamics of Intersecting
Quadratic Surfaces .
The Lorenz model was invented as a three Fourier mode truncation of the basic
eqs. for heat convection in fluids in Reyleigh-Benard type of experiments [12]
The time evolution eqns. in the space of three Fourier modes $x,y,z$ which we
identify as phase-space are :
$\displaystyle\dot{x}\ $ $\displaystyle=$ $\displaystyle\ \sigma(y-x)$
$\displaystyle\dot{y}\ $ $\displaystyle=$ $\displaystyle\ x(r-z)-y$
$\displaystyle\dot{z}\ $ $\displaystyle=$ $\displaystyle\ xy-bz$ (20)
where $\sigma$ is the Prandtl number, r is the relative Reynolds number and b
the geometric aspect ratio .
The standard values for $\sigma$, b are $\sigma=10,b=\frac{8}{3}$ with r
taking values in $1\leq r<\infty$. There are dramatic changes of the system as
r passes through various critical values which follow the change of stability
character of the three critical points of the system $P_{1}$ :
$x=y=z=0,P_{\pm}:x=y=\pm\sqrt{b(r-1)},z=r-1$
Lorenz discovered the non-periodic deterministic chaotic orbit for the value
$r=28$, which is today identified as a Strange Attractor with a Hausdorff
dimension of ($d=2.06$)[13]. Standard reference for an exhaustive numerical
investigation of the Lorenz system is the book by Sparrow [14].
There have been various attempts made to localize the Lorenz attractor, by
convex surfaces, in order to get information about Hausdorff dimensions [13]
and other characteristics [15].
We will proceed to exhibit Localization of the full Lorenz attractor from the
Nambu surfaces which we will determine shortly. At the Quantum level the
existence of an attracting ellipsoid in a matrix model formulation of the
Lorenz system is manifest [3] as one gets attracting ellipsoids in higher
dimensional phase spaces.
We now proceed to describe the Lorenz system in the framework of section 2.
The flow vector field $\vec{v}$ is analyzed into its dissipative and non-
dissipative parts as follows:
$\vec{v}_{D}\ =\ (-\sigma x,-y,-bz)\ =\ \vec{\nabla}D$ (21)
with the ”Dissipation” function
$D\ =\ -\frac{1}{2}\ (\sigma x^{2}\ +\ y^{2}\ +\ bz^{2})$ (22)
and
$\vec{v}_{ND}\ =\ (\sigma y,x(r-z),xy)\ =\ (0,y,z-r)\times(-x,0,\sigma)$ (23)
The two Hamiltonians or Clebsch-Monge potentials $H_{1},H_{2}$ are determined
by
$\vec{\nabla}H_{1}\times\vec{\nabla}H_{2}\ =\ \vec{v}_{ND}$ (24)
or equivalently
$H_{1}\ =\ \frac{1}{2}[y^{2}+(z-r)^{2}]$ (25)
and
$H_{2}\ =\ \sigma z\ -\ \frac{x^{2}}{2}$ (26)
The Lorenz system (20) can thus be written in the equivalent form in terms of
$\vec{r}=(x,y,z)$:
$\dot{\vec{r}}\ =\ \vec{\nabla}H_{1}\times\vec{\nabla}H_{2}\ +\ \vec{\nabla}D$
(27)
In the Non-Dissipative part(ND) of the dynamical system
$\dot{\vec{r}}\ =\ \vec{\nabla}H_{1}\times\vec{\nabla}H_{2}$ (28)
the Hamiltonians $H_{1},H_{2}$ are conserved and their intersection defines
the ND orbit. Moreover if we get the reduced Poisson structure (sect.2) from
$H_{2}$ we obtain the 2-dim phase space $\Sigma_{2}$ to be the family of
parabolic cylinders with symmetry axis the y-axis:
$H_{2}=\mbox{constant}=H_{2}(\vec{r_{o}})$ (29)
$\Sigma_{2}$ is thus given by
$z\ =\ z_{o}+\frac{x^{2}-x_{o}^{2}}{2\sigma}$ (30)
with $x_{o},z_{o}$ the initial condition for $x,z$. The induced Poisson
algebra (rel.19) is given by
$\displaystyle\\{x,y\\}_{H_{2}}\ $ $\displaystyle=$ $\displaystyle\
\partial_{z}H_{2}\ =\sigma$ $\displaystyle\\{y,z\\}_{H_{2}}\ $
$\displaystyle=$ $\displaystyle\ \partial_{x}H_{2}\ =-x$
$\displaystyle\\{z,x\\}_{H_{2}}\ $ $\displaystyle=$ $\displaystyle 0$ (31)
The dynamics on the 2d-phase space $\Sigma_{2}$ is given by $H_{1}$
$\displaystyle\dot{x}\ $ $\displaystyle=$ $\displaystyle\
\\{x,H_{1}\\}_{H_{2}}\ \ \ \ $ $\displaystyle\dot{y}\ $ $\displaystyle=$
$\displaystyle\ \ \\{y,H_{1}\\}_{H_{2}}$ $\displaystyle\dot{z}\ $
$\displaystyle=$ $\displaystyle\ \ \\{z,H_{1}\\}_{H_{2}}$ (32)
and $H_{1}$ is an anharmonic oscillator Hamiltonian with $(x/\sigma,y)$
conjugate canonical variables. Using rel.(25-26) we get on $\Sigma_{2}$:
$H_{1}\ =\ \frac{1}{2\sigma^{2}}\ [y^{2}\ +\ \frac{1}{2}\ (x^{2}\ -\
a^{2})^{2}]$ (33)
with
$a^{2}\ =\ x_{o}^{2}\ -\ 2\sigma(z_{o}\ -\ r)\ =\ -2H_{2}\ +\ 2\sigma r$ (34)
where $\frac{1}{\sigma^{2}}$ plays the role of the mass. Depending on the
initial conditions we may have a single well $(a^{2}\leq 0,H_{2}\geq\sigma r)$
or a double well potential ( $a^{2}>0,H_{2}<\sigma r$ ) respectively. The
trajectories , the intersections of the two cylinders , $H_{1}$ and $H_{2}$
with orthogonal symmetry axes $(x,y)$ may either have one lobe left/right or
may be running from the right to the left lobe. This is reminiscent of the
topology structure of the orbits of the Lorenz chaotic attractor.
The full Lorenz system does not conserve $H_{1},H_{2}$ and there is a random
motion of the two surfaces against each other. Their intersection is time
varying. In effect at every moment the system jumps from periodic to periodic
orbit of the non-dissipative sector. Moreover the motion of the non-
dissipative system around the two lobes, either left or right, can now jump
from time to time from one lobe to the other.
## 4 The Rössler Attractor from Dissipative Dynamics of a Cylinder
Intersecting with an Helicoid
Rössler introduced a simpler than Lorenz’s nonlinear ODE system with a 3d-
phase space, in order to study in more detail the characteristics of chaos,
which is motivated by simple chemical reactions [7].
The Rössler system is given by the evolution eqns:
$\displaystyle\dot{x}\ $ $\displaystyle=$ $\displaystyle\ -y-z$
$\displaystyle\dot{y}\ $ $\displaystyle=$ $\displaystyle\ x+ay$
$\displaystyle\dot{z}\ $ $\displaystyle=$ $\displaystyle\ b+z(x-c)$ (35)
with parameters $a,b,c$ usually taking standard values $a=b=0.2,c=5.1$ or
$a=b=0.1,c=14$ for the appearance of the chaotic attractor.
We turn now to the study of the Rössler system as a Dissipative Nambu-
Hamiltonian dynamical system.
The key difference with the Lorenz attractor is that the dynamics of the
system is simpler. Chaos appears as random jumps outwards and inwards the
single lob attractor.
In order to get the three scalars, the two generalized Hamiltonians
$H_{1},H_{2}$ which are conserved and characterize the non-dissipative part
and D the dissipation term :
$\dot{\vec{r}}\ =\ \vec{\nabla}H_{1}\times\vec{\nabla}H_{2}\ +\ \vec{\nabla}D$
(36)
we checked after some guess work, that we must subtract and add a new term in
the first equation. Indeed we find for the two parts
$\vec{v}_{ND}\ =\ (-y-z-\frac{z^{2}}{2},x,b)$ (37) $\vec{v}_{D}\ =\
(\frac{z^{2}}{2},ay,z(x-c))$ (38)
satisfying accordingly $\vec{\nabla}\cdot\vec{v}_{ND}\ =0$ and
$\vec{\nabla}\times\vec{v}_{D}\ =\ 0$
We must determine $H_{1},H_{2}$ and D such that
$\vec{\nabla}H_{1}\times\vec{\nabla}H_{2}\ =\ \vec{v}_{ND}$ (39)
and
$\vec{\nabla}D\ =\ \vec{v}_{D}$ (40)
For D we find easily
$D\ =\ \frac{1}{2}\ [ay^{2}+(x-c)z^{2}]$ (41)
To get $H_{1},H_{2}$ we must integrate first the Non-dissipative system:
$\displaystyle\dot{x}\ $ $\displaystyle=$ $\displaystyle\
-y-z-\frac{z^{2}}{2}\ \ \ \ \ $ $\displaystyle\dot{y}\ $ $\displaystyle=$
$\displaystyle\ x\ \ \ \ \ \ \ $ $\displaystyle\dot{z}\ $ $\displaystyle=$
$\displaystyle\ b$ (42)
The general solution is :
$x(t)\ =\ -b(1+z(t))\ +\ (x_{o}+b(1+z_{o}))\mbox{
cos(t)}-(y_{o}+z_{o}+\frac{z_{o}^{2}}{2}-b^{2})\mbox{ sin (t)}$ (43) $y(t)\ =\
b^{2}-z(t)-\frac{z^{2}(t)}{2}+(y_{o}+z_{o}+\frac{z^{2}_{o}}{2}-b^{2})\mbox{
cos(t)}\ +\ (x_{o}+b+bz_{o})\mbox{ sin (t)}$ (44) $z(t)\ =\ bt+z_{o}$ (45)
To uncover $H_{1},H_{2}$ we introduce the complex variable
$w(t)\ =\ w_{1}(t)+iw_{2}(t)$ (46)
with
$\displaystyle w_{1}(t)\ $ $\displaystyle=$ $\displaystyle\ x(t)\ +\
b(1+z(t))$ $\displaystyle w_{2}(t)\ $ $\displaystyle=$ $\displaystyle\
y(t)+z(t)+\frac{z^{2}(t)}{2}-b^{2}$ (47)
We obtain
$w(t)\ =\ w_{o}\cdot e^{it}$ (48)
with
$w_{o}\ \equiv\ w(t=0)$ (49)
We see that there are two constants of motion, the first one being :
$\mid w(t)\mid\ =\ \mid w_{o}\mid$ (50)
and we define correspondingly,
$H_{1}\ =\ \frac{1}{2}\mid w(t)\mid^{2}\ =\ \frac{1}{2}\ (x+b(1+z))^{2}\ +\
\frac{1}{2}\ (y+z+\frac{z^{2}}{2}-b^{2})^{2}$ (51)
The second integral of motion is obtained through the phase
$w(t)\ =\ \mid w_{o}\mid\cdot e^{\imath\varphi_{o}}\cdot e^{\imath t}\ =\ \mid
w_{o}\mid e^{\imath\varphi(t)}$ (52)
or from (4.14)
$\varphi(t)\ -\ \frac{z}{b}\ =\ \varphi_{o}-\frac{z_{o}}{b}$ (53)
and we define appropriately the second constant surface $H_{2}$
$H_{2}\ =\ b\ \ \mbox{arctg}\frac{y+z+\frac{z^{2}}{2}-b^{2}}{1+b(1+z)}\ \ -\
z$ (54)
We easily check that rel.(41) is satisfied:
$\vec{\nabla}H_{1}\times\vec{\nabla}H_{2}\ =\ (-y-z-\frac{z^{2}}{2},x,b)$ (55)
The family of surfaces $H_{1}$ and $H_{2}$ are a quadratic deformation of a
cylinder and respectively a quadratic deformation of a right helicoid. Their
intersection is the trajectory (43-44).
## 5 Conclusions-Open Problems
The main result of our present work is the demonstration of Dissipative Nambu-
Hamiltonian mechanics of intersecting surfaces as the conceptual framework
that underlies strange chaotic attractors both in their classical as well as
quantum-noncommutative incarnation. It reproduces the familiar and well
studied attractor dynamics of Lorenz and Rössler in a very intuitive manner
accounting of their gross topological aspects (double lobe Butterfly for the
Lorenz system ) or single lobe for the Rössler attractor.
Quantum Nambu Dynamics of Surfaces [3, 8, 9], raises also the issue of
possible existence of Quantum Strange Attractors. Their Quantum behavior was
built systematically through fuzzifying the classical intersecting surfaces
of the ND sector. We demonstrated this for the simplest case of the Lorenz
system with a linear dissipation.
## References
* [1] P. Cvitanovic Ed., Universality in Chaos Adam Holger, Bristol 1984.
* [2] J.P. Eckmann, Roads to Turbulence in Dissipative Dynamical Systems Rev. Mod. Phys. 53 no.4 (1981) 643.
* [3] M. Axenides and E. Floratos, Strange Attractors in Dissipative Nambu Mechanics: Classical and Quantum Aspects JHEP 1004 (2010) 036 [arXiv:0910.3881 [nlin.CD]].
* [4] Y. Nambu, Generalized Hamiltonian Dynamics Phys. Rev. D 7, (1973) 2403.
* [5] L. Takhtajan, On Foundation Of The Generalized Nambu Mechanics (Second Version) Commun. Math. Phys. 160 , (1994) 295.
* [6] E.N. Lorenz, Deterministic Non-Periodic Flow J.Atm.Sci. 20 (1963), 130.
* [7] O.E. Rössler, An Equation for Continuous Chaos Phys. Lett.57A, (1976) 397.
* [8] M. Axenides and E. Floratos, Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds JHEP 0902 (2009) 039 [arXiv:0809.3493 [hep-th]].
* [9] M. Axenides, E. G. Floratos and S. Nicolis, Nambu Quantum Mechanics on Discrete 3-Tori J. Phys. A 42 (2009) 275201 [arXiv:0901.2638 [hep-th]].
* [10] U. Weiss, Quantum Dissipative Systems World Scientific- Series in Modern Condensed Matter Physics, vol.13 2008; C-I. Um, K-H. Yeon, T.F. George, The Quantum Damped Harmonic Oscillator Phys. Rep. 362 (2002) 63; M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics Springer-Verlag (1990), New York.
* [11] A. Clebsch , J.Reine Angew. Math. 56 (1859) 1 .
* [12] M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction Wiley-Interscience (1989).
* [13] D. Farmer, E. Ott and J.A. Yorke, Hausdorff Dimension computation Physica 7D (1983) 153.
* [14] C. Sparrow, The Lorenz Equation, Bifurcations, Chaos and the Strange Attractors, Springel-Verlag, New York 1987.
* [15] C. Doering and J. Gibbon, On the shape and Dimension of the Lorenz Attractor Dynamics and Stability of Systems , vol10, No.3, (1995), 255 ; ibid, vol13, No.3,(1998)299.
|
arxiv-papers
| 2011-09-02T15:09:58 |
2024-09-04T02:49:21.890153
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Minos Axenides",
"submitter": "Minos Axenides",
"url": "https://arxiv.org/abs/1109.0470"
}
|
1109.0596
|
# Discrete Wigner Function Reconstruction and Compressed Sensing
Jia-Ning Zhang galileo@mail.nankai.edu.cn Lei Fang Mo-Lin Ge Theoretical
physics division, Chern Institute of Mathematics, Nankai University, Tianjin
300071, China
###### Abstract
A new reconstruction method for Wigner function is reported for quantum
tomography based on compressed sensing. By analogy with computed tomography,
Wigner functions for some quantum states can be reconstructed with less
measurements utilizing this compressed sensing based method.
###### pacs:
03.65.Wj, 42.30.Wb
Wigner function(WF), a quasi-probability distribution in phase space, was
first introduced to describe quantum state in quantum mechanics by E. P.
Wignerwigner . And later, it was extended to classical optics and signal
processing. Since its birth, a great number of applications have been
conducted in different fields. As for the original quantum case, like a
continuous one-dimensional quantum system, the Wigner function is defined as
$\displaystyle
W(x,p)\equiv\frac{1}{\pi}\int_{-\infty}^{\infty}d\xi\exp(-2ip\xi)\langle
x+\xi|\hat{\rho}|x-\xi\rangle$ (1)
where $x$ and $\xi$ are positions and $p$ is momentum, $\hat{\rho}$ denotes
the density operator, and $\hbar$ is set to 1 for simplicity.
One of the advantage for the Wigner formalism is that a tomographic scheme
which could be used to reconstruct the WF. On the other hand, in medical
tomography, like X-ray computed tomography, X-ray photons transmitted through
the patient along projection lines from one side, and detectors measured the
number of transmitted photons on the other side, then from the distribution of
X-ray photons in different incident angles, linear attenuation coefficients in
the slice being imaged are recovered. Likewise, in phase space tomography, the
WF tomographic reconstruction is more or less the same, based on a set of
intensity measurementsvogelpra . This is because both of the two tomographic
schemes share the same mathematical basis — Radon transform(Eqn.(2)) and its
inverse(Eqn.(3)).
$\displaystyle pr(x_{\phi},\phi)=\int\int
f(x,y)\delta(x_{\phi}-x\cos\phi-y\sin\phi)dxdy$ (2)
$\displaystyle
f(x,y)=-\frac{1P}{2\pi^{2}}\int_{0}^{\pi}\int_{-\infty}^{+\infty}\frac{pr(x_{\phi},\phi)dx_{\phi}d\phi}{(x\cos\phi+y\sin\phi-
x_{\phi})^{2}}$ (3)
Where $P$ denotes the Cauchy’s principle value, $pr(x_{\phi},\phi)$ denotes
the measured distribution, in $f(x,y)$ is the quantity in tomography to be
recovered. In tomography, objective function is reconstructed using back-
projection algorithm for the inverse Radon transformation(Eqn.(3)). However,
such a algorithm has some drawbacks, for example, many measurements are needed
to achieve a relatively good recovery. To reduce the dose of radiation on the
patient in CT or make a more efficient estimate of quantum state in phase
space tomography, the amount of measurements should be as small as possible
but guarantee a good enough recovery. Nevertheless, according to the well-
known Nyquist-Shannon sampling theoremunser , it seems impossible to depress
the measurements without giving a negative impact on the resolution.
Recently, with the arise of a novel theory called compressed sensing
(compressive sampling or CS) candes , people find a fundamentally new approach
to acquire data. It demonstrates that if the images or signals are sparse, we
can recover them from what was believed to be highly incomplete measurements.
In fact, it is just the application to medical tomography that started the
seminal paper in CS. For example, in CT, X-ray images of human body are sparse
in the wavelet and gradient representation, then compressed sensing is applied
to reduce the irradiation and obtain clearer medical imagessidky ; Chen . And
then an immediate question would be is it possible to reconstruct WF in phase
space tomography via compressed sensing? In fact, there exists infinite family
of quasi-probability distribution in phase, including Husimi-Kando Q-Function,
Glauber-Sudarshan P-Function and so on, but WF is chose for it is directly
related to the quadrature histograms and can be easily measured through
experiments.
In this brief report, we explore the similarity between phase space tomography
and computed tomography, and report a new discrete WF reconstruction method
based on compressed sensing. We mention that some compressed sensing protocols
have been proposed to reconstruct density matrix though quantum
tomographyGrossprl .
Suppose that a signal $x\in\mathbb{C}^{N}$ is sparse in some basis $x=\Psi
x^{\prime}$, which means $x^{\prime}$ has $k(k\ll N)$ nonzero entries. Then a
measurement matrix $\Phi\in\mathbb{C}^{m\times N}$ is used to sense $x$ and
obtain a measurement vector $y\in\mathbb{C}^{m}$. The central idea of
compressed sensing is that it is possible to reconstruct sparse signals of
scientific interest accurately and sometimes exactly by a number of incoherent
sampling which is far smaller than N. In general, such a recovery of $x$ by
performing $\ell_{0}$-minimization is a NP-hard problem. An alternative
procedure called $\ell_{1}$-minimization is usually used which says
$\min_{x^{\prime}}\|x^{\prime}\|_{\ell_{1}}\qquad\textrm{subject to}\qquad
y=\Phi\Psi x^{\prime}=\Upsilon x^{\prime},$ (4)
This is a convex optimization problem, and many numerical algorithms apply for
its solution.
Before showing the reconstruction method, we note that there are two main
differences between conventional medical tomography and the quantum
tomography. First, WF is a pseudo probability distribution, may have negative
values, while the attenuation coefficients in CT images are always
nonnegative. Second, for a given angle, a complete probability density for the
quadrature operator is measured due to the quantum mechanical property, but in
medical tomography, detectors’ number can be adjusted. On the other hand,
similar to the medical case, at the heart of a possible CS application to
phase space tomography is to find a suitable sparse representation basis. In
real computed tomography, images are generally extended distributions which
violate the prerequisite of $\ell_{1}$-based algorithms. But other sparseness
property (gradient sparse) and reconstruction algorithm(TV-algorithm)sidky
are found to reconstruct certain tomographic images which are relatively
constant over extended areas. While due to its configuration’s flexibility ,
it is hard even impossible to find a unitive sparse basis for WF in general.
However, for some special applications, WF might share the same sparse basis.
And in contrary to the medical images, some quantum states have finite values
on a small region in phase space with zero or nearly zero values elsewhere, so
have been sparse in this pseudo-pixel representation. Other situations may be
completely different, for WF is not sparse in phase space representation. A
basis transform like Fourier transform, wavelet transform may bring WF to a
sparse representation.
Although it is also possible to reconstruct continuous Wigner function, but
discretization must be taken before it can be tackled in practise. So in this
report we adopt a discrete quantum tomography model. But we note that it
should be straightforward to transport our idea to an continuous one. And
indeed such an attempt has been taken in information scienceChen . The quantum
state tomographic model for discrete Wigner functions first proposed by U.
Leonhardtleonhardt . We note that continuous and discrete quantum tomography
are the same in principle. In finite quantum systems, position and momentum
take discrete values in $\mathbb{Z}_{d}$. In Leonhardt’s discrete quantum
tomography picture, a WF in a $d$-dimensional(odd) Hilbert space was
proposed(Eqn.(6)) and phase was quantized to take place of momentum in
continuous variable quantum systems.
$\displaystyle W(m,\mu)=\frac{1}{d}\sum_{n}\exp(-\frac{4\pi i}{d}\mu n)\langle
m-n|\hat{\rho}|m+n\rangle$ (5)
Here, $m,n$ denotes the spins or particle numbers, $\mu,\nu$ denotes the
quantized phases. Similar to the continuous case, the observable distribution
is the overlap of measured state’s WF and the observable state’s WF in phase
space.
$\displaystyle
pr(\mu_{0},\tau)=\sum_{\mu,m}W(m,\mu)\delta(\mu-\mu_{0}+2m\tau;d)$
$\displaystyle
pr(m_{0},\tau)=\sum_{\mu,m}W(m,\mu)\delta(m-m_{0};d)\delta(m-m_{0};a).$ (6)
Here, $m_{0},\mu_{0}\in\mathbb{Z}_{d},pr(\mu_{0},\tau)$ denotes the
probability distribution, $W(m,\mu)$ denotes the WF to be recovered,
$\tau\in\mathbb{Z}_{d}$, $m,m_{0}$ represent the spins or particle numbers,
$\mu,\mu_{0}$ represent the quantum phases and $\delta(k;a)$ is the modular
Kronecker symbol(equals 1 when $k$=0 (mod a), and zero otherwise)
In our simulation, a WF for finite dimensional coherent state
$||\alpha|\textrm{e}^{i\phi}\rangle_{s}$ (s=d-1)Miranowicz is used as an
example to illustrate the CS based reconstruction algorithm.
$\displaystyle W(m,\mu)$ $\displaystyle=$ $\displaystyle\sum_{k=M+1}^{s}$ (7)
$\displaystyle\times$
$\displaystyle\frac{\exp[i(2k-M-s-1)(2\pi\mu/d-\phi+\pi/2)]}{[k!(M-k-s-1)!]^{1/2}}G_{1k}$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{M}\frac{\exp[i(2k-M)(2\pi\mu/d-\phi+\pi/2)]}{[k!(M-k)!]^{1/2}}G_{0k}$
where
$\displaystyle G_{\eta k}$ $\displaystyle=$
$\displaystyle\frac{(s!)^{2}}{(s+1)^{3}}\sum_{p=0}^{s}\sum_{q=0}^{s}$
$\displaystyle\times$
$\displaystyle\exp[i(x_{q}-x_{p})|\alpha|]\frac{\textrm{He}_{k}(x_{p})\textrm{He}_{M-k+\eta(s+1)}(x_{q})}{[\textrm{He}_{s}(x_{p})\textrm{He}_{s}(x_{q})]^{2}}$
with $\eta=0,1$, $M\equiv 2n\textrm{mod}(d)$, $\textrm{He}_{s+1}(x)$ is the
Hermite polynomial ($\textrm{He}_{n}(x)\equiv 2^{-n/2}H_{n}(x/\sqrt{2})$) and
$x_{k}$ is the $k$th root of $\textrm{He}_{s+1}(x)=0$.
It should be pointed out that different from classical signal processing (the
sensing matrix elements can be independently selected from a random
distribution), only the rows of the sensing matrix can be randomly measured in
quantum tomography. The sensing matrix was randomly selected from the rows of
full measurement matrix (modular Kronecker in Eqn.(7)). Then, we get the
precise probability distribution by multiplying the sensing matrix and the WF.
The problem considered now is to reconstruct a state from measured
insufficient data. Computer simulation has been conducted to demonstrate the
effectiveness of the aforementioned scheme. All the recovery processes were
implemented in Java and C. For $\ell_{1}$-minimization, we adopt a linearized
bergman iteration algorithmyin .
To simulate the sampling process, we assign a number to each row of the full
measurement matrix, then computer generated a series of random numbers
representing the rows we select to form the sensing matrix. For simplicity,
the experiment error was totally ignored. So in this pseudo pixel
representation, WF can be just roughly reconstructed. This is partly due to
measurement process itself, for only a small fraction of the WF is sensed
once, which means the sensing matrix is not dense enough. To overcome this
obstacle, two approaches might help: finding another sparse representation or
using a denser sensing matrix(measurement quantum state, for example, coherent
state). For example, for some type of mixed state ensemble, WF varies along a
particular direction when we rearrange the two dimensional WF into a one
dimensional object carefully, it is sparse in a discrete cosine Fourier
representation, which means this basis is its sparse basis.
FIG. 1: Darker regions correspond to higher values, the regions between dark
and white are dominated by zero or nearly zero values .(a)Accurate WF for
finite dimensional coherent state(Eqn.(8), $d=19$, $|\alpha=1.472\rangle$).
(b) Recovered WF via compressed sensing, 285 rows have been taken from total
measurement matrix as the sensing matrix.
In spite of some drawbacks, when we deal with a high dimensional quantum state
reconstruction or what we need is just a rough knowledge of WF, our
reconstruction algorithm would show its advantage, it makes the measurements
more economical. And any other prior knowledge could also be helpful to
recovery process, for example, the symmetry of the WF. By exploring the
correspondence between compressed sensing and tomography, we have shown the
possibility of a CS-based WF reconstruction method. In addition, most convex
optimization algorithms related to CS theory deal with real signal. And in our
case, WF is real function. But when we need to reconstruct complex signals or
images, for example, ambiguity function(AF)testorf which is complex in
general, problems appear. Here, we mention that G.Zweig proposed an
algorithmzweig processing complex signals, which converted the the usual
$\ell_{1}$-minimization in complex space to a linear programming problem in
real space.
In conclusion, we have described a reconstruction method based on compressed
sensing to recover the WF in phase space tomography. We regard the Wigner
function as an image in phase space. The values of WF on different sites
corresponds to pixels’ gray values. Our numerical simulation has shown good
recovery for a sparse discrete WF with usually supposed insufficient
measurements. Such a method can also be easily extended to reconstruct
electromagnetic field’s WF with just some straightforward revisions. We hope
that our scheme can serve as a motivation for other quantum information or
phase space optics researches. Possible relations to the popular maximum-
likelihood reconstruction in quantum tomography still need to be explored. By
combination with other reconstruction ideas, we believe a more efficient and
precise phase recovery should be possible.
This work is supported by the NSF of China(Grant No. 11075077) and also partly
by the SRFDP of China(Grant No. 200800550015).
## References
* (1) E. P. Wigner, Phys. Rev. 40 749 (1932).
* (2) K. Vogel, and H. Risken, Phys. Rev. A 40 2847 (1989).
* (3) M. Unser, Proc. IEEE 88 569 (2000).
* (4) D. Donoho, IEEE Trans. Inf. Theory 52 1289 (2006).
E. J. Candès, J. Romberg, and T. Tao, Comm. Pure Appl. Math. 59 1207 (2006);
* (5) E. Y. Sidky, C.-M.Kao, and X. Pan, J. X-ray Sci. Tech. 14 119 (2006)
* (6) G. H. Chen, J. Tang, and S. Leng, Med. Phys. 35 660 (2008)
* (7) D. Gross, Y.-K.Liu, S. T. Flammia, S. Becker, and J. Eisert Phys. Rev. Lett. 105 150401 (2010).
* (8) D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Phys. Rev. Lett. 70 1244 (1993).
* (9) A. I. Lvovsky, and M. G. Raymer, Rev. Mod. Phys. 81 299 (2009).
* (10) P. Flandrin, P. Borgnat IEEE Trans. Signal Process. 58 2974 (2010).
* (11) U. Leonhardt Phys. Rev. Lett. 74 4101 (1995).
* (12) A. Miranowicz, W. Leonski and N. Imoto Adv. Chem. Phys. 119 155 (2001).
* (13) W. Yin, S. Osher, D. Goldfarb, and J. Darbon, J. Imaging Sci. 1 143 (2008)
* (14) M. Testorf, B. M. Hennelly and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Application McGraw-Hill, New York, 2009.
* (15) G. Zweig, IEE Proc.-Radar Sonar Navig. 150 247 (2003).
|
arxiv-papers
| 2011-09-03T06:35:48 |
2024-09-04T02:49:21.896788
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jia-Ning Zhang, Lei Fang, Mo-Lin Ge",
"submitter": "Jia-Ning Zhang",
"url": "https://arxiv.org/abs/1109.0596"
}
|
1109.0918
|
11institutetext: Department of Chemistry, University of York, YO10 5DD, York,
UK 22institutetext: Department of Computer Science, University of York, YO10
5DD, York, UK
# Boolean logic gate design principles in unconventional computers: an NMR
case study
Matthias Bechmann 11 matthias.bechmann@york.ac.uk Angelika Sebald 11
angelika.sebald@york.ac.uk Susan Stepney 22 susan@cs.york.ac.uk
###### Abstract
We present a general method for analysing novel computational substrates to
determine which of their parameters can be manipulated to exhibit the complete
set of 2-input boolean logical operations. We demonstrate this approach with
an NMR-based case study, showing which NMR parameters can be used to perform
boolean logic.
###### keywords:
NMR, boolean logic, canalising functions, universal gates, spin
## 1 Introduction
There is much work on _in materio_ computing: exploiting unconventional
material substrates and their dynamical properties to perform computation, and
examining their computational properties and abilities. One of the more
advanced is optical computing [33, 34]. Other, more exotic, substrates include
nuclear spins in NMR experiments [16, 17, 24, 6], liquid crystals [12, 13,
14], conductive media [21, 22], reaction-diffusion chemical systems [19, 23,
27, 32], DNA [3, 4], and even slime moulds [2, 31].
The aim is to allow the material to do what comes naturally, under control of
physical laws, and to exploit this natural dynamics as an efficient form of
computation [29, 30, 18]. The question arises: given some novel substrate, how
to analyse its properties to determine whether it is a suitable medium for
computation, and, if so, how it can be manipulated to perform computation in a
manner best suited to that substrate.
Here we tackle a part of that problem: analysing a novel substrate to
determine how it can be used to perform boolean logical operations. We present
the design principles through a case study of using bulk nuclear spins, in the
context of NMR (nuclear magnetic resonance) experiments.
## 2 Theory and Background
### 2.1 Boolean functions and logic gates
The NAND and NOR logic gates are both _universal_ , in that either can be used
to construct all the other boolean 2-input logic gates. However, to do so,
several such gates may be required. In an implementation, it is often more
important to minimise the circuitry (the number of actual gates) than to
minimise the number of types of gates.
Logic gates can be classified in terms of their symmetry properties [28, 11].
In terms of minimising circuitry, gates can be classified in terms of
equivalence classes on permutations (rewiring) of inputs, and on negation of
inputs and/or outputs (i.e. adding inverters, important in cases where they
are significantly _cheaper_ to implement than binary gates [9]). There are
four such equivalence classes for the 16 boolean 2-input logic gates (Tables 1
and 2).
class | members
---|---
0 | T, F
1 | $A$, NOT $A$, $B$, NOT $B$
2 | $A$ AND $B$; $A$ NAND $B$; NOT $A$ AND $B$, $A$ AND NOT $B$, …
3 | $A$ XOR $B$, NOT ($A$ XOR $B$)
Table 1: The four equivalence classes of the 16 boolean 2-input logic gates. These correspond to equivalence classes under permutations of inputs and negation of the inputs and/or outputs [9], and to the different kinds of canalising functions [10]. $A$ | $B$ | T | $B$ | $A$ NAND $B$ | $A$ XOR $B$
---|---|---|---|---|---
0 | 0 | 1 | 0 | 1 | 0
0 | 1 | 1 | 1 | 1 | 1
1 | 0 | 1 | 0 | 1 | 1
1 | 1 | 1 | 1 | 0 | 0
Table 2: Truth tables of examples of the four classes of boolean logic gates,
for each possible input value of $A$ and $B$. Class 0 (example T) is
independent of either input value. Class 1 (example $B$) is independent of one
of the input values (here, $A$) for all possible inputs. Class 2 (example $A$
NAND $B$) is independent of one of the input values when the other has a
particular value (here, it it independent of the value of $B$ when $A$ has the
value $0$). Class 3 (example $A$ XOR $B$) depends on both input values for all
inputs.
The same classification is found by considering canalising functions [10].
These are functions where the output is independent of one (or more) of the
inputs, for one (or more) input values. Class 0 consists of the constant
functions: the output is completely independent of the inputs. Class 1
consists of the strongly canalising functions: the output is independent of
one of the inputs (for example, the function $B$ is independent of the value
of input $A$). Class 2 consists of the weakly canalising functions: the output
is independent of one of the inputs when the other input has a specific value
(for example, the function $A$ NAND $B$ is independent of the value of input
$B$ when the value of input $A$ is false). Class 3 consists of the non-
canalising functions: the output is determined by both inputs.
We use these symmetries/canalising properties of the gates (Figure 1) to look
for analogous properties exhibited by the substrate that indicates a natural
implementation route.
Figure 1: Logic-gates patterns and their different symmetries as visible from
the two-dimensional representations of the canalising functions gate classes
0, 1, 2 and 3.
### 2.2 Nuclear spin dynamics
The most common application of NMR spectroscopy is that as a tool for
structure elucidation of condensed matter in general, and of molecules in the
liquid state in particular. The quantised nature of nuclear spin can also be
assigned to the notion of a qubit in quantum computations [26]. Nuclear spins
have also been used as a substrate to implement classical computation
paradigms such as binary or continuous logic gates and circuits [24, 6].
A major reason for these computational applications to choose nuclear magnetic
spins as the implementation platform is the rigour by which nuclear spin
dynamics are described by quantum mechanics. In addition, the accuracy by
which the macroscopically measured signal in an NMR experiment can be related
to the underlying microscopic spin dynamics, using a density matrix approach,
is nearly unrivalled by any other spectroscopic technique. In order to develop
the formal relationship between the properties of logic gates and the quantum
mechanical expressions describing the spin dynamics in a NMR experiment, we
give a short summary of the necessary formalisms. For more details see
textbooks on NMR [20]. Here we consider only uncoupled spins $S=1/2$ in the
liquid state.
The time evolution of an initial spin state vector
$\Ket{\psi\left(t_{a}\right)}$ is described as
$\displaystyle\Ket{\psi\left(t\right)}$ $\displaystyle=$
$\displaystyle\mathcal{U}\left(t,t_{a}\right)\Ket{\psi\left(t_{a}\right)}$ (1)
where $\mathcal{U}\left(t,t_{a}\right)$ is the time propagator describing the
spin dynamics at every given point in time. The general orientation of the
spin vector $\Ket{\psi}$ is [20]
$\Ket{\psi}=\left(\begin{array}[]{c}\cos\frac{\theta_{\mathrm{s}}}{2}\mathrm{e}^{-\mathrm{i}\frac{1}{2}\phi_{\mathrm{s}}}\\\
\sin\frac{\theta_{\mathrm{s}}}{2}\mathrm{e}^{+\mathrm{i}\frac{1}{2}\phi_{\mathrm{s}}}\end{array}\right)$
(2)
where $\theta_{\mathrm{s}}$ and $\phi_{\mathrm{s}}$ are the polar and azimuth
angles. Its three-dimensional representation is given by
$\left(\begin{array}[]{c}\Braket{\psi}{I_{x}}{\psi}\\\
\Braket{\psi}{I_{y}}{\psi}\\\
\Braket{\psi}{I_{z}}{\psi}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{c}\sin\theta_{\mathrm{s}}\cos\phi_{\mathrm{s}}\\\
\sin\theta_{\mathrm{s}}\sin\phi_{\mathrm{s}}\\\
\cos\theta_{\mathrm{s}}\end{array}\right)$ (3)
describing a general Cartesian vector orientation (see Figure 2), where
$I_{x}$, $I_{y}$ and $I_{z}$ are basis spin operators [25].
Figure 2: Spin state vector orientation of $\Ket{\psi}$ in its Cartesian
representation
The time propagation of the macroscopic NMR signal is described by the density
matrix $\rho(t)=\overline{\Ket{\psi\left(t\right)}\Bra{\psi\left(t\right)}}$
as
$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\mathcal{U}\left(t,t_{a}\right)\rho(t_{a})\mathcal{U}^{\dagger}\left(t,t_{a}\right)$
(4)
where the bar signals the ensemble average. The thermal equilibrium density
matrix is [20]
$\rho_{z}^{\mathrm{eq}}=\frac{1}{2}\mathbbm{1}+\frac{1}{2}\lambda_{\mathrm{B}}I_{z}$
(5)
where $\lambda_{\mathrm{B}}=\hbar\gamma B_{0}/(k_{\mathrm{B}}T)$ defines a
Boltzmann factor, scaling the separation of spin energy levels, that can be
interpreted as a spin polarisation along the $z$-axis.
The NMR system Hamiltonian and generator of the time evolution propagators for
isolated, uncoupled spins is
$\mathcal{H}=\mathcal{H}_{\mathrm{CS}}+\mathcal{H}_{\mathrm{\mathrm{rf}}}$ (6)
where $\mathcal{H}_{\mathrm{CS}}$ and $\mathcal{H}_{\mathrm{rf}}$ are the
chemical shielding and radio frequency (r.f.) Hamiltonian, respectively. The
explicit representation of the r.f. Hamiltonian is particularly simple in the
rotating reference frame (RRF) as
$\displaystyle\mathcal{H}_{\mathrm{rf}}$ $\displaystyle=$
$\displaystyle\omega_{\mathrm{p}}I_{z}+\kappa_{\mathrm{p}}\left(I_{x}\cos\phi_{\mathrm{p}}+I_{y}\sin\phi_{\mathrm{p}}\right)$
(7)
where $\omega_{\mathrm{p}}$ is a frequency offset (relative to the frequency
of RRF), and $\kappa_{\mathrm{p}}$ and $\phi_{\mathrm{p}}$ are the amplitude
and phase of the pulse. Figure 3(a) summarises the behaviour of the
magnetisation vector $\boldsymbol{M}$ (Eqs. (1)–(43)) under the influence of a
r.f. pulse; Figure 3(b) depicts the relevant parameters.
Figure 3: NMR single-pulse experiment: (a) rotation of the magnetisation
vector $\boldsymbol{M}$ by a r.f. pulse
$R_{\phi_{\mathrm{p}}}\left(\beta\right)$ with
$\phi_{\mathrm{p}}=\frac{\pi}{2}$ and
$\beta=\kappa_{\mathrm{p}}\tau_{\mathrm{p}}=\frac{\pi}{2}$; (b) the relevant
parameters describing the single-pulse NMR experiment (pulse amplitude
$\kappa_{\mathrm{p}}$, pulse duration $\tau_{\mathrm{p}}$, pulse phase
$\phi_{\mathrm{p}}$, pulse frequency $\omega_{\mathrm{p}}$, free evolution
delay $\tau{}_{\mathrm{d}}$ and acquisition phase $\phi_{\mathrm{a}}$).
Pulses applied to spins resonating at the rotation frequency of the RRF are
described by the system Hamiltonian
$\displaystyle\mathcal{H}=\mathcal{H}_{\mathrm{rf}}$ $\displaystyle=$
$\displaystyle\kappa_{\mathrm{p}}\left(I_{x}\cos\phi_{\mathrm{p}}+I_{y}\sin\phi_{\mathrm{p}}\right)$
(8)
and the overall system Hamiltonian for uncoupled isolated spins $S=1/2$ is
$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\mathrm{CS}}+\mathcal{H}_{\mathrm{\mathrm{rf}}}:\mbox{during
pulse}$ $\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\mathrm{CS}}\,:\mbox{during free evolution}$ (9)
including a chemical shielding offset $\mathcal{H}_{\mathrm{CS}}$ from the RRF
frequency.
The general form of the time evolution propagator in NMR is
$\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)$ $\displaystyle=$
$\displaystyle\mathcal{T}\exp\left\\{-\mathrm{i}\int_{t_{a}}^{t_{b}}\mathcal{H}\left(t\right)\mathrm{d}t\right\\}$
(10)
where $\mathcal{T}$ is the Dyson time-ordering operator [25].
Dependent on the symmetry of the Hamiltonian, different (simpler) expressions
for the propagator can be formulated
$\displaystyle\mathcal{H}\left(t\right)=\mathcal{H}$
$\displaystyle\rightarrow$
$\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)=\exp\left\\{-\mathrm{i}\mathcal{H}\left(t_{b}-t_{a}\right)\right\\}$
$\displaystyle\mathrm{time\ independent}$
$\displaystyle\left[\mathcal{H}\left(t^{\prime}\right),\mathcal{H}\left(t^{\prime\prime}\right)\right]=0$
$\displaystyle\rightarrow$
$\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)=\exp\\{-\mathrm{i}{\textstyle\int}_{t_{a}}^{t_{b}}\mathrm{d}t^{\prime}\mathcal{H}\left(t^{\prime}\right)\\}$
$\displaystyle\mathrm{self-commuting}$
$\displaystyle\mathcal{H}\left(t_{a}+\tau\right)=-\mathcal{H}\left(t_{b}-\tau\right)$
$\displaystyle\rightarrow$
$\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)=1\>\mbox{; anti-symmetric}$
(14) $\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)=\pm 1\>\mbox{; cyclic}$
In the case of a hard pulse, a so-called $\delta$ pulse is usually a valid
description of the time propagator
$\displaystyle\mathcal{U}\left(t_{b},t_{a}\right)$ $\displaystyle=$
$\displaystyle\exp\left\\{-\mathrm{i}\mathcal{H}_{\mathrm{rf}}\tau_{\mathrm{p}}\right\\};\,\tau_{\mathrm{p}}=t_{b}-t_{a}$
(15) $\displaystyle=$
$\displaystyle\exp\left\\{-\mathrm{i}\kappa_{\mathrm{p}}\tau_{\mathrm{p}}\left(I_{x}\cos\phi_{\mathrm{p}}+I_{y}\sin\phi_{\mathrm{p}}\right)\right\\}$
$\displaystyle=$ $\displaystyle
R_{\phi_{\mathrm{p}}}\left(\beta\right);\,\beta=\kappa_{\mathrm{p}}\tau_{\mathrm{p}}$
which is equivalent to the rotation operation
$R_{\phi_{\mathrm{p}}}\left(\beta\right)$. It generates a rotation by an angle
$\beta$ about an axis in the $xy$-plane with azimuth angle
$\phi_{\mathrm{p}}$. For a sequence of $\delta$ pulses the propagator can be
factorised as
$\displaystyle\mathcal{U}\left(t_{c},t_{a}\right)$ $\displaystyle=$
$\displaystyle\mathcal{U}_{2}\left(t_{c},t_{b}\right)\mathcal{U}_{1}\left(t_{b},t_{a}\right);\,\tau_{\mathrm{p1}}=t_{b}-t_{a}\,\mbox{and
}\tau{}_{\mathrm{p2}}=t_{c}-t_{b}$ (16) $\displaystyle=$ $\displaystyle
R_{\phi{}_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p}1}}\left(\beta_{1}\right);\,\beta_{1}=\kappa_{\mathrm{p}1}\tau_{\mathrm{p}1};\,\beta_{2}=\kappa{}_{\mathrm{p}2}\tau{}_{\mathrm{p2}}$
The explicit form of these propagators is equivalent to rotation operators in
two dimensions and is [20]
$\displaystyle R_{x}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-\mathrm{i}\beta
I_{x}}=\left(\begin{array}[]{cc}\cos\frac{\beta}{2}&-\mathrm{i}\sin\frac{\beta}{2}\\\
-\mathrm{i}\sin\frac{\beta}{2}&\cos\frac{\beta}{2}\end{array}\right)$ (19)
$\displaystyle R_{y}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-\mathrm{i}\beta
I_{y}}=\left(\begin{array}[]{cc}\cos\frac{\beta}{2}&-\sin\frac{\beta}{2}\\\
\sin\frac{\beta}{2}&\cos\frac{\beta}{2}\end{array}\right)$ (22) $\displaystyle
R_{z}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-\mathrm{i}\beta
I_{z}}=\left(\begin{array}[]{cc}\mathrm{e}^{-\mathrm{i}\frac{\beta}{2}}&0\\\
0&\mathrm{e}^{+\mathrm{i}\frac{\beta}{2}}\end{array}\right)$ (25)
$\displaystyle R_{\phi}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-\mathrm{i}\beta\left(I_{x}\cos\phi+I_{y}\sin\phi\right)}=\left(\begin{array}[]{cc}\cos\frac{\beta}{2}&-\mathrm{i}\sin\frac{\beta}{2}\mathrm{e}^{-\mathrm{i}\phi}\\\
-\mathrm{i}\sin\frac{\beta}{2}\mathrm{e}^{+\mathrm{i}\phi}&\cos\frac{\beta}{2}\end{array}\right)$
(28) $\displaystyle=$ $\displaystyle
R_{z}\left(\phi\right)R_{x}\left(\beta\right)R_{z}\left(-\phi\right)$ (29)
After a pulse $R_{\phi_{\mathrm{p}}}\left(\beta\right)$ with phase
$\phi_{\mathrm{p}}=\frac{\pi}{2}$ and flip angle $\beta=\frac{\pi}{2}$ the
thermal equilibrium density matrix $\rho_{z}^{\mathrm{eq}}$ is transformed
according to
$\displaystyle
R_{\frac{\pi}{2}}\left(\frac{\pi}{2}\right)\rho_{z}^{\mathrm{eq}}R_{\frac{\pi}{2}}^{\dagger}\left(\frac{\pi}{2}\right)$
$\displaystyle=\frac{1}{2}\mathbbm{1}+\frac{\lambda_{\mathrm{B}}}{2}$
$\displaystyle\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbbm{1}+\frac{1}{2}\lambda_{\mathrm{B}}I_{x}$
$\displaystyle=$ $\displaystyle\rho_{x}$ (33)
yielding spin polarisation along the $+x$ direction.
A r.f. pulse $R_{\phi_{\mathrm{p}}}\left(\beta\right)$ applied to an initial
state $\rho^{\left(i\right)}$ generates a spin density matrix that can be
expressed in terms of its final polar coordinates
$\rho\left(\phi_{\mathrm{s}},\theta_{s}\right)=\overline{\Ket{\psi}\Bra{\psi}}$
(Eq. (2)) or as a function of the pulse parameters rotating an initial state
$\rho^{\left(i\right)}\left(\phi_{\mathrm{p}},\beta\right)$. For example when
starting from $\rho^{\left(z\right)}=\rho_{z}^{\mathrm{eq}}$ the final state
is
$\displaystyle\rho^{\left(z\right)}\left(\phi_{\mathrm{p}},\beta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbbm{1}+\frac{\lambda_{\mathrm{B}}}{2}\left(\begin{array}[]{cc}\cos\beta&-\mathrm{i}\sin\beta\mathrm{e}^{-\mathrm{i}\phi_{\mathrm{p}}}\\\
-\mathrm{i}\sin\beta\mathrm{e}^{+\mathrm{i}\phi_{\mathrm{p}}}&-\cos\beta\end{array}\right)$
(36)
and the relation
$\rho\left(\phi_{\mathrm{s}},\theta_{s}\right)=\rho^{\left(z\right)}\left(\phi_{\mathrm{p}},\beta\right)$
holds for
$\left(\phi_{\mathrm{s}},\theta_{s}\right)=\left(\phi_{\mathrm{p}}-\frac{\pi}{2},\beta\right)$.
Equivalently to the calculation of the three-dimensional spin-vector
orientation in Eq. (3) from the two-dimensional spin state $\Ket{\psi}$ (Eq.
(2)), the three-dimensional magnetisation vector $\boldsymbol{M}$ can then be
calculated from any given two-dimensional density matrix $\rho$ as
$\displaystyle\boldsymbol{M}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\overline{\Braket{I_{x}}}\\\
\overline{\Braket{I_{y}}}\\\
\overline{\Braket{I_{z}}}\end{array}\right)=\left(\begin{array}[]{c}\mbox{Tr}\left\\{\rho
I_{x}\right\\}\\\ \mbox{Tr}\left\\{\rho I_{y}\right\\}\\\
\mbox{Tr}\left\\{\rho I_{z}\right\\}\end{array}\right)$ (43)
In the case of single isolated spins this also implies the equivalence in
interpretation of the spin vector orientation in Figure 2 and the
magnetisation vector $\boldsymbol{M}$.
In general a NMR pulse sequence can consist of any number and parametrisation
of r.f. pulses and free evolution delays. This provides for numerous options
for the implementation of e.g. logic gates. The most useful parametrisation in
the following is the one describing the NMR signal as a function of the pulse
sequence, $\rho^{\left(i\right)}\left(\phi_{\mathrm{p}},\beta\right)$.
## 3 Logic Gates from NMR parameters
### 3.1 Classification of NMR parameters
The most common parameters describing a NMR experiment, open for use as
control parameters to implement logic operations, are
$\left\\{\kappa_{\mathrm{p}},\phi_{\mathrm{p}},\tau_{\mathrm{p}},\omega_{\mathrm{p}},\tau_{\mathrm{d}},\phi_{\mathrm{a}}\right\\}$:
pulse amplitude, pulse phase, pulse duration, pulse frequency, free evolution
delay, and receiver phase, respectively (see Figure 3).
A heuristic classification rule has been given elsewhere [24], separating
logic gates based on the relation between the experimental NMR parameters
chosen:
1. 1.
if the effect of the first parameter cannot be compensated for by setting the
second parameter (canalising input value), then
$\mbox{AND},>,<\mbox{NOR},\mbox{OR},\leq,\geq,\mbox{NAND}$ gates can be
constructed (asymmetric gates);
2. 2.
if the effect of the first parameter can be compensated by a setting of the
second parameter, then XOR and XNOR gates can be constructed (symmetric
gates).
This classification of logic gates seamlessly integrates in a generalised
description by using the concept of canalising functions [10] and equivalence
classes [9]. Fixing one of the inputs of a logic gate ($A$ or $B$) is a
_canalising input value_ if changing the other (unfixed) input does not alter
the gate output. The number of all possible canalising input values for a
given logic gate for all possible inputs states is used as a criterion to
assign logic gates to one of four classes (see Table 3):
class | canalising input values for fixed: | $A$ | $B$
---|---|---|---
0 | T | 2 | 2
1 | $B$ | 0 | 2
2 | $A$ NAND $B$ | 1 | 1
3 | $A$ XOR $B$ | 0 | 0
Table 3: Number of canalising input values for the four gate classes.
Demonstrated for one representative member of every class: T, $B$, $A$ NAND
$B$, $A$ XOR $B$. The respective truth tables for these four gates are shown
in Table 2.
class 0
two canalising input values for either of the two inputs $A$ and $B$
class 1
two canalising input values for exactly one of the inputs $A$ or $B$
class 2
one canalising input values for either of the two inputs $A$ and $B$
class 3
no canalising input values for either inputs $A$ and $B$
These four classes also show unique symmetry patterns in their two-dimensional
representations (see Figure 1). Rows and columns of equal coloured squares
signal the presence of a canalising input. These patterns can be used to map
functions $f(x1,x2)$ to corresponding canalising functions and, therefore, the
logic gate(s) they can naturally implement are readily identified.
According to our previous definitions [24] a strong parameter would generate a
canalising function for at least one of its states (class 0, 1 and 2), because
altering the other parameter causes no change in the NMR output. On the other
hand, for a weak parameter the output would always change (class 3).
In the following we demonstrate how the abstract concept of canalising
functions relates to the symmetries and commutation relations of the NMR
Hamiltonian, the time evolution propagator, and the experimental output of a
NMR experiment.
### 3.2 Canalising input values in NMR context
Here we consider a system Hamiltonian that consists only of the
$\mathcal{H}_{\mathrm{rf}}$ term. Hence, we are in the single spin, strong
pulse, on-resonant regime, and for simplicity we only consider the
experimental parameters
$\kappa_{\mathrm{p}},\phi_{\mathrm{p}},\tau_{\mathrm{p}}$ (r.f. pulse
amplitude, phase and duration) for logic gate generation.
The NMR r.f. pulse propagator
$R_{\phi_{\mathrm{p}}}\left(\beta\right)=R_{z}\left(\phi_{\mathrm{p}}\right)R_{x}\left(\beta\right)R_{z}\left(-\phi_{\mathrm{p}}\right)$
(Eq. (29)) is the general rotation operator about an axis in the traverse
$xy$-plane of the RRF. It therefore commutes with a particular spin state
($I_{\phi_{\mathrm{p}}}$) in the transverse plane for a given value
$\phi_{\mathrm{p}}$ (e.g. for $\phi_{\mathrm{p}}=0$;
$\left[R_{0}(\beta),I_{x}\right]=0$ independent of the value of $\beta$).
However, $R_{\phi_{\mathrm{p}}}\left(\beta\right)$ will never commute with the
thermal equilibrium state $\rho_{z}^{\mathrm{eq}}$ ($\sim I_{z}$) (Eq. (5)),
which is perpendicular to the $xy$-plane. For example, a
$R_{0}\left(\frac{\pi}{2}\right)$ pulse applied to $z$-magnetisation ($\sim
I_{z}$) will flip it to the $-y$ direction, and therefore change the spin
state and its orientation. A subsequent pulse applied using
$R_{-\frac{\pi}{2}}\left(\frac{\pi}{2}\right)$ will leave it unaltered, since
the system is in the $-I_{y}$ spin state, an eigenstate of
$R_{-\frac{\pi}{2}}\left(\frac{\pi}{2}\right)$ and, therefore, a canalising
input is generated in this second step. One can postulate now that a
_canalising input_ is generated by an r.f. pulse if:
1. 1.
$R_{\phi_{\mathrm{p}}}\left(\beta\right)$ leaves the system state unaltered
(eigenstate;$\left[R_{\phi_{\mathrm{p}}}(\beta),\rho\right]=0$),
2. 2.
$R_{\phi_{\mathrm{p}}}\left(\beta\right)=\mathbbm{1}$, the unity operator.
In the following this behaviour and the possibility of generating canalising
input values is analysed. First we consider single-pulse experiments assuming
different initial system-state preparations $\rho^{\left(i\right)}$. Second we
consider experiments composed of more than one pulse.
#### 3.2.1 Single-pulse gates
The signal as detected during a NMR experiment is not the effect of the full
spin magnetisation vector $\boldsymbol{M}$ as given in Eq. (43) but only its
projection into the $xy$-plane [20]. Hence, the physically meaningful
quantities that can be measured by NMR are the magnetisation vector components
$M_{x}=\overline{\Braket{I_{x}}}$ and $M_{y}=\overline{\Braket{I_{y}}}$,
describing the orientational distribution of this projection, and the
magnitude $M_{xy}=\sqrt{\Braket{I_{x}}^{2}+\Braket{I_{y}}^{2}}$ describing the
total amount of magnetisation present in the $xy$-plane. The functional
structure of $M_{x}$, $M_{y}$ and $M_{xy}$ is determined by the r.f. pulse
$R_{\phi_{\mathrm{p}}}\left(\beta\right)$ and the initial spin state
$\rho^{\left(i\right)}$ to which the r.f. pulse is applied. $M_{x}$, $M_{y}$
and $M_{xy}$ are therefore functions of pulse phase $\phi_{\mathrm{p}}$, pulse
flip angle $\beta=\kappa_{\mathrm{p}}\tau_{\mathrm{p}}$ and the initial
direction of the spin magnetisation.
The spin magnetisation can assume every possible orientation, while the r.f.
pulse and therefore the rotation axis of the spin magnetisation is restricted
to the $xy$-plane. It is instructive to examine scenarios where the initial
spin state $\rho^{\left(i\right)}$ is either perpendicular to the $xy$-plane
and never commutes with the r.f. pulse operator
$R_{\phi_{\mathrm{p}}}\left(\beta\right)$, or where it is coplanar to it and
therefore can commute.
##### Starting from thermal equilibrium state
The most simple and natural initial spin state is the one the system assumes
at thermal equilibrium $\rho^{\left(i\right)}=\rho_{z}^{\mathrm{eq}}$ (Eq.
(5)). Here the spin polarisation is pointing along the $z$-axis, perpendicular
to the $xy$-plane and $R_{\phi_{\mathrm{p}}}(\beta)$ never commutes with it.
$\rho_{z}^{\mathrm{eq}}$ transforms under the influence of a pulse
$R_{\phi_{\mathrm{p}}}(\beta)$ as
$\displaystyle\rho^{\left(z\right)}\left(\phi_{\mathrm{p}},\beta\right)$
$\displaystyle=$ $\displaystyle
R_{\phi_{\mathrm{p}}}(\beta)\rho_{z}^{\mathrm{eq}},R_{\phi_{\mathrm{p}}}^{\dagger}(\beta)$
(44)
After this single pulse the spin magnetisation vector $\boldsymbol{M}$ (Eq.
(43)) is a function of pulse flip angle $\beta$ and pulse phase
$\phi_{\mathrm{p}}$
$\displaystyle\left(\begin{array}[]{c}M_{x}\\\ M_{y}\\\
M_{z}\end{array}\right)=\left(\begin{array}[]{c}\mathrm{Tr}\left\\{\rho^{\left(z\right)}(\phi_{\mathrm{p}},\beta)I_{x}\right\\}\\\
\mathrm{Tr}\left\\{\rho^{\left(z\right)}(\phi_{\mathrm{p}},\beta)I_{y}\right\\}\\\
\mathrm{Tr}\left\\{\rho^{\left(z\right)}(\phi_{\mathrm{p}},\beta)I_{z}\right\\}\end{array}\right)=\frac{\lambda_{\mathrm{B}}}{4}\left(\begin{array}[]{c}\sin\phi_{\mathrm{p}}\sin\beta\\\
-\cos\phi_{\mathrm{p}}\sin\beta\\\ \cos\beta\end{array}\right)$ (54)
and the NMR measurable quantities $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)=\frac{\lambda_{\mathrm{B}}}{4}\left|\sin\beta\right|$
are readily determined. These quantities are functions of the two pulse
parameters $\phi_{\mathrm{p}}$ and $\beta$ just like the binary logic gates
are functions of the two inputs $A$ and $B$ (Table 2). An instructive way to
analyse $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$, as to which logic gates can be
implemented by them, is by examining their representation as two-dimensional
contour plots. These are shown in Figure 4.
Figure 4: Contour plots of the NMR detectable quantities (a)
$M_{x}=\overline{\Braket{I_{x}}}$, (b)$M_{y}=\overline{\Braket{I_{y}}}$ and
(c)
$M_{xy}=\sqrt{\overline{\Braket{I_{x}}}^{2}+\overline{\Braket{I_{y}}}^{2}}$
(Eq. (54)) as a function of the pulse parameters $\phi_{\mathrm{p}}$ and
$\beta$. The initial spin polarisation has been along the $z$-axis
($\rho_{z}^{\mathrm{eq}}$) and $\lambda_{\mathrm{B}}=1$ has been assumed. The
ranges for $\phi_{\mathrm{p}}$ and $\beta$ have been chosen such that the only
symmetry operation necessary to generate a full, infinite plot are horizontal
and vertical translations.
A comparison with the two-dimensional sketches of 2-input logic gates in
Figure 1 immediately reveals agreements and disagreements in symmetries.
In principle the functions $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ are continuous in
$\phi_{\mathrm{p}}$ and $\beta$ whereas logic gates are boolean functions of
$A$ and $B$ that can only assume the discrete values of
$\left\\{0,1\right\\}$. In order to map the continuous functions in Figure 4
to the four possible discrete sets ($A$,$B$) of a logic gate one has to find
four discrete pairs $\left(\phi_{\mathrm{p}},\beta\right)$ at which to
evaluate the continuous functions and where the symmetry pattern of the
desired logic gate results.
For example from Figure 4(a) depicting $M_{x}$, one can directly identify the
symmetry pattern corresponding to a class 3 gate in Figure 1. In order to
implement the class 3 (XOR) gate (Table 2) one needs to define four discrete
value sets
$\left(\phi_{\mathrm{p}}=\phi_{\mathrm{p}}^{A},\beta=\beta^{B}\right)$ (with
$A$,$B$ $\in\\{0,1\\}$). Inspection of Figure 4(a) immediately suggests the
positive and negative extrema as possible candidates: Selecting the pair
$\left(\phi_{\mathrm{p}}^{A=0},\beta^{B=0}\right)=\left(\frac{\pi}{2},-\frac{\pi}{2}\right)$
the detected output is
$M_{x}\left(\phi_{\mathrm{p}}^{A=0}=\frac{\pi}{2},\beta^{B=0}=-\frac{\pi}{2}\right)=-0.25$.
In the same fashion, $M_{x}\left(\frac{\pi}{2},\frac{\pi}{2}\right)=0.25$,
$M_{x}\left(\frac{3\pi}{2},-\frac{\pi}{2}\right)=0.25$ and
$M_{x}\left(\frac{3\pi}{2},\frac{\pi}{2}\right)=-0.25$ are calculated for the
remaining XOR gate input configurations. The values of
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ in Figure 4 are calculated
assuming $\lambda_{\mathrm{B}}=1$. Mapping these values to the final boolean
values $\left\\{0,1\right\\}$ is achieved by scaling them by a factor of 4.
class | $\phi_{\mathrm{p}}^{A}$ | $\beta^{B}$ | Output=$M_{x}\left(\phi_{p}^{A},\beta^{B}\right)$ | gate
---|---|---|---|---
0 | 1 | 0 | 1 | 00 | 01 | 10 | 11
0 | $\frac{\pi}{2}$ | $\frac{5\pi}{2}$ | $\frac{\pi}{2}$ | $\frac{5\pi}{2}$ | 0.25 | 0.25 | 0.25 | 0.25 | T
1 | $\frac{\pi}{2}$ | $\frac{5\pi}{2}$ | $-\frac{\pi}{2}$ | $\frac{\pi}{2}$ | -0.25 | 0.25 | -0.25 | 0.25 | $B$
2 | $\pi$ | $\frac{3\pi}{2}$ | $0$ | $\frac{\pi}{2}$ | 0 | 0 | 0 | -0.25 | $A$ NAND $B$
3 | $\frac{\pi}{2}$ | $\frac{3\pi}{2}$ | $-\frac{\pi}{2}$ | $\frac{\pi}{2}$ | -0.25 | 0.25 | 0.25 | -0.25 | $A$ XOR $B$
Table 4: Some selected values $\phi_{p}$ and $\beta$ to generate one
representative gate of every gate class using
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ as shown in Figure 4(a).
A class 3 gate is characterised by the absence of any canalising input (Table
3). A change in $\phi_{\mathrm{p}}^{A}$ or $\beta^{B}$ will therefore always
change the value $M_{x}\left(\phi_{\mathrm{p}}^{A},\beta^{B}\right)$. In order
to change one of the parameters $\phi_{\mathrm{p}}^{A}$ or $\beta^{B}$ and
achieve the same result in $M_{x}\left(\phi_{\mathrm{p}}^{A},\beta^{B}\right)$
one can take advantage of the periodicity of the trigonometric functions in
Eq. (54) and in this way
$M_{x}\left(\frac{\pi}{2},-\frac{\pi}{2}\right)=M_{x}\left(\frac{5\pi}{2},-\frac{\pi}{2}\right)=-0.25$
and
$M_{x}\left(\frac{\pi}{2},\frac{\pi}{2}\right)=M_{x}\left(\frac{5\pi}{2},\frac{\pi}{2}\right)=0.25$
can be implemented. This gate is evaluating $\phi_{\mathrm{p}}^{A}$ at values
separated by $2\pi$ and the gate is therefore independent of a change in
$\phi_{\mathrm{p}}^{A}$ no matter which of its two permitted values
$\beta^{B}$ assumes. A change in $\beta^{B}$, however, will always change
$M_{x}\left(\phi_{\mathrm{p}}^{A},\beta^{B}\right)$. This behaviour
corresponds to a class 1 gate with two canalising input values for one
parameter. In order to implement class 2 gates a comparison of the logic gate
patterns and the contour plot for
$M_{x}\left(\phi_{\mathrm{p}}^{A},\beta^{B}\right)$ suggests a restriction of
the parameter range to smaller intervals. For example, one may choose the
interval $\phi_{\mathrm{p}}\in\left[\pi,\frac{3\pi}{2}\right]$ and
$\beta\in\left[0,\frac{\pi}{2}\right]$ to implement a NAND gate representing a
class 2 (NAND) gate (see Table 4). The implementation of a class 0 gate
requires two canalising input values for both $\phi_{\mathrm{p}}^{A}$ and
$\beta^{B}$. As in the class 1 ($B$) gate scenario the periodicity of
$M_{x}\left(\phi_{\mathrm{p}}^{A},\beta^{B}\right)$ can be used to achieve
canalising inputs. However, this time both parameters have to undergo the
$2\pi$ value changes. Alternatively one can use pulses corresponding to
rotation operators $R_{\phi_{\mathrm{p}}}\left(\beta\right)=\pm\mathbbm{1}$
for generating canalising inputs for any parameter configuration of
$\phi_{\mathrm{p}}^{A}$ or $\beta^{B}$.
Having demonstrated how to implement one member of every gate class it is now
a trivial task to generate implementations of all the members of a given gate
class. By using the symmetry operations of permuting the input assignment of
$A$ and $B$ to $\phi_{\mathrm{p}}$ and $\beta$, inverting the input or the
output, the remaining gates are obtained directly.
This shows that all 2-input logic gates can be implemented using the functions
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$.
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ are both products of two linear
trigonometric functions, differing only by a phase shift of $\frac{\pi}{2}$ in
the factor depending on $\phi_{\mathrm{p}}$
($\sin\left(\phi_{\mathrm{p}}+\frac{\pi}{2}\right)=\cos\phi_{\mathrm{p}}$).
This phase shift can be seen in Figure 4(a) and (b) as a horizontal shift by
$\frac{\pi}{2}$, otherwise both plots are identical. Their overall symmetry
pattern is that of a class 3 gate. Hence, everything said about
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ implementations equally holds for
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$.
However the same can not be said about the magnetisation magnitude
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ (Figure 4(c)).
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)=M_{xy}\left(\beta\right)=\frac{\lambda_{\mathrm{B}}}{4}\left|\sin\beta\right|$
is only a function of the pulse flip angle $\beta$ and, therefore, displays a
contour plot with a symmetry corresponding to a class 1 gate.
The behaviour of the functions under permutation of the variables
$\phi_{\mathrm{p}}$ and $\beta$ and/or their inversion distinguishes
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ from each other.
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ is invariant to permutation of the
variables while $M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ is not. In
contrast, $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ changes under inversion
of its variables $\phi_{\mathrm{p}}\rightarrow-\phi_{\mathrm{p}}$ or
$\beta\rightarrow-\beta$, while $M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$
is invariant to inversion of $\phi_{\mathrm{p}}\rightarrow-\phi_{\mathrm{p}}$.
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ shows the same behaviour as
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ under these transformations.
However, a simple shift in $\phi_{\mathrm{p}}=\frac{\pi}{2}$ gives
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)=M_{y}\left(\phi_{\mathrm{p}}+\frac{\pi}{2},\beta\right)$,
an operation which is not possible for
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ since it is independent of
$\phi_{\mathrm{p}}$. The differences in behaviour arise because for
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ both variables $\phi_{\mathrm{p}}$
and $\beta$ are arguments of products of trigonometric functions, which
generates function values in the range of $\left[-1,1\right]$, whereas
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ has trigonometric function values
in $\left[-1,1\right]$ only for the variable $\beta$.
In principle, the task of implementing all 2-input logic gates by NMR
spectroscopy is already accomplished by using only the simplest of all NMR
experiments: starting from thermal equilibrium and only using a single r.f.
pulse. This scenario can therefore serve as a Universal Logic Module (ULM)
[35]. There are, however, good reasons why one needs to explore other starting
conditions, and more complicated NMR pulse sequences. NMR is uniquely suitable
as a single testbed for the implementation of classical as well as quantum
computations. In common formulations of quantum algorithms the initial state
of the computation is not the thermal equilibrium state but a superposition
state [16]. For valid comparisons between classical and quantum algorithm NMR
implementations the same initial (superposition) state should be used.
Further, it may be desirable to construct more extended circuitry than just a
single logic gate [15]. Then gate implementations that can deal with initial
states other than $\rho_{z}^{\mathrm{eq}}$ are attractive as an efficient
means of taking advantage of the output of a logic gate without having to
restore the initial state $\rho_{z}^{\mathrm{eq}}$ before the next logic gate
in a circuit can be executed.
##### Starting from superposition state
A r.f. pulse $R_{\frac{\pi}{2}}(\frac{\pi}{2})$ applied to
$\rho_{z}^{\mathrm{eq}}$ generates the superposition state
$\rho_{x}=\frac{1}{2}\mathbbm{1}+\frac{1}{2}\mathbb{\lambda_{\mathrm{B}}}I_{x}$,
which corresponds to spin polarisation pointing along the $+x$-axis. Here we
take $\rho_{x}$ as the initial superposition state. The commutator between
$R_{\phi_{\mathrm{p}}}(\beta)$ and $\rho_{x}$ is
$\left[R_{\phi_{\mathrm{p}}}(\beta),\rho_{x}\right]=-\lambda_{\mathrm{B}}I_{z}\sin\phi_{\mathrm{p}}\sin\frac{\beta}{2}$
(55)
and is zero for values $\phi_{\mathrm{p}}=n\pi$. These angles correspond to
rotations around the positive and negative $x$-axis (the commutator is also
zero for $\beta=2n\pi$, but these angles represent trivial $2\pi$ rotations).
$\rho_{x}$ transforms under the influence of a r.f. pulse
$R_{\phi_{\mathrm{p}}}(\beta)$ as
$\displaystyle\rho^{\left(x\right)}\left(\phi_{\mathrm{p}},\beta\right)$
$\displaystyle=$ $\displaystyle
R_{\phi_{\mathrm{p}}}(\beta)\rho_{x}R_{\phi_{\mathrm{p}}}^{\dagger}(\beta)$
(56)
The r.f. pulse generates a magnetisation vector $\boldsymbol{M}$ (Eq. (43)) as
a function of pulse flip angle $\beta$ and pulse phase $\phi_{\mathrm{p}}$
according to
$\displaystyle\left(\begin{array}[]{c}M_{x}\\\ M_{y}\\\
M_{z}\end{array}\right)=\left(\begin{array}[]{c}\mathrm{Tr}\left\\{\rho^{\left(x\right)}(\phi_{\mathrm{p}},\beta)I_{x}\right\\}\\\
\mathrm{Tr}\left\\{\rho^{\left(x\right)}(\phi_{\mathrm{p}},\beta)I_{y}\right\\}\\\
\mathrm{Tr}\left\\{\rho^{\left(x\right)}(\phi_{\mathrm{p}},\beta)I_{z}\right\\}\end{array}\right)=\frac{\lambda_{\mathrm{B}}}{4}\left(\begin{array}[]{c}1-2\sin^{2}\phi_{\mathrm{p}}\sin^{2}\frac{\beta}{2}\\\
\sin 2\phi_{\mathrm{p}}\sin^{2}\frac{\beta}{2}\\\
-\sin\phi_{\mathrm{p}}\sin\beta\end{array}\right)$ (66)
All the quantities $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)=\frac{\lambda_{\mathrm{B}}}{4}\sqrt{1-\sin^{2}\phi_{\mathrm{p}}\sin^{2}\beta}$
are functions of $\phi_{\mathrm{p}}$ and $\beta$. Contour plots of these three
functions are shown in Figure 5.
Figure 5: Contour plots of the NMR detectable quantities (a)
$M_{x}=\overline{\Braket{I_{x}}}$, (b)$M_{y}=\overline{\Braket{I_{y}}}$ and
(c)
$M_{xy}=\sqrt{\overline{\Braket{I_{x}}}^{2}+\overline{\Braket{I_{y}}}^{2}}$
(Eq. (66)) as a function of the pulse parameters $\phi_{\mathrm{p}}$ and
$\beta$. The initial spin polarisation has been along the $x$-axis
($\rho_{x}$) and $\lambda_{\mathrm{B}}=1$ has been assumed. The ranges for
$\phi_{\mathrm{p}}$ and $\beta$ have been chosen such that the only symmetry
operation necessary to generate a full, infinite plot are horizontal and
vertical translations.
Again, agreements and disagreements in symmetries are found by comparison with
Figure 1. Inspection of Figures 4 and 5 immediately reveals differences in the
transformation behaviour of $\rho_{x}$ and $\rho_{z}^{\mathrm{eq}}$.
As shown earlier it is possible to implement any logic gate from a NMR
measurable quantity that has the symmetry of a class 3 gate in Figure 1 (e.g.
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ in Figure 4(a)). However, none of
the three contour plots in Figure 5 displays this symmetry and no class 3
gates can be implemented. The symmetries present belong to those of class 0, 1
and 2 gates and can be found in any of the three contour plots. Implementing
class 1 gates requires taking advantage of the periodicity of the functions
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ in order to generate canalising
inputs. Class 2 gates use a reduced parameter range for $\phi_{\mathrm{p}}$
and $\beta$.
Permutation of $\phi_{\mathrm{p}}$ and $\beta$ changes
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$, while
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ stays invariant under this
operation. $M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ are invariant under input
inversion $\phi_{\mathrm{p}}\rightarrow-\phi_{\mathrm{p}}$ or
$\beta\rightarrow-\beta$ while $M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ is
only invariant under $\beta\rightarrow-\beta$. All functions are now periodic
with $\pi$ in $\phi_{\mathrm{p}}$ because of their dependence on
$\sin^{2}\phi_{\mathrm{p}}$ or $\sin 2\phi_{\mathrm{p}}$.
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ have a $2\pi$ periodicity interval
in $\beta$ and $\pi$ periodicity for
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$. All the functions
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ are the products of two
trigonometric functions of which at least one is quadratic. The squaring
reduces the trigonometric function values to the interval $\left[0,1\right]$
and therefore, it is not possible to have a symmetry behaviour required for
class 3 gates.
The rotation operator $R_{\phi_{\mathrm{p}}}(\beta)$ _not commuting_ with the
initial density matrix $\rho_{z}^{\mathrm{eq}}$ leads to the functions
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ described by products of linear
trigonometric functions and the ability to implement _all_ logic gates. The
rotation operator $R_{\phi_{\mathrm{p}}}(\beta)$ _commuting_ with the initial
density matrix $\rho_{x}$, however, leads to the functions
$M_{x}\left(\phi_{\mathrm{p}},\beta\right)$,
$M_{y}\left(\phi_{\mathrm{p}},\beta\right)$ and
$M_{xy}\left(\phi_{\mathrm{p}},\beta\right)$ characterised by quadratic
trigonometric factors and _does not_ permit the implementation of all logic
gates. Achieving universality when starting from $\rho_{x}$ therefore requires
a different rotation operator that never commutes with $\rho_{x}$. This could
be achieved by choosing different pairs of parameters from the set of
parameters describing the single-pulse experiment (Figure 3) such that a non-
commuting rotation operator for $\rho_{x}$ results. Alternatively, one can
keep the pair of parameters $\phi_{\mathrm{p}}$ and $\beta$, and use more than
one r.f. pulse, with rotation operators such as
$R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)$.
This latter option we consider next.
#### 3.2.2 Two-pulse gates
The initial density matrix $\rho_{x}$ transforms under the influence of two
r.f. pulses
$R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)$
as
$\displaystyle\rho^{\left(x\right)}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)=R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)\rho_{x}R_{\phi_{\mathrm{p1}}}^{\dagger}\left(\beta_{1}\right)R_{\phi_{\mathrm{p}2}}^{\dagger}\left(\beta_{2}\right)$
(67)
Thus, the NMR detectable quantities
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$,
$M_{y}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
and
$M_{xy}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
are functions of the four variables $\phi_{\mathrm{p2}}$, $\beta_{2}$,
$\phi_{\mathrm{p1}}$ and $\beta_{1}$. Any two of these variables can serve as
logic gate input parameters with the remaining two fixed. This offers a high
degree of flexibility in assigning parameters as variable (and to control the
logic gate), and as fixed.
A trivial example is based on the fact that all 2-input gates can be
implemented when starting from thermal equilibrium magnetisation by a single
pulse. Starting from $\rho_{x}$, a first r.f. pulse
$R_{\frac{\pi}{2}}\left(-\frac{\pi}{2}\right)$ generates
$\rho_{z}^{\mathrm{eq}}$ and yields $\phi_{\mathrm{p1}}=\frac{\pi}{2}$ and
$\beta_{1}=-\frac{\pi}{2}$. From there on any second r.f. pulse
$R_{\phi_{\mathrm{p2}}}\left(\beta_{2}\right)$ generates a functional
behaviour identical to that of the single-pulse scenario in Eq. (54),
generating universality.
Another strategy uses the commutation properties of the rotation operators. In
the single-pulse scenario, starting from $\rho_{z}^{\mathrm{eq}}$ all logic
gates can be implemented because the overall rotation operator never commutes
with the initial density matrix. In the two-pulse scenario,
$R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)$
must not commute with $\rho_{x}$. This implies further that the first r.f.
pulse $R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)$ must not commute with
$\rho_{x}$, otherwise Eq. (67) simplifies to
$\rho^{\left(x\right)}\left(\phi_{\mathrm{p2}},\beta_{2}\right)=R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)\rho_{x}R_{\phi_{\mathrm{p}2}}^{\dagger}\left(\beta_{2}\right)$
(identical to Eq. (56)) which _can not_ implement all logic gates. To avoid
commutation, $\phi_{\mathrm{p1}}$ should never be fixed to values $n\pi$ (Eq.
(55)). The second r.f. pulse $R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)$
must not commute with the spin basis operators $I_{x}$ or $I_{y}$ when
calculating
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)=\overline{\Braket{I_{x}}}$
or
$M_{y}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)=\overline{\Braket{I_{y}}}$
(Eq. (43)), otherwise
$R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right)R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)\rho_{x}R_{\phi_{\mathrm{p1}}}^{\dagger}\left(\beta_{1}\right)I_{x}R_{\phi_{\mathrm{p}2}}^{\dagger}\left(\beta_{2}\right)$
is just a similarity transform of
$R_{\phi_{\mathrm{p1}}}\left(\beta_{1}\right)\rho_{x}R_{\phi_{\mathrm{p1}}}^{\dagger}\left(\beta_{1}\right)I_{x}$
under which the trace is invariant. A commutation
$\left[R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right),I_{x}\right]=0$ can be
avoided by setting $\phi_{\mathrm{p}2}\neq n\pi$. For
$M_{y}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
a commutator
$\left[R_{\phi_{\mathrm{p}2}}\left(\beta_{2}\right),I_{y}\right]=0$ is avoided
for $\phi_{\mathrm{p}}\neq\left(2n+1\right)\frac{\pi}{2}$ . These constraints
on the parameters of type phase $\phi_{\mathrm{p}}$ implement the functions
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
such as those shown in Table 5 and Figure 6: these examples were deliberately
chosen such that the two fixed variables are taken as having equal values of
$\frac{\pi}{2}$.
| $\phi_{\mathrm{p2}}$ | $\beta_{2}$ | $\phi_{\mathrm{p1}}$ | $\beta_{1}$ | $M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
---|---|---|---|---|---
(a) | $A$ | $\frac{\pi}{2}$ | $B$ | $\frac{\pi}{2}$ | $\frac{\lambda_{\mathrm{B}}}{4}\left(\cos\phi_{\mathrm{p2}}\cos\phi_{\mathrm{p1}}\cos\left(\phi_{\mathrm{p2}}-\phi_{\mathrm{p1}}\right)-\sin\phi_{\mathrm{p2}}\sin\phi_{\mathrm{p1}}\right)$
(b) | $A$ | $\frac{\pi}{2}$ | $\frac{\pi}{2}$ | $B$ | $\frac{\lambda_{\mathrm{B}}}{4}\left(\cos^{2}\phi_{\mathrm{p2}}\cos\beta_{1}-\sin\phi_{\mathrm{p2}}\sin\beta_{1}\right)$
(c) | $\frac{\pi}{2}$ | $A$ | $\frac{\pi}{2}$ | $B$ | $\frac{\lambda_{\mathrm{B}}}{4}\cos\left(\beta_{2}+\beta_{1}\right)$
(d) | $\frac{\pi}{2}$ | $A$ | $B$ | $\frac{\pi}{2}$ | $\frac{\lambda_{\mathrm{B}}}{4}\left(\cos^{2}\phi_{\mathrm{p1}}\cos\beta_{2}-\sin\phi_{\mathrm{p1}}\sin\beta_{2}\right)$
(e) | $A$ | $B$ | $\frac{\pi}{2}$ | $\frac{\pi}{2}$ | $-\frac{\lambda_{\mathrm{B}}}{4}\sin\phi_{\mathrm{p2}}\sin\beta_{2}$
(f) | $\frac{\pi}{2}$ | $\frac{\pi}{2}$ | $A$ | $B$ | $-\frac{\lambda_{\mathrm{B}}}{4}\sin\phi_{\mathrm{p1}}\sin\beta_{1}$
Table 5: Examples of logic gate implementations from
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
using any possible pair of variables as gate input $A$ and $B$, and fixing the
remaining two parameters to $\frac{\pi}{2}$. Apart from a change in sign,
examples (e) and (f) are identical to the case in Figure 4(a). Example (d) is
identical to (b). Figure 6: Contour plots of
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
corresponding to examples (a)–(c) in Table 5.
One can see that class 0, 1 and 3 gates can be implemented by all possible
permutations of fixed pairs of variables of identical value (here
$\frac{\pi}{2}$). The non-commutation constraint provides the non-canalising
behaviour of class 3 gates, while the inherent periodicity of the system is
sufficient to provide the canalising input configurations necessary for class
0 and 1 gates. However, periodicity does not suffice for a class 2 gate
implementation. Comparing the single-pulse scenario (Figure 4(a)) with the
contour plots in Figure 6 demonstrates that class 2 gate implementations
require horizontal and vertical traces of constant value zero. Such traces are
absent in the contour plots in Figure 6. In short, arbitrarily choosing pairs
of fixed parameters as equal valued (not just for $\frac{\pi}{2}$) leads to
the loss of universality.
Let us examine if universality is regained if the fixed parameters are no
longer taken as equal valued. For example, taking $\beta_{1}=\frac{\pi}{2}$
and $\beta_{2}=\pi$ gives
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)=\frac{\lambda_{\mathrm{B}}}{4}\cos\left(2\phi_{\mathrm{p2}}-\phi_{\mathrm{p1}}\right)\cos\left(\phi_{\mathrm{p1}}\right)$.
The corresponding contour plot is shown in Figure 7(a) highlighting the
presence of constant horizontal zero-valued traces. Allowing for unequal-
valued pairs of parameters $\beta_{1}$ and $\beta_{2}$ (or
$\phi_{\mathrm{p1}}$ and $\phi_{\mathrm{p2}}$ ) permits implementation of
class 0, 1 and 2 gates but not class 3.
Universality is regained if pairs of parameters of _different_ _types_ , e.g.
$\phi_{\mathrm{p1}}$ and $\beta_{2}$ are fixed and do _not_ assume equal
values. Figure 7(b) shows that $\phi_{\mathrm{p1}}=\frac{\pi}{2}$,
$\beta_{2}=\pi$ gives
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)=\frac{\lambda_{\mathrm{B}}}{4}\cos
2\phi_{\mathrm{p2}}\cos\beta_{1}$. Both horizontal and vertical zero-valued
traces are re-established in the corresponding contour plot (Figure 7(b)); all
2-input logic gates can be obtained.
Figure 7: Contour plot of
$M_{x}\left(\phi_{\mathrm{p2}},\beta_{2},\phi_{\mathrm{p1}},\beta_{1}\right)$
for (a) $\beta_{1}=\frac{\pi}{2}$, $\beta_{2}=\pi$ and for (b)
$\phi_{\mathrm{p1}}=\frac{\pi}{2}$, $\beta_{2}=\pi$.
The flexibility and the ease of implementation as seen from our illustrative
examples are good indicators for the richness of the natural computational
potential of this system. Computational operations more complicated than just
2-input boolean logics are therefore well within the grasp of this system.
This includes multi-input logic gates, continuous logic, and analogue
computing implementations. The ubiquitous occurrence of function values
$\left\\{-1,0,1\right\\}$, due to the trigonometric system functions,
especially holds promise for the implementation of a balanced ternary logic as
being natural to this particular system [7]. Note also, that we can reverse
the operation of a logic gate if we know the pulse(s) that originally
generated the output [20]. This does not match with the usual predictions
about a computation where e.g. NAND gates are not reversible [8]. Obviously
our raw output stores additional information that can be exploited for more
sophisticated computations.
## 4 Summary and Conclusions
We have provided a design approach to analysing novel substrates in order to
determine which of their parameters can be used to implement boolean logic
gates. We have illustrated this with a case study drawn from NMR-based
classical computation. The design process requires cataloguing the parameters
that are naturally used to describe and manipulate the target system,
analysing their behaviours in combinations, and then matching the resulting
patterns of behaviour with the corresponding behaviour patterns of the target
gate classes. Our case study here focusses on two particular parameters
($\beta=\kappa_{\mathrm{p}}\tau_{\mathrm{p}}$ and $\phi_{\mathrm{p}}$); a full
design study would assess other combinations of other parameters, since the
aim is not simply to find some solution, but to analyse the ‘natural’
computational capabilities of the substrate. For example, in our work on
continuous gates [6], we focussed on $\omega_{\mathrm{p}}$ and
$\tau_{\mathrm{p}}$.
The design approach is not restricted to 2-input gates; combining several
parameters can produce patterns corresponding to more complex gates. More
sophisticated experiments could directly exploit symmetry properties of the
Hamiltonian (Eq. (2.2)–(14)). Eventually, however, specific behaviours will be
more easily achieved by combining simpler gates in circuits than by directly
designing complex gates.
Circuit design requires additional analysis to determine how individual gates
can be combined in a circuit in a manner ‘natural’ for the substrate in
question. Circuit design requires determining techniques for: sequencing
parameter manipulations to implement the sequence of gate operations in a
circuit; combining manipulations to implement multiple gate operations in
parallel; routing and transforming the output of one gate to the appropriate
input of the next. We will address these circuit design issues in a companion
paper [5].
This work addresses only how to implement ‘classical’ boolean logic gates in
unconventional substrates, yet it is clear that the example system has further
information available, that is thrown away when viewing it as a boolean gate.
If this extra information is instead retained and exploited, more powerful
computation becomes available [1]. The real computational power of these novel
substrates will come from not viewing them as merely alternative ways of
implementing classical logic gates, but from exploiting them to implement non-
classical forms of computation: quantum, continuous (analogue), hybrid, and
more. The design approach described here forms a first step in a principled
approach for analysing substrates with a view to performing a specified form
of computation.
## 5 Acknowledgements
Support of this work by the Deutsche Forschungsgemeinschaft and Leverhulme
grant FM/00224/AM is gratefully acknowledged. We thank Alastair Abbott for
constructive comments.
## References
* [1] Alastair A. Abbott, Matthias Bechmann, Christian S. Calude, and Angelika Sebald. (May 2011). A nuclear magnetic resonance implementation of a classical Deutsch-Jozsa algorithm. submitted.
* [2] Andrew Adamatzky. (2007). Physarum machines: encapsulating reaction-diffusion to compute spanning tree. Naturwissenschaften, 94(12):975–980.
* [3] Leonard M. Adleman. (1994). Molecular computation of solutions to combinatorial problems. Science, 266(5187):1021–1024.
* [4] Martyn Amos. (2005). Theoretical and Experimental DNA Computation. Springer.
* [5] Matthias Bechmann, Angelika Sebald, and Susan Stepney. Direct wiring of multi-gate NMR logic circuits. In preparation.
* [6] Matthias Bechmann, Angelika Sebald, and Susan Stepney. (2010). From binary to continuous gates—and back again. In ICES 2010, York, UK, September 2010, volume 6274 of LNCS, pages 335–347. Springer.
* [7] Matthias Bechmann, Angelika Sebald, and Susan Stepney. (2011). Design of ternary logic gates: a nucler magnetic resonance case study. In preparation.
* [8] Charles H. Bennett. (1982). The thermodynamics of computation—a review. Int. J. Theor. Phys., 21:905–940.
* [9] Vinícius P. Correia, André I. Reis, Cep Porto, and Alegre Rs Brasil. (2001). Classifying n-input boolean functions. In Proc. VII Workshop de Iberchip, IWS, Montevideo, page 58.
* [10] Barbara Drossel. (2009). Random Boolean Networks, pages 69–110. Wiley.
* [11] Solomon Golomb. (1959). On the classification of boolean functions. IRE Transactions on Information Theory, 5(5):176–186.
* [12] Simon L. Harding and Julian F. Miller. (2004). A tone discriminator in liquid crystal. In CEC 2004, pages 1800–1807. IEEE Press.
* [13] Simon L. Harding and Julian F. Miller. (2005). Evolution in materio: A real-time robot controller in liquid crystal. In Proc. NASA/DoD Conference on Evolvable Hardware, pages 229–238. IEEE Press.
* [14] Simon L. Harding, Julian F. Miller, and Edward A Rietman. (2008). Evolution in materio: Exploiting the physics of materials for computation. Int. J. Unconventional Computing, 4(2):155–194.
* [15] Yorik Hardy and Willi H. Steeb. (2001). Classical and Quantum Computing. Birkhäuser Verlag, Basel.
* [16] Jonathan A. Jones. (2001). NMR quantum computation. Prog. Nucl. Magn. Reson. Spectrosc., 38(4):325–360.
* [17] Jonathan A. Jones. (2011). Quantum computing with nmr. Progress in Nuclear Magnetic Resonance Spectroscopy, 59(2):91–120.
* [18] Viv Kendon, Angelika Sebald, Susan Stepney, Matthias Bechmann, Peter Hines, and Robert C. Wagner. (2011). Heterotic computing. In Unconventional Computation 2011, number 6714 in LNCS, pages 113–124. Springer.
* [19] Lothar Kuhnert, Konstantin Agladze, and Valentin Krinsky. (1989). Image processing using light-sensitive chemical waves. Nature, 337:244–247.
* [20] Malcolm. H. Levitt. (April 2008). Spin Dynamics: Basics of nuclear magnetic resonance. John Wiley & Sons, Ltd, Chichester, 2nd edition.
* [21] Jonathan W. Mills. (2008). The nature of the extended analog computer. Phys. D, 237(9):1235–1256.
* [22] Jonathan W. Mills, Matt Parker, Bryce Himebaugh, Craig Shue, Brian Kopecky, and Chris Weilemann. (2006). “Empty Space” computes: The evolution of an unconventional supercomputer. In Proc. of the 3rd conference on Computing frontiers, pages 115–126.
* [23] Ikuko N. Motoike and Andrew Adamatzky. (2005). Three-valued logic gates in reaction-diffusion excitable media. Chaos, Solitons & Fractals, 24(1):107–114.
* [24] Marta Roselló-Merino, Matthias Bechmann, Angelika Sebald, and Susan Stepney. (2010). Classical computing in nuclear magnetic resonance. Int. J. Unconventional Computing, 6(3–4):163–195.
* [25] Jun J. Sakurai. (1994). Modern Quantum Mechanics. Addison-Wesley Publishing Company, Reading, MA, Revised edition.
* [26] Benjamin Schumacher. (Apr 1995). Quantum coding. Phys. Rev. A, 51(4):2738–2747.
* [27] Jakub Sielewiesiuk and Jerzy Gorecki. (2001). Logical functions of a cross-junction of excitable chemical media. J. Phys. Chem. A, 105(35):8189–8195.
* [28] David Slepian. (1953). On the number of symmetry types of boolean functions of $n$ variables. Can. J. Math., 5(2):185–193.
* [29] Susan Stepney. (July 2008). The neglected pillar of material computation. Physica D, 237(9):1157–1164.
* [30] Susan Stepney, Samson Abramsky, Andy Adamatzky, Colin Johnson, and Jon Timmis. (2008). Grand challenge 7: Journeys in non-classical computation. In Visions of Computer Science, pages 407–421. BCS.
* [31] Atsushi Tero, Seiji Takagi, Tetsu Saigusa, Kentaro Ito, Dan P. Bebber, Mark D. Fricker, Kenji Yumiki, Ryo Kobayashi, and Toshiyuki Nakagaki. (2010). Rules for biologically inspired adaptive network design. Science, 327(5964):43–9–442.
* [32] Ágota Tóth and Kenneth Showalter. (1995). Logic gates in excitable media. J. Chem. Phys., 103:2058–2066.
* [33] Damien Woods and Thomas J. Naughton. (2008). Parallel and sequential optical computing. In Optical SuperComputing, volume 5172 of LNCS, pages 70–86. Springer.
* [34] Damien Woods and Thomas J. Naughton. (2009). Optical computing. Appl. Math. Comput., 215(4):1417–1430.
* [35] Stephen S. Yau and Calvin K. Tang. (feb. 1970). Universal logic modules and their applications. IEEE Transactions on Computers, C-19(2):141–149.
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2024-09-04T02:49:21.911678
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthias Bechmann, Angelika Sebald, Susan Stepney",
"submitter": "Matthias Bechmann",
"url": "https://arxiv.org/abs/1109.0918"
}
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1109.0963
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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-013 CERN-PH-EP-2011-135
Observation of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pair
production in pp collisions at $\sqrt{s}=7~{}\mathrm{TeV}$
The LHCb Collaboration 111Authors are listed on the following pages.
The production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
pairs in proton-proton collisions at a centre-of-mass energy of
$7~{}\mathrm{TeV}$ has been observed using an integrated luminosity of
$37.5~{}\mathrm{pb}^{-1}$ collected with the LHCb detector. The production
cross-section for pairs with both
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ in the rapidity
range $2<y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}<4.5$ and
transverse momentum
$p_{\mathrm{T}}^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<10~{}\mathrm{GeV}/c$ is
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}=5.1\pm 1.0\pm
1.1~{}\mathrm{nb},$
where the first uncertainty is statistical and the second systematic.
(Submitted to Physics Letters B.)
R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z.
Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P.
Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,
R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A.
Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11,
J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C.
Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A.
Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8,
G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M.
van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T.
Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V.
Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52,
T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J.
Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H.
Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J.
Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7, M. Calvi20,j,
M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho
Akiba23, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N.
Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M.
Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P.
Collins37, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42,
G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P.
David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C. Deplano15, O. Deschamps5,
F. Dettori15,d, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, S. Donleavy48,
A. Dosil Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R.
Dzhelyadin34, C. Eames49, S. Easo45, U. Egede49, V. Egorychev30, S.
Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L.
Eklund47, Ch. Elsasser39, D.G. d’Enterria35,o, D. Esperante Pereira36, L.
Estève43, A. Falabella16,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C.
Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S.
Filippov32, C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M.
Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J.
Garofoli52, J. Garra Tico43, L. Garrido35, C. Gaspar37, N. Gauvin38, M.
Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C.
Göbel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M.
Grabalosa Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34,
T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-
Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7,
V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks48, W. Hofmann10, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40, M.
Karacson37, T.M. Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38,
B. Khanji6, Y.M. Kim46, M. Knecht38, S. Koblitz37, P. Koppenburg23, A.
Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25,
R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J.
Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37,
J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N.
Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A.
Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A.
Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B.
Maynard52, A. Mazurov32,16,37, G. McGregor50, R. McNulty12, C. Mclean14, M.
Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A.
Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P. Morawski25, R.
Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26,
M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, J.
Nardulli45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-
Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N. Nikitin31, A. Novoselov34, A.
Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41,
R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, B. Pal52,
J. Palacios39, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b,
C. Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49,
S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A.
Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du
Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H.
Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24,
S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36,
G.J. Rogers43, S. Roiser37, V. Romanovsky34, J. Rouvinet38, T. Ruf37, H.
Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B.
Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, R. Santinelli37,
E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I.
Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O.
Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P.
Skottowe43, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, K. Sobczak5, F.J.P.
Soler47, A. Solomin42, F. Soomro49, B. Souza De Paula2, B. Spaan9, A.
Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S.
Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, N.
Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T.
Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51,45, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, A. Ukleja27, P. Urquijo52, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36,
S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42,
A. Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25,37, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51,
Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, L. Zhang52, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The mechanism of heavy quarkonium production is a long-standing problem in
QCD. An effective field theory, non-relativistic QCD (NRQCD), provides the
foundation for much of the current theoretical work. According to NRQCD, the
production of heavy quarkonium factorizes into two steps: a heavy quark-
antiquark pair is first created perturbatively at short distances and
subsequently evolves non-perturbatively into quarkonium at long distances. The
NRQCD calculations depend on the colour-singlet (CS) and colour-octet (CO)
matrix elements, which account for the probability of a heavy quark-antiquark
pair in a particular colour state to evolve into heavy quarkonium.
Leading order (LO) calculations in the CS model [1, 2, 3] were first used to
describe experimental data. However, they underestimate the observed cross-
section for single ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
production at high $p_{\mathrm{T}}$ at the Tevatron [4]. To resolve this
discrepancy the CO mechanism was introduced [5]. The corresponding matrix
elements were determined from the large-$p_{\mathrm{T}}$ data as the CO cross-
section falls more slowly than the CS one. However, recent calculations [6, 7,
8, 9] close the gap between the CS predictions and the experimental data [10]
reducing the need for large CO contributions. Thus, further experimental tests
are needed. Pair production of quarkonium can cast light on this problem as
this process depends heavily on the production mechanism. For both the CS and
CO models, contributions from double parton scattering [11, 12, 13] could
potentially be significant.
The only observation of charmonia pair production in hadronic collisions to
date was by the NA3 collaboration, who found evidence for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pair production in
multi-muon events in pion-platinum interactions at pion momenta of 150 and 280
GeV$/c$ [14] and in proton-platinum interactions at a proton momentum of 400
GeV$/c$ [15]. The cross-section ratio
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}/\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ was
measured to be $(3\pm 1)\times 10^{-4}$ for pion-induced production, where
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is the
inclusive ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production
cross-section. At NA3 energies the main contribution to the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pair cross-section
arises from the quark-antiquark annihilation channel [16]. This is not the
case for proton-proton collisions at the LHC, where the gluon-gluon fusion
process dominates [17, 18].
Theoretical calculations based on the LO production of CS-states predict that
the total cross-section for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$-pair production in proton-proton interactions at $\sqrt{s}=7$ TeV is
equal to $24$ nb [19, 20]. These calculations take into account
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$,
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uppsi(2S)$ and
$\uppsi(2S)\uppsi(2S)$ production but do not include the possible contribution
from double parton scattering. In the rapidity interval
$2.0<y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}<4.5$,
relevant to the LHCb experiment, the expected value is $4~{}\mathrm{nb}$ with
an uncertainty of around 30%. At small invariant masses of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pair a tetraquark
state, built from four $c$-quarks, may exist [20] and would be visible as a
narrow resonance in the mass spectrum.
## 2 The LHCb detector and dataset
The LHCb detector is a forward spectrometer [21] providing charged particle
reconstruction in the pseudorapidity range $1.9<\eta<4.9$. The detector
elements are placed along the beam line of the LHC starting with the Vertex
Locator, a silicon strip device that surrounds the proton-proton interaction
region. This reconstructs precisely the locations of interaction vertices, the
locations of decays of long-lived hadrons and contributes to the measurement
of track momenta. Other detectors used to measure track momenta comprise a
large area silicon strip detector located upstream of a dipole magnet with
bending power around 4 Tm and a combination of silicon strip detectors and
straw drift-tubes placed downstream. Two ring imaging Cherenkov detectors are
used to identify charged hadrons. Further downstream, an electromagnetic
calorimeter is used for photon and electron identification, followed by a
hadron calorimeter and a muon system consisting of alternating layers of iron
and chambers (MWPC and triple-GEM) that distinguishes muons from hadrons. The
calorimeters and muon system provide the capability of first-level hardware
triggering.
The LHCb trigger system consists of three levels. The first level (L0) is
designed to reduce the LHC bunch crossing rate of 40 MHz to a maximum of 1
MHz, at which the complete detector is read out. This is the input to the
first stage of the software trigger, which performs a partial event
reconstruction to confirm or discard the L0 trigger decision. The second stage
of the software trigger performs a full event reconstruction to further
discriminate signal events from other $\mathrm{pp}$ collisions. To avoid that
a few events with high occupancy dominate the CPU time, a set of global event
cuts is applied on the hit multiplicities of each sub-detector used by the
pattern recognition algorithms. These cuts were chosen to reject high-
multiplicity events with a large number of $\mathrm{pp}$ interactions with
minimal loss of luminosity.
The data used for this analysis comprise an integrated luminosity of
$37.5~{}\mathrm{pb}^{-1}$ of $\mathrm{pp}$ collisions at a centre-of-mass
energy of 7 $\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb experiment
between July and November 2010. This number includes the dead-time of trigger
and data acquisition systems. During this period all detector components were
fully operational and in a stable condition. The mean number of visible
proton-proton collisions per bunch crossing was up to 2.5.
The simulation samples used are based on the Pythia 6.4 generator [22]
configured with the parameters detailed in Ref. [23]. The EvtGen [24] and
Geant4 [25] packages are used to generate hadron decays and simulate
interactions in the detector, respectively. Prompt charmonium production is
generated in Pythia according to the leading order CS and CO mechanisms.
## 3 Event selection and signal yield
In this analysis the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
is reconstructed through its decay into a pair of muons. Events with at least
four muons are selected. ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ candidates are formed from pairs of
oppositely-charged particles identified as muons that each have a transverse
momentum greater than $650~{}\mathrm{MeV}/c$ and that originate from a common
vertex. Track quality is ensured by requiring that the
$\chi^{2}_{\rm{tr}}/\mathrm{ndf}$ provided by the track fit is less than five.
Well identified muons are selected by requiring that the difference in
logarithms of the global likelihood of the muon hypothesis, provided by the
particle identification detectors [26], with respect to the hadron hypothesis,
$\Delta\ln\mathcal{L}^{\mu-\mathrm{h}}$, be greater than zero. To suppress the
contribution from duplicate particles created by the reconstruction procedure,
if two muon candidates have a symmetrized Kullback-Leibler divergence [27]
less than 5000, only the particle with the best track fit is considered.
Selected $\upmu^{+}\upmu^{-}$ candidates with an invariant mass in the range
$3.0<m_{\upmu^{+}\upmu^{-}}<3.2~{}\mathrm{GeV}/c^{2}$ are paired to form
$(\upmu^{+}\upmu^{-})_{1}(\upmu^{+}\upmu^{-})_{2}$ combinations. A fit of the
four-muon candidate is performed [28] that requires the four tracks to be
consistent with originating from a common vertex and that this vertex is
compatible with one of the reconstructed $\mathrm{pp}$ collision vertices. To
reject background where two ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ candidates originate from different $\mathrm{pp}$ collisions, the
reduced $\chi^{2}$ of this fit, $\chi^{2}_{\mathrm{}}/\mathrm{ndf}$, is
required to be less than five.
The number of events with two
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons is extracted
from the single ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass
spectra. The invariant mass distributions of the first muon pair are obtained
in bins of the invariant mass of the second pair.222 The $\upmu^{+}\upmu^{-}$
pair with lower transverse momentum is chosen to be the first pair. The single
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass spectrum is
modelled empirically using simulated events. This exhibits non-Gaussian tails
on either side of the peak. The tail on the left-hand side is dominated by
radiative effects in ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
decay, while the right-hand side tail is due to non-Gaussian effects in the
reconstruction. The shape of the distribution is described by a function that
is similar to a Crystal Ball function [29, 30], but with the power-law tails
on both sides of the core Gaussian component. The position of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ peak, the effective
mass resolution and the tail parameters of this double-sided Crystal Ball
function are fixed to the values determined from an analysis of the signal
shape in the inclusive ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ sample. Combinatorial background is modelled using an exponential
function. This model is used to extract the yield of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\left(\upmu^{+}\upmu^{-}\right)_{1}$ in bins of the
$(\upmu^{+}\upmu^{-})_{2}$ invariant mass. The extracted yield is shown in
Fig. 1a together with the result of a fit according to the model described
above. The yield of events with double
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production given by
the fit is $N^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}=141\pm 19$,
where the statistical significance of this signal exceeds $6\sigma$. A fit
with position and resolution of the signal peak left free was also performed
and gave consistent results.
$m_{\left(\upmu^{+}\upmu^{-}\right)_{2}}$$m_{\left(\upmu^{+}\upmu^{-}\right)_{2}}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$$\begin{array}[]{l}\text{LHCb}\\\
\sqrt{s}=7~{}\mathrm{TeV}\end{array}$$\begin{array}[]{l}\text{LHCb}\\\
\sqrt{s}=7~{}\mathrm{TeV}\end{array}$
$\frac{\mathrm{d}N^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\left(\upmu^{+}\upmu^{-}\right)_{1}}}{\mathrm{d}m_{\left(\upmu^{+}\upmu^{-}\right)_{2}}}\left[\frac{1}{20~{}\mathrm{MeV}/c^{2}}\right]$
$\frac{\mathrm{d}N^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\left(\upmu^{+}\upmu^{-}\right)_{1}}}{\mathrm{d}m_{\left(\upmu^{+}\upmu^{-}\right)_{2}}}\left[\frac{1}{20~{}\mathrm{MeV}/c^{2}}\right]$
a) b) Figure 1: The fitted yields of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\left(\upmu^{+}\upmu^{-}\right)_{1}$ in bins of
$(\upmu^{+}\upmu^{-})_{2}$ invariant mass: (a) the raw signal yield observed
in the data; (b) the efficiency corrected yield (Sect. 6). The result of a fit
with a double-sided Crystal Ball function for the signal and an exponential
background is superimposed.
Studies of single ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
production indicate that the detector acceptance and efficiency is high for
the fiducial range $2<y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<4.5$ and
$p_{\mathrm{T}}^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<10~{}\mathrm{GeV}/c$. The raw yield of events with both
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons within this
range is $139\pm 18$. The yield of events with both
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons in the
fiducial range and explicitly triggered by one of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ candidates through
the single muon or dimuon trigger lines [31], is found to be $116\pm 16$. This
sample is considered for the determination of the production cross-section.
The contribution to the yield from the pileup of two interactions each
producing a single ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
meson is estimated using simulation together with the measured
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production cross-
section [32]. This study shows that for the 2010 data-taking conditions the
background from this source is expected to be less than 1.5 events and hence
can be neglected. In a similar way the contribution to the yield from events
with ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons
originating from the decays of beauty hadrons is found to be negligible.
## 4 Efficiency evaluation
The per-event efficiency for a
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$-pair event,
$\varepsilon^{\mathrm{tot}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$, is decomposed
into three factors,
$\varepsilon^{\mathrm{tot}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\times\varepsilon^{\mathrm{\mu
ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\times\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}},$ (1)
where
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is the product
of the (geometrical) acceptance with reconstruction and selection efficiency,
$\varepsilon^{\mathrm{\mu
ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is the
efficiency for muon identification and
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is the trigger
efficiency for reconstructed and selected events.
The efficiency for the acceptance, reconstruction and selection for the two
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons is factorized
into the product of efficiencies for the first and second
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$,
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\left(p^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{1}}_{\mathrm{T}},y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{1}},|\cos\vartheta^{*}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{1}}|\right)\times\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\left(p^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{2}}_{\mathrm{T}},y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{2}},|\cos\vartheta^{*}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{2}}|\right).$ (2)
The single ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
efficiency
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ is a function of the transverse momentum $p_{\mathrm{T}}$, rapidity
$y$ and $\left|\cos\vartheta^{*}\right|$, where $\vartheta^{*}$ is the angle
between the $\upmu^{+}$ momentum in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ centre-of-mass frame
and the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ flight
direction in the laboratory frame. It is evaluated using simulation. The
validity of the factorization hypothesis of Eq. (2) is checked with simulation
and based on these studies a correction factor of 0.975 is applied to
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$. For the
simulated data of single prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production the cut
on the muon likelihood is not applied and that on
$\chi^{2}_{\mathrm{}}({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu})/\mathrm{ndf}$
is replaced with a similar cut on the single
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$,
$\chi^{2}_{\mathrm{}}({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu})/\mathrm{ndf}<5$. The efficiency of the cut on
$\chi^{2}_{\mathrm{}}/\mathrm{ndf}$ is estimated from the data and compared to
the simulation. Based on these studies a correction factor of 1.026 is applied
to
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$, and a
systematic uncertainty of 3% is assigned. The efficiency
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ is also corrected by a factor $1.024\pm 0.011$ [32], that accounts
for the ratio of the reconstruction efficiency of the muon detector observed
in the data compared to the simulation.
The muon identification efficiency is extracted from the analysis of the
inclusive ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ sample.
Two efficiencies are evaluated: the single muon identification efficiency
$\varepsilon_{\mu}^{\mathrm{\mu ID}}$ and the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ efficiency
$\varepsilon_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}^{\mathrm{\mu ID}}$. Both are measured as a function of the value of
the cut made on $\Delta\ln\mathcal{L}^{\mu-\mathrm{h}}$. The squared
efficiency $(\varepsilon_{1\mu}^{\mu\mathrm{ID}})^{2}$ and
$\varepsilon_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}^{\mu\mathrm{ID}}$ are found to be equal to better than one per mille.
The value of $\varepsilon^{\mathrm{\mu
ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=(\varepsilon^{\mathrm{\mu
ID}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}})^{2}=(91.0\pm
0.1)\%$ has been used as a global factor for the evaluation of the total
efficiency using Eq. (1). As a cross-check, the efficiency of the muon
identification for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
pairs has been estimated from the signal itself. Though statistically limited,
the value is consistent with that given above.
The trigger efficiency is calculated to be
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=1-\left(1-\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\left(p^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{1}}_{\mathrm{T}},y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{1}}\right)\right)\times\left(1-\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\left(p^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{2}}_{\mathrm{T}},y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}_{2}}\right)\right).$
The trigger efficiency for a single
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$,
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$, is determined directly on data from the inclusive prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ sample as a function
of $p_{\mathrm{T}}$ and rapidity $y$. The efficiency is determined by
classifying an event which would also have been triggered without the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ as TIS (Trigger
Independent of Signal), and/or classifying the event where the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ alone is sufficient
to trigger the event as a TOS(Trigger On Signal) event [33, 34]. The LHCb
trigger system records all the information needed for such classification.
Events can be classified as TIS and TOS simultaneously (TIS&TOS), which allows
the extraction of the trigger efficiency relative to the off-line
reconstructed and selected events from data alone
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=\frac{N^{\mathrm{TIS\&TOS}}}{N^{\mathrm{TIS}}},$
where $N^{\mathrm{TIS}}$ is the number of TIS events, and
$N^{\mathrm{TIS\&TOS}}$ is the number of events that are simultaneously TIS
and TOS. The method has been cross checked using Monte Carlo simulation.
The effect of the global event cuts applied in the trigger has been studied in
detail for inclusive ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
events in Ref. [32]. Since the subdetector hit multiplicity observed in single
and double ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ events is
similar, the efficiency of the global event cuts, $(93\pm 2)\%$, is taken from
that study and applied to
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$.
For selected ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$-pair
events the mean value of
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is 31% and it
varies from 0.8% to 70%. The mean value for
$\varepsilon^{\mathrm{trg}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is 85% and it
varies from 61% to 93%.
## 5 Systematic uncertainties
Systematic uncertainties affecting the cross-section measurement have been
evaluated properly taking correlations into account where appropriate. The
dominant source of systematic uncertainty is due to the knowledge of the
track-finding efficiency. An uncertainty of 4% per track is assigned based on
studies comparing the reconstruction efficiency in data and simulation using a
tag and probe approach [34].
The second major source of uncertainty is due to the evaluation of the trigger
efficiency. The method discussed in Sect. 4 has been cross-checked in several
ways, in particular, by using events triggered by the first or second
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ only. The observed
differences lead to the assignment of an 8% systematic uncertainty.
A further source of uncertainty is the determination of the per-event
efficiency defined by Eq. (1). This is estimated to be $3\%$ by varying the
uncertainties of the various factors entering into Eq. (1).
The unknown polarization of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ mesons affects the acceptance, reconstruction and selection efficiency
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ [32]. In this analysis the effect is reduced by explicitly taking
into account the dependence of the acceptance on
$\varepsilon^{\mathrm{reco}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$ on $\left|\cos\vartheta^{*}\right|$ in the efficiency determination
(Eq. 2). The remaining dependence results in a systematic uncertainty of 5%
per ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$.
Additional systematic uncertainties arise due to the difference observed
between the data and simulation for the behaviour of the cut on
$\chi^{2}_{\mathrm{}}$ ($3\%$), the global event cuts ($2\%$), and uncertainty
of 1.1% per ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
associated with the efficiency for muon identification, as discussed in Sect.
4. The systematic uncertainties associated with the other selection criteria
and the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ lineshape
parametrization are negligible.
Table 1: Relative systematic uncertainties on the cross-section measurement. The total uncertainty is calculated as the quadratic sum of the individual components. Source | Systematic uncertainty $\left[\%\right]$
---|---
Track finding efficiency | $4\times 4$
Trigger efficiency | 8
Per-event efficiency | 3
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ polarization | $2\times 5$
Data/simulation difference for $\chi^{2}_{\mathrm{}}/{\mathrm{ndf}}$ | 3
Global event cuts | 2
Muon identification | $2\times 1.1$
Luminosity | 3.5
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ branching ratio | $2\times 1$
Total | 21
The luminosity was measured at specific periods during the data taking using
both van der Meer scans [35] and a beam-gas imaging method [36]. The
instantaneous luminosity determination is then based on a continuous recording
of the multiplicity of tracks reconstructed in Vertex Locator, which has been
normalized to the absolute luminosity scale. Consistent results are found for
the absolute luminosity scale with a precision of 3.5%, dominated by the beam
current uncertainty [37, 38].
The relative systematic uncertainties are summarized in Table 1, where the
total uncertainty is defined as the quadratic sum of the individual
components.
## 6 Cross-section determination
The efficiency-corrected yield for events with both
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ candidates in the
fiducial region is extracted using the procedure discussed in Sect. 3. To
account for the efficiency a weight $\omega$, defined as
$\omega^{-1}=\varepsilon^{\mathrm{tot}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$
where
$\varepsilon^{\mathrm{tot}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is the total
efficiency defined in Eq. (1), is applied to each candidate in the sample.
The corrected yield of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\left(\upmu^{+}\upmu^{-}\right)_{1}$ in bins of
$(\upmu^{+}\upmu^{-})_{2}$ invariant mass is shown in Fig. 1b. As previously
described, to extract the yield a fit with a double-sided Crystal Ball
function for the signal, together with an exponential function for the
background component, is performed. Again, the position of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ peak and the
effective mass resolution are fixed to the values found in the inclusive
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ sample. The event
yield after the efficiency correction is
$N^{\mathrm{corr}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}=672\pm 129.$
The cross-section for double ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ production in the fiducial range
$2<y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}<4.5$ and
$p_{\mathrm{T}}^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<10~{}\mathrm{GeV}/c$ is computed as
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=\frac{N^{\mathrm{corr}}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}{\mathcal{L}\times{\cal B}_{\upmu^{+}\upmu^{-}}^{2}},$
where $\mathcal{L}=37.5\pm 1.3~{}\mathrm{pb}^{-1}$ is the integrated
luminosity and ${\cal B}_{\upmu^{+}\upmu^{-}}=(5.93\pm 0.06)\%$ [39] is the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ branching ratio. The result is
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}=5.1\pm 1.0\pm
1.1~{}\mathrm{nb},$
where the first uncertainty is statistical and the second systematic.
Using the measured prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ production cross-section in the same fiducial region [32] and taking
into account the correlated uncertainties, the ratio of cross-sections
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}/\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ is
calculated to be
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}/\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=\left(5.1\pm 1.0\pm 0.6\,^{+1.2}_{-1.0}\right)\times 10^{-4},$
where the first error is statistical, the second systematic and the third is
due to the unknown polarization of the prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ from pair
production.
The differential production cross-section of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pairs as a function
of the invariant mass of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ system is shown
in Fig. 2. The whole analysis chain has been repeated for each bin of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ invariant mass
to get the differential production cross-section. The bulk of the distribution
is concentrated in the low invariant mass region. A theoretical prediction for
the shape of this distribution taking into account both direct production and
feeddown from $\uppsi(2S)$ decays [20] is overlaid. Within the available
statistics the agreement between the data and the prediction is reasonable.
$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}$$\left[\mathrm{GeV}/c^{2}\right]$$\begin{array}[]{l}\text{LHCb}\\\
\sqrt{s}=7~{}\mathrm{TeV}\end{array}$
$\frac{\mathrm{d}\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}{\mathrm{dm}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}\left[\frac{\mathrm{nb}}{1~{}\mathrm{GeV}/c^{2}}\right]$
Figure 2: Differential production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pairs as a function
of the invariant mass of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ system. The
points correspond to the data. Only statistical uncertainties are included in
the error bars. The shaded area corresponds to prediction by the model
described in Ref. [20].
## 7 Conclusions
The production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
pairs in proton-proton collisions at a centre-of-mass energy of
$7~{}\mathrm{TeV}$ has been observed with a statistical significance in excess
of $6\sigma$. The data are consistent with the predictions given in Refs. [19,
20]. The higher statistics that will be collected during the 2011 data-taking
period will allow the kinematic properties of these events to be studied and
different production models to be probed.
## Acknowledgments
We would like to thank A.K. Likhoded for many fruitful discussions. We express
our gratitude to our colleagues in the CERN accelerator departments for the
excellent performance of the LHC. We thank the technical and administrative
staff at CERN and at the LHCb institutes, and acknowledge support from the
National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China);
CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN
(Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS (Romania); MinES of
Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7 and the Region
Auvergne.
## References
* [1] V.G. Kartvelishvili, A.K. Likhoded and S.R. Slabospitsky, “D meson and $\uppsi$ meson production in hadronic interactions”, Sov. J. Nucl. Phys. 28, 678 (1978) [Yad. Fiz. 28, 1315 (1978)].
* [2] E.L. Berger and D.L. Jones, “Inelastic photoproduction of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$ by gluons”, Phys. Rev. D23, 1521, (1981).
* [3] R. Baier and R. Ruckl, “Hadronic production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$: transverse momentum distributions”, Phys. Lett. B 102, 364 (1981).
* [4] F. Abe et al., “Inclusive ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$, $\uppsi(2S)$ and $\mathrm{b}$-quark production in $\bar{\mathrm{p}}\mathrm{p}$-collisions at $\sqrt{s}=1.8$ TeV”, Phys. Rev. Lett. 69, 3704 (1992).
* [5] E. Braaten and S. Fleming, “Color octet fragmentation and the $\uppsi^{\prime}$-surplus at the Tevatron”, Phys. Rev. Lett. 74, 3327 (1995).
* [6] J.M. Campbell, F. Maltoni and F. Tramontano, “QCD corrections to ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$ production at hadron colliders”, Phys. Rev. Lett. 98, 252002 (2007).
* [7] B. Gong and J.X. Wang, “Next-to-leading-order QCD corrections to ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ polarization at Tevatron and Large-Hadron-Collider energies”, Phys. Rev. Lett. 100, 232001 (2008).
* [8] P. Artoisenet, J.M. Campbell, J.-P. Lansberg, F. Maltoni and F. Tramontano, “$\Upsilon$ production at Fermilab Tevatron and LHC energies”, Phys. Rev. Lett. 101, 152001 (2008).
* [9] J.-P. Lansberg, “On the mechanisms of heavy-quarkonium hadroproduction”, Eur. Phy. J. C61, 693 (2009).
* [10] N. Brambilla et al., “Heavy quarkonium: progress, puzzles, and opportunities”, Eur. Phy. J. C71, 1534 (2011).
* [11] C.H. Kom, A. Kulesza and W.J. Stirling, “Pair production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ as a probe of double parton scattering at LHCb”, Phys. Rev. Lett. 107, 092002 (2011).
* [12] S.P. Baranov, A.M. Snigirev and N.P. Zotov, “Double heavy meson production through double parton scattering in hadronic collisions”, arXiv:1105.6276 [hep-ph].
* [13] A. Novoselov, “Double parton scattering as a source of quarkonia pairs in LHCb”, arXiv:1106.2184 [hep-ph].
* [14] J. Badier et al., “Evidence for $\uppsi\uppsi$ production in $\pi^{-}$ interactions at 150 GeV/$c$ and 280 GeV/$c$”, Phys. Lett. B 114, 457 (1982).
* [15] J. Badier et al., “$\uppsi\uppsi$ production and limits on beauty meson production from 400 GeV/$c$ protons”, Phys. Lett. B 158, 85 (1985).
* [16] V.G. Kartvelishvili and S.M. Esakiya, “On hadron induced production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson pairs”, Yad. Fiz. 38, 722 (1983).
* [17] B. Humpert and P. Mery, “ $\uppsi\uppsi$ production at collider energies”, Z. Phys. C 20, 83 (1983).
* [18] V.V. Kiselev, A.K. Likhoded, S.R. Slabospitsky and A.V. Tkabladze, “Hadronic pair production of $\uppsi$ particles with large $M(\uppsi\uppsi)$,” Sov. J. Nucl. Phys. 49, 1041 (1989) [Yad. Fiz. 49, 1681 (1989)].
* [19] C.F. Qiao, L.P. Sun and P. Sun, “Testing charmonium production mechanism via polarized ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$-pair production at the LHC”, J. Phys. G 37, 075019 (2010).
* [20] A.V. Berezhnoy, A.K. Likhoded, A.V. Luchinsky and A.A. Novoselov, “Double ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson production at LHC and $4c$-tetraquark state”, arXiv:1101.5881 [hep-ph].
* [21] A.A. Alves et al., “The LHCb detector at LHC”, JINST 3, S08005 (2008).
* [22] T. Sjöstrand, S. Mrenna and P. Z. Skands, “Pythia 6.4 physics and manual”, JHEP 05, 026 (2006), version 6.422, arXiv:hep-ph/0603175.
* [23] I. Belyaev et al., “Handling of the generation of primary events in Gauss, the LHCb simulation framework”, Nuclear Science Symposium Conference Record (NSS/MIC), IEEE, 1155 (2010).
* [24] D. Lange, “The EvtGen particle decay simulation package”, Nucl. Instrum. Meth. A 462, 152 (2001).
* [25] S. Agostinelli et al., “Geant4: a simulation toolkit”, Nucl. Instrum. Meth. A 506, 250 (2003).
* [26] A. Powell, “Particle identification at LHCb”, PoS ICHEP2010, 020 (2010).
* [27] S. Kullback and R.A. Leibler, “On information and sufficiency”, Annals of Mathematical Statistics 22, 79 (1951);
S. Kullback, “Letter to editor: the Kullback-Leibler distance”, The American
Statistician 41, 340 (1987).
* [28] W.D. Hulsbergen, “Decay chain fitting with a Kalman filter”, Nucl. Instrum. Meth. A 552, 566 (2005).
* [29] J.E. Gaiser, “Charmonium spectroscopy from radiative decays of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\uppsi^{\prime}$”, Ph.D. Thesis, SLAC-R-255 (1982).
* [30] T. Skwarnicki, “A study of the radiative cascade transitions between the $\Upsilon^{\prime}$ and $\Upsilon$ resonances”, Ph.D. Thesis, DESY-F31-86-02 (1986).
* [31] R. Aaij et al., “Search for the rate decays $\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upmu^{+}\upmu^{-}$ and $\mathrm{B}^{0}_{\mathrm{d}}\rightarrow\upmu^{+}\upmu^{-}$”, Phys. Lett. B 699, 330 (2011).
* [32] R. Aaij et al., “Measurement of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production in pp collisions at $\sqrt{s}=7~{}\mathrm{TeV}$”, Eur. Phy. J. C 71, 1645 (2011).
* [33] E. Lópes Azamar et al., “Measurement of trigger efficiencies and biases”, CERN-LHCb-2008-073, (2008).
* [34] R. Aaij et al., “Prompt $\mathrm{K}^{0}_{\mathrm{S}}$ production in $\mathrm{pp}$ collisions at $\sqrt{s}=0.9~{}\mathrm{TeV}$”, Phys. Lett. B 693, 69 (2010).
* [35] S. van der Meer, “Calibration of the effective beam height in the ISR”, ISR-PO/68-31 (1968).
* [36] M. Ferro-Luzzi, “Proposal for an absolute luminosity determination in colliding beam experiments using vertex detection of beam-gas interactions”, Nucl. Instrum. Meth. A 553, 388 (2005).
* [37] G. Anders et al., “LHC bunch current normalization for the October 2010 luminosity calibration measurements”, CERN-ATS-Note-2011-016 PERF (2011).
* [38] R. Aaij et al., “Absolute luminosity measurements with the LHCb detector at the LHC”, LHCB-PAPER-2011-015, CERN-PH-EP-2011-157, arXiv:1110.2866, submitted to JINST.
* [39] K. Nakamura et al., “Review of particle physics”, J. Phys. G 37, 075021 (2010).
|
arxiv-papers
| 2011-09-05T17:05:16 |
2024-09-04T02:49:21.919447
|
{
"license": "Public Domain",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Anderson, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A. Artamonov, M.\n Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad,\n T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, S. Brisbane, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea,\n A. Bursche, J. Buytaert, S. Cadeddu, J.M. Caicedo Carvajal, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo,\n M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, P. Collins, F. Constantin, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, S. Donleavy, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L. Est\\`eve, A.\n Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, N. Gauvin, M. Gersabeck,\n T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, W. Hofmann, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse,\n R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei,\n M. Karacson, T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, Y.M. Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, S. Kukulak, R. Kumar, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, J. Luisier, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde,\n R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A.\n Milanes, M.-N. Minard, S. Monteil, D. Moran, P. Morawski, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli, I. Nasteva, M.\n Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, V. Niess, N.\n Nikitin, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, B. Pal, J. Palacios, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, B. Pie Valls, B.\n Pietrzyk, T. Pilar, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, T. du Pree, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, K. Rinnert, D.A. Roa Romero,\n P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S.\n Roiser, V. Romanovsky, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov,\n Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A.\n Shires, R. Silva Coutinho, H.P. Skottowe, T. Skwarnicki, A.C. Smith, N.A.\n Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B.\n Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica,\n S. Stone, B. Storaci, M. Straticiuc, U. Straumann, N. Styles, V.K. Subbiah,\n S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Topp-Joergensen, M.T. Tran, A. Tsaregorodtsev, N. Tuning, A.\n Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, K. Vervink, B. Viaud,\n I. Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Voong, A.\n Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D.R. Ward, A.D. Webber,\n D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P.\n Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A.\n Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zavertyaev, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev,\n A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1109.0963"
}
|
1109.1234
|
Matrix Quantization of Turbulence111Based on invited talks delivered at: Fifth
Aegean Summer School, “From Gravity to Thermal Gauge theories and the AdS/CFT
Correspondance”, September 2009, Milos, Greece; the Intern. Conference on
Dynamics and Complexity, Thessaloniki, Greece, 12 July 2010; Workshop on
“AdS4/CFT3 and the Holographic States of Matter”, Galileo Galilei Institute,
Firenze, Italy, 30 October 2010.
Emmanuel Floratos
Department of Physics, Univ. of Athens, GR-15771 Athens, Greece
Institute of Nuclear Physics, N.C.S.R. Demokritos, GR-15310, Athens, Greece
mflorato@phys.uoa.gr
Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos
(2009)], we provide an explicit quantization of the Lorenz chaotic attractor
through the introduction of Non-commutative phase space coordinates as
Hermitian $N\times N$ matrices in $R^{3}$. For the volume preserving part,
they satisfy the commutation relations induced by one of the two Nambu
Hamiltonians, the second one generating a unique time evolution. Dissipation
is incorporated quantum mechanically in a self-consistent way having the
correct classical limit without the introduction of external degrees of
freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Matrix Lorenz system develop fast decoherence to N
independent Lorenz attractors. On the other hand there is a weak dissipation
regime, where the quantum mechanical properties of the volume preserving non-
dissipative sector survive for long times.
## 1 Motivation
The study of turbulence is a very difficult and interesting subject in various
fields of science and engineering since it is very common in nature [Landau &
Lifshitz(1987)]. In modern fundamental physics it has been discussed in
extremes scales and temperatures ranging from cosmology for its important role
in the determination of the distribution of voids and the distribution of
peculiar velocities of galactic clusters around them [Gibson & Schild (2010)]
to the thermalization process of quark-gluon plasma in heavy ion collisions
[Asakawa et al.(2007)] as well as to the non-gaussianities in the velocity
spectrum of liquid helium vortex interactions [Van Sciver (2009)].
We are interested at the question of turbulence in quantum dissipative
systems. There, dissipation deforms the non-commutative phase space to
classical phase space and the basic question is about the spectra of
relaxation times to classical behavior. From the point of view of dissipation
the problem is equivalent to the one of open Hamiltonian [Weiss (2008)] or
non-Hamiltonian systems [Tarasov (2008)]. Moreover turbulent behavior
presupposes a fluid structure for the system with a superimposed mechanism for
the creation and annihilation of vortices [Landau & Lifshitz(1987)].
The phase space noncommutativity in Quantum mechanics at times $t=0$ is given
by the Canonical Commutation Relations(CCR)
$\left[Q_{i},P_{j}\right]\ =\ \imath\hbar\delta_{ij}\ \ \ \ \ \ \ t=0\ \ \ \ $
(1)
where $Q_{i},P_{i}\ \ \ i=1,2\cdots$ are conjugate canonical pairs of position
and momentum for a system with n degrees of freedom. In the Heisenberg picture
the operators develop in time according to the Heisenberg-Hamilton eqs. of
motion
$\displaystyle\imath\hbar\dot{Q}_{i}\ $ $\displaystyle=$ $\displaystyle\
\left[Q_{i},H\right]\ \ \ \ \ \ i=1,\ldots,n$
$\displaystyle\imath\hbar\dot{P}_{i}\ $ $\displaystyle=$ $\displaystyle\
\left[P_{i},H\right]$ (2)
with the CCRs preserved in time.
If dissipation is added linearly by hand then the Heisenberg eqs. become
[Weiss (2008)]
$\displaystyle\imath\hbar\dot{Q}_{i}\ $ $\displaystyle=$ $\displaystyle\
\left[Q_{i},H\right]\ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3)
$\displaystyle\imath\hbar\dot{P}_{i}\ $ $\displaystyle=$ $\displaystyle\
\left[P_{i},H\right]\ \ -\ \ \imath\hbar\Gamma P_{i}\ \ \ \ \ \ i=1,\ldots,n$
with the commutator decaying exponentially in time
$\left[Q_{i}(t),P_{j}(t)\right]\ =\ \exp(-\Gamma t)\ \ \
\left[Q_{i}(0),P_{j}(0)\right]$ (4)
One of the most well studied toy models,as mechanisms for the onset of chaos,
is the Lorenz chaotic attractor [Eckmann & Ruelle (1985)]. Through the work of
Ruelle and Takens an explosion of theoretical and experimental work followed
on the general roads to turbulence through the study of strange attractors in
dynamical systems [Viana (2000)].
If one is interested in the problem of turbulence in quantum systems such as,
liquid Helium, Bose-Einstein condensates(BEC) or the quark-gluon plasma, or
more generally in the quantum regime of the Big Bang and its later epochs, one
must create a framework to quantize the Navier-Stokes equation as well as its
relativistic generalization [Landau (1941)].
In the present talk we address this question in the case of a toy model for
turbulence , the Lorenz attractor. Nevertheless we can pose the problem for
more general dynamical systems. We now proceed to tackle the following issues
[Tarasov (2008)]
a) Is it possible to quantize volume preserving flows in fluid dynamics of the
form
$\frac{dx^{i}}{dt}\ =\ f^{i}(\vec{x})\ \ \ \ \ \ \ i=1,\cdots,n$ (5)
for general non-Hamiltonian dynamical systems which upon introduction of
dissipation lead to strange attractors?
b) The same problem but with the inclusion of dissipation that renders the
system chaotic with strange attractors.
In the present talk, we will answer both questions in the affirmative by
expressing incompressible flow equations in terms of the Nambu n-brackets with
n being the dimensionality of the dynamical phase space
$x^{1},x^{2},\cdots,x^{n}$ . For the case of the Lorenz attractor $n=3$. We
will show how to quantize the volume preserving Nambu flow equations, which
are associated to the Lorenz attractors and as a second step we consider also
the inclusion of dissipation. For more information on the Lorenz and
R$\ddot{o}$ssler attractors as well as their equivalents we refer the reader
to our recent work[Axenides & Floratos (2010), Axenides (2011)].
The quantization scheme we propose is analogous to the standard Heisenberg
quantization of classical mechanics leading to a unique time evolution for the
system. As a concrete example we will treat the quantization of the Lorenz
strange attractor by N $\times$ N Hermitian matrices implementing our flow
decomposition methodology into nondissipative-dissipative components.
## 2 Nambu vs Hamilton’s Mechanics
In Hamilton’s mechanics phase space is an even dimensional symplectic manifold
$(M_{2n})$ $n=1,2,\cdots$ thus posssessing the structure of a Poissonian
manifold[Arnold (1978)]. We consider two such examples:
A) The Real Plane $M_{2}\ =\ R^{2}$ with Poisson bracket
$\\{f,g\\}\ =\ \epsilon_{ij}\partial^{i}f\partial^{j}g\ \ \ \ \ \ \ f,g\in
C^{\infty}(R^{2})$ (6)
which for the coordinates $x_{1},x_{2}$ gives
$\\{x_{i},x{j}\\}=\epsilon_{ij}\ \ \ \ \ i,j=1,2$ (7)
Given a Hamiltonian $H\in C^{\infty}(R^{2})$ a flow vector field is defined
$L_{H}\ =\ \frac{\partial H}{\partial x_{2}}\frac{\partial}{\partial
x_{1}}-\frac{\partial H}{\partial x_{1}}\frac{\partial}{\partial x_{2}}\ \ ,\
\ \dot{f}=L_{H}f$ (8)
For different Hamiltonians, we have the integrability condition
$\left[L_{H_{1}},L_{H_{2}}\right]\ =\ L_{\\{H_{1},H_{2}\\}}\ \ \ \ \ \ \ \ $
(9)
B) $M=S^{2}$ the 2-dimensional sphere embedded isometrically in $R^{3}$,
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$ $i=1,2,3$
with Poisson bracket
$\\{f,g\\}_{S^{2}}\ \ =\ \ \epsilon_{ijk}\ \partial^{i}f\ \partial^{j}g\
x_{k}\ \ \ \ \ $ (10)
For the coordinates $x^{i}$ it gives the $SO(3)$ algebra
$\\{x_{i},x_{j}\\}_{S^{2}}\ \ =\ \ \epsilon_{ijk}x_{k}\ \ \ \ \ \ i,j,k=1,2,3$
(11)
The corresponding Hamiltonian flows are, $H\in C^{\infty}(S^{2})$,
$L_{H}\ =\ \epsilon^{ijk}\ x^{i}\ \frac{\partial H}{\partial
x^{j}}\frac{\partial}{\partial x^{k}}\ \ \ \ \ \ \ \ i,j,k=1,2,3$ (12)
with Lie algebra the area preserving diffeomorphisms of the sphere
$\mbox{SDiff}(S^{2})$,
$\left[L_{H_{1}},L_{H_{2}}\right]\ =\ L_{\\{H_{1},H_{2}\\}}$ (13)
With the choice of basis functions for $C^{\infty}(S^{2})$, the spherical
harmonics $Y_{lm}(\theta,\phi)$, we get the algebra of $\mbox{SDiff}(S^{2})$
[Arnold (1978), Hoppe (1982)].
$\\{H_{lm},H_{l^{\prime}m^{\prime}}\\}_{S^{2}}\ =\
f^{mm^{\prime}m^{\prime\prime}}_{ll^{\prime}l^{\prime\prime}}\ \
H_{l^{\prime\prime}m^{\prime\prime}}\ \ \ \ \ \ $ (14)
Y.Nambu in 1973 [Nambu (1973)] proposed a generalization of Hamiltonian
mechanics for volume preserving flows. The simplest example is for the
$M=R^{3}$ case. For a given set $f,g,h\in C^{\infty}(R^{3})$ he introduced the
3-bracket
$\displaystyle\\{f,g,h\\}\ $ $\displaystyle=$ $\displaystyle\ \epsilon_{ijK}\
\partial^{i}f\ \partial^{j}g\ \partial^{k}h$ (15) $\displaystyle=$
$\displaystyle\vec{\nabla}f\cdot(\vec{\nabla}g\ \times\ \vec{\nabla}h)$
which satisfies the integrability equation-Fundamental Identity(FI)
$\displaystyle\\{f_{1},f_{2},\\{f_{3},f_{4},f_{5}\\}\\}$ $\displaystyle=$
$\displaystyle\\{\\{f_{1},f_{2},f_{3}\\},f_{4},f_{5}\\}$ (16) $\displaystyle+$
$\displaystyle\\{f_{3},\\{f_{1},f_{2},f_{4}\\},f_{5}\\}+\\{f_{3},f_{4},\\{f_{1},f_{2},f_{5}\\}\\},$
L.Takhtajan in [Takhtajan (1994)] showed that the n-bracket defines a
hierarchy of $n-1,\cdots,2$ brackets.
For the $n=3$ for example by fixing the third function h in (15) we get
$\\{f,g\\}_{h}\ =\ \\{f,g,h\\}\ \ \ \ \ \ \ \ \ \ \ \forall f,g\in
C^{\infty}(R^{3})$ (17)
which from (16) satisfies the relation
$\\{\\{f_{1},f_{2}\\}_{h}\ +\ \\{\\{f_{2},f_{3}\\}_{h},f_{1}\\}_{h}\ +\
\\{\\{f_{3},f_{1}\\}_{h},f_{2}\\}_{h}\ =\ 0$ (18)
which is the Poisson-Jacobi identity. In this way we may obtain a Poisson
bracket for any surface in $R^{3}$, which is defined by the level set function
$h(x_{1},x_{2},x_{3})=constant$.
If we restrict the variables $x^{1},x^{2},x^{3}$ to be on the surface h then
rel. (17) defines on it a non-degenerate Poisson bracket. In effect this
surface gets promoted to a phase space with dynamics. As an example we
consider the plane $R^{2}$ embedded in $R^{3}$.
1. 1.
The Plane $h=\alpha^{i}\dot{x}^{i}$ , $\alpha^{i}\in R$
$\\{x^{i},x^{j}\\}_{h}\ =\ \epsilon^{ijk}\alpha^{k}$ (19)
2. 2.
The Sphere $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$ , $S^{2}\in R^{3}$
$\\{x^{i},x^{j}\\}_{S^{2}}\ =\ \epsilon^{ijk}x^{k}\ \ \ \ \ \ \ SO(3)\ on\
S^{2}$ (20)
Nambu introduced his mechanics for volume preserving flows in $R^{3}$ by
specifying $H_{1},H_{2}\in C^{\infty}(R^{3})$ two Hamiltonians inducing the
following eq. of motion
$\dot{x}^{i}\ =\ \\{x^{i},H_{1},H_{2}\\},\ \ \ \ \ \ \ \ \ \ \ \ i=1,2,3$ (21)
or equivalently
$\dot{\vec{x}}\ =\ \vec{\nabla}H_{1}\ \times\ \vec{\nabla}H_{2}$ (22)
These equations imply the conservation of the “ Hamiltonians ”
$\displaystyle\dot{H}_{1}\ $ $\displaystyle=$ $\displaystyle\ \dot{\vec{x}}\
\cdot\ \vec{\nabla}H_{1}\ =\ 0\ \ \ \ \ \ $ $\displaystyle\dot{H}_{1}\ $
$\displaystyle=$ $\displaystyle\ \dot{\vec{x}}\ \cdot\ \vec{\nabla}H_{1}\ =\
0$ (23)
The ideal example right from Nambu’s paper is the Euler’s top
$\displaystyle H_{1}\ \ $ $\displaystyle=$ $\displaystyle\ \
\frac{1}{2}\left(l_{1}^{2}\ +\ l_{2}^{2}\ +\ l_{3}^{2}\right)\ \ \ \ \ $
$\displaystyle H_{2}\ \ $ $\displaystyle=$ $\displaystyle\ \
\frac{1}{2}\left(\frac{l_{1}^{2}}{I_{1}}\ +\ \frac{l_{1}^{2}}{I_{1}}\ +\
\frac{l_{1}^{2}}{I_{1}}\right)$ (24)
with equations of motion
$\displaystyle\dot{l}_{1}\ $ $\displaystyle=$ $\displaystyle\
\left(\frac{1}{I_{2}}\ -\ \frac{1}{I_{3}}\right)l_{2}\ l_{3}\ \ \ \ \ \ \ $
$\displaystyle\dot{l}_{2}\ $ $\displaystyle=$ $\displaystyle\
\left(\frac{1}{I_{3}}\ -\ \frac{1}{I_{1}}\right)l_{3}\ l_{1}\ \ \ \ \ \ \ $
$\displaystyle\dot{l}_{3}\ $ $\displaystyle=$ $\displaystyle\
\left(\frac{1}{I_{1}}\ -\ \frac{1}{I_{2}}\right)l_{1}\ l_{2}$ (25)
The generalization of the Hamiltonian flow vector fields are the flow vector
fields
$\displaystyle L_{H_{1},H_{2}}\ \ $ $\displaystyle=$ $\displaystyle\
\epsilon^{ijk}\ \frac{\partial H_{1}}{\partial x_{i}}\ \frac{\partial
H_{2}}{\partial x_{j}}\ \frac{\partial}{\partial x_{k}}\ \ \ \ \ \ \ \ \ \ $
(26) $\displaystyle=$ $\displaystyle\ \ \left(\vec{\nabla}H_{1}\ \times\
\vec{\nabla}H_{2}\right)\cdot\vec{\nabla}\ \ =\ \ \ \vec{v}\cdot\vec{\nabla}$
where
$\dot{\vec{x}}\ =\ \vec{v}\ =\ \vec{\nabla}H_{1}\ \times\ \vec{\nabla}H_{2}\ \
\ $ (27)
which are volume preserving and therefore describe flows of incompressible
fluids. The question, which we pose now, is whether the inverse also holds.
Namely if for any flow
$\dot{\vec{x}}\ =\ \vec{v}(\vec{x})$ (28)
which is incompressible
$\vec{\nabla}\cdot\vec{v}\ =\ 0$ (29)
there exist two “ Hamiltonians ” $H_{1},H_{2}$ with
$v^{i}\ =\ \\{x^{i},H_{1},H_{2}\\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\ $ (30)
Since a long time ago [Lamb (1932)] the answer to this problem has been
settled to the affirmative locally in $R^{3}$. Indeed for any incompressible
vector flows $\vec{v}$ , there is a vector potential $\vec{A}$ such that
$\vec{v}=\vec{\nabla}\times\vec{A}$ which, moreover, satisfies the Clebsch-
Monge decomposition (CM- stream potentials),
$\vec{A}\ \ =\ \ \vec{\nabla}\alpha\ +\ \beta\vec{\nabla}\gamma$ (31)
from which we can deduce that
$\vec{v}\ =\ \vec{\nabla}\beta\ \times\ \vec{\nabla}\gamma\ \ \ \ \ \
\beta=H_{1},\gamma=H_{2}$ (32)
The corresponding infinite dimensional algebra of volume preserving
diffeomorphisms in the Clebsch-Monge gauge is [Axenides & Floratos (2007)]
$\left[L_{H_{1},H_{2}}\ ,\ L_{H_{3},H_{4}}\ \right]\ \ =\ \
L_{\\{H_{1},H_{2},H_{3}\\},H_{4}}\ +\ L_{H_{3},\\{H_{1},H_{2},H_{4}\\}}\ \ \ \
\ \ \ \ \ \ \ \ \ $ (33)
So the full structure of the infinite dimensional group is determined by the
3-bracket algebra of a basis of functions $H_{i}\in C^{\infty}(R^{3}),\ \ \ \
i=1,2,\cdots$ [Axenides & Floratos (2009)]
$\\{H_{i},H_{j},H_{k}\\}\ =\ f_{ijk}^{l}\ H_{l}\ \ \ \ \ \ \ \ \ \ \ \ \ \
i,j,k,l=1,2,\cdots$ (34)
As an example we present the Nambu 3-algebra on $S^{3}$.
Consider $S^{3}$ isometricallly embedded in $R^{4}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1$ (35)
where $x_{i}\in R,\ \ i=1,2,3,4$ The Nambu bracket on $R^{4}$
$\\{x_{i},x_{j},x_{k},x_{l}\\}_{R^{4}}\ =\ \epsilon_{ijkl}\ \ \ \ \ \ \ \ \ \
\ i,j,k,l=1,,2,3,4$ (36)
induces a 3-bracket on $S^{3}$. Define : $\forall\ f_{i}\ \in\ C^{\infty}\
(S^{3})\ \ \ \ \ i=1,2,3$
$\\{f_{1},f_{2},f_{3}\\}_{S^{3}}\ =\ \epsilon^{ijkl}\ x^{i}\
\partial^{j}f_{1}\ \partial^{k}f_{2}\ \partial^{l}f_{3}$ (37)
which on the $R^{4}$ coordinates gives
$\\{x_{i},x_{j},x_{k}\\}_{S^{3}}\ =\ \epsilon_{ijkl}\ x_{l}\ \ \ \ \ \ \
i,j,k,l=1,2,3,4$ (38)
This is the celebrated Baggert-Lambert algebra for $SO(4)$ [Bagger & Lambert
(2007)]. For any $f,g\in C^{\infty}(S^{3})$ we consider the flow vector field
$L_{f,g}\ \ =\ \ \\{f,g,\cdot\\}_{S^{3}}\ \ \ \ \ \ \ \ \ \ \ L_{f,g}h\ =\
\\{f,g,h\\}_{S^{3}}$ (39)
and thus the Nambu equations on $S^{3}$ are given as :
$\dot{x}^{i}\ \ =\ \ L_{H_{1},H_{2}}\ x^{i}\ \ =\ \
\\{x^{i},H_{1},H_{2}\\}_{S^{3}}\ \ \ \ \ \ \ \ \ \ \forall\ H_{1},H_{2}\ \in\
C^{\infty}(S^{3})$ (40)
## 3 The Lorenz Attractor as a Dissipative Nambu Mechanics
We are not going to repeat the history of the famous Lorenz-Saltzman system
[Saltzman(1962), Lorenz(1963)] of nonlinear equations, which is the Galerkin
trancation of the Fourier modes of the temperature gradient and stream
potential for the Rayleigh-Benard experiment. We will nevertheless present
their dynamical system of three variables
$\displaystyle\dot{X}\ $ $\displaystyle=$ $\displaystyle\ \sigma\ (Y\ -\ X)\ \
\ \ $ (41) $\displaystyle\dot{Y}\ $ $\displaystyle=$ $\displaystyle\ X\ (r\ -\
Z)\ -\ Y\ \ \ \ \ $ $\displaystyle\dot{Z}\ $ $\displaystyle=$ $\displaystyle\
X\ Y\ -\ b\ Z\ \ \ \ \ $
with $\sigma$ being the Prandl Number, r the relative Reynolds number and b is
the aspect ratio.
The system has as critical points
1. 1.
$X_{1}\ =\ Y_{1}\ =\ Z_{1}\ =0$
2. 2.
$X_{2}\ =\ Y_{2}\ =\ \sqrt{b(r-1)}\ ,\ \ Z_{2}=r-1$
3. 3.
$X_{3},\ Y_{3}\ =-\sqrt{b(r-1)}\ ,\ \ Z_{3}\ =\ r-1$
Using stability analysis for fixed $\sigma$ and b, their behaviour is
controled by the relative Reynolds number r. Turbulent behavior starts for
values of
$r=r_{H}\ =\ \sigma\ \frac{\sigma+b+3}{\sigma-b-1}$ (42)
with $\sigma$ = 10 , $b=\frac{8}{3}$ ($\rightarrow r_{H}=24.73)$). Lorenz’s
choice is $r=28>r_{H}$ in order to obtain chaotic behaviour [Sparrow (1987)]
where Hopf bifurcation appears with $\vec{X}_{1}$ a saddle point with
$\vec{X}_{2}$ and $\vec{X}_{3}$ going repelling-unstable.
In our recent work [Axenides & Floratos (2010)] we implemented a decomposition
of the Lorenz vector flow into a volume preserving (non-dissipative) and a
dissipative components
$\vec{v}\ =\ \vec{v}_{ND}\ +\ \vec{v}_{D}$ (43)
The non-dissipative component gives a volume preserving flow
$\vec{\nabla}\cdot\vec{v}_{ND}=0$
$\vec{v_{ND}}\ \ =\ \ (\sigma Y,\ X(r-Z),\ XY)$ (44)
with
$\displaystyle\dot{X}\ $ $\displaystyle=$ $\displaystyle\ \sigma Y\ =\
\\{X,H_{1},H_{2}\\}$ $\displaystyle\dot{Y}\ $ $\displaystyle=$ $\displaystyle\
X(r-Z)\ =\ \\{Y,H_{1},H_{2}\\}\ \ \ $ (45) $\displaystyle\dot{Z}\ $
$\displaystyle=$ $\displaystyle\ XY\ =\ \\{Z,H_{1},H_{2}\\}\ \ \ $
with
$\vec{V}_{ND}\ =\ \vec{\nabla}H_{1}\ \times\ \vec{\nabla}H_{2}$ (46)
We have identified the conserved Hamiltonians (see also [Nevir & Blender
(1994)])
$\displaystyle H_{1}\ $ $\displaystyle=$ $\displaystyle\ \frac{1}{2}\
\left(Y^{2}\ +\ (Z-r)^{2}\right)\ \ \ \ \ \ $ $\displaystyle H_{2}\ $
$\displaystyle=$ $\displaystyle\ \sigma Z\ -\ \frac{X^{2}}{2}\ \ \ \ \ \ \ $
(47)
The corresponding Poisson brackets on the surface $\Sigma$, $H_{2}=$constant
are
$\displaystyle\\{X,Y\\}_{\Sigma}\ $ $\displaystyle=$ $\displaystyle\
\partial_{Z}H_{2}\ =\ \sigma\ \ \ \ $ $\displaystyle\\{Y,Z\\}_{\Sigma}\ $
$\displaystyle=$ $\displaystyle\ \partial_{X}H_{2}\ =\ -X\ \ \ \ $
$\displaystyle\\{Z,X\\}_{\Sigma}\ $ $\displaystyle=$ $\displaystyle\ 0$ (48)
By eliminating the variables Y,Z from eq.(45) we get
$\ddot{X}\ +\ (H_{2}\ -\ \sigma r)\ X\ +\ \frac{X^{3}}{2}\ =\ 0$ (49)
which is just the equation of motion for the one dimensional Anharmonic
Oscillator. Depending on the initial conditions we identify its two familiar
symmetry phases as:
* •
$H_{2}\ -\ \sigma r\ \geq 0$ Single well-Symmetric Phase
* •
$H_{2}\ -\ \sigma r\ \leq 0$ Double Well-Broken Phase
By including dissipation we have
$\dot{\vec{X}}\ =\ \vec{\nabla}H_{1}\ \times\ \vec{\nabla}H_{2}\ -\
\vec{\nabla}D\ =\ \vec{V}_{ND}\ +\ \vec{V}_{D}$ (50)
where the dissipation potential D is given below
$\displaystyle H_{1}\ $ $\displaystyle=$ $\displaystyle\
\frac{1}{2}((z-r)^{2}+y^{2})\ \ \ \ $ $\displaystyle H_{2}\ \ $
$\displaystyle=$ $\displaystyle\ \ \sigma Z\ -\ \frac{X^{2}}{2}\ \ \ \ \ \ \ $
(51) $\displaystyle D\ \ $ $\displaystyle=$ $\displaystyle\ \ \frac{1}{2}\
(\sigma X^{2}\ +\ Y^{2}\ +\ bZ^{2})\ \ \ $
Now, of course, $H_{1},H_{2}$ along with $D$ are not any more conserved. But
still $H_{2}$ is a useful quantity since by eliminating Y,Z we find the
dynamical system for X, $H_{2}$
$\displaystyle\ddot{X}\ +\ (1+\sigma)\ \dot{X}+X(\frac{X^{2}}{2}\ +\ H_{2}\ -\
\sigma(r-1))\ $ $\displaystyle=$ $\displaystyle\ 0\ \ \ $
$\displaystyle\dot{H}_{2}\ +\ bH_{2}\ -\ \sigma(1\ -\ \frac{b}{2\sigma})\
X^{2}\ \ =\ \ 0$ (52)
The anharmonic oscillator moerover develops a damping term and since $H_{2}$
is not conserved, we obtain a sequence of symmetry breaking and symmetry
restoring phases. By solving for $H_{2}$ we obtain the well known Takeyama
memory term in the anharmonic potential [Takeyama (1978), Takeyama (1980)]
$H_{2}(t)\ =\ H_{2}(0)e^{-bt}\ +\ \sigma(1-\frac{b}{2\sigma})\
e^{-bt}\int_{0}^{t}\ d\xi\ e^{b\xi}\ X^{2}(\xi)$ (53)
## 4 Matrix Quantization of the Lorenz Attractor
We will quantize the volume preserving part, in the Nambu form, of the Lorenz
attractor system. Firstly we write the Nambu equations,
$\displaystyle\dot{X}\ \ $ $\displaystyle=$ $\displaystyle\ \
\\{X,H_{1}\\}_{H_{2}}\ =\\{X,H_{1},H_{2}\\}\ \ \ \ \ \ $
$\displaystyle\dot{Y}\ \ $ $\displaystyle=$ $\displaystyle\ \
\\{Y,H_{1}\\}_{H_{2}}\ =\\{Y,H_{1},H_{2}\\}\ \ \ \ \ \ \ $ (54)
$\displaystyle\dot{Z}\ \ $ $\displaystyle=$ $\displaystyle\ \
\\{Z,H_{1}\\}_{H_{2}}\ =\\{Z,H_{1},H_{2}\\}\ \ $
Using the Poisson algebra of equations (48).
$\displaystyle\\{X,Y\\}_{H_{2}}\ \ $ $\displaystyle=$ $\displaystyle\ \
\sigma\ \ =\ \ \partial_{Z}\ H_{2}\ \ \ \ \ \ \ $
$\displaystyle\\{Y,Z\\}_{H_{2}}\ \ $ $\displaystyle=$ $\displaystyle\ \ -X\ \
=\ \ \partial_{X}H_{2}\ \ \ \ \ \ \ \ \ \ $ (55)
$\displaystyle\\{Z,X\\}_{H_{2}}\ \ $ $\displaystyle=$ $\displaystyle\ \
\partial_{Y}H_{2}\ \ =\ 0\ \ \ \ \ $
The quantization of the system is carried through the quantization of the
Poisson algebra, i.e. by lifting it to commutators with Weyl-ordered operators
$\displaystyle\left[\widehat{X},\widehat{Y}\right]\ \ $ $\displaystyle=$
$\displaystyle\ \ \imath\hbar\sigma\ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\left[\widehat{Y},\widehat{Z}\right]\ \ $ $\displaystyle=$
$\displaystyle\ \ -\imath\hbar\widehat{X}\ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (56)
$\displaystyle\left[\widehat{Z},\widehat{X}\right]\ \ $ $\displaystyle=$
$\displaystyle\ \ \ 0\ \ \ \ \ $
We must find either infinite dim. matrices or differential operators that
satisfy rel. (56) The Quantum Nambu equations for the volume preserving part
follow,
$\displaystyle\imath\hbar\dot{\widehat{X}}\ $ $\displaystyle=$ $\displaystyle\
\ \left[\widehat{X},\widehat{H}_{1}\right]_{H_{2}}\ \ \ \ \ \ \ \ $
$\displaystyle\imath\hbar\dot{\widehat{Y}}\ $ $\displaystyle=$ $\displaystyle\
\ \left[\widehat{Y},\widehat{H}_{1}\right]_{H_{2}}\ \ \ \ \ \ \ $ (57)
$\displaystyle\imath\hbar\dot{\widehat{Z}}\ $ $\displaystyle=$ $\displaystyle\
\ \left[\widehat{Z},\widehat{H}_{1}\right]_{H_{2}}\ \ \ \ \ \ \ \ $
where the $H_{2}$ index implies that we evaluate the commutators by using the
algebra (56) which has as Casimir the second Hamiltonian $\widehat{H}_{2}$(
parabolic-cylinder phase space)
$\widehat{H}_{2}\ =\ \frac{\widehat{X}^{2}}{2}\ -\ \sigma\widehat{Z}$ (58)
We obtain
$\displaystyle\dot{\widehat{X}}\ $ $\displaystyle=$ $\displaystyle\
\sigma\widehat{Y}\ \ \ \ $ $\displaystyle\dot{\widehat{Y}}\ $ $\displaystyle=$
$\displaystyle\ -\frac{1}{2}\
\left(\widehat{X}\widehat{Z}+\widehat{Z}\widehat{X}\right)\ +\ r\widehat{X}$
$\displaystyle\dot{\widehat{Z}}\ $ $\displaystyle=$ $\displaystyle\
\frac{1}{2}\ \left(\widehat{X}\widehat{Y}+\widehat{Y}\widehat{X}\right)$ (59)
from where we get the conservation of $\widehat{H_{1}},\widehat{H_{2}}$ and a
unique time evolution
$\displaystyle\widehat{X}(t)\ $ $\displaystyle=$ $\displaystyle\
e^{-\frac{i}{\hbar}tH_{1}}\ \widehat{X}(0)\ e^{\frac{i}{\hbar}tH_{1}}\ \ \ \ \
\ \ $ $\displaystyle\widehat{Y}(t)\ $ $\displaystyle=$ $\displaystyle\
e^{-\frac{i}{\hbar}tH_{1}}\ \widehat{Y}(0)\ e^{\frac{i}{\hbar}tH_{1}}\ \ \ \ \
\ \ $ (60) $\displaystyle\widehat{Z}(t)\ $ $\displaystyle=$ $\displaystyle\
e^{-\frac{i}{\hbar}tH_{1}}\ \widehat{Z}(0)\ e^{\frac{i}{\hbar}tH_{1}}\ \ \ $
By eliminating $\widehat{Y},\widehat{Z}$ from rels. (59) we get the anharmonic
oscillator quantum eqs.
$\ddot{\widehat{X}}\ +\ \left[(\widehat{H}_{2}\ -\ \sigma r)\widehat{X}\ +\
\frac{\widehat{X}^{3}}{2}\right]\ =\ 0$ (61)
The $H_{2}$ Casimir values determine superselection sectors for rel.(61) with
$H_{2}<\sigma r(\mbox{Double Well})$, where quantum tunneling appears. For
$H_{2}\geq\sigma r(\mbox{Single Well})$ anharmonic potential governs quantum
dynamics. Both cases possess discrete spectra.
For the non-dissipative part of the quantum Lorenz system, we could also work
in the Schrondinger picture for the one dimensional anharmonic potential.
## 5 Inclusion of Dissipation; Decoherence of the Quantum Lorenz Attractor
The Quantum Lorenz equations including dissipation are [Axenides & Floratos
(2010)]
$\displaystyle\dot{\widehat{X}}\ $ $\displaystyle=$ $\displaystyle\
\sigma\left(\widehat{Y}-\widehat{X}\right)\ \ \ \ \ $
$\displaystyle\dot{\widehat{Y}}\ $ $\displaystyle=$ $\displaystyle\
-\frac{1}{2}\ \left(\widehat{X}\widehat{Z}\ +\ \widehat{Z}\widehat{X}\right)\
+r\widehat{X}-\widehat{Y}\ $ (62) $\displaystyle\dot{\widehat{Z}}\ $
$\displaystyle=$ $\displaystyle\
\frac{1}{2}\left(\widehat{Y}\widehat{X}+\widehat{X}\widehat{Y}\right)\ -\
b\widehat{Z}$
It is obvious that the quantum algebra(56) does not hold anymore for all times
and $\widehat{H}_{1},\widehat{H}_{2}$ are not conserved. Still we can
eliminate $\widehat{Y},\widehat{Z}$ and obtain a dynamical anharmonic system
with quantum Takeyama memory term. The Schrondinger picture is not anymore
convenient due to the time nonlocality of the memory term. Qualitatively one
expects that weak dissipation will induce a broadening of the discrete energy
levels. Moreover for the case of strong dissipation the quantum system is
expected to collapse to the classical one.
We can approximate the quantum system through the usage of finite dimensional
$N\times N$ Hermitian matrices for the representation of
$\widehat{X},\widehat{Y},\widehat{Z}$. This is, for example, the case with a
spin system with $N=2s+1$. See, for example, the case in [Axenides & Floratos
(2007)] of an Euler top interpretation for the Lorenz attractor. Numerical
experiments for the non-dissipative systems, show integrable behaviour while
for weak dissipation we obtain N weakly interacting Lorenz attractors. On the
other hand for strong dissipation the quantum system quickly undergoes
decoherence to a classical system of N independent Lorenz attractors ( as
$\widehat{X},\widehat{Y},\widehat{Z}$ become asymptotically mutually
commuting).
We evaluated numerically the time evolution of the N $\times$ N matrix Lorenz
system with the standard parameter values $(r=28,\sigma=10,b=\frac{8}{3})$.
We found that the commutators of $\widehat{X},\widehat{Y},\widehat{Z}$ undergo
rapid decoherence unless dissipation is weak.
$\displaystyle\left[\widehat{X}(t),\widehat{Y}(t)\right]\ $
$\displaystyle\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}$
$\displaystyle\ \ e^{-\Gamma_{12}\cdot t}\
\left[\widehat{X}(0),\widehat{Y}(0)\right]$
$\displaystyle\left[\widehat{Y}(t),\widehat{Z}(t)\right]\ $
$\displaystyle\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}$
$\displaystyle e^{\Gamma_{23}t}\ \left[\widehat{Y}(0),\widehat{Z}(0)\right]$
(63) $\displaystyle\left[\widehat{Z}(t),\widehat{X}(t)\right]\ $
$\displaystyle\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}$
$\displaystyle e^{-\Gamma_{31}t}\left[\widehat{Z}(0),\widehat{X}(0)\right]$
For completeness we also present the equations for the closed systems of
operators $\widehat{H}_{2},\widehat{X}$
$\dot{\widehat{H}_{2}}\ +\ b\widehat{H}_{2}\ =\ \sigma\left(1\ -\
\frac{b}{2\sigma}\right)\widehat{X}^{2}$ (64)
and
$\ddot{\widehat{X}}\ +\ \left(1+\sigma\right)\dot{\widehat{X}}\ +\
\left(\frac{\widehat{X}\widehat{H}_{2}+\widehat{H}_{2}\widehat{X}}{2}\ -\
\sigma(r-1)\widehat{X}\ +\ \frac{\widehat{X}^{3}}{2}\right)\ =\ 0$ (65)
Currently we are in the process of performing numerical calculations for large
symmetric or Hermitian marices for the case of weak dissipation where
interesting behaviour appears between the quantum and classical overlap
regimes. For discussions of the Lorenz attractor quantization in the physical
framework of laser instabilities see [Elgin & Sarkar (1984)].
## 6 Conclusions-Open Questions
Applications
We presented a framework to quantize non-Hamiltonian dissipative flows in
$R^{3}$. We applied the method to the very interesting and rich example of the
Lorenz Strange attractor. This we did in order to check the robustness of
quantum mechanics against the robustness of strange attractors. The
qualitative behavior for a finite matrix approximations of the quantum system,
exhibits interesting results for the case of weak dissipation. This is not the
case of a typical Lorenz attractor, which exhibits very strong dissipation and
undergoes quick decoherence. We believe the study of quantum dynamical systems
with soft turbulence [Elgin & Sarkar (1984)] is useful to probe the classical
$\leftrightarrow$ quantum regime transition for dissipative dynamical systems.
Applications of quantum turbulence are awaiting for exciting new discoveries
and new paradigm shifts in the behaviour of quantum matter and radiation in
extreme conditions of high temperatures and densities where the Lagrangian
particle concept looses its importance and the Eulerian fluid interpretation
takes the lead. In all of these, chaotic attractor mechanisms should prevail.
Their quantum mechanical survival will be important to our understanding of
quantum turbulence.
In this direction important developments come from recent studies of
holographic hydrodynamics in its relation with black hole horizon physics,
where the problem of quantization of gravity resurfaces from a new
hydrodynamic point of view [Eling et al.(2009)].
## References
* [Asakawa et al.(2007)] Asakawa, S. and Bass, A. and Muller, B. [2007] Anomalous transport processes in anisotropically expanding quark-gluon plasmas, Prog. Theor. Phys. 116 p.725 [arXiv:hep-ph/0608270].
* [ Arnold (1978)] Arnold, V.I. Mathematical Methods of Classical Mechanics [1978] Springer Science.
* [ Axenides & Floratos (2007) ] Axenides, M. and Floratos, E. [2007] Euler top dynamics of Nambu-Goto p-branes JHEP 0703 p. 093, [arXiv:hep-th/0608017].
* [ Axenides & Floratos (2009)] Axenides, M and Floratos, E [2009] Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds JHEP 0902 p. 039, [arXiv:0809.3493 [hep-th]].
* [ Axenides & Floratos (2010)] Axenides, M. and Floratos, E. [2010] Strange Attractors in Dissipative Nambu Mechanics: Classical and Quantum Aspects, JHEP 1004 p.036, [arXiv:0910.3881 [nlin.CD]].
* [ Axenides & Floratos (2009)] Axenides, E. and Floratos, E. and Nicolis, S. [2009] Nambu Quantum Mechanics on Discrete 3-Tori J. Phys. A 42 p. 275201, [ arXiv:0901.2638 [hep-th]].
* [ Axenides (2011) ] Axenides, M.[2011] Non-Hamiltonian Chaos from Nambu Dynamics of Surfaces in Chaos Theory: Modeling, Simulation and Applications, C.H.Skiadas et.al.(Eds), World Scientific Publ.Co., pp 110, [arXiv:1109.0470 [nlin.CD]].
* [Bagger & Lambert (2007)] Bagger, J. and Lambert, N. [2007] Gauge Symmetry and Supersymmetry of Multiple M2-Branes Phys. Rev. D 77 p. 065008 , [arXiv:0711.0955 [hep-th]].
* [ Eckmann & Ruelle (1985)] Eckmann, J.P. and Ruelle, D. [1985] Ergodic Theory of Chaos and Strange Attractors Review of Modern Physics , vol.57, p. 617.
* [ Elgin & Sarkar (1984) ] Elgin, J.N. and Sarkar, S. [1984] Quantum Fluctuations and the Lorenz Strange Attractor, Phys. Rev. Lett. 52 p. 1215 .
* [Eling et al.(2009)] Eling,C. and Fouxon,I. and Oz,Y. [2009] The Incompressible Navier-Stokes Equations From Membrane Dynamics Phys. Lett. B 680 p.496, [arXiv:0905.3638 [hep-th]].
* [Gibson & Schild (2010)] Gibson, C.H. and Schild, R.H. [2010] Turbulent behavior of protogalaxies at the end of the Plasma Epoch: theory and observation, Jour. of Cosmology 6, p. 1351. [arXiv:1004.2016[astro-ph.CO]]
* [Hoppe (1982)] Hoppe, J. [1982] Quantum theory of a massless relativistic surface and a two dimensional bound state problem Ph.D. Thesis MIT , Aachen preprint PITHA-86/24.
* [Lamb (1932)] Lamb, H. [1932] Hydrodynamics Cambridge U. Press, p.248.
* [Landau (1941) ] Landau, L. [1941] J.Phys.5, p.71.
* [Landau & Lifshitz(1987)] Landau, L.D. and Lifshitz, E.M. [1987] Fluid Mechanics, 2nd edit. (Elsevier).
* [Lorenz(1963)] Lorenz,E.N. [1963] Deterministic Non-Periodic Flow J.Atm.Sci. 20, p. 130.
* [ Nambu (1973)] Nambu, Y. [1973] Generalized Hamiltonian Dynamics Phys. Rev. D 7, p.2405.
* [ Nevir & Blender (1994)] Nevir, P. and Blender, R. [1994] Hamiltonian and Nambu Representation of the Non-Dissipative Lorenz Equation Beitr. Phys. Atmosph. 67, p.133.
* [Saltzman(1962)] Saltzman, B. [1962] Finite Amplitude Free Convection as an Initial Value Problem J. Atm. Sci. 19 p.329.
* [Sparrow (1987)] Sparrow, C. [1987] The Lorenz Equation, Bifurcations, Chaos and the Strange Attractors, Springel-Verlag, New York.
* [Takhtajan (1994)] Takhtajan, L. [1994] On Foundation Of The Generalized Nambu Mechanics (Second Version) Commun. Math. Phys. 160 , p. 295.
* [Takeyama (1978)] Takeyama, K. [1978] Dynamics of the Lorenz Model of Convective Instabilities, Prog.Theor.Phys. 60 p. 613.
* [Takeyama (1980)] Takeyama, K. [1980] , Prog. Theor. Phys. 63 p.91 .
* [ Tarasov (2008)] Tarasov, V.E. [2008] Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, Elsevier.
* [Van Sciver (2009)] Van Sciver, S.W. and Barenghi, C.F.[2009] Visualization of Quantum Turbulence, Prog.Low Temp.Phys, edt. W.P.Halperin and M.Tsubota .
* [ Viana (2000) ] Viana, M. [2000] What’s New Lorenz Strange Attractors?, The Mathematical Intelligencer, pp. 6-19.
* [Weiss (2008)] Weiss, U. [2008] Quantum Dissipative Systems World Scientific-Series, vol.13.
|
arxiv-papers
| 2011-09-06T17:16:22 |
2024-09-04T02:49:21.933468
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Emmanuel Floratos",
"submitter": "Emmanuel Floratos",
"url": "https://arxiv.org/abs/1109.1234"
}
|
1109.1357
|
# Wire network behavior in superconducting Nb films with diluted triangular
arrays of holes
S. K. He W. J. Zhang H. F. Liu G. M. Xue B. H. Li H. Xiao Z. C. Wen X.
F. Han S. P. Zhao C. Z. Gu X. G. Qiu xgqiu@aphy.iphy.ac.cn Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Science, Beijing 100190, China
###### Abstract
We present transport measurement results on superconducting Nb films with
diluted triangular arrays (honeycomb and kagomé) of holes. The patterned films
have large disk-shaped interstitial regions even when the edge-to-edge
separations between nearest neighboring holes are comparable to the coherence
length. Changes in the field interval of two consecutive minima in the field
dependent resistance $R(H)$ curves are observed. In the low field region, fine
structures in the $R(H)$ and $T_{c}(H)$ curves are identified in both arrays.
Comparison of experimental data with calculation results shows that these
structures observed in honeycomb and kagomé hole arrays resemble those in wire
networks with triangular and $T_{3}$ symmetries, respectively. Our findings
suggest that even in these specified periodic hole arrays with very large
interstitial regions, the low field fine structures are determined by the
connectivity of the arrays.
###### pacs:
74.25.F-, 74.78.Na, 74.81.Fa
## I INTRODUCTION
Superconducting films with periodic arrays of artificial pinning sites have
been extensively studied for a long time.Baert et al. (1995); Martin et al.
(1999); Hoffmann et al. (2000); Karapetrov et al. (2005) It is found that at
the fields where the number of superconducting flux quantum $\Phi_{0}=hc/2e$
in unit area is an integer multiple of the pinning sites, the so called
commensurate effects such as peaks in the $I_{c}(H)$ and dips in the $R(H)$
curves can be observed.Hoffmann et al. (2000); Van Look et al. (2002); Martin
et al. (1997) The most prevailing explanation for these effects is that the
vortex lattice is commensurate with the underlying array and pinned
efficiently at the matching fields.
However, similar phenomena have also been observed in other systems including
superconducting wire networks and Josephson junction arrays.Pannetier et al.
(1984); Ling et al. (1996); Tinkham et al. (1983) In wire networks, when the
magnetic flux through single plaquette does not equal to integer multiple of
$\Phi_{0}$, supercurrents must be induced along the loops to satisfy the
fluxoid quantization condition. There is an energy cost due to the induced
supercurrents and thus results in a reduced $T_{c}$. From this point of view,
in wire networks, the resistance oscillations are caused by $T_{c}$
suppression at non-matching fields rather than enhanced pinning at the
matching fields.Patel et al. (2007) Interestingly, transitions from the
pinning regime to the wire network regime can be observed in some
superconducting film with pinning arrays. For example, with increasing hole
diameter, the width of the strips between neighboring holes in a hole array
becomes comparable to the coherence length at temperatures close to $T_{c0}$
(zero resistance transition temperature) and the system behaves like a wire
network.Bruynseraede et al. (1997); Moshchalkov et al. (1998); Hoffmann et al.
(2000)
Figure 1: Scanning electron microscopy (SEM) image of the superconducting Nb
film with a kagomé array of holes. The center-to-center distance between
nearest neighbor is 400 nm and the hole diameter is about 340 nm.
We notice that these findings are based on squareBruynseraede et al. (1997);
Moshchalkov et al. (1998); Hoffmann et al. (2000) and triangularPatel et al.
(2007) lattices. In those cases, with small edge-to-edge separation, the
remaining geometrical structures of the patterned films, are indeed like wire
networks with uniform width and small nodes. Thus the transition from pining
array to wire network is straightforward. However, other series of hole arrays
with large hole diameter may not have such direct geometrical simplification.
Investigations of those hole arrays are needed to better understand the
physics related to superconductors with micro/nano structures.
In this work, superconducting Nb films with honeycomb and kagomé arrays of
holes are studied. The arrays can be viewed as diluted triangular arrays, for
they can be constructed with 1/3 and 1/4 of the sites removed form the
original lattice, respectively.Reichhardt and Reichhardt (2007) The edge-to-
edge separations between neighboring holes in these samples are comparable to
the coherence length at temperature close to $T_{c0}$. Even though, unlike
square and triangular arrays, the patterned films still have large
interstitial regions. This distinct character makes the system different from
wire networks, for in ideal wire networks, both the width of the stripes and
the radius of the nodes are smaller than the coherence length and a uniform
order parameter in the cross section of any stripe is expected. Large
interstitial regions, on the other hand, can facilitate the nucleation of
Abrikosov vorticesMartin et al. (1999); Harada et al. (1996); Karapetrov et
al. (2005); Metlushko et al. (1999); Wu et al. (2005); Kramer et al. (2009);
Grigorenko et al. (2003) , resulting in the appearance of normal cores. In our
experiments, a series of minima are observed in the $R(H)$ curves. While the
oscillation period at low fields is in good agreement with the value derived
from the hole density, the periodicity in higher fields is much larger. It is
found that the transition of the two regions is due to the presence of
interstitial vortices in the high field region. Surprisingly, in the low field
region, wire network behaviors are observed in both samples.
We identify the wire network behavior by the fractional matchings (fine
structures) in the $R(H)$ and $T_{c}(H)$ curves. The positions and the
relative values of the matching minima are studied in detail. We notice that
the connectivity of a wire network determines the characters of the fine
structures, therefore we highlight the connectivity of our hole arrays and
simplify each of them to a wire network. The comparison of the results in the
holes arrays with those of their corresponding wire networks, including
reported experimental data as well as calculation results based on the
Alexander model,Alexander (1983) demonstrates that the hole arrays studied in
this work are well described by wire networks when subjected to small field.
## II EXPERIMENT
The nano-structured superconducting films were prepared as follows. First, the
superconducting Nb film with a thickness of 100 nm was deposited by magnetron
sputtering on Si substrate with SiO2 buffer layer. Next, a micro-bridge for
four terminal transport measurement was fabricated by ultraviolet
photolithography followed by reactive ion etching. Then the desired arrays
covering the whole bridge area of $60\times 60~{}\mu$m2 was patterned by
electron-beam lithography on a polymethyl metacrylate (PMMA) resist layer.
Finally, the pattern was transferred to the Nb film by magnetically enhanced
reactive ion etching. In both the honeycomb and kagomé samples, the value of
the center-to-center distance between nearest neighbor ($a$) is 400 nm and the
hole diameter ($d$) is about 340 nm. The scanning electron micrograph image of
the kagomé sample is shown in Fig. 1. The smallest width of the stripes
between the adjacent holes is about 60 nm.
The transport measurements were carried out in a commercial Physical
Properties Measurement System (PPMS) manufactured by Quantum Design. The
magnetic field was applied perpendicular to the film surface. During the
measurements, the temperature stability was better than 2 mK. The zero field
transition temperatures $T_{c}(0)$ are 8.713 K for the honeycomb sample and
8.755 K for the kagomé sample, using a criterion of half the normal state
resistance $R_{N}$ at 9 K, which are 8.18 $\Omega$ and 9.02 $\Omega$,
respectively. The transition width of these two samples is about 0.15 K. The
reference film without any pattern has a slightly higher $T_{c}$ of 8.87 K and
transition width of 50 mK. We have measured the $T_{c}(H)$ phase boundary of
the reference sample and obtained the zero-temperature coherence length
$\xi(0)=9.9$ nm.Tinkham (1975)
## III RESULTS AND DISCUSSIONS
### III.1 Reconfiguration
Figure 2 shows the $R(H)$ curve measured at 8.55 K with a current of 50
$\mu$A. A series of local minima and maxima are observed. Among these, seven
integer matching minima can be clearly identified. The field interval between
two consecutive integer minima versus the corresponding index number are shown
in the inset. At fields below 300 Oe, the observed intervals are about 96.5
Oe, is in agreement with the value 99.6 Oe derived from the hole density. When
the field is larger than the third matching field, the field spacing has a
larger value of about 146.5 Oe. This value is very close to the matching field
of a triangular lattice with the same lattice constant which is 149.4 Oe. One
exception is that the spacing between the 4th and 5th minima is 100 Oe.
Figure 2: (Color online) $R(H)$ curve of Nb film with honeycomb array of holes
measured at 8.55 K with a current of 50 $\mu$A. f is the filling ratio
indicating the flux quanta per unit cell. The inset shows the field spacing of
two consecutive minima as a function of the minima index.
Figure 3 is the $R(H)$ curve of the kagomé sample measured at 8.60 K with a
current of 10 $\mu$A. Several resistance minima are identified. As can be seen
more clearly from the inset of Fig. 3, the field intervals have two sets of
values too. The field intervals of the first three minima are about 108.7 Oe,
again in good agreement with the first matching field ($H_{1}=$112.1 Oe) of
the underlying kagomé lattice. However, the value of the intervals jumps to
about 148 Oe when the field is larger than the third matching field.
Figure 3: (Color online) $R(H)$ curve of Nb film with kagomé array of holes
measured at 8.60 K with a current of 10 $\mu$A. f is the filling ratio
indicating the flux quanta per unit cell. The inset shows the field spacing of
two consecutive minima as a function of the minima index.
The change in the field interval of two consecutive minima have been observed
in rectangular arrays and been explained in terms of reconfiguration of the
vortex lattice from rectangular to square.Martin et al. (1999, 2000) The
interstitial regions in rectangular arrays are stripes which lead to the
channeling effect of vortices.Velez et al. (2002); Villegas et al. (2003)
While in honeycomb and kagomé arrays the interstitial regions have disk-like
shape. At the typical measurement temperatures 8.60 K, the diameters of the
interstitial regions, approximately 520 nm, are about six times larger than
the coherence length. Thus, Abrikosov vortices can be easily accommodated by
the interstitial sites.
Form the field spacing values between two consecutive integer matchings, the
overall feature of the $R(H)$ curves can be explained as follows. When the
field is small, supercurrents are generated around the holes in order to
satisfy the fluxoid quantization condition. At the integer matching field, it
is the arrangement of the fluxoids that is commensurate with the hole lattice.
When the applied magnetic field is larger than the third matching field,
Abrikosov vortices are generated in the interstitial region to minimize the
free energy.Wu et al. (2005) These interstitial vortices are effectively
pinned by the repulsive interactions from the fluxoids in the holes, which
create a caging potential.Berdiyorov et al. (2006a); Reichhardt and Olson
Reichhardt (2009) The overall flux lattice, which is composed of interstitial
vortices and the fluxoids in the holes, would have a triangular symmetry. That
is why the field intervals of the minima observed in high field region in both
lattices are in good agreement with the value derived form a vortex lattice
with triangular arrangement. The saturation number $n_{s}$, namely, the
maximum number of fluxoids trapped by one hole without the entry of
interstitial vortices are three in both samples. This value exceeds the limit
$n_{s}=d/(4\xi(T))\simeq 1$ given by Mkrtchyan and Schmidt,Mkrtchyan and
Schmidt (1972) where $d$ is the diameter of the defect and $\xi(T)$ is the
temperature dependent coherence length. This fact indicates that the proximity
of the holes is prominent.Doria et al. (2000)
However, the exact vortex configurations and their evolution are strongly
dependent on the density of flux quanta and the confinement of the mesoscopic
structure. Both image experiment Karapetrov et al. (2005) and simulation Olson
Reichhardt et al. (2006); Berdiyorov et al. (2006a, b) have revealed the
complex nature and the transition process between various configurations of
the composite vortex lattice. This may also responsible for the observation of
a field interval of 100 Oe between the 4th and 5th minima in Fig. 2.
### III.2 Fractional matching and wire network behavior
In the low field region, fine and repeatable sub-minima are observed as can be
seen from Fig. 2 and Fig. 3. Because no interstitial vortex is involved in
this region and the proximity of the holes, we can related the systems to wire
networks despite the large disk-shaped interstitial sites.
We notice that a wire network is obtained by assigning nodes to the center of
the interstitial regions in the original hole array and connecting them. In
Fig. 4, the centers of the interstitial regions are regarded as nodes. Then,
the superconducting stripes between neighboring holes are viewed as wires
(dotted lines) connecting the nodes. Following this procedure, the
connectivity of the patterned films are highlighted and the honeycomb and
kagomé hole arrays are transformed to triangular and $T_{3}$ wire networks,
respectively. Although C. C. Abilio et al. have found that superconducting Al
film with a square array of holes can be described as a square wire
network,Abilio et al. (2000) the simplification we made here is more radical
for the diameter of the disk-shaped interstitial regions is even larger than
that of the holes.
Figure 4: Illustration of the transformation from hole arrays (dark circles)
to wire networks (dotted lines). (a) Honeycomb hole array to triangular wire
network. (b) Kagomé hole array to $T_{3}$ wire network. $a$: the distance
between the centers of the neighboring holes. $l$: the side length of the
corresponding wire networks.
Assuming that the distance between the centers of the neighboring holes is $a$
in the hole array, then in the dual lattices the side length $l$ of the
elementary triangles will be $\sqrt{3}a$ and the side length of the rhombus
tile in the $T_{3}$ geometry will be $2\sqrt{3}a/3$ as can been seen from Fig.
4(a) and Fig. 4(b), respectively. If the matching effects of the hole arrays
originate from $T_{c}$ suppression at non-matching fields, the experimental
results should show similar features with what have been observed in their
corresponding wire networks.
Figure 5: (Color online) (a)Low field $R(H)$ curve for the sample with a
honeycomb hole array measured at 8.60 K and $I=$10 $\mu$A. The field is
normalized by the first matching field and the resistance is normalized by
$R_{N}$ at 9 K. (b)The field dependent transition temperature $T_{c}(H)$
curves of the honeycomb sample with different criteria $r=R/R_{N}$. The
parabolic background has been subtracted. The top one is the theoretical curve
for triangular wire network in unit of $l^{2}/\xi(0)^{2}$.
Figure 5(a) shows the fine structures of the $R(H)$ curve of the honeycomb
sample measured at 8.60 K with a current of 10 $\mu$A. The $x$ axis is given
as filling ratio, $f=\Phi/\Phi_{0}$, where $\Phi=Ha^{2}3\sqrt{3}/4$ is the
magnetic flux per elementary triangle of the corresponding wire network. In
the field range $0\leq f\leq 1$, resistance minima are observed at $f=$ 1/4,
1/3, 1/2, 2/3 and 3/4. The magnitude of the oscillation is about eight percent
of $R_{N}$. At the field $f=$1/2 where the commensurate effect is most
pronounced, the resistance drops to a value comparable to that at zero field.
At $f=$ 1/4 and 3/4, the resistances are about three percent of $R_{N}$.
Figure 5(b) shows the $T_{c}(H)$ curves in reduced units, $\Delta
T_{c}/T_{c}(0)$ vs $f$, using different resistance criteria $r=R/R_{N}$. The
curves were obtained with a current of 10 $\mu$A. The parabolic
backgroundPannetier et al. (1984) which reflects the contribution of the
finite width of the strands to the critical field has been subtracted. For
$r=$0.05 and 0.1, dips are observed at filling ratios $f=$ 1/4, 1/3, 1/2, 2/3
and 3/4. With the criterion of $r=$ 0.5, only dips at $f=$ 1/4, 1/2 and 3/4
are visible and those at $f=$ 1/3 and 2/3 are missing. For $r=$ 0.8, $T_{c}$
changes smoothly with field and no features can be identified. Thus, comparing
Fig. 5(a) and (b), one can see that minima in the $R(H)$ curve is located at
the same fields where dips are observed in the $T_{c}(H)$ curve. The fine
structures are more pronounced in $R(H)$ than in $\Delta T_{c}/T_{c}(0)$ vs
$f$. This is because the resistance is measured at temperatures slightly above
$T_{c0}$ in the superconducting transition region where resistance drops
sharply with decreasing temperature, a small change in $T_{c}$ will result in
an enhanced effect in resistance, and thus a more pronounced commensurate
effect.
To our knowledge, fractional matching effects in honeycomb arrays have only
been observed at $f=$ 1/2 in previous studies.Wu et al. (2005); Reichhardt and
Reichhardt (2007) At the typical measuring temperature $T=$ 8.6K (0.987
$T_{c}$) in this work, $\xi(T)=\xi(0)/\sqrt{1-T/{T_{c}}}=$ 86.9 nm is slightly
larger than the width of the narrow stripes. This fact makes the samples
investigated here be qualitatively different from those in previous works on
hole arrays and be better described as wire networks. Rich structures in the
$T_{c}(H)$ curves of wire networks may serve as fingerprints to differentiate
one array from another, since for a given geometry the fine structures are
only expected at a particular series of filling ratios.Pannetier et al.
(1984); Lin and Nori (2002); Erdős and Zheng (2010) From the linearized
Ginzburg-Landau equations, Alexander had derived the relation of the order
parameters at the nodes subjected to non-integer flux.Alexander (1983) Based
on those equations, the task of finding the field dependent transition
temperature is reduced to eigenvalue problems. The mathematical treatment is
very similar to that of the tight bounding electrons in 2D arrays subjected to
an external field which leads to the famous Hofstadter butterfly energy
spectrum.Hofstadter (1976) The top curve plotted in Fig. 5(b) is the
theoretical values of $T_{c}(H)$ of the triangular wire network, which is the
dual structure of honeycomb hole array. The curve is calculated by
$\frac{{{T_{c}}(0)-{T_{c}}(f)}}{{{T_{c}}(0)}}=\frac{{\xi{{(0)}^{2}}}}{{{l^{2}}}}{\left({\arccos\frac{{{\varepsilon_{t}}}}{6}}\right)^{2}},$
(1)
where $l$ is the side length of the wires, $\varepsilon_{t}$ is the
eigenvalueAlexander (1983) of the following equation obtained by using Landau
gauge and periodic boundary conditions,
$\displaystyle\varepsilon_{t}{\psi_{n}}$ $\displaystyle=$ $\displaystyle
2\cos\left[{\pi(2n-1)f-{k_{y}}/2}\right]{e^{-i{k_{y}}/2}}{\psi_{n-1}}$ (2)
$\displaystyle+$ $\displaystyle 2\cos\left[{4\pi nf-{k_{y}}}\right]{\psi_{n}}$
$\displaystyle+$ $\displaystyle
2\cos\left[{\pi(2n+1)f-{k_{y}}/2}\right]{e^{i{k_{y}}/2}}{\psi_{n+1}},$
where $k_{y}=2\pi\frac{{k-1}}{N},\quad k=1,...,N$ implies the periodic
condition, $n$ denotes the node index and $\psi_{n}$ is the order parameter at
node $n$. The fine structures of the $T_{c}(H)$ of triangular wire network
have also been studied by analytical approach based on multiple-loop Aharonov-
Bohm Feynman path integrals.Lin and Nori (2002) The main features in
$T_{c}(H)$ such as the position and relative strength of the most prominent
dips are the same as those in the Alexander’s treatment. The most pronounced
dips occur at $f=$ 1/4, 1/3, 1/2 etc., in good agreement with the experimental
data. At the fields where dips are observed, the order parameter at the nodes
interferences constructively and form different locked-in states corresponding
to local maxima in $T_{c}$.Higgins et al. (2000) In the fabrication process,
contamination and damage of the samples are most significant to the narrow
strips, resulting in a lower $T_{c}$ and a widening in the transition width
compared to the reference film. The phase coherence between adjacent nodes is
fully established when all the stripes are in superconducting state.
Therefore, the interference effects is strong near the zero resistance state
$r=$ 0.05 and significantly reduced for $r>0.1$.
Figure 6(a) is the low field $R(H)$ curve of the kagomé sample measured at
8.67 K. With kagomé geometry, $f$ corresponds to the magnetic flux through a
rhombic tile with side length $2\sqrt{3}a/3$:
$f={\Phi}/{{\Phi_{0}}}={2\sqrt{3}{a^{2}}H}/{(3\Phi_{0})}$. Again we focus on
the results for $f$ in the interval between 0 and 1. Dips are identified at
$f$=1/6, 2/9, 7/9 and 5/6. These fine structures are observed for the first
time in kagomé hole array, perhaps because of the specified geometrical
parameters. Anomalies are also visible at $f=$ 1/3 and 2/3, but they are more
like kinks than dips. Most strikingly, in contrast to the dip observed in the
honeycomb sample, a remarkable peak is observed at $f=$ 1/2. The resistance is
far above the background, reaching 16.9 percent of the normal state
resistance.
Figure 6: (Color online) (a)Low field $R(H)$ curve for the sample with a
kagomé array measured at 8.67 K and $I$=10 $\mu$A. (b) The field dependent
transition temperature $T_{c}(H)$ of the kagomé sample with different criteria
$r=R/R_{N}$. The parabolic background has been subtracted. The top one is the
theoretical curve for $T_{3}$ wire network (Ref. Abilio et al., 1999) in unit
of $l^{2}/\xi(0)^{2}$.
Figure 6(b) shows the $T_{c}(H)$ curves in reduced units, $\Delta
T_{c}/T_{c}(0)$ vs $f$, of the kagomé sample determined by using different
criteria of $r$. Again, all the measurement were carried out with a current of
10 $\mu$A. For small value of $r$ (0.05 and 0.1), dips present at 1/6, 2/9,
1/3, 2/3 and 5/6, although the one at 2/3 is relatively weak (see the labels
in Fig. 6(b)). For larger values of $r$ (0.5 and 0.8), the dips at 1/6, 2/9
and 1/3 gradually disappear but the peak at $f=$ 1/2 remains pronounced.
Most of the fractional matchings observed here are absent in previous works
performed on kagomé lattice.Morgan and Ketterson (1998); Laguna et al. (2001);
Reichhardt and Reichhardt (2007); Reichhardt and Olson Reichhardt (2009);
Cuppens et al. (2011) In a recent work on hole arrays with small hole size
which is in the pinning regime limit, clear fractional matchings were only
observed at $f=$ 1/3 and 2/3.Cuppens et al. (2011) Contrasting to their
findings, in our work with edge-to-edge separation comparable to the coherence
length, dips at $f=$ 1/6 and 5/6 are most pronounced and a distinct peak
anomaly at $f=$ 1/2 is observed. The results agree well with what have been
observed in $T_{3}$ wire networks,Abilio et al. (1999); Vidal et al. (1998)
the dual structure of the kagomé hole array as seen in Fig. 4(b). The top
curve of Fig. 6(b) shows the theoretical curve of $T_{3}$ networks.Abilio et
al. (1999) This is done by the following equation which relates the
eigenvalues of $T_{3}$ ($\varepsilon(f)$) at filling ratio $f$ to the
eigenvalues for triangular lattice at $3f/2$.Abilio et al. (1999)
${\varepsilon^{2}}(f)-6=2\cos(\pi f){\varepsilon_{t}}(3f/2)$ (3)
The comparison with the theoretical curve shows that only the dips at $f$=7/9
and other weaker ones are absent in the experimental curves. For $f$=1/6, 2/9
and 1/3, constructive interference occurs among the superconducting order
parameters of the interstitial sites and the fluxoids establish long range
locked-in commensurate order. While for $f$=1/2, superconductivity is
localized in single tiles and long range coherence between network sites
cannot be established.Abilio et al. (1999) This kind of fully destructive
quantum interference has also been observed in kagomé wire networks.Higgins et
al. (2000); Xiao et al. (2002) The oscillation of $R(H)$ curve with a peak at
half a flux per tile is similar to the single-loop Little-Parks effect. In
that case, the supercurrent density reaches the largest value at half flux in
order to satisfy the fluxoid quantization condition.Little and Parks (1962)
The results obtained from these two lattices confirms the validity of the
transform from a hole array to a wire network. However, in contrast to the
small nodes in ideal wire networks or square arrays of holes, the geometries
studied here possess very large superconducting disk-shaped nodes which can
trap Abrikosov vortices easily. The formation of Abrikosov vortices implies
spatial variation and presence of zero points of the order parameters in the
interstitial region. Then the simplification of the large interstitial region
to a node in a wire network is no longer appropriate. Therefore,
transformation from hole arrays to wire networks is only valid for small field
values.
## IV CONCLUSIONS
In conclusion, we have studied the commensurate effects in superconducting
films with honeycomb and kagomé arrays of holes with small edge-to-edge
separation. We found that the magnetoresistance curves have two regions with
different oscillation periods. At small fields, there are no Abrikosov
vortices presents in the interstitial regions. The large disk-shaped
interstitial region can be simplified as a single node and a one-to-one
correspondence between hole arrays and wire networks is established.
Comparison of experimental data with calculation shows that the simplification
works well. Our results suggest that at low fields, the behavior of these
specified periodic hole arrays are determined by the connectivity of the
systems.
## V ACKNOWLEDGEMENTS
We thank Q. Niu, X. C. Xie, V. V. Moshchalkov, B. Y. Zhu and Y. Yeshurun for
fruitful discussion. This work is supported by National Basic Research Program
of China(No.2009CB929100, 2011CBA00107, 2012CB921302) and National Science
Foundation of China (No. 91121004, 10974241, 11104335).
## References
* Baert et al. (1995) M. Baert, V. V. Metlushko, R. Jonckheere, V. V. Moshchalkov, and Y. Bruynseraede, Phys. Rev. Lett. 74, 3269 (1995).
* Martin et al. (1999) J. I. Martin, M. Velez, A. Hoffmann, I. K. Schuller, and J. L. Vicent, Phys. Rev. Lett. 83, 1022 (1999).
* Hoffmann et al. (2000) A. Hoffmann, P. Prieto, and I. K. Schuller, Phys. Rev. B 61, 6958 (2000).
* Karapetrov et al. (2005) G. Karapetrov, J. Fedor, M. Iavarone, D. Rosenmann, and W. K. Kwok, Phys. Rev. Lett. 95, 167002 (2005).
* Van Look et al. (2002) L. Van Look, B. Y. Zhu, R. Jonckheere, B. R. Zhao, Z. X. Zhao, and V. V. Moshchalkov, Phys. Rev. B 66, 214511 (2002).
* Martin et al. (1997) J. I. Martin, M. Velez, J. Nogues, and I. K. Schuller, Phys. Rev. Lett. 79, 1929 (1997).
* Pannetier et al. (1984) B. Pannetier, J. Chaussy, R. Rammal, and J. C. Villegier, Phys. Rev. Lett. 53, 1845 (1984).
* Ling et al. (1996) X. S. Ling, H. J. Lezec, M. J. Higgins, J. S. Tsai, J. Fujita, H. Numata, Y. Nakamura, Y. Ochiai, C. Tang, P. M. Chaikin, et al., Phys. Rev. Lett. 76, 2989 (1996).
* Tinkham et al. (1983) M. Tinkham, D. W. Abraham, and C. J. Lobb, Phys. Rev. B 28, 6578 (1983).
* Patel et al. (2007) U. Patel, Z. L. Xiao, J. Hua, T. Xu, D. Rosenmann, V. Novosad, J. Pearson, U. Welp, W. K. Kwok, and G. W. Crabtree, Phys. Rev. B 76, 020508 (2007).
* Bruynseraede et al. (1997) Y. Bruynseraede, T. Puig, E. Rosseel, M. Baert, M. Van Bael, K. Temst, V. Moshchalkov, and R. Jonckheere, J. Low Temp. Phys. 106, 173 (1997).
* Moshchalkov et al. (1998) V. V. Moshchalkov, M. Baert, V. V. Metlushko, E. Rosseel, M. J. Van Bael, K. Temst, Y. Bruynseraede, and R. Jonckheere, Phys. Rev. B 57, 3615 (1998).
* Reichhardt and Reichhardt (2007) C. Reichhardt and C. J. O. Reichhardt, Phys. Rev. B 76, 064523 (2007).
* Harada et al. (1996) K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, and V. V. Moshchalkov, Science 274, 1167 (1996).
* Metlushko et al. (1999) V. Metlushko, U. Welp, G. W. Crabtree, R. Osgood, S. D. Bader, L. E. DeLong, Z. Zhang, S. R. J. Brueck, B. Ilic, K. Chung, et al., Phys. Rev. B 60, R12585 (1999).
* Wu et al. (2005) T. C. Wu, J. C. Wang, L. Horng, J. C. Wu, and T. J. Yang, J.Appl.Phys. 97, 10B102 (2005).
* Kramer et al. (2009) R. B. G. Kramer, A. V. Silhanek, J. Van de Vondel, B. Raes, and V. V. Moshchalkov, Phys. Rev. Lett. 103, 067007 (2009).
* Grigorenko et al. (2003) A. N. Grigorenko, S. J. Bending, M. J. Van Bael, M. Lange, V. V. Moshchalkov, H. Fangohr, and P. A. J. de Groot, Phys. Rev. Lett. 90, 237001 (2003).
* Alexander (1983) S. Alexander, Phys. Rev. B 27, 1541 (1983).
* Tinkham (1975) M. Tinkham, _Introduction to Superconductivity_ (McGraw-Hill, New York, 1975).
* Martin et al. (2000) J. I. Martin, M. Velez, A. Hoffmann, I. K. Schuller, and J. L. Vicent, Phys. Rev. B 62, 9110 (2000).
* Velez et al. (2002) M. Velez, D. Jaque, J. I. Martin, M. I. Montero, I. K. Schuller, and J. L. Vicent, Phys. Rev. B 65, 104511 (2002).
* Villegas et al. (2003) J. E. Villegas, E. M. Gonzalez, M. I. Montero, I. K. Schuller, and J. L. Vicent, Phys. Rev. B 68, 224504 (2003).
* Berdiyorov et al. (2006a) G. R. Berdiyorov, M. V. Milosevic, and F. M. Peeters, Phys. Rev. Lett. 96, 207001 (2006a).
* Reichhardt and Olson Reichhardt (2009) C. Reichhardt and C. J. Olson Reichhardt, Phys. Rev. B 79, 134501 (2009).
* Mkrtchyan and Schmidt (1972) G. Mkrtchyan and V. V. Schmidt, Soviet Physics JETP 34, 195 (1972).
* Doria et al. (2000) M. M. Doria, S. C. B. de Andrade, and E. Sardella, Physica C: Superconductivity 341-348, 1199 (2000).
* Olson Reichhardt et al. (2006) C. J. Olson Reichhardt, A. Libál, and C. Reichhardt, Phys. Rev. B 73, 184519 (2006).
* Berdiyorov et al. (2006b) G. R. Berdiyorov, M. V. Milošević, and F. M. Peeters, Phys. Rev. B 74, 174512 (2006b).
* Abilio et al. (2000) C. Abilio, L. Amico, R. Fazio, and B. Pannetier, J. Low Temp. Phys. 118, 23 (2000).
* Lin and Nori (2002) Y.-L. Lin and F. Nori, Phys. Rev. B 65, 214504 (2002).
* Erdős and Zheng (2010) P. Erdős and W. Zheng, Phys. Rev. B 82, 134532 (2010).
* Hofstadter (1976) D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
* Higgins et al. (2000) M. J. Higgins, Y. Xiao, S. Bhattacharya, P. M. Chaikin, S. Sethuraman, R. Bojko, and D. Spencer, Phys. Rev. B 61, R894 (2000).
* Abilio et al. (1999) C. C. Abilio, P. Butaud, T. Fournier, B. Pannetier, J. Vidal, S. Tedesco, and B. Dalzotto, Phys. Rev. Lett. 83, 5102 (1999).
* Morgan and Ketterson (1998) D. J. Morgan and J. B. Ketterson, Phys. Rev. Lett. 80, 3614 (1998).
* Laguna et al. (2001) M. F. Laguna, C. A. Balseiro, D. Domínguez, and F. Nori, Phys. Rev. B 64, 104505 (2001).
* Cuppens et al. (2011) J. Cuppens, G. W. Ataklti, W. Gillijns, J. Van de Vondel, V. V. Moshchalkov, and A. V. Silhanek, J Spercond Nov Magn 24, 7 (2011).
* Vidal et al. (1998) J. Vidal, R. Mosseri, and B. Douçot, Phys. Rev. Lett. 81, 5888 (1998).
* Xiao et al. (2002) Y. Xiao, D. A. Huse, P. M. Chaikin, M. J. Higgins, S. Bhattacharya, and D. Spencer, Phys. Rev. B 65, 214503 (2002).
* Little and Parks (1962) W. A. Little and R. D. Parks, Phys. Rev. Lett. 9, 9 (1962).
|
arxiv-papers
| 2011-09-07T05:37:14 |
2024-09-04T02:49:21.943912
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. K. He, W. J. Zhang, H. F. Liu, G. M. Xue, B. H. Li, H. Xiao, Z. C.\n Wen, X. F. Han, S. P. Zhao, C. Z. Gu, and X. G. Qiu",
"submitter": "Shikun He",
"url": "https://arxiv.org/abs/1109.1357"
}
|
1109.1395
|
# The Goldman bracket characterizes homeomorphisms
Siddhartha Gadgil Department of Mathematics,
Indian Institute of Science,
Bangalore 560012, India gadgil@math.iisc.ernet.in
###### Abstract.
We show that a homotopy equivalence between compact, connected, oriented
surfaces with non-empty boundary is homotopic to a homeomorphism if and only
if it commutes with the Goldman bracket.
Nous montrons que d’une équivalence d’homotopie entre les compacte, con-
connect s, les surfaces orient es avec bord non vide est homotope à un homéo-
morphisme si et seulement si elle commute avec le Goldman bracket.
###### 1991 Mathematics Subject Classification:
Primary 57M99; Secondary 17B99
## 1\. Introduction
Given a homotopy equivalence between manifolds of the same dimension, a
fundamental question in topology is whether it is homotopic to a
homeomorphism. Basic examples of exotic homotopy equivalences – ones that are
not homotopic to the homeomorphisms, are maps between surfaces with boundary.
For example, the three-holed sphere and the one-holed torus are homotopy
equivalent but not homeomorphic.
We assume throughout that all surfaces we consider are oriented. In this note,
we show that there is a simple and natural characterisation of when a homotopy
equivalence $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ between compact
surfaces is homotopic to a homeomorphism in terms of the _Goldman bracket_ ,
which is a Lie Algebra structure associated to a surface. More precisely, if
$\hat{\pi}(\Sigma)$ denotes the set of free homotopy classes of closed curves
in $\Sigma$, then the Goldman bracket (whose definition we recall below) is a
bilinear map
$[\cdot,\cdot]\colon\thinspace\mathbb{Z}[\hat{\pi}(\Sigma)]\times\mathbb{Z}[\hat{\pi}(\Sigma)]\to\mathbb{Z}[\hat{\pi}(\Sigma)],$
which is skew-symmetric and satisfies the Jacobi identity.
###### Theorem 1.1.
A homotopy equivalence $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ between
compact, connected, oriented surfaces with non-empty boundary is homotopic to
a homeomorphism if and only if it commutes with the Goldman bracket, i.e., for
all $x,y\in\mathbb{Z}[\hat{\pi}(\Sigma_{1})]$, we have,
(1) $[f_{*}(x),f_{*}(y)]=f_{*}([x,y]),$
where
$f_{*}\colon\thinspace\mathbb{Z}[\hat{\pi}(\Sigma_{1})]\to\mathbb{Z}[\hat{\pi}(\Sigma_{1})]$
is the function induced by $f$.
For closed surfaces, every homotopy equivalence is homotopic to a
homeomorphism. So the corresponding result holds trivially in this case.
Homotopy equivalences that are homotopic to homeomorphisms can be
characterised as those that preserve the so called _peripheral structure_.
However, our result has the advantage that the characterization is in terms of
a structure defined without reference to the boundary. Furthermore, the
Goldman bracket is the simplest instance of so called _string topology_ [2],
which in turn is related to the Floer homology of the cotangent bundle [1].
## 2\. Preliminaries
### 2.1. The Goldman bracket
We recall the definition of the Goldman bracket [4] on a connected surface
$\Sigma$. Let $\hat{\pi}(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. For a closed curve $\alpha$, let
$\langle\alpha\rangle\in\hat{\pi}(\Sigma)$ denote its homotopy class.
Given homotopy classes $x$ and $y$ of closed curves, we consider smooth,
oriented representatives $\alpha$ and $\beta$ that intersect transversally in
double points. The bracket of $x$ and $y$ is then defined as the sum
(2) $[x,y]=\sum_{p\in\alpha\cap\beta}\varepsilon_{p}\
\langle\alpha*_{p}\beta\rangle,$
where, for $p\in\alpha\cap\beta$, $\varepsilon_{p}$ is the sign of the
intersection between $\alpha$ and $\beta$ at $p$ and $\alpha*_{p}\beta$ is the
product of the loops $\alpha$ and $\beta$ viewed as based at $p$.
This expression is well defined by the first part of the following remarkable
theorem of Goldman. A completely topological proof of this has been given by
Chas [3].
###### Theorem 2.1 (Goldman [4]).
The bracket defined by Equation 2 has the following properties.
1. (1)
Equation 2 gives a well-defined bilinear map
$\mathbb{Z}[\hat{\pi}(\Sigma)]\times\mathbb{Z}[\hat{\pi}(\Sigma)]\to\mathbb{Z}[\hat{\pi}(\Sigma)]$,
i.e., the right hand side of the equation depends only on the homotopy classes
of $\alpha$ and $\beta$.
2. (2)
The bracket is skew-symmetric and satisfies the Jacobi identity.
3. (3)
If $\alpha$ is a simple curve and $x=\langle\alpha\rangle$, then, for
$y\in\hat{\pi}(\Sigma)$, $[x,y]=0$ if and only if $y=\langle\beta\rangle$ for
a closed curve $\beta$ such that $\beta$ is disjoint from $\alpha$.
### 2.2. Peripheral classes
For a connected surface $\Sigma$, if $p\in\Sigma$ is a point,
$\hat{\pi}(\Sigma)$ is the set of conjugacy classes in $\pi_{1}(\Sigma,p)$.
The power operations $\pi_{1}(\Sigma,p)\to\pi_{1}(\Sigma,p)$,
$\alpha\mapsto\alpha^{n}$, $n\in\mathbb{Z}$, and the inverse function
$\alpha\mapsto\alpha^{-1}$ on $\pi_{1}(\Sigma,p)$ induce well-defined
functions on $\hat{\pi}(\Sigma)$. A class $x\in\hat{\pi}$ that is not a non-
trivial power is called _primitive_.
Suppose henceforth that $\Sigma$ is a compact, connected surface with non-
empty boundary. An element of $\hat{\pi}$ is said to be _peripheral_ if it can
be represented by a curve $\alpha\subset\partial\Sigma$.
Assume further that $\Sigma$ has negative Euler characteristic, i.e., $\Sigma$
is not a disc or annulus. Then each component of $\partial\Sigma$ corresponds
to a pair of primitive peripheral classes (one for each orientation of the
boundary curve) which are inverses of each other. Further, the primitive
peripheral classes corresponding to different boundary components are
different.
## 3\. Proof of Theorem 1.1
We now turn to the proof of Theorem 1.1. It is clear from the definition that
a homeomorphism commutes with the Goldman bracket. As homotopic maps induce
the same function on $\hat{\pi}$, it follows that if a homotopy equivalence
$f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ is homotopic to a homeomorphism,
then $f_{*}$ commutes with the Goldman bracket.
The rest of the paper is devoted to proving the converse. Assume henceforth
that $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ is a map between connected,
compact, oriented surfaces with non-empty boundary so that for all
$x,y\in\mathbb{Z}[\hat{\pi}(\Sigma_{1})]$, we have,
$[f_{*}(x),f_{*}(y)]=f_{*}([x,y]).$
###### Lemma 3.1.
For a compact surface $\Sigma$ with boundary, a non-trivial class
$x\in\hat{\pi}(\Sigma)$ is peripheral if and only if for all
$y\in\hat{\pi}(\Sigma)$, we have $[x,y]=0.$
###### Proof.
Assume $x$ is peripheral, i.e., $x=\langle\alpha\rangle$ with
$\alpha\subset\partial\Sigma$. As any closed curve in $\Sigma$ is freely
homotopic to one in the interior of $\Sigma$, every element
$y\in\hat{\pi}(\Sigma)$ can be represented by a curve $\beta\subset
int(\Sigma)$. Hence $\alpha\cap\beta=\phi$, which implies by Equation 2 that
$[x,y]=0$.
Conversely, let $x=\langle\alpha\rangle$ be such that $[x,y]=0$ for all
$y\in\hat{\pi}(\Sigma)$. If $x$ is not peripheral, it is well known that there
is a simple closed curve $\beta$ so that the geometric intersection number of
$\alpha$ and $\beta$ is non-zero, i.e., $\alpha$ is not homotopic to a curve
that is disjoint from $\beta$. This follows from the fact that there is a pair
of simple curves in $\Sigma$ that _fill_ $\Sigma$ (i.e., all curves with zero
geometric intersection with each of them is peripheral), or alternatively by
considering the geodesic representative for $x$ with respect to a hyperbolic
metric and using the result that geodesics intersect minimally.
Now, by Goldman’s theorem (Theorem 2.1, part (3)), as the simple closed curve
$\beta$ has non-zero geometric intersection number with $\alpha$ and
$y=\langle\beta\rangle$, we have $[x,y]\neq 0$, a contradiction.
∎
Recall that we assume that we have a map
$f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ such that $f_{*}$ commutes with the
Goldman bracket. In case $\Sigma_{1}$ (hence $\Sigma_{2}$) is a disc or an
annulus, it is easy to see that any homotopy equivalence is homotopic to a
homeomorphism. We can hence assume henceforth that $\Sigma_{1}$ and
$\Sigma_{2}$ have negative Euler characteristic.
###### Lemma 3.2.
Suppose $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ induces a surjection
$f_{*}\colon\thinspace\hat{\pi}(\Sigma_{1})\to\hat{\pi}(\Sigma_{2})$. Then if
$x\in\hat{\pi}(\Sigma_{1})$ is a peripheral class, so is $f_{*}(x)$.
###### Proof.
As $x$ is peripheral, by Lemma 3.1 $[x,y]=0$ for all
$y\in\hat{\pi}(\Sigma_{1})$. As $f_{*}$ commutes with the Goldman bracket, it
follows that $[f_{*}(x),f_{*}(y)]=0$ for all $y\in\hat{\pi}(\Sigma_{1})$. As
$f_{*}$ is surjective, $[f_{*}(x),z]=0$ for all $z\in\hat{\pi}(\Sigma_{2})$.
It follows by Lemma 3.1 that $f_{*}(x)$ is peripheral. ∎
###### Lemma 3.3.
Suppose $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ induces an injection
$f_{*}\colon\thinspace\hat{\pi}(\Sigma_{1})\to\hat{\pi}(\Sigma_{2})$. Then,
for $x\in\hat{\pi}(\Sigma_{1})$, if $f_{*}(x)$ is peripheral then $x$ is
peripheral.
###### Proof.
As $f_{*}(x)$ is peripheral, by Lemma 3.1 $[f_{*}(x),f_{*}(y)]=0$ for all
$y\in\hat{\pi}(\Sigma_{1})$. As $f_{*}$ is injective and commutes with the
Goldman bracket, it follows that $[x,y]=0$ for all
$y\in\hat{\pi}(\Sigma_{1})$. Thus, $x$ is peripheral by Lemma 3.1. ∎
###### Lemma 3.4.
Suppose $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ is a homotopy equivalence
and $C_{1}$, $C_{2}$, …$C_{n}$ are the boundary components of $\Sigma_{1}$.
Then there are boundary components $C^{\prime}_{1}$, $C_{2}^{\prime}$,…,
$C^{\prime}_{n}$ of $\Sigma_{2}$ such that $f(C_{i})$ is homotopic to
$C_{i}^{\prime}$ for all $i$, $1\leq i\leq n$. Further the boundary components
$C_{i}^{\prime}$ are all distinct and $\partial\Sigma_{2}=C_{1}^{\prime}\cup
C_{2}^{\prime}\cup\dots\cup C_{n}^{\prime}$.
###### Proof.
As $f$ is a homotopy equivalence,
$f_{*}\colon\thinspace\hat{\pi}(\Sigma_{1})\to\hat{\pi}(\Sigma_{2})$ is both
surjective and injective and maps primitive classes to primitive classes. For
$1\leq i\leq n$, let $\alpha_{i}$ be a closed curve parametrizing $C_{i}$. By
Lemma 3.2, $\alpha_{i}^{\prime}=f(\alpha_{i})$ represents a primitive
peripheral class for $1\leq i\leq n$, and hence represents a boundary
component $C^{\prime}_{i}$. Furthermore, as $\Sigma_{1}$ has negative Euler
characteristic, for $i\neq j$, $\alpha_{i}\neq\alpha_{j}^{\pm 1}$, and hence
$\alpha^{\prime}_{i}\neq{\alpha^{\prime}}_{j}^{\pm 1}$. It follows that the
curves $\alpha_{i}^{\prime}$ are homotopic to curves representing distinct
boundary components $C_{i}^{\prime}$, $1\leq i\leq n$.
Finally, we see that $\partial\Sigma_{2}=C_{1}^{\prime}\cup
C_{2}^{\prime}\cup\dots\cup C_{n}^{\prime}$. Namely, if $\gamma$ is a curve
parametrizing a boundary component $C^{\prime\prime}$ of $\partial\Sigma_{2}$,
$\langle\gamma\rangle=f_{*}(x)$ for some class $x$ (as $f_{*}$ is surjective).
The class $x$ must be peripheral by Lemma 3.3 and primitive as $\gamma$ is
primitive. Hence $x$ is represented by either $\alpha_{i}$ or
$\alpha_{i}^{-1}$ for some $i$, $1\leq i\leq n$. In either case,
$C^{\prime\prime}=C_{i}^{\prime}$. ∎
###### Lemma 3.5.
Suppose $f\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ is a homotopy equivalence.
Then $f$ is homotopic to a map $g\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ so
that $g(\partial\Sigma_{1})=\partial\Sigma_{2}$ and
$g|_{\partial\Sigma_{1}}\colon\thinspace\partial\Sigma_{1}\to\partial\Sigma_{2}$
is a homeomorphism.
###### Proof.
By Lemma 3.4, the restriction of $f$ to $\partial\Sigma_{1}$ is homotopic in
$\Sigma_{2}$ to a homeomorphism
$\varphi\colon\thinspace\partial\Sigma_{1}\to\partial\Sigma_{2}$. Let
$h\colon\thinspace\partial\Sigma_{1}\times[0,1]\to\partial\Sigma_{2}$ be such
a homotopy, i.e., $h$ is a map so that $h(\cdot,0)=f(\cdot)$ and
$h(\cdot,1)=\varphi(\cdot)$.
Let
$\Sigma_{1}^{\prime}=\Sigma_{1}\coprod_{\partial\Sigma_{1}=(\partial\Sigma_{1}\times\\{0\\})}\partial\Sigma_{1}\times[0,1]$
be the space obtained from the disjoint union of $\Sigma_{1}$ and
$\partial\Sigma_{1}\times[0,1]$ by identifying points
$x\in\partial\Sigma_{1}\subset\Sigma_{1}$ with the corresponding points
$(x,0)\in\partial\Sigma_{1}\times[0,1]$. Define a map
$g^{\prime}\colon\thinspace\Sigma_{1}^{\prime}\to\Sigma_{2}$ by
$g^{\prime}(x)=f(x)$ for $x\in\Sigma_{1}$ and $g^{\prime}(x,t)=h(x,t)$ for
$(x,t)\in\partial\Sigma_{1}\times[0,1]$.
Observe that $\partial\Sigma_{1}^{\prime}=\partial\Sigma_{1}\times\\{1\\}$.
The inclusion map $\Sigma_{1}\to\Sigma_{1}^{\prime}$ is homotopic to a
homeomorphism $\psi$ so that, for $x\in\partial\Sigma_{1}$, $\psi(x)=(x,1)$.
Let $H^{\prime}\colon\thinspace\Sigma_{1}\to\Sigma_{1}^{\prime}$ be such a
homotopy, i.e., $H^{\prime}$ is a map so that $H^{\prime}(\cdot,0)$ is the
inclusion map and $H^{\prime}(\cdot,1)=\psi(\cdot)$. Let
$g\colon\thinspace\Sigma_{1}\to\Sigma_{2}$ be given by
$g(x)=g^{\prime}(H^{\prime}(x,1))=g^{\prime}(\psi(x)).$
Then, for $x\in\partial\Sigma_{1}$,
$g(x)=g^{\prime}(\psi(x))=g^{\prime}((x,1))=h(x,1)=\varphi(x)$, i.e.
$g|_{\partial\Sigma_{1}}=\varphi$. Hence $g$ is a map from
$\Sigma_{1}\to\Sigma_{2}$ so that $g(\partial\Sigma_{1})=\partial\Sigma_{2}$
and
$g|_{\partial\Sigma_{1}}\colon\thinspace\partial\Sigma_{1}\to\partial\Sigma_{2}$
is a homeomorphism. Finally, for $x\in\Sigma_{1}$,
$g^{\prime}(H^{\prime}(x,0))=g^{\prime}(x)=f(x)$. Hence, a homotopy from $f$
to $g$ is given by
$H(x,t)=g^{\prime}(H(x,t)).$
∎
By a theorem of Nielsen, $g$, and hence $f$, is homotopic to a homeomorphism
(see Lemma 1.4.3 of [5] – as
$g\colon(\Sigma_{1},\partial\Sigma_{1})\to(\Sigma_{2},\partial\Sigma_{2})$
induces an isomorphism of fundamental groups, it follows that $g$ is homotopic
to a homeomorphism). This completes the proof of Theorem 1.1.∎
## References
* [1] Abbondandolo, A. and Schwarz, M., Floer homology of cotangent bundles and the loop product, Geom. Topol. 14 (2010), 1569 1722.
* [2] M. Chas and D. Sullivan, String Topology, Annals of Mathematics (to appear).
* [3] Chas, M., Combinatorial Lie bialgebras of curves on surfaces, Topology 43, (2004), 543-568.
* [4] Goldman, W. M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85, No.2, (1986), 263-302.
* [5] Waldhausen, F., On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 (1968) 56 88.
|
arxiv-papers
| 2011-09-07T09:08:20 |
2024-09-04T02:49:21.955631
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Siddhartha Gadgil",
"submitter": "Siddhartha Gadgil",
"url": "https://arxiv.org/abs/1109.1395"
}
|
1109.1414
|
# A theoretical analysis of inertia-like switching in magnets: applications to
a synthetic antiferromagnet
Satadeep Bhattacharjee, Anders Bergman, Andrea Taroni, Johan Hellsvik, Biplab
Sanyal and Olle Eriksson Department of Physics and Astronomy, Box 516, 75120,
Uppsala University, Uppsala, Sweden
###### Abstract
The magnetization dynamics of a synthetic antiferromagnet subject to a short
magnetic field pulse, has been studied by using a combination of first-
principles and atomistic spin dynamics simulations. We observe switching
phenomena on the time scale of tens of picoseconds, and inertia-like behavior
in the magnetization dynamics. We explain the latter in terms of a dynamic
redistribution of magnetic energy from the applied field pulse to other
possible energy terms, such as the exchange interaction and the magnetic
anisotropy, without invoking concepts such as inertia of an antiferromagnetic
vector. We also demonstrate that such dynamics can also be observed in a
ferromagnetic material where the incident field pulse pumps energy to the
magnetic anisotropy.
###### pacs:
75.78.-n, 75.50.Ee, 75.50.Ss, 75.75.-c
## I Introduction
Studies of magnetization reversal by an external magnetic field is a central
paradigm which relates to magnetic data storage.beaurepaire The magnetic
random access memory (MRAM) devicemram based on magneto-resistance (MR),
where magnetic bits can be described in terms high (1) or low (0) resistance
of MR units, is found to be one of the most promising memory devices for the
near future. The magneto-resistance unit consists of a non-magnetic layer
sandwiched between two magnetic metallic layers. The magneto-resistance could
be either giant magneto-resistance (GMR)gmr1 ; gmr2 or tunnel magneto-
resistance (TMR)tmr1 ; tmr2 depending on the sandwiched layer, which could be
either metallic or insulating. Such memory devices, with a very dense
architecture, capable of performing with low power consumption is proposed to
be a Universal memory device with advantages of other memory devices such as
static RAM (SRAM), dynamic RAM (DRAM) and flash memorybib2 . An MRAM, which is
capable of performing at the speed of SRAM with dense architecture comparable
to DRAM and non-volatility, (i.e. there is no loss of information when the
power is switched off) is to some extent limited by it’s writing speed. This
speed depends on the magnetization reversal time, where the magnetization of a
given magnetic unit is reversed.
The potential for applications of magnetization dynamics in technology,
combined with the new knowledge of fundamental magnetic interactions in nano-
sized materials, has caused a large increase in the interest in this field.
Several mechanisms for shorter switching speeds have been proposed, e.g.toggle
switchingbib2 , all optical switching allopt etc. One recent suggestion is
the so called inertia driven switching.ultra Here a short magnetic field
pulse, optically generated, of 100 fs, was found to cause a dynamic behavior
of the magnetism of HoFeO3, which went on for tens of picoseconds. In the
paper of Ref. ultra, it was pointed out that this behavior is expected only
for materials with antiferromagnetic exchange interactions, either for bulk
compounds with intrinsic antiferromagnetic exchange (e.g. like HoFeO3) or in
artificial magnetic tri- or multilayers, with antiferromagnetic interlayer
exchange. It was argued that for materials with antiferromagnetic exchange
interactions, the switching can be described by a second order differential
equation of the so called anti-ferromagnetic unit vector, and hence have
features associated to an inertia. It was in particular argued that the
inertia accumulated during the short pulse, was responsible for the
magnetization dynamics that lasted even after the pulse was switched off.
In this paper we address the microscopic mechanisms behind this kind of
switching for a magnetic tri-layer, using atomistic spin-dynamics
simulations.antropov ; skubic We have chosen for our studies a Fe/Cr/Fe model
system, with antiferromagnetic interlayer exchange interaction (IEC). A
schematic figure of the system considered is shown in Fig.1. The exchange
constants were calculated from first principles by mapping the ground state
electronic structure to a Heisenberg Hamiltonian, using the Liechtenstein-
Katnelson-Gubanov method.sasha We have calculated the time-evolution of the
magnetizations of a Fe/Cr/Fe trilayer using spin dynamics simulation based on
Landau-Lifshitz-Gilbert method, using the UppASD package.skubic In our
simulations we have analyzed the different terms that play a role in the
switching behavior of materials with antiferromagnetic exchange, which are
exposed to ultra short pulses. Although our simulations reproduce much of the
observations reported in Ref.ultra, , our interpretation of the microscopic
mechanisms is different, and we argue that a more appropriate description of
the mechanism behind this switching is described as due to a redistribution of
the energy from the ultra fast magnetic pulse to the antiferromagnetic
exchange interaction and magnetic anisotropy of the material. After the pulse
is switched off the dynamics is governed by an effective field, now composed
only of the exchange and anisotropy fields, that temporally evolves for much
longer time scales than the initial pulse. The dynamically evolving
magnetization carries the effect of the field pulse applied at the initial
phase of the simulation and the effect persists through the course of
simulation.
## II Method
The temporal evolution of sub-lattice magnetizations are calculated using the
dynamical equation of Landau-Lifshitz-Gilbert form expressed in terms of
atomic momentssd
$\frac{d{\bf m_{i}}(t)}{dt}=-\frac{\gamma}{1+\alpha^{2}}{\bf
m_{i}}(t)\times[{\bf H^{i}_{eff}}+\frac{\alpha}{M_{s}}~{}({\bf
m_{i}}(t)\times\bf H^{i}_{eff})]$ (1)
where ${\bf m_{i}}(t)$ is the atomic moment on the ith site at time t.
$\gamma$ is the gyro-magnetic ratio and $\alpha$ is the Gilbert damping
factor. The effective field $\bf H^{i}_{eff}$ on the $i^{th}$ atom is
calculated from the effective magnetic Hamiltonian given by
$H_{Mag}=H_{ex}+H_{MA}+H_{a},$ (2)
through
${\bf H^{i}_{eff}}=-\frac{\partial H_{Mag}}{\partial{\bf m_{i}}(t)}.$
The sub-lattice magnetization for a given Fe or Cr sublattice is simply
${\bf M}(t)=\sum_{i}{\bf m_{i}}(t).$
The first term in the Hamiltonian describes the interatomic exchange
interaction between between all atoms of the simulation cell. It can be
separated into two parts; (i) pair-exchange between the atomic moments in the
same layer (intra-layer exchange coupling) and (ii) the exchange coupling
between the moments residing in different layers (inter-layer exchange
coupling). The second term of Eqn.2 represents the magnetic anisotropy. The
form of magnetic anisotropy Hamiltonian we have considered is
$H_{MA}=Ksin^{2}\theta$, where $\theta$ is the angle which the sublattice
magnetization makes with anisotropy axis. K is the anisotropy energy density
and can be written as the sum of two contributions $K=K_{mca}+K_{shape}$.
Furthermore, $K_{mca}$ is the energy density which corresponds to magneto-
crystalline anisotropy and $K_{shape}$ corresponds to the energy density for
the shape anisotropy arising from dipole-dipole interaction. We choose K such
a way that the hard axis is considered to be along the z-direction and the xy-
plane is considered as easy plane, due to the shape anisotropy of thin
magnetic films and due to the possible intrinsic uniaxial anisotropy. The last
term of Eqn.2 is interaction due to the applied magnetic field. For the
present case it describes the pulsed field we apply during a short time of the
simulations. Equation 1 was solved numerically using the procedure of the Ref.
mentink, .
For T= 0 K we consider the macrospin approximation, i.e. all atomistic spins
within one layer are assumed to co-rotate. For finite temperature simulations,
one has to deal with this problem in an atomistic way, considering a large
number of atoms in the simulations.skubic As mentioned, the exchange
constants were calculated from first principles by mapping the ground state
electronic structure to a Heisenberg Hamiltonian. The ground state electronic
structure was calculated using the Korringa-Kohn-Rostocker Green’s function
method within the atomic sphere approximation.kkr1 ; kkr2 In reality the
interlayer exchange between the Fe layers through the Cr spacer layer are
influenced by the interface roughness of the Fe-Cr interface, and to simulate
this we rescaled our interlayer exchange interactions with a factor of 10,
which is according to previous analysis.erikPNAS1 ; erikPNAS2
We consider each Fe sublattice has magnetization ${\bf
M}=M_{s}{\bf\hat{n}}(t)$, where ${\bf\hat{n}}(t)$ is the unit vector. The
magnitude of the effective anisotropy field is given by
${F_{MA}}=\frac{2K}{M_{s}}cos\theta$. $M_{s}$ is the saturation moment for the
Fe sublattice. The initial value of a Fe sublattice magnetization is along the
positive or negative x-direction, as shown in Fig.1, i.e. ${\bf M}=\pm
M_{s}{\bf\hat{e_{x}}}$. For all simulations we used a square shaped pulse of
the external field. At time t=0, we apply a field of constant value in the x-z
plane with constant value described by ${\bf{H}_{a}}=[H_{x}~{}0~{}H_{z}]$. We
studied the cases where the magnetization was in-plane, due to the shape
anisotropy, without any further in-plane anisotropy.
Figure 1: (Color online) (Color online) The considered geometry of the
atomistic simulations. The red slabs represent Fe and the blue slab Cr. The
arrows in the slabs represent directions of magnetization.
## III Results
Figure 2: (Color online) The time-evolution of the x-, y- and z-component of
the top-most Fe sublattice magnetization of the tri-layer shown in Fig.1. We
have used Hx=3T, Hz=5T, $|$K$|$=0.06 mRy for $\alpha=0.02$
We start by showing in Fig.2 an example of the magnetization dynamics of the
tri-layer in Fig.1. Note that the magnetization shown in the figure is only
for the top Fe-slab in Fig.1 (normalized to unity). The results shown are for
a field pulse with Hx= 3 T and Hz=5 T, for a duration of 5 picoseconds (ps).
The Gilbert damping constant used was 0.02 and the strength of the uniaxial
anisotropy constant was 0.06 mRy/atom, providing an easy plane magnetization.
At time t=0, we applied a field pulse. From the figure, we see that as soon as
the pulse is applied, the magnetization is brought out of plane, and all the
components of magnetizations are non-zero. The magnetization then precess
under the influence of the external field, as well as the internal fields,
provided by the anisotropy field and the exchange field due to the IEC. From
the figure it is clear that even after the external magnetic field is switched
of (i.e. after 5 ps) the system continues to evolve in time, and the
magnetization dynamics continues over the entire time-interval shown in the
figure. At sufficiently long simulation time it seems however that the
x-component of the magnetism approaches the negative x-direction. Hence for
this choice of external pulse a switching of the magnetism occurs, and the
direction of all magnetic moments illustrated in Fig.1 have become reversed.
The data in Fig. 2 are similar to those demonstrated experimentally in Ref.
ultra, , in that the magnetization dynamics continue long after the external
pulse has been first applied and then removed from the system.
By sweeping the strength of the x-component of the applied field we can
realize different situations, in which the sublattice magnetization of one of
the Fe slabs in Fig.1 rotates half a turn (180 degrees), a whole turn (360
degrees) or even more than one turn. In Fig.3 we illustrate this result, in a
phase diagram of the switching behaviors. In the simulations shown in Fig.3 we
have used the same z-component of the applied field, the same Gilbert damping
factor, pulsing time, and the same uniaxial anisotropy constant, as used in
Fig.2. The figure only contains information about the Fe magnetization of the
top Fe slab of Fig.1. In the phase diagram of Fig.3 the time of the
simulations is shown on the y-axis and the magnitude of the x-component of the
applied field is shown on the x-axis. The switched and unswitched regions are
represented in terms of a color coding. The red color specifies that the
magnetization has turned $n\cdot 360$ degrees (n=0,1,2,…), whereas blue color
specifies that the magnetization has turned $n\cdot 360+180$ degrees
(n=0,1,2,…). Taking as an example Hx= 5T, Fig.3a shows that when the pulse
arrives the magnetism of the top Fe slab is along the positive x-direction
(red color), but after 5-6 ps the magnetization has reversed to the negative
x-direction, and after 10 ps a rotation of 360 degrees to the positive
x-direction has occurred. In between, the magnetism of course points along the
positive or negative y-direction (this component is not shown), and hence
exhibits a rotational behavior. After some 30 ps the magnetism finally settles
down along the negative x-direction. Overall, the data in Fig.3 shows that the
magnetization dynamics continues long after the external pulse has been turned
off, in a similar fashion as observed experimentally.ultra
Another interesting feature of the data in Fig.3a is that the results are not
symmetric with respect to positive or negative x-component to the applied
field. This might at first sight seem puzzling, since the geometry shown in
Fig.1 is very symmetric with respect to positive and negative x-direction.
However, the data in Fig.3a depict only the magnetization of the topmost Fe
layer, and for this layer alone there is no obvious symmetry for cases when
the applied pulse is along the positive or negative x-direction, and the
patterns in Fig.3a for positive and negative values of Hx are different.
However, for all values of Hx we find that after sufficiently long time the
magnetism has settled down into a given orientation.
Figure 3: (Color online) The switching phase diagram of an Fe/Cr/Fe-trilayer
at T=0 K. The duration of the magnetic pulse is 5 ps, and the x-component of
the magnetic field is shown on the x-axis, whereas the z-component is 5 T. A
uniaxial anisotropy of 0.06 mRy is considered in the calculations, providing
an easy plane magnetization. The x-component of the magnetization of the top
Fe slab of Fig.1 is shown in a color scheme. The blue/red areas indicate the
regions where switching has occurred/not occurred (see also text for
description). In a) we show simulations for a damping constant of 0.02 and in
b) the damping constant is 0.002. Hz= 5T, $|$K$|$=0.06 mRy
There are two torques that in general are relevant for the magnetization
dynamics, a precessional torque and a damping torque, and the influence of the
latter is provided in the simulations by the strength of the Gilbert damping
parameter. Hence, one may expect that the phase diagram should depend
critically on this parameter and to investigate this possibility, we show in
Fig. 3b a simulation with all parameters kept exactly the same as for the
simulation shown in Fig.3a, with the one difference that the Gilbert damping
parameter was set to 0.002, i.e. ten times lower than that used for Fig.3a. In
this case the rotation of the magnetism continues throughout the entire period
of the simulations, at least for positive values of Hx, which shows that it is
the damping parameter that primarily determines when the magnetization stops
rotating.
As the pulse hits the sample (simulation box) the angle between the two Fe
slabs start to deviate from the 180 degree antiferromagnetic coupling. To
illustrate this fact we show in Fig.4 a contour plot for the angle between two
Fe sublattice magnetizations, in which red coloration signifies 180 degree
antiferromagnetic coupling. This plot corresponds to the simulations shown in
Fig.3a and Fig.3b. As is clear from Fig.4a,b, the magnetization of the two Fe
sublattices are initially at 180 degree with each other, but after the
external field is switched on and time evolves, the angle between the
sublattices is reduced from 180 degrees and oscillates with time, especially
when H${}_{x}\neq 0$. Fig.4a,b shows data for two different damping
parameters, and we note that these two different damping parameters result in
slightly different magnetization dynamics, in that it takes longer time for
the angle between the two sublattices to approach 180 degrees for the weakly
damped system. However, for both simulations a 180 degree coupling is obtained
when t $\rightarrow\infty$. The minimum angle is seen when the magnitude of
the x-component of the applied field is largest, after 5 ps (i.e. the duration
of the pulse), where the two sublattice magnetizations are approximately
perpendicular to each other. This means that, with the applied field as a
driving force, the angle between the two sublattices is increasingly
deviating, albeit in an oscillating fashion, from antiferromagnetic coupling.
When the angle between two sublattices is less than 180 degrees an internal
exchange field is build up in each Fe slab, which causes a dynamical response.
In addition, a non-zero z-component to the magnetization (e.g. as shown in
Fig.2) builds up an anisotropy field. These two contributions to the effective
field, which drives the magnetization dynamics, are non-zero long after the
external field has been switched of. In fact, they are zero only after the
magnetization has settled down to a constant value. Hence, the effective field
is in this case dynamic, and the torques due to this field has as usual two
components; the damping part and the precessional term.
The data in Fig.4 demonstrate an important result, namely that during the
short time that the external field is applied, a significant energy is
established both in the exchange interaction term, $H_{ex}$, as well as in the
anisotropy energy, $H_{MAE}$, of Eqn.2. In a way one can view the whole
process as being due to energy provided by the $H_{a}$-term of Eqn.2, that
during the pulse duration pumps energy into the other two terms, $H_{ex}$ and
$H_{MAE}$. Even if $H_{a}$ is non-zero only for a short time, the other two
terms will be non-zero for a much longer period, and the effective field
composed of contributions due to these two terms depending on mi(t), carries
some sort of information of the field pulse applied at an earlier stage.
To demonstrate this in a more formal way, we plot in Fig.5 the temporal
evolution of the different components of magnetic energy during one of the
simulations.
Figure 4: (Color online) Evolution of angle between the sublattice
magnetizations (a) for damping factor of 0.02 (b) with a damping factor of
0.002. Hz= 5 T, $|$K$|$=0.06 mRy,$\alpha$=0.02
From Fig.5 one can see that the Zeeman energy is negative during the duration
of the pulse whereas the exchange and anisotropy energy are positive. The sum
of the three components, i.e. the total energy is negative during the duration
of the pulse, which means that during the time that the pulse is applied, the
energy is lowered and the system tries to relax into a new ground state
configuration. When the external pulse is switched off, Fig.5 shows that there
is still positive energy in the exchange and anisotropy contribution to the
energy and the total energy is now positive. In this configuration, there is
both a z-component of the magnetization as well non-collinear coupling between
the moments in the two Fe-slabs. The system then relaxes back to the original
configuration of collinear antiferromagnetic coupling of the Fe-slabs without
a z-component to the magnetism and it does this in the anisotropy and exchange
field that has been build up during the duration of the external field . The
relaxation back to collinear configuration takes considerably longer time than
the duration of the field as Fig.5 shows. From Fig.5 it can be seen that the
energy due to applied field (Zeeman energy) acts as a reservoir for the other
two terms (exchange and anisotropy) of the magnetic Hamiltonian. Hence, in
order to analyze the magnetization dynamics provided in Figs.2 and Fig. 3, one
does not need to invoke concepts like inertia of antiferromagnetic unit
vectorsultra . We suggest a simpler interpretation based on a redistribution
of energy among the three terms in Eqn.2, in particular a redistribution of
energy from $H_{a}$ to $H_{ex}$ and $H_{MAE}$, during the time period that the
external field is applied.
Figure 5: (Color online) Time evolution of different components of magnetic
energy (Exchange, Anisotropy, Applied), plotted alongside with total magnetic
energy. Hz= 5 T, $\alpha$=0.02
The analysis in the preceding paragraph indicates that also the strength of
the anisotropy influenced the magnetization dynamics and to investigate this
further, we plot in Fig.6, simulated data as a function of uniaxial anisotropy
constant, K. All other parameters were the same as for the simulation of
Fig.3a. First we note that when K=0, the tendency of switching is reduced,
since in this case the dynamics is provided by the exchange field alone, via
the IEC. However, when K increases to 0.01-0.04 mRy, the tendency for
switching increases and the magnetization is seen to precess two times before
stabilizing along the positive x-direction. For even larger values of
anisotropy energy, K larger than $\approx$ 0.05 mRy, only one revolution is
found in the simulations, before the magnetization is stabilized. Hence there
is a critical value of K which provides an optimally large amount of
revolutions in the magnetization.
We have also investigated the case when the inter-layer exchange coupling is
zero, in Eqn.2, by performing simulations to only the top-most Fe slab, of
Fig. 1, i.e. a purely ferromagnetic system having an easy-plane anisotropy.
The simulated results are shown in Fig.7 and we note that also in this case
can the magnetization behave as if it had ’inertia’, i.e. it proceeds after an
external field is switched off. An analysis relying on antiferromagnetic unit
vectors becomes cumbersome in this case, but the pumping of energy from
$H_{a}$ to the two other components given in Eqn.2, in this case the
anisotropy energy, provides an easy interpretation. The data in Fig.7, are
actually consistent with the experiments performed in Ref. stamm, , where a
short pulse was applied to a ferromagnet thin film.
Figure 6: (Color online) Switching times as function of out-of-plane
anisotropy. The duration of the magnetic pulse is 5 ps, the z-component of the
applied magnetic field is 5T and the x-component is 3T. (All other parameters
are same as Fig.2) Figure 7: (Color online) Switching phase diagram for Fe3
layers with the same set of parameters as indicated in Fig.3 for Fe3/Cr4/Fe3
trilayer, except the IEC. Hz= 5 T, $|$K$|$=0.06 mRy,$\alpha$=0.02
## IV Conclusions
In conclusion we have performed atomistic spin dynamics simulations for a
synthetic antiferromagnet subjected to a short magnetic pulse. We have
demonstrated that the magnetization dynamics continues long after the pulse
has been applied and removed, in a fashion that is consistent with the
experimental results of Ref. ultra, .
Several materials parameters, like the damping and strength of the magnetic
anisotropy, are found to influence the details of the magnetization dynamics.
We suggest that such magnetization dynamics could be understood in terms of a
redistribution of energy, during the time that the pulse is applied, so that
the magnetic anisotropy energy and interlayer exchange energy become in an
out-of-equilibrium state. The magnetic energy due to the incident pulse hence
becomes distributed dynamically to the effective interlayer exchange coupling
between the Fe layers and it also modifies the magnetic anisotropy energy. The
latter term ensures that the observed behavior of a magnetization dynamics
that progresses long after an external field has been applied and switched
off, is possible also in ferromagnetic materials.
## V Acknowledgment
We are grateful to the Swedish Research Council for support. O.E. acknowledges
also support from the KAW foundation and the ERC (grant 247062 - ASD).
Simulations performed on the Swedish Supercomputer Central NSC, via a grant
from SNAC. Valuable discussions with the late Prof. H.-C. Siegmann, Prof. J.
Stöhr and A.Kirilyuk are acknowledged.
## References
* (1) E. Beaurepaire, J.-C.Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).
* (2) Y Zheng, Yuankai Wu, Yihong Li, Kebin Qiu, Jinjun Han, Guchang Guo, Zaibing Luo, Ping An, Lihua Liu, Zhiyong Wang, Li Tan, Seng Ghee Zong, Baoyu Liu, Bo, J. of Nanosci. and Nanotechno., 7, 117 (2007).
* (3) M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, and F. Petroff,P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett 61 2472 (1988).
* (4) G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989).
* (5) W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, Phys. Rev. B 63, 054416 (2001).
* (6) J. S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, and R. Meservey, Phys. Rev. Lett. bf 74, 3273 (1995).
* (7) J Akerman, Science, 308, 508 (2005)
* (8) C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 99, 047601 (2007).
* (9) A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk, and Th. Rasing, Nature Phys. 5, 727 (2009).
* (10) V. P. Antropov, M. I. Katsnelson, B. N. Harmon ,M. van Schilfgaarde and D.Kusnezov, Phys. Rev. B54, 1019 (1996).
* (11) B.Skubic, J.Hellsvik, L.Nordström, O.Eriksson, Journal of Physics Condensed Matter, 20 315203 (2008). http://www.fysik.uu.se/cmt/asd/.
* (12) A I Liechtenstein, M I Katnelson and V A Gubanov, J.Phys. F:Met.Phys. 14, L125 (1984); A I Liechtenstein, M I Ketnelson V.P. Antropov and V A Gubanov, J. Mag.Mater. 67, 65 (1987).
* (13) Jacques Miltat, Goncalo Albuquerque and Andre Thiavill, Spin dynamics in confined magnetic structures, (Springer Verlag, more info here e.g. year of publication etc.)
* (14) J H Mentink, M V Tretyakov, A Fasolino, M I Katsnelson and Th. Rasing, J. Phys. Cond. Matter. 22 176001 (2010).
* (15) J Korringa, Physica, 13, 392 (1947).
* (16) W Kohn W and N Rostoker, Phys. Rev. 94, 1111 (1954).
* (17) J. Kudrnovsky, V. Drchal, I. Turek, M. Sob and P. Weinberger, Phys. Rev. B 53, 5125 (1996).
* (18) E. Holmström, L.Bergqvist, B. Skubic, B.Hjörvarsson, L.Nordström, I.Abrikosov, P.Svedlindh and O. Eriksson, PNAS 101, 4742 (2004).
* (19) C. Stamm, I. Tudosa, H. C. Siegmann, and J. Stöhr, A. Yu. Dobin, G. Woltersdorfand B. Heinrich, A. Vaterlaus, Phys. Rev. Lett. 94, 197603 (2005).
|
arxiv-papers
| 2011-09-07T10:41:22 |
2024-09-04T02:49:21.961762
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Satadeep Bhattacharjee, Anders Bergman, Andrea Taroni, Johan Hellsvik,\n Biplab Sanyal and Olle Eriksson",
"submitter": "Satadeep Bhattacharjee",
"url": "https://arxiv.org/abs/1109.1414"
}
|
1109.1520
|
# Heavy quarkonium production and polarization
Zhong-Bo Kang1, Jian-Wei Qiu2,3 and George Sterman3 1RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, NY 11973, USA 2Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA 3C.N. Yang
Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY
11794, USA
###### Abstract
We present a perturbative QCD factorization formalism for the production of
heavy quarkonia of large transverse momentum $p_{T}$ at collider energies,
which includes both the leading power (LP) and next-to-leading power (NLP)
contributions to the cross section in the $m_{Q}^{2}/p_{T}^{2}$ expansion for
heavy quark mass $m_{Q}$. We estimate fragmentation functions in the non-
relativistic QCD formalism, and reproduce the bulk of the large enhancement
found in explicit NLO calculations in the color singlet model. Heavy quarkonia
produced from NLP channels prefer longitudinal polarization.
###### pacs:
12.38.Bx, 12.39.St, 13.87.Fh, 14.40.Pq
††preprint: YITP-SB-11-32
Introduction — More than thirty-five years after the discovery of the $J/\psi$
jpsi-disc , the production of heavy quarkonia remains a key subject in strong
interaction physics Brambilla:2010cs . The inclusive production of pairs of
charm or bottom quarks, with masses $m_{Q}\gg\Lambda_{\rm QCD}$, is an
essentially perturbative process, while the subsequent evolution of the pair
into a physical quarkonium is nonperturbative. Different treatments of the
transformation from heavy quark pair to bound quarkonium are given in various
formalisms, most notably, the color singlet model (CSM), the color evaporation
model (CEM), and non-relativistic QCD (NRQCD). The current status of theory
and experiment has been summarized very recently in Ref. Brambilla:2010cs .
For the NRQCD formalism Caswell:1985ui ; Bodwin:1994jh , small, color-octet
production matrix elements can provide good fits to high $p_{T}$ inclusive
hadron collider cross sections for $J/\psi$ and $\Upsilon$, but a complete
description remains elusive. Polarization in particular remains a challenge,
along with the surprisingly high rate of associated production in electron-
positron annihilation Brambilla:2010cs . Although plausible arguments for the
use of NRQCD at leading power ($p_{T}^{-4}$) have been around since the
beginning of the formalism Bodwin:1994jh , issues of gauge invariance and
infrared cancellation are still not completely settled Nayak:2005rw ;
Nayak:2007mb . Meanwhile, large next-to-leading order (NLO) and potentially
large next-to-next-to leading order (NNLO) corrections to high-$p_{T}$ cross
sections Campbell:2007ws in the CSM have attracted attention. The size of
these color singlet cross sections seems to upset long-held expectations for
gluon fragmentation/color octet dominance, and indeed, they appear in diagrams
that fall off like $p_{T}^{-6}$ in $d\sigma/dp_{T}^{2}$, compared to the
leading, $p_{T}^{-4}$ behavior associated with gluon fragmentation. These
developments suggest that we must widen the formalism for quarkonium
production beyond leading power.
Figure 1: $J/\psi$ production through a single parton (a gluon) fragmentation
(left) and fragmentation of a heavy quark pair (right).
In this letter, we emphasize the expansion of the production rate of the heavy
quark pairs first in the large scale $p_{T}$ and then in the coupling constant
$\alpha_{s}$. We present a perturbative QCD factorization formalism, accurate
to the first nonleading power in $m_{Q}^{2}/p_{T}^{2}$ that incorporates both
leading power gluon fragmentation and direct production of heavy quarks at
short distances with subsequent fragmentation, as illustrated in Fig. 1
Kang:2011zz . These figures are shown in cut diagram notation, in which the
amplitude and complex conjugate are combined into a forward scattering diagram
in which the final state is identified by a vertical line. When $p_{T}\gg
m_{Q}$, the rate to produce a gluon (or in general a single parton) at
distance scale $1/p_{T}$, which fragments into a heavy quark pair and thence
into a physical quarkonium, is characterized by a $p_{T}^{-4}$ behavior. We
refer to this as the leading power (LP) contribution. The perturbative
production of a collinear heavy quark pair directly at the short distance
scale $1/p_{T}$ is suppressed by $1/p_{T}^{2}$ relative to the production rate
of a single parton at the same $p_{T}$. We therefore refer to it as a next-to-
leading power (NLP) contribution. As we will see below, however, the
probability for a such heavy quark pair to evolve into a heavy quarkonium is
naturally enhanced compared to that of a single parton. This can promote the
NLP channel to phenomenological interest, despite its suppression by two
powers of $p_{T}$.
The physical heavy quarkonium is likely formed long after the heavy quark pair
was produced Brodsky:1988xz . Both the LP and NLP production channels in Fig.
1 can therefore be considered as fragmentation contributions Brambilla:2010cs
. Similar to factorization at LP Nayak:2005rw , the NLP contribution to the
production rate is factorized into perturbatively calculable short-distance
coefficient functions for producing the heavy quark pair convoluted with non-
perturbative but universal long-distance fragmentation functions for the pair
to become a heavy quarkonium Kang:2011zz . The universality of the
fragmentation functions can in principle be tested when we compare the data of
heavy quarkonium production from processes with different short-distance
coefficients. As a consequence of this perturbative factorization, the short-
distance coefficient functions at both the LP and NLP capture the dynamics at
the distance scale $1/p_{T}$, and are insensitive to the details of the
produced heavy quarkonium states.
Factorization formalism — For the production of a heavy quarkonium state $H$
of momentum $p$, $A(P_{1})+B(P_{2})\to H(p)+X$, the leading contribution from
the channels in Fig. 1 can be summarized in an extended factorization formula
Kang:2011zz ; KQSfac:2011 , given schematically by
$\displaystyle d\sigma_{A+B\to H+X}(p)$ $\displaystyle\approx$
$\displaystyle\sum_{f}d\hat{\sigma}_{A+B\to f+X}(p_{f}=p/z)\otimes
D_{H/f}(z,m_{Q})$
$\displaystyle+\sum_{[Q\bar{Q}(\kappa)]}d\hat{\sigma}_{A+B\to[Q\bar{Q}(\kappa)]+X}(p(1\pm\zeta)/2z,p(1\pm\zeta^{\prime})/2z)$
$\displaystyle\otimes{\cal
D}_{H/[Q\bar{Q}(\kappa)]}(z,\zeta,\zeta^{\prime},m_{Q})\,,$
where the factorization scale dependence is suppressed. The first (second)
term on the right-hand side gives the contribution of LP (NLP) in
$m_{Q}/p_{T}$. In the first term, $d\hat{\sigma}_{A+B\to f+X}(p_{f})$ is the
cross section to produce an on-shell parton of flavor $f$ at short distances,
which contains all of the information about the incoming state and includes
convolutions with parton distributions when $A$ or $B$ is a hadron. The sum
over $f$ runs over all parton flavors, and $D_{H/f}(z,m_{Q})$ is the
fragmentation function for a heavy quarkonium state $H$ from parton $f$ with
momentum fraction $z$ Nayak:2005rw . For $H$ a $J/\psi$ or other heavy
quarkonium, the dominant channel at hadron colliders is gluon fragmentation,
$f=g$. The second term on the right-hand side in Eq. (Heavy quarkonium
production and polarization) is suppressed by $p_{T}^{-2}$ relative to the
first, and the quark-pair fragmentation function, ${\cal
D}_{H/[Q\bar{Q}(\kappa)]}(z,\zeta,\zeta^{\prime},m_{Q})$, has units of mass
squared, which are compensated by large invariants from the hard-scattering
function,
$d\hat{\sigma}_{A+B\to[Q\bar{Q}(\kappa)]+X}(p(1\pm\zeta)/2z,p(1\pm\zeta^{\prime})/2z)$,
which describes production of an on-shell, collinear heavy quark pair. The
momentum fractions $z$, $\zeta$ and $\zeta^{\prime}$ are defined as
$\displaystyle p_{Q}^{+}$ $\displaystyle=$ $\displaystyle
p^{+}\,\frac{1+\zeta}{2z}\,,\quad\quad
p_{\bar{Q}}^{+}=p^{+}\,\frac{1-\zeta}{2z}\,,$ $\displaystyle
p_{Q}^{{}^{\prime}+}$ $\displaystyle=$ $\displaystyle
p^{+}\,\frac{1+\zeta^{\prime}}{2z}\,,\quad\quad
p_{\bar{Q}}^{{}^{\prime}+}=p^{+}\,\frac{1-\zeta^{\prime}}{2z}\,.$ (2)
By analogy to the single-parton case, $z$ measures the fractional momentum of
the pair carried by the observed quarkonium in this leading region, which is
the same on both sides of the cut in Fig. 1. Parameters $\zeta$ and
$\zeta^{\prime}$ characterize the sharing of the pair’s momentum between the
heavy quark and antiquark on either side of the cut in the figure. In
principle, these need not be the same. The $\otimes$ in Eq. (Heavy quarkonium
production and polarization) represents the convolution over the partons’
momentum fractions KQSfac:2011 .
In Eq. (Heavy quarkonium production and polarization), the perturbative hard
parts at both the LP and NLP capture the QCD dynamics at distance scale of
$1/p_{T}$, are independent of heavy quark mass, and are the same for the
production of all heavy quarkonium states. For the LP, the hard parts are
effectively the same as those for inclusive production of any single hadron at
high $p_{T}$ and are available to both leading order (LO) and next-to-leading
order (NLO) in $\alpha_{s}$ Aversa:1988vb .
For the hard parts at NLP, we only need to calculate the rate to produce a
heavy quark pair with zero relative transverse momentum, since the effect of
relative transverse momentum will be further suppressed in $1/p_{T}$. The hard
parts can be perturbatively calculated order-by-order in powers of
$\alpha_{s}$ by applying the factorization formula in Eq. (Heavy quarkonium
production and polarization) to the production of an asymptotic state of a
heavy quark pair $[Q\bar{Q}(\kappa)]$ of momentum $p$ with various spin and
color quantum numbers. For example, the hard part for the subprocess,
$q+\bar{q}\to[Q\bar{Q}(\kappa)]+g$, can be derived by applying Eq. (Heavy
quarkonium production and polarization) to heavy pair production in quark-
antiquark scattering,
$\displaystyle d\hat{\sigma}^{(3)}_{q+\bar{q}\to[Q\bar{Q}(\kappa)]+g}(p)$
$\displaystyle=$ $\displaystyle
d\sigma^{(3)}_{q+\bar{q}\to[Q\bar{Q}(\kappa)]+g}(p)$
$\displaystyle\hskip-72.26999pt-d\hat{\sigma}^{(2)}_{q+\bar{q}\to
g+g}(p_{g}=p/z)\otimes{\cal D}^{(1)}_{[Q\bar{Q}(\kappa)]/g}(z,m_{Q})\,,$
where the superscript “(n)” indicates the order in $\alpha_{s}$. The first
term on the right of Eq. (Heavy quarkonium production and polarization) is the
differential cross section for the subprocess given by the diagrams in Fig.
2(a) and (b) plus three more diagrams. When $p_{T}\gg m_{Q}$, the square of
diagram (a) clearly contributes to the NLP, while the the square of diagram in
(b) contributes to both LP and NLP. We note that the interference between the
two contributes only to NLP. The second term in Eq. (Heavy quarkonium
production and polarization) exactly removes the LP piece of this subprocess,
where $d\hat{\sigma}^{(2)}_{q+\bar{q}\to g+g}(p_{g})$ is the LO differential
cross section for $q+\bar{q}\to g(p_{g})+g$ and ${\cal
D}^{(1)}_{[Q\bar{Q}(\kappa)]/g}(z,m_{Q})$ is the LO fragmentation function
given by the (cut) diagram in Fig. 2(c). Since the second term on the right of
Eq. (Heavy quarkonium production and polarization) removes a power mass
singularity $\sim 1/m_{Q}^{2}$, it is important to keep the heavy quark mass
when evaluating partonic diagrams and to set $m_{Q}\to 0$ (only) after
carrying out the subtraction KQSfac:2011 .
(a)
(b)
(c)
Figure 2: Sample Feynman diagrams for $q+\bar{q}\to[Q\bar{Q}(\kappa)](p)+g$
subprocess. The diagram in (a) contributes to the short-distance coefficient
at NLP while the diagram in (b) contributes to both the LP and NLP hard parts.
The cut diagram in (c) gives the LO gluon to heavy quark pair fragmentation
function.
The predictive power of the factorization formula in Eq. (Heavy quarkonium
production and polarization) relies on the universality of the fragmentation
functions. The single parton fragmentation functions $D_{H/f}(z,m_{Q})$ are
defined in Ref. Nayak:2005rw . The operator definition for ${\cal
D}_{H/[Q\bar{Q}(\kappa)]}(z,\zeta,\zeta^{\prime},m_{Q})$ depends on $\kappa$,
which represents the pair’s color and spin. For the heavy quark pair moving in
the “$+z$” direction with light-cone momentum components,
$p^{\mu}=(p^{+},(2m_{Q})^{2}/2p^{+},0_{\perp})$, there are singlet ($1$) and
octet ($8$) color states, and four spin states described by relativistic Dirac
spin projection operators: $\gamma^{+}\gamma_{5}/4p^{+}$, $\gamma^{+}/4p^{+}$,
and $\gamma^{+}\gamma^{i}/4p^{+}$ with $i=1,2$, for effective axial vector
($a$), vector ($v$), and tensor ($t$) “currents”, respectively KQSfac:2011 .
As an example, the operator definition of the axial vector/octet fragmentation
function can be written as
$\displaystyle{\cal D}_{H/[Q\bar{Q}(a8)]}(z,\zeta,\zeta^{\prime})$
$\displaystyle=$ $\displaystyle\sum_{X}\int\frac{p^{+}dy^{-}}{2\pi}\,{\rm
e}^{-i(p^{+}/z)y^{-}}$ (4)
$\displaystyle\times\int\frac{p^{+}dy_{1}^{-}p^{+}dy_{2}^{-}}{(2\pi)^{2}}{\rm
e}^{i(p^{+}/2z)(1-\zeta)y_{1}^{-}}{\rm
e}^{-i(p^{+}/2z)(1-\zeta^{\prime})y_{2}^{-}}$
$\displaystyle\times\frac{4}{(N^{2}-1)}\langle
0|\,\psi_{i}(0)\frac{\gamma^{+}\gamma_{5}}{4p^{+}}\,(t^{a})_{ij}\,\overline{\psi}_{j}(y_{2}^{-})|H(p^{+})X\rangle$
$\displaystyle\times\langle
H(P^{+})X|\psi_{l}(y^{-}+y_{1}^{-})\frac{\gamma^{+}\gamma_{5}}{4p^{+}}\,(t^{a})_{lk}\,\overline{\psi}_{k}(y^{-})\,|0\rangle\,,$
where we have suppressed dependence on a factorization scale. We also suppress
gauge links along the minus light cone, inserted between repeated color
indices, which provide a gauge invariant definition of the operator
KQSfac:2011 . The links are in adjoint representation for index $a$, and
fundamental representation for $i,j,l,k$. (We note that the color matrices
$t^{a}$ may be taken at any points along the light cone.) Overall, the pair
fragmentation functions are given by matrix elements of nonlocal operators,
and the form is very similar to operator definitions of single parton
fragmentation functions, simply replacing the parton field by the product of
quark fields. They are also reminiscent of hadronic wave functions that
connect multiple partons to the hard scattering in the factorized expressions
for elastic amplitudes Brodsky:1989pv .
Similar to the single parton fragmentation functions, heavy quark pair
fragmentation functions like the one defined in Eq. (4) are nonperturbative,
but, universal. They are boost invariant, and require renormalization. They
thus evolve in the usual sense, and depend upon a factorization scale, chosen
to match the short-distance scale of the problem. It is natural to think of
the choice of factorization scale as the same for NLP as for LP, and in
general, a change of scale could mix them. This leads to a closed set of
general evolution equations for both LP and NLP fragmentation functions, which
will be addressed elsewhere KQSfac:2011 .
We note that there are further additive corrections at the power of
$1/p_{T}^{2}$, such as those involving twist-4 parton distributions, or
twist-4 light-parton fragmentation functions. These corrections, however, can
be considered small since they do not introduce natural enhancements that
scale with $m_{Q}$ in the probability to form a quarkonium, to compensate the
suppression of $1/p_{T}^{2}$.
Cross section and polarization — It is the fragmentation functions that
determine the absolute normalization of perturbative calculations using the
factorization formula in Eq. (Heavy quarkonium production and polarization),
as well as the differences in the production rate between various quarkonium
states and their polarizations. Since only pairs with small relative momentum
are likely to form bound quarkonia, we may apply the basic NRQCD factorization
hypothesis for heavy quarkonium production to these fragmentation functions to
reduce the unknown functions to a few universal constants in the form of NRQCD
matrix elements,
$\displaystyle
D_{H/f}(z,m_{Q},\mu)=\sum_{c}d_{f\to[Q\bar{Q}(c)]}(z,m_{Q},\mu)\langle
O_{[Q\bar{Q}(c)]}^{H}\rangle$ $\displaystyle{\cal
D}_{H/[Q\bar{Q}(\kappa)]}(z,\zeta,\zeta^{\prime},m_{Q},\mu)\,,$
$\displaystyle\
=\sum_{c}d_{[Q\bar{Q}(\kappa)]\to[Q\bar{Q}(c)]}(z,\zeta,\zeta^{\prime},m_{Q},\mu)\langle
O_{[Q\bar{Q}(c)]}^{H}\rangle\,,$ (5)
where the functions $d$ are calculable, $\mu$ is a factorization scale, and
$\langle O_{[Q\bar{Q}(c)]}^{H}\rangle$ are local NRQCD matrix elements
Bodwin:1994jh . Although we cannot provide a full proof for the NRQCD
factorization in Eq. (5), it is reasonable to evaluate the coefficient
functions in Eq. (5) to estimate the properties of fragmentation functions
Nayak:2005rw .
We can use this formalism to help understand the source of the surprisingly
large corrections to $J/\psi$ production in the CSM at NLO and NNLO with
predominantly longitudinal polarization, in contrast to the small and
transversely polarized LO Gong:2008sn ; Lansberg:2008gk . To do so, we use Eq.
(Heavy quarkonium production and polarization) to calculate NLP $J/\psi$ cross
sections at LO in $\alpha_{s}$ from a color singlet ${}^{3}{\rm S}_{1}$ pair.
We estimate the ${\cal O}(\alpha_{s})$ fragmentation functions from the
diagrams in Fig. 3, where the upper lines are fixed at $p/2$ and Dirac indices
are contracted with an NRQCD singlet projection with matrix element
essentially equivalent to the CSM. At order of $\alpha_{s}$, only the
$[Q\bar{Q}(a8)]$ state fragments into a color singlet ${}^{3}{\rm S}_{1}$
heavy quark pair, while both $[Q\bar{Q}(v8)]$ and $[Q\bar{Q}(t8)]$ states give
vanishing contributions due to charge conjugation symmetry KQSfac:2011 .
Defining $r(z)\equiv z^{2}\mu^{2}/(4m_{c}^{2}(1-z)^{2})$, we have
$\displaystyle{\cal D}^{L}_{[Q\bar{Q}(a8)]\to
J/\psi}(z,\zeta,\zeta^{\prime},m_{Q},\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2N^{2}}\frac{\langle O^{J/\psi}_{1(^{3}{\rm
S}_{1})}\rangle}{3m_{c}}\Delta(\zeta,\zeta^{\prime})$
$\displaystyle{\hskip-108.405pt}\times\frac{\alpha_{s}}{2\pi}z(1-z)\left[\ln\left(r(z)+1\right)-\left(1-\frac{1}{1+r(z)}\right)\right],$
$\displaystyle{\cal D}^{T}_{[Q\bar{Q}(a8)]\to
J/\psi}(z,\zeta,\zeta^{\prime},m_{Q},\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2N^{2}}\frac{\langle O^{J/\psi}_{1(^{3}{\rm
S}_{1})}\rangle}{3m_{c}}\Delta(\zeta,\zeta^{\prime})$ (6)
$\displaystyle{\hskip-108.405pt}\times\frac{\alpha_{s}}{2\pi}z(1-z)\left[1-\frac{1}{1+r(z)}\right],$
where ${\cal D}^{L}$ (${\cal D}^{T}$) is the pair-fragmentation function with
the pair longitudinally (transversely) polarized, and
$\displaystyle\Delta(\zeta,\zeta^{\prime})=\frac{1}{4}\sum_{a,b}\delta(\zeta-a(1-z))\delta(\zeta^{\prime}-b(1-z)),$
(7)
with $a,b=-1,1$, and total unpolarized contribution, ${\cal
D}_{[Q\bar{Q}(a8)]\to J/\psi}=2\,{\cal D}^{T}_{[Q\bar{Q}(a8)]\to J/\psi}+{\cal
D}^{L}_{[Q\bar{Q}(a8)]\to J/\psi}$. In deriving Eq. (6), we renormalized the
UV divergence by a cutoff $\mu$ on transverse momenta. Quark masses make an IR
cutoff on transverse momenta unnecessary. Other renormalization schemes give
similar results and will be discussed elsewhere KQSfac:2011 . With these
fragmentation functions, and LO hard scattering functions KQSfac:2011 , we can
compute NLP cross sections that can be compared to NLO CSM cross sections
Campbell:2007ws ; Lansberg:2008gk ; Butenschoen:2010rq ; Ma:2010yw . Using the
CTEQ6L parton distributions Pumplin:2002vw , $\mu=p_{T}$ for factorization and
renormalization scales, $m_{c}=1.5$ GeV, and $\langle O^{J/\psi}_{1(^{3}{\rm
S}_{1})}\rangle=1.32$ GeV3, we calculate and plot the unpolarized cross
section as a function $p_{T}$ in Fig. 4 (upper panel). We see that the
factorization-based cross section shows much the same enhancement above the LO
CSM as the full NLO result in CSM.
+
\+ …
Figure 3: Leading order Feynman diagrams represent the fragmentation of a
heavy quark pair to another heavy quark pair. Figure 4: $J/\psi$ cross section
(upper panel) and polarization (lower panel) as a function of $p_{T}$. The
solid lines are computed from the NLP term in Eq. (Heavy quarkonium production
and polarization) with fragmentation functions given by Eq. (6); the dashed
line is LO CSM cross section as in Ref. Butenschoen:2010rq , and all in
[nb/GeV].
Actually, our calculated NLP cross section with the estimated ${\cal
O}(\alpha_{s})$ fragmentation functions reproduces as much as eighty percent
of the full NLO CSM result. The difference should be due mainly to ${\cal
O}(\alpha_{s})$ corrections to the hard scattering functions from the color
singlet $[Q\bar{Q}(v1)]$ channel. With higher order fragmentation functions
from other $[Q\bar{Q}(\kappa)]$ states, the full NLP cross section at LO in
$\alpha_{s}$ could be larger than the solid line in Fig. 4. Similarly, we
calculate the $J/\psi$ polarization (lower panel), as measured by the
parameter $\alpha=(\sigma^{T}-\sigma^{L})/(\sigma^{T}+\sigma^{L})$ in terms of
transverse (longitudinal) cross section $\sigma^{T}(\sigma^{L})$. Our result
is consistent with that in Refs. Gong:2008sn ; Lansberg:2008gk . We regard
these results as compelling evidence for the phenomenological relevance of the
power expansion.
Summary — We have argued that a practical strategy for the phenomenology of
heavy quarkonium production at high $p_{T}$ is to expand the cross sections
first in the large scale $p_{T}$, and then in the coupling, $\alpha_{s}$, and
have presented a new perturbative QCD factorization formalism for quarkonium
production including the first non-leading powers in $m_{Q}/p_{T}$. This
approach enables us to resum perturbative logarithms into the fragmentation
functions, to analyze systematically the influence of the larger color singlet
matrix elements despite their suppressed $p_{T}$-dependence, and to resolve
some of the mystery associated with the discovery of large high order
corrections to color singlet cross sections. We have found that heavy
quarkonia produced from pair fragmentations are likely to be longitudinally
polarized, in contrast to single parton fragmentation. The observed quarkonium
polarization should be a consequence of the competition of these two leading
production channels.
We thank G.T. Bodwin, E. Braaten, J.-P. Lansberg, B. Pire, and O. Teryaev for
helpful discussions. This work was supported in part by the U.S. Department of
Energy, contract number DE-AC02-98CH10886, and by the National Science
Foundation, grants PHY-0653342 and PHY-0969739.
## References
* (1) J. J. Aubert et al. [E598 Collaboration], Phys. Rev. Lett. 33, 1404 (1974); J. E. Augustin et al. [SLAC-SP-017 Collaboration], Phys. Rev. Lett. 33, 1406 (1974).
* (2) N. Brambilla, et al., Eur. Phys. J. C71, 1534 (2011), and references therein.
* (3) W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).
* (4) G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum-ibid. D 55, 5853 (1997)].
* (5) G. C. Nayak, J.-W. Qiu, G. F. Sterman, Phys. Lett. B613, 45-51 (2005); Phys. Rev. D72, 114012 (2005); Phys. Rev. D 74, 074007 (2006).
* (6) G. C. Nayak, J.-W. Qiu, G. F. Sterman, Phys. Rev. Lett. 99, 212001 (2007); Phys. Rev. D 77 (2008) 034022.
* (7) J. Campbell, F. Maltoni and F. Tramontano, Phys. Rev. Lett. 98, 252002 (2007).
* (8) Z.-B. Kang, J.-W. Qiu, G. Sterman, Nucl. Phys. Proc. Suppl. 214, 39-43 (2011).
* (9) S. J. Brodsky and A. H. Mueller, Phys. Lett. B 206, 685 (1988).
* (10) Z.-B. Kang, J.-W. Qiu, G. Sterman, in preparation.
* (11) F. Aversa, P. Chiappetta, M. Greco and J. P. Guillet, Nucl. Phys. B 327, 105 (1989).
* (12) S. J. Brodsky, G. P. Lepage, Adv. Ser. Direct. High Energy Phys. 5, 93 (1989).
* (13) B. Gong, J.-X. Wang, Phys. Rev. Lett. 100, 232001 (2008).
* (14) J. P. Lansberg, Eur. Phys. J. C61, 693-703 (2009). [arXiv:0811.4005 [hep-ph]].
* (15) M. Butenschoen and B. A. Kniehl, Phys. Rev. Lett. 106, 022003 (2011).
* (16) Y. Q. Ma, K. Wang and K. T. Chao, Phys. Rev. Lett. 106, 042002 (2011).
* (17) J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung, JHEP 0207, 012 (2002).
|
arxiv-papers
| 2011-09-07T18:03:00 |
2024-09-04T02:49:21.973608
|
{
"license": "Public Domain",
"authors": "Zhong-Bo Kang, Jian-Wei Qiu and George Sterman",
"submitter": "Jianwei Qiu",
"url": "https://arxiv.org/abs/1109.1520"
}
|
1109.1608
|
On Levi-flat hypersurfaces tangent to
holomorphic webs
Arturo Fernández-Pérez 111Work supported by CNPq-Brazil.
Keywords: Levi-flat hypersurfaces - Holomorphic webs
Mathematical subjet classification: 32V40 - 14C21
Abstract. We investigate germs of real analytic Levi-flat hypersurfaces
tangent to germs of codimension one holomorphic webs. We introduce the notion
of first integrals for local webs. In particular, we prove that a $k$-web with
finitely many invariant analytic subvarieties through the origin tangent to a
Levi-flat hypersurface has a holomorphic first integral.
Résumé. Nous étudions les germes d’hypersurfaces réelles analytiques Levi-
plate tangente à les germes d’webs holomorphes codimension un. Nous
introduisons la notion des intégrales premières des webs locales. En
particulier, nous montrons que une $k$-web avec un nombre fini de feuilles
invariant analytique par l’origine, tangente à une hypersurface Levi-plate
possède une intégrale première holomorphe.
## 1 Introduction
In very general terms, a germ of codimension one $k$-web is a collection of
$k$ germs of codimension one holomorphic foliations in “general position”. The
study of webs was initiated by Blaschke and his school in the late 1920 s. For
a recent account of the theory, we refer the reader to [12].
For instance, take $\omega\in Sym^{k}\Omega^{1}(\mathbb{C}^{2},0)$ defined by
$\omega=(dy)^{k}+a_{k-1}(dy)^{k-1}dx+\ldots+a_{0}(dx)^{k},$
where $a_{j}\in\mathcal{O}_{2}$ for all $0\leq j\leq k-1$. Then
$\mathcal{W}:\omega=0$, define a non-trivial $k$-web on $(\mathbb{C}^{2},0)$.
In this paper we study webs and its relation with Levi-flat hypersurfaces.
Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real codimension one analytic
irreducible analytic set. Since $M$ is real analytic of codimension one, it
can be decomposed into $M_{reg}$ and $\textsf{Sing}(M)$, where $M_{reg}$ is a
germ of smooth real analytic hypersurface in $\mathbb{C}^{n}$ and
$\textsf{Sing}(M)$, the singular locus, is contained in a proper analytic
subvariety of lower dimension. We shall say that $M$ is Levi-flat if the
complex distribution $L$ on $M_{reg}$
$\displaystyle L_{p}:=T_{p}M\cap iT_{p}M\subset T_{p}M,\,\,\,\,\text{for
any}\,\,p\in M_{reg}$ (1.1)
is integrable, in Frobenius sense. It follows that $M_{reg}$ is smoothly
foliated by immersed complex manifolds of complex dimension $n-1$. The
foliation defined by $L$ is called the Levi foliation and will be denoted by
$\mathcal{L}_{M}$.
If $M$ is a real analytic smooth Levi-flat hypersurface, by a classic result
of E. Cartan there exists a local holomorphic coordinates
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ such that $M$ can be represented by
$M=\\{\mathcal{I}m(z_{n})=0\\}$. The situation if different if the
hypersurface have singularities. Singular Levi-flat real analytic
hypersurfaces have been studied by Burns and Gong [1], Brunella [2], Lebl [9],
the author [6], [7] and many others.
Recently D.Cerveau and A. Lins Neto [5] have studied codimension one
holomorphic foliations tangent to singular Levi-flat hypersurfaces. A
codimension one holomorphic foliation $\mathcal{F}$ is tangent to $M$, if any
leaf of $\mathcal{L}_{M}$ is also a leaf of $\mathcal{F}$. In [5] it is proved
that a germ of codimension one holomorphic foliation tangent to a real
analytic Levi-flat hypersurface has a non-constant meromorphic first integral.
In the same spirit, the authors propose a problem for webs, which is as
follows:
Problem.- Let $M$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 2$, of real
analytic hypersurface Levi-flat. Assume that there exists a singular
codimension one $k$-web, $k\geq 2$, such that any leaf of the Levi foliation
$\mathcal{L}_{M}$ on $M_{reg}$ is also a leaf of the web. Does the web has a
non-constant meromorphic first integral?.
By a meromorphic first integral we mean something like
$f_{0}(x)+z.f_{1}(x)+\ldots+z^{k}.f_{k}(x)=0$, where
$f_{0},f_{1},\ldots,f_{k}\in\mathcal{O}_{n}$. In this situation, the web is
obtained by the elimination of $z$ in the system given by
$\left\\{\begin{array}[]{ll}f_{0}+z.f_{1}+z^{2}.f_{2}+\ldots+z^{k}.f_{k}=0&\\\
df_{0}+z.df_{1}+z^{2}.df_{2}+\ldots+z^{k}.df_{k}=0.&\end{array}\right.$
In this work, we organize some results on singular Levi-flat hypersurfaces and
holomorphic foliations which provide a best approach to study of webs and
Levi-flats. Concerning the problem, we obtain an interesting result in a case
very special (Theorem 1), the problem remains open in general.
### 1.1 Local singular webs
It is customary to define a germ of singular holomorphic foliation as an
equivalence class $[\omega]$ of germs of holomorphic 1-forms in
$\Omega^{1}(\mathbb{C}^{n},0)$ modulo multiplication by elements of
$\mathcal{O}^{*}(\mathbb{C}^{n},0)$ such that any representative $\omega$ is
integrable ( $\omega\wedge d\omega=0$ ) and with singular set
$\textsf{Sing}(\omega)=\\{p\in(\mathbb{C}^{n},0):\omega(p)=0\\}$ of
codimension at least two.
An analogous definition can be made for codimension one $k$-webs. A germ at
$(\mathbb{C}^{n},0)$, $n\geq 2$ of codimension one $k$-web $\mathcal{W}$ is an
equivalence class $[\omega]$ of germs of $k$-symmetric 1-forms, that is
sections of $Sym^{k}\Omega^{1}(\mathbb{C}^{n},0)$, modulo multiplication by
$\mathcal{O}^{*}(\mathbb{C}^{n},0)$ such that a suitable representative
$\omega$ defined in a connected neighborhood $U$ of the origin satisfies the
following conditions:
1. 1.
The zero set of $\omega$ has codimension at least two.
2. 2.
The 1-form $\omega$, seen as a homogeneous polynomial of degree $k$ in the
ring $\mathcal{O}_{n}[dx_{1},\ldots,dx_{n}]$, is square-free.
3. 3.
(Brill’s condition) For a generic $p\in U$, $\omega(p)$ is a product of $k$
linear forms.
4. 4.
(Frobenius’s condition) For a generic $p\in U$, the germ of $\omega$ at $p$ is
the product of $k$ germs of integrable 1-forms.
Both conditions $(3)$ and $(4)$ are automatic for germs at
$(\mathbb{C}^{2},0)$ of webs and non-trivial for germs at $(\mathbb{C}^{n},0)$
when $n\geq 3$.
We can think $k$-webs as first order differential equations of degree $k$. The
idea is to consider the germ of web as a meromorphic section of the
projectivization of the cotangent bundle of $(\mathbb{C}^{n},0)$. This is a
classical point view in the theory of differential equations, which has been
recently explored in Web-geometry. For instance see [3], [4], [14].
### 1.2 The contact distribution
Let us denote $\mathbb{P}:=\mathbb{P}T^{*}(\mathbb{C}^{n},0)$ the
projectivization of the cotangent bundle of $(\mathbb{C}^{n},0)$ and
$\pi:\mathbb{P}T^{*}(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ the
natural projection. Over a point $p$ the fiber $\pi^{-1}(p)$ parametrizes the
one-dimensional subspaces of $T^{*}_{p}(\mathbb{C}^{n},0)$. On $\mathbb{P}$
there is a canonical codimension one distribution, the so called contact
distribution $\mathcal{D}$. Its description in terms of a system of
coordinates $x=(x_{1},\ldots,x_{n})$ of $(\mathbb{C}^{n},0)$ goes as follows:
let $dx_{1},\ldots,dx_{n}$ be the basis of $T^{*}(\mathbb{C}^{n},0)$
associated to the coordinate system $(x_{1},\ldots,x_{n})$. Given a point
$(x,y)\in T^{*}(\mathbb{C}^{n},0)$, we can write
$y=\sum_{j=1}^{n}y_{j}dx_{j}$, $(y_{1},\ldots,y_{n})\in\mathbb{C}^{n}$. In
this way, if $(y_{1},\ldots,y_{n})\neq 0$ then we set
$[y]=[y_{1},\ldots.y_{n}]\in\mathbb{P}^{n-1}$ and
$(x,[y])\in(\mathbb{C}^{n},0)\times\mathbb{P}^{n-1}\cong\mathbb{P}$. In the
affine coordinate system $y_{n}\neq 0$ of $\mathbb{P}$, the distribution
$\mathcal{D}$ is defined by $\alpha=0$, where
$\displaystyle\alpha=dx_{n}-\sum_{j=1}^{n-1}p_{j}dx_{j},\,\,\,\,\,\,\,\,p_{j}=-\frac{y_{j}}{y_{n}}\,\,\,\,\,\,\,\,\,(1\leq
j\leq n-1).$ (1.2)
The 1-form $\alpha$ is called the contact form.
### 1.3 Webs as closures of meromorphic multi-sections
Let us consider $X\subset\mathbb{P}$ a subvariety, not necessarily
irreducible, but of pure dimension $n$. Let
$\pi_{X}:X\rightarrow(\mathbb{C}^{n},0)$ be the restriction to $X$ of the
projection $\pi$. Suppose also that $X$ satisfies the following conditions:
1. 1.
The image under $\pi$ of every irreducible component of $X$ has dimension $n$.
2. 2.
The generic fiber of $\pi$ intersects $X$ in $k$ distinct smooth points and at
these the differential $d\pi_{X}:T_{p}X\rightarrow
T_{\pi(p)}(\mathbb{C}^{n},0)$ is surjective. Note that $k=\deg(\pi_{X})$.
3. 3.
The restriction of the contact form $\alpha$ to the smooth part of every
irreducible component of $X$ is integrable. We denote $\mathcal{F}_{X}$ the
foliation defined by $\alpha|_{X}=0$.
We can define a germ $\mathcal{W}$ at $0\in\mathbb{C}^{n}$ of $k$-web as a
triple $(X,\pi_{X},\mathcal{F}_{X})$. This definition is equivalent to one
given in Section 1.1. In the sequel, $X$ will always be the variety associated
to $\mathcal{W}$, the singular set of $X$ will be denoted by
$\textsf{Sing}(X)$ and its the smooth part will be denoted by $X_{reg}$.
###### Definition 1.1.
Let $R$ be the set of points $p\in X$ where
* •
either $X$ is singular,
* •
or the differential $d\pi_{X}:T_{p}X_{reg}\rightarrow
T_{\pi(p)}(\mathbb{C}^{n},0)$ is not an isomorphism.
The analytic set $R$ is called the criminant set of $\mathcal{W}$ and
$\Delta_{\mathcal{W}}=\pi(R)$ the discriminant of $\mathcal{W}$. Note that
$\textsf{dim}(R)\leq n-1$.
###### Remark 1.2.
Let $\omega\in Sym^{k}\Omega_{1}(\mathbb{C}^{n},0)$ and assume that it defines
a $k$-web $\mathcal{W}$ with variety $X$. Then $X$ is irreducible if, and only
if, $\omega$ is irreducible in the ring
$\mathcal{O}_{n}[dx_{1},\ldots,dx_{n}]$. In this case we say that the web is
irreducible.
Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real analytic Levi-flat
hypersurface.
###### Definition 1.3.
We say that $M$ is tangent to $\mathcal{W}$ if any leaf of the Levi foliation
$\mathcal{L}_{M}$ on $M_{reg}$ is also a leaf of $\mathcal{W}$.
### 1.4 First integrals for webs
###### Definition 1.4.
We say that $\mathcal{W}$ a $k$-web has a meromorphic first integral if, and
only if, there exists
$P(z)=f_{0}+z.f_{1}+\ldots+z^{k}.f_{k}\in\mathcal{O}_{n}[z],$
where $f_{0},\ldots,f_{k}\in\mathcal{O}_{n}$, such that every irreducible
component of the hypersurface $(P(z_{0})=0)$ is a leaf of $\mathcal{W}$, for
all $z_{0}\in(\mathbb{C},0)$.
###### Definition 1.5.
We say that $\mathcal{W}$ a $k$-web has a holomorphic first integral if, and
only if, there exists
$P(z)=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}\in\mathcal{O}_{n}[z],$
where $f_{0},\ldots,f_{k-1}\in\mathcal{O}_{n}$, such that every irreducible
component of the hypersurface $(P(z_{0})=0)$ is a leaf of $\mathcal{W}$, for
all $z_{0}\in(\mathbb{C},0)$.
We will prove a result concerning the situation of definitions 1.3 and 1.5.
###### Theorem 1.
Let $\mathcal{W}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$ of $k$-web
defined by
$\omega=\underset{i_{1},\ldots,i_{n}\geq
0}{\sum_{i_{1}+\ldots+i_{n}=k}}a_{i_{1},\ldots,i_{n}}(z)dz_{1}^{i_{1}}\ldots
dz_{n}^{i_{n}},$
where $a_{i_{1},\ldots,i_{n}}\in\mathcal{O}_{n}$ and
$a_{0,0,\ldots,0,k}(0)\neq 0$. Suppose that $\mathcal{W}$ is tangent to a germ
at $0\in\mathbb{C}^{n}$ of an irreducible real-analytic Levi-flat hypersurface
$M$. Furthermore, assume that $\mathcal{W}$ is irreducible and has finitely
many invariant analytic subvarieties through the origin. Let $X$ be the
variety associated to $\mathcal{W}$. Then $\mathcal{W}$ has a non-constant
holomorphic first integral, if one of the following conditions is fulfilled :
1. 1.
If $n=2$.
2. 2.
If $n\geq 3$ and $cod_{X_{reg}}(\textsf{Sing}(X))\geq 2$.
Moreover, if
$P(z)=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}\in\mathcal{O}_{n}[z]$ is a
holomorphic first integral for $\mathcal{W}$, then $M=(F=0)$, where $F$ is
obtained by the elimination of $z$ in the system given by
$\left\\{\begin{array}[]{ll}f_{0}+z.f_{1}+z^{2}.f_{2}+\ldots+z^{k-1}.f_{k-1}+z^{k}=0&\\\
\bar{f}_{0}+z.\bar{f}_{1}+z^{2}.\bar{f}_{2}+\ldots+z^{k-1}.\bar{f}_{k-1}+z^{k}=0.&\end{array}\right.$
###### Remark 1.6.
Under the hypotheses of Theorem 1, if $n=2$ and $k=1$, $\mathcal{W}$ is a non-
dicritical holomorphic foliation at $(\mathbb{C}^{2},0)$ tangent to a germ of
an irreducible real analytic Levi-flat hypersurface $M$, then Theorem 1 given
by Cerveau and Lins Neto [5] assures that $\mathcal{W}$ has a non-constant
holomorphic first integral. In this sense, our theorem is a generalization of
result of Cerveau and Lins Neto.
###### Remark 1.7.
Let $\mathcal{W}$ a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$, of a smooth
$k$-web tangent to a germ at $0\in\mathbb{C}^{n}$ of an irreducible real
codimension one submanifold $M$. In other words,
$\mathcal{W}=\mathcal{F}_{1}\boxtimes\ldots\boxtimes\mathcal{F}_{k}$ is a
generic superposition of $k$ germs at $0\in\mathbb{C}^{n}$ of smooth
foliations $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$. In this case the
irreducibility and tangency conditions to $M$ implies the existence of a
unique $i\in\\{1,\ldots,k\\}$ such that $\mathcal{F}_{i}$ is tangent to $M$.
Therefore we can find a coordinates system $z_{1},\ldots,z_{n}$ of
$\mathbb{C}^{n}$ such that $\mathcal{F}_{i}$ is defined by $dz_{n}=0$ and
$M=(\mathcal{I}m(z_{n})=0)$.
## 2 The foliation associated to a web
In this section, we prove a key lemma which will be used in the proof of main
theorem.
Since the restriction of $\mathcal{D}$ to $X_{reg}$ is integrable, it defines
a foliation $\mathcal{F}_{X}$, which in general is a singular foliation. Given
$p\in(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$,
$\pi_{X}^{-1}(p)=\\{q_{1},\ldots,q_{k}\\}$, where $q_{i}\neq q_{j}$, if $i\neq
j$, ($\deg(\pi_{X})=k$), denote by $\mathcal{F}_{X}^{i}$ the germ of
$\mathcal{F}_{X}$ at $q_{i}$, $i=1,\ldots,k$.
The projections $\pi_{*}(\mathcal{F}_{X}^{i}):=\mathcal{F}_{p}^{i}$ define $k$
germs of codimension one foliations at $p$.
###### Definition 2.1.
A leaf of the web $\mathcal{W}$ is, by definition, the projection on
$(\mathbb{C}^{n},0)$ of a leaf of $\mathcal{F}_{X}$.
###### Remark 2.2.
Given $p\in(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$, and
$q_{i}\in\pi_{X}^{-1}(p)$, the projection $\pi_{X}(L_{i})$ of the leaf $L_{i}$
of $\mathcal{F}_{X}$ through $q_{i}$, gives rise to a leaf of $\mathcal{W}$
through $p$. In particular, $\mathcal{W}$ has at most $k$ leaves through $p$.
We will use the following proposition (cf. [8] Th. 5, pg. 32). Let
$\mathcal{O}(X)$ denote the ring of holomorphic functions on $X$.
###### Proposition 2.3.
Let $V$ be an analytic variety. If $\pi:V\rightarrow W$ is a finite branched
holomorphic covering of pure order $k$ over an open subset
$W\subseteq\mathbb{C}^{n}$, then to each holomorphic function
$f\in\mathcal{O}(V)$ there is a canonically associated monic polynomial
$P_{f}(z)\in\mathcal{O}_{n}[z]\subseteq\mathcal{O}(V)[z]$ of degree $k$ such
that $P_{f}(f)=0$ in $\mathcal{O}(V)$.
We have now the following lemma.
###### Lemma 2.4.
Suppose that $(X,\pi_{X},\mathcal{F}_{X})$ defines a $k$-web $\mathcal{W}$ on
$(\mathbb{C}^{n},0)$, $n\geq 2$, where $X$ is an irreducible subvariety of
$\mathbb{P}$. If $\mathcal{F}_{X}$ has a non-constant holomorphic first
integral then $\mathcal{W}$ also has a holomorphic first integral.
###### Proof.
Let $g\in\mathcal{O}(X)$ be the first integral for $\mathcal{F}_{X}$. By
Proposition 2.3, there exists a monic polynomial
$P_{g}(z)\in\mathcal{O}_{n}[z]$ of degree $k$ such that $P_{g}(g)=0$ in
$\mathcal{O}(X)$. Write
$P_{g}(z)=g_{0}+z.g_{1}+\ldots+z^{k-1}.g_{k-1}+z^{k},$
where $g_{0},\ldots,g_{k-1}\in\mathcal{O}_{n}$.
Assertion.– $P_{g}$ define a holomorphic first integral for $\mathcal{W}$.
Let $U\subseteq(\mathbb{C}^{n},0)\backslash\Delta_{\mathcal{W}}$ be an open
subset and let $\varphi:X\rightarrow(\mathbb{C}^{n},0)\times\mathbb{C}$ be
defined by $\varphi=(\pi_{X},g)$. Take a leaf $L$ of $\mathcal{W}|_{U}$. Then
there is $z\in\mathbb{C}$ such that the following diagram
---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi^{-1}_{X}(U)\cap\varphi^{-1}(L\times\\{z\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\varphi}$$\textstyle{L\times\\{z\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$$\textstyle{L}$
is commutative, where $pr_{1}$ is the projection on the first coordinate. It
follows that $L$ is a leaf of $\mathcal{W}$ if and only if $g$ is constant
along of each connected component of $\pi_{X}^{-1}(L)$ contained in
$\varphi^{-1}(L\times\\{z\\})$.
Consider now the hypersurface
$G=\varphi(X)\subset(\mathbb{C}^{n},0)\times\mathbb{C}$ which is the closure
of set
$\\{(x,s)\in
U\times\mathbb{C}:g_{0}(x)+s.g_{1}(x)+\ldots+s^{k-1}.g_{k-1}(x)+s^{k}=0\\}.$
Let $\psi:(\mathbb{C}^{n},0)\times\mathbb{C}\rightarrow(\mathbb{C}^{n},0)$ be
the usual projection and denote by $Z\subset(\mathbb{C}^{n},0)$ the analytic
subset such that the restriction to $G$ of $\psi$ not is a finite branched
covering. Notice that for all $x_{0}\in(\mathbb{C}^{n},0)\backslash Z$, the
equation
$g_{0}(x)+s.g_{1}(x)+\ldots+s^{k-1}.g_{k-1}(x)+s^{k}=0$
defines $k$ analytic hypersurfaces pairwise transverse in $x_{0}$ and
therefore correspond to leaves of $\mathcal{W}$. ∎
## 3 Examples
This section is devoted to give some examples of Levi-flat hypersurfaces
tangent to holomorphic foliations or webs.
###### Example 3.1.
Take a non constant holomorphic function
$f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ and set
$M=(\mathcal{I}m(f)=0)$. Then $M$ is Levi-flat and $M_{sing}$ is the set of
critical points of $f$ lying on $M$. Leaves of the Levi foliation on $M_{reg}$
are given by $\\{f=c\\}$, $c\in\mathbb{R}$. Of course, $M$ is tangent to a
singular holomorphic foliation generated by the kernel of $df$.
###### Example 3.2.
([5]) Let $f_{0},f_{1},\ldots,f_{k}\in\mathcal{O}_{n}$, $n\geq{2}$, be
irreducible germs of holomorphic functions, where $k\geq{2}$. Consider the
family of hypersurfaces
$G:=\\{G_{s}:=f_{0}+sf_{1}+\ldots+s^{k}f_{k}/s\in\mathbb{R}\\}.$
By eliminating the real variable $s$ in the system $G_{s}=\bar{G}_{s}=0$, we
obtain a real analytic germ $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$
such that any complex hypersurface $(G_{s}=0)$ is contained in the real
hypersurface $(F=0)$. For instance, in the case $k=2$, we obtain
$F=det\left(\begin{array}[]{cccc}f_{0}&f_{1}&f_{2}&0\\\ 0&f_{0}&f_{1}&f_{2}\\\
\bar{f}_{0}&\bar{f}_{1}&\bar{f}_{2}&0\\\
0&\bar{f}_{0}&\bar{f}_{1}&\bar{f}_{2}\end{array}\right)=$
$=f^{2}_{0}.\bar{f}^{2}_{2}+\bar{f}^{2}_{0}.f^{2}_{2}+f_{0}.f_{2}.\bar{f}^{2}_{1}+\bar{f}_{0}.\bar{f}_{2}.f_{1}^{2}-|f_{1}|^{2}(f_{0}.\bar{f}_{2}+\bar{f}_{0}.f_{2})-2|f_{0}|^{2}.|f_{2}|^{2}.$
(3.1)
which comes from the elimination of $s$ in the system
$f_{0}+s.f_{1}+s^{2}.f_{2}=\bar{f}_{0}+s.\bar{f}_{1}+s^{2}.\bar{f}_{2}=0.$
We would like to observe that the examples of this type are tangent to
singular webs. The web is obtained by the elimination of $s$ in the system
given by
$\left\\{\begin{array}[]{ll}f_{0}+s.f_{1}+s^{2}.f_{2}+\ldots+s^{k}.f_{k}=0&\\\
df_{0}+s.df_{1}+s^{2}.df_{2}\ldots+s^{k}.df_{k}=0&\end{array}\right.$
In the case we get a $2$-web given by the implicit differential equation
$\Omega=0$, where
$\Omega=det\left(\begin{array}[]{cccc}f_{0}&f_{1}&f_{2}&0\\\
0&f_{0}&f_{1}&f_{2}\\\ df_{0}&df_{1}&df_{2}&0\\\
0&df_{0}&df_{1}&df_{2}\end{array}\right)$
This example shows that, although $\mathcal{L}_{M}$ is a foliation on
$M_{reg}\subset M=(F=0)$, in general it is not tangent to a germ of
holomorphic foliation at $(\mathbb{C}^{n},0)$.
###### Example 3.3.
[Clairaut’s equations] Clairaut’s equations are tangent to Levi-flat
hypersurfaces. Consider the first-order implicit differential equation
$y=xp+f(p),$ (3.2)
where $(x,y)\in\mathbb{C}^{2}$, $p=\frac{dy}{dx}$ and $f\in\mathbb{C}[p]$ is a
polynomial of degree $k$, the equation (3.2) define a $k$-web $\mathcal{W}$ on
$(\mathbb{C}^{2},0)$. The variety $S$ associated to $\mathcal{W}$ is given by
$(y-xp-f(p)=0)$ and the foliation $\mathcal{F}_{S}$ is defined by
$\alpha|_{S}=0$, where $\alpha=dy-pdx$. In the chart $(x,p)$ of $S$, we get
$\alpha|_{S}=(x+f^{\prime}(p))dp$. The criminant set of $\mathcal{W}$ is given
by
$R=(y-xp-f(p)=x+f^{\prime}(p)=0).$
Observe that $\mathcal{F}_{S}$ is tangent to $S$ along $R$ and has a non-
constant first integral $g(x,p)=p$. Denote by
$\pi_{S}:S\rightarrow(\mathbb{C}^{2},0)$ the restriction to $S$ of the usual
projection $\pi:\mathbb{P}\rightarrow(\mathbb{C}^{2},0)$, then the leaves of
$\mathcal{F}_{S}$ project by $\pi_{S}$ in leaves of $\mathcal{W}$. Those
leaves are as follows
$\displaystyle-y+s.x+f(s)=0,$ (3.3)
where $s$ is a constant. By the elimination of the variable $s\in\mathbb{R}$
in the system
$\left\\{\begin{array}[]{ll}-y+s.x+f(s)=0&\\\
-\bar{y}+s.\bar{x}+\overline{f(s)}=0,&\end{array}\right.$
we obtain a Levi-flat hypersurface tangent to $\mathcal{W}$. In particular,
Clairaut’s equation has a holomorphic first integral.
## 4 Lifting of Levi-flat hypersurfaces to the cotangent bundle
In this section we give some remarks about the lifting of a Levi-flat
hypersurface to the cotangent bundle of $(\mathbb{C}^{n},0)$.
Let $\mathbb{P}$ be as before, the projectivized cotangent bundle of
$(\mathbb{C}^{n},0)$ and $M$ an irreducible real analytic Levi-flat at
$(\mathbb{C}^{n},0)$, $n\geq 2$. Note that $\mathbb{P}$ is a
$\mathbb{P}^{n-1}$-bundle over $(\mathbb{C}^{n},0)$, whose fiber
$\mathbb{P}T_{z}^{*}\mathbb{C}^{n}$ over $z\in\mathbb{C}^{n}$ will be thought
of as the set of complex hyperplanes in $T_{z}^{*}\mathbb{C}^{n}$. Let
$\pi:\mathbb{P}\rightarrow(\mathbb{C}^{n},0)$ be the usual projection.
The regular part $M_{reg}$ of $M$ can be lifted to $\mathbb{P}$: just take,
for every $z\in M_{reg}$, the complex hyperplane
$T_{z}^{\mathbb{C}}M_{reg}=T_{z}M_{reg}\cap i(T_{z}M_{reg})\subset
T_{z}\mathbb{C}^{n}.$ (4.1)
We call
$M^{\prime}_{reg}\subset\mathbb{P}$ (4.2)
this lifting of $M_{reg}$. We remark that it is no more a hypersurface: its
(real) dimension $2n-1$ is half of the real dimension of
$\mathbb{P}T^{*}\mathbb{C}^{n}$. However, it is still “Levi-flat”, in a sense
which will be precised below.
Take now a point $y$ in the closure $\overline{M^{\prime}_{reg}}$ projecting
on $\mathbb{C}^{n}$ to a point $x\in\overline{M}$. Now, we shall consider the
following results, which are adapted from [2].
###### Lemma 4.1.
There exist, in a germ of neighborhood
$U_{y}\subset\mathbb{P}T^{*}(\mathbb{C}^{n},0)$ of $y$, a germ of real
analytic subset $N_{y}$ of dimension $2n-1$ containing $M^{\prime}_{reg}\cap
U_{y}$.
###### Proposition 4.2.
Under the above conditions, in a germ of neighborhood $V_{y}\subset U_{y}$ of
$y$, there exists a germ of complex analytic subset $Y_{y}$ of (complex)
dimension $n$ containing $N_{y}\cap V_{y}$.
## 5 Proof of Theorem 1
The proof will be divided in two parts. First, we give the proof for $n=2$.
The proof in dimension $n\geq 3$ will be done by reduction to the case of
dimension two.
First of all, we recall some results (cf. [5]) about foliations and Levi-
flats. Let $M$ and $\mathcal{F}$ be germs at $(\mathbb{C}^{2},0)$ of a real
analytic Levi-flat hypersurface and of a holomorphic foliation, respectively,
where $\mathcal{F}$ is tangent to $M$. Assume that:
1. (i)
$\mathcal{F}$ is defined by a germ at $0\in\mathbb{C}^{2}$ of holomorphic
vector field $X$ with an isolated singularity at $0$.
2. (ii)
$M$ is irreducible.
Let us assume that $0$ is a reduced singularity of $X$, in the sense of
Seidenberg [13]. Denote the eigenvalues of $DX(0)$ by
$\lambda_{1},\lambda_{2}$.
###### Proposition 5.1.
Suppose that $X$ has a reduced singularity at $0\in\mathbb{C}^{2}$ and is
tangent to a real analytic Levi-flat hypersurface $M$. Then
$\lambda_{1},\lambda_{2}\neq 0$, $\lambda_{2}/\lambda_{1}\in\mathbb{Q}_{-}$
and $X$ has a holomorphic first integral.
In particular, in a suitable coordinates system $(x,y)$ around
$0\in\mathbb{C}^{2}$, $X=\phi.Y$, where $\phi(0)\neq 0$ and
$\displaystyle Y=q.x\partial_{x}-p.y\partial_{y}\,\,,g.c.d(p,q)=1.$ (5.1)
In this coordinate system, $f(x,y):=x^{p}.y^{q}$ is a first integral of $X$.
We call this type of singularity of $\mathcal{F}$ a saddle with first
integral, (cf. [10], pg. 162). Now we have the following lemma.
###### Lemma 5.2.
For any $z_{0}\in M_{reg}$, the leaf $L_{z_{0}}$ of $\mathcal{L}_{M}$ through
$z_{0}$ is closed in $M_{reg}$.
### 5.1 Planar webs
A $k$-web $\mathcal{W}$ on $(\mathbb{C}^{2},0)$ can be written in coordinates
$(x,y)\in\mathbb{C}^{2}$ by
$\omega=a_{0}(x,y)(dy)^{k}+a_{1}(x,y)(dy)^{k-1}(dx)+\ldots+a_{k}(x,y)(dx)^{k}=0,$
where the coefficients $a_{j}\in\mathcal{O}_{2}$, $j=1,\ldots,k$. We set
$U=\\{(x,y,[adx+bdy])\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):a\neq 0\\}$
and
$V=\\{(x,y,[adx+bdy])\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):b\neq 0\\}.$
Note that $\mathbb{P}T^{*}(\mathbb{C}^{2},0)=U\cup V$. Suppose that
$(S,\pi_{S},\mathcal{F}_{S})$ define $\mathcal{W}$, in the coordinates
$(x,y,p)\in U$, where $p=\frac{dy}{dx}$, we have
1. 1.
$S\cap U=\\{(x,y,p)\in\mathbb{P}T^{*}(\mathbb{C}^{2},0):F(x,y,p)=0\\},$ where
$F(x,y,p)=a_{0}(x,y)p^{k}+a_{1}(x,y)p^{k-1}+\ldots+a_{k}(x,y).$
Note that $S$ is possibly singular at $0$.
2. 2.
$\mathcal{F}_{S}$ is defined by $\alpha|_{S}=0$, where $\alpha=dy-pdx$.
3. 3.
The criminant set $R$ is defined by the equations
$F(x,y,p)=F_{p}(x,y,p)=0.$
In $V$ the coordinate system is $(x,y,q)\in\mathbb{C}^{3}$, where
$q=\frac{1}{p}$, the equations are similar.
### 5.2 Proof in dimension two
Let $\mathcal{W}$ be a $k$-web tangent to $M$ Levi-flat and let us consider
$S$, $\pi$ be as before. The idea is to use Lemma 2.4, assume that
$\mathcal{W}$ is defined by
$\omega=a_{0}(x,y)(dy)^{k}+a_{1}(x,y)(dy)^{k-1}dx+\ldots+a_{k}(x,y)(dx)^{k}=0,$
(5.2)
where the coefficients $a_{j}\in\mathcal{O}_{2}$, $j=1,\ldots,k$ and
$a_{0}(0,0)=1$.
###### Lemma 5.3.
Under the hypotheses of Theorem 1 and above conditions, the surface $S$ is
irreducible and $S\cap\pi^{-1}(0)$ contains just a number finite of points.
See figure 1.
###### Proof.
Since $\mathcal{W}$ is irreducible so is $S$. On the other hand,
$S\cap\pi^{-1}(0)$ is finite because $\mathcal{W}$ has a finite number of
invariant analytic leaves through the origin and is defined as in 5.2. ∎
Figure 1: $S\cap\pi^{-1}(0)$
We can assume without lost of generality that $S\cap\pi^{-1}(0)$ contains just
one point, in the case general, the idea of the proof is the same. Then in the
coordinate system $(x,y,p)\in\mathbb{C}^{3}$, where $p=\frac{dy}{dx}$, we have
$\pi^{-1}(0)\cap S=\\{p_{0}=(0,0,0)\\}$, which implies that $S$ must be
singular at $p_{0}\in\mathbb{P}T^{*}(\mathbb{C}^{2},0)$. In particular,
$(S,p_{0})$ the germ of $S$ at $p_{0}$ is defined by $F^{-1}(0)$, where
$F(x,y,p)=p^{k}+a_{1}(x,y)p^{k-1}+\ldots+a_{k}(x,y),$
and $a_{1},\ldots,a_{k}\in\mathcal{O}_{2}$. Let $\mathcal{F}_{S}$ be the
foliation defined by $\alpha|_{S}=0$. The assumptions implies that
$\mathcal{F}_{S}$ is a non-dicritical foliation with an isolated singularity
at $p_{0}$.
Recall that a germ of foliation $\mathcal{F}$ at $p_{0}\in S$ is dicritical if
it has infinitely many analytic separatrices through $p_{0}$. Otherwise it is
called non-dicritical.
Let $M^{\prime}_{reg}$ be the lifting of $M_{reg}$ by $\pi_{S}$, and denote by
$\sigma:(\tilde{S},D)\rightarrow(S,p_{0})$ the resolution of singularities of
$S$ at $p_{0}$. Let $\tilde{\mathcal{F}}=\sigma^{*}(\mathcal{F}_{S})$ be the
pull-back of $\mathcal{F}_{S}$ under $\sigma$. See figure 2.
###### Lemma 5.4.
In the above situation. The foliation $\tilde{\mathcal{F}}$ has only
singularities of saddle with first integral type in $D$.
###### Proof.
Let $y\in\overline{M^{\prime}_{reg}}$, it follows from Lemma 4.1 the
existence, in a neighborhood $U_{y}\subset\mathbb{P}T^{*}(\mathbb{C}^{2},0)$
containing $y$, of a real analytic subset $N_{y}$ of dimension $3$ containing
$M^{\prime}_{reg}\cap U_{y}$. Then by Proposition 4.2, there exists, in a
neighborhood $V_{y}\subset U_{y}$ of $y$, a complex analytic subset $Y_{y}$ of
(complex) dimension $2$ containing $N_{y}\cap V_{y}$. As germs at $y$, we get
$Y_{y}=S_{y}$ then $N_{y}\cap V_{y}\subset S_{y}$, we have that $N_{y}\cap
V_{y}$ is a real analytic hypersurface in $S_{y}$, and it is Levi-flat because
each irreducible component contains a Levi-flat piece (cf. [1], Lemma 2.2).
Let us denote $M^{\prime}_{y}=N_{y}\cap V_{y}$. The hypotheses implies that
$\mathcal{F}_{S}$ is tangent to $M^{\prime}_{y}$. These local constructions
are sufficiently canonical to be patched together, when $y$ varies on
$\overline{M^{\prime}_{reg}}$: if $S_{y_{1}}\subset V_{y_{1}}$ and
$S_{y_{2}}\subset V_{y_{2}}$ are as above, with $M^{\prime}_{reg}\cap
V_{y_{1}}\cap V_{y_{2}}\neq\emptyset$, then $S_{y_{2}}\cap(V_{y_{1}}\cap
V_{y_{2}})$ and $S_{y_{1}}\cap(V_{y_{1}}\cap V_{y_{2}})$ have some common
irreducible components containing $M^{\prime}_{reg}\cap V_{y_{1}}\cap
V_{y_{2}}$, so that $M^{\prime}_{y_{1}}$, $M^{\prime}_{y_{2}}$ can be glued by
identifying those components. In this way, we obtain a Levi-flat hypersurface
$N$ on $S$ tangent to $\mathcal{F}_{S}$.
By doing additional blowing-ups if necessary, we can suppose that
$\tilde{\mathcal{F}}$ has reduced singularities. Since $\mathcal{F}_{S}$ is
non-dicritical, all irreducible components of $D$ are
$\tilde{\mathcal{F}}$-invariants. Let $\tilde{N}$ be the strict transform of
$N$ under $\sigma$, then $\tilde{N}\supset D$. In particular, $\tilde{N}$
contains all singularities of $\tilde{\mathcal{F}}$ in $D$. It follows from
Proposition 5.1 that all singularities of $\tilde{\mathcal{F}}$ are saddle
with first integral. ∎
Figure 2: Resolution of singularities of $S$ at $p_{0}$.
### 5.3 End of the proof of Theorem 1 in dimension two
The idea is to prove that $\mathcal{F}_{S}$ has a holomorphic first integral.
Since $D$ is invariant by $\tilde{\mathcal{F}}$, i.e., it is the union of
leaves and singularities of $\tilde{\mathcal{F}}$, we have
$S:=D\backslash\textsf{Sing}(\tilde{\mathcal{F}})$ is a leaf of
$\tilde{\mathcal{F}}$. Now, fix $p\in S$ and a transverse section $\sum$
through $p$. By Lemma 5.4, the singularities of $\tilde{\mathcal{F}}$ in $D$
are saddle with first integral types. Therefore the transverse section $\sum$
is complete, (see [10], pg. 162). Let $G\subset\textsf{Diff}(\sum,p)$ be the
Holonomy group of the leaf $S$ of $\tilde{\mathcal{F}}$. It follows from Lemma
5.2 that all leaves of $\mathcal{F}_{S}$ through points of $N_{reg}$ are
closed in $N_{reg}$. This implies that all transformations of $G$ have finite
order and $G$ is linearizable. According to [11], $\mathcal{F}_{S}$ has a non-
constant holomorphic first integral. Finally from Lemma 2.4, $\mathcal{W}$ has
a first integral as follows:
$P(z)=f_{0}(x,y)+z.f_{1}(x,y)+\ldots+z^{k-1}.f_{k-1}(x,y)+z^{k},$
where $f_{0},f_{1},\ldots,f_{k-1}\in\mathcal{O}_{2}$.
### 5.4 Proof in dimension $n\geq 3$
Let us give an idea of the proof. First of all, we will prove that there is a
holomorphic embedding $i:(\mathbb{C}^{2},0)\rightarrow(\mathbb{C}^{n},0)$ with
the following properties:
1. (i)
$i^{-1}(M)$ has real codimension one on $(\mathbb{C}^{2},0)$.
2. (ii)
$i^{*}(\mathcal{W})$ is a $k$-web on $(\mathbb{C}^{2},0)$ tangent to
$i^{-1}(M)$.
Set $E:=i(\mathbb{C}^{2},0)$. The above conditions and Theorem 1 in dimension
two imply that $\mathcal{W}|_{E}$ has a non-constant holomorphic first
integral, say $g=f_{0}+z.f_{1}+\ldots+z^{k-1}.f_{k-1}+z^{k}$, where
$f_{0},\ldots,f_{k-1}\in\mathcal{O}_{2}$. After that we will use a lemma to
prove that $g$ can be extended to a holomorphic germ $g_{1}$, which is a first
integral of $\mathcal{W}$.
Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$, of a
holomorphic codimension one foliation, tangent to a real analytic hypersurface
$M$. Let us suppose that $\mathcal{F}$ is defined by $\omega=0$, where
$\omega$ is a germ at $0\in\mathbb{C}^{n}$ of an integrable holomorphic 1-form
with $cod_{\mathbb{C}^{n}}(\textsf{Sing}(\omega))\geq 2$. We say that a
holomorphic embedding $i:(\mathbb{C}^{2},0)\rightarrow(\mathbb{C}^{n},0)$ is
transverse to $\omega$ if $cod_{\mathbb{C}^{n}}(\textsf{Sing}(\omega))=2$,
which means in fact that, as a germ of set, we have
$\textsf{Sing}(i^{*}(\omega))=\\{0\\}$. Note that the definition is
independent of the particular germ of holomorphic 1-form which represents
$\mathcal{F}$. Therefore, we will say that the embedding $i$ is transverse to
$\mathcal{F}$ if it is transverse to some holomorphic 1-form $\omega$
representing $\mathcal{F}$.
We will use the following lemma of [5].
###### Lemma 5.5.
In the above situation, there exists a 2-plane $E\subset\mathbb{C}^{n}$,
transverse to $\mathcal{F}$, such that the germ at $0\in E$ of $M\cap E$ has
real codimension one.
We say that a embedding $i$ is transverse to $\mathcal{W}$ if it is transverse
to all $k$-foliations which defines $\mathcal{W}$. Now, one deduces the
following
###### Lemma 5.6.
There exists a 2-plane $E\subset\mathbb{C}^{n}$, transverse to $\mathcal{W}$,
such that the germ at $0\in E$ of $M\cap E$ has real codimension one.
###### Proof.
First of all, note that outside of the discriminant set of $\mathcal{W}$, we
can suppose that
$\mathcal{W}=\mathcal{F}_{1}\boxtimes\ldots\boxtimes\mathcal{F}_{k}$, where
$\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$ are germs of codimension one smooth
foliations. Since $\mathcal{W}$ is tangent to $M$, there is a foliation
$\mathcal{F}_{j}$ such that is tangent to a Levi foliation $\mathcal{L}_{M}$
on $M_{reg}$. Lemma 5.5 implies that we can find a 2-plane $E_{0}$ transverse
to $M$ and to $\mathcal{F}_{j}$. Clearly the set of linear mappings transverse
to $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$ simultaneously is open and dense
in the set of linear mappings from $\mathbb{C}^{2}$ to $\mathbb{C}^{n}$, by
Transversality theory, there exists a linear embedding $i$ such that
$E=i(\mathbb{C}^{2},0)$ is transverse to $M_{reg}$ and to $\mathcal{W}$
simultaneously. ∎
Let $E$ be a 2-plane as in Lemma 5.6. It easy to check that $\mathcal{W}|_{E}$
satisfies the hypotheses of Theorem 1. By the two dimensional case
$\mathcal{W}|_{E}$ has a non-constant first integral:
$g_{0}+z.g_{1}+\ldots+z^{k-1}.g_{k-1}+z^{k},$ (5.3)
where $g_{0},\ldots,g_{k-1}\in\mathcal{O}_{2}$.
Let $X$ be the variety associated to $\mathcal{W}$ and set $S$ be the surface
associated to $\mathcal{W}|_{E}$. Observe that $\mathcal{F}_{S}$ has a non-
constant holomorphic first integral $g$ defined on $S$.
###### Lemma 5.7.
In the above situation, we have $\mathcal{F}_{X}|_{S}=\mathcal{F}_{S}$ and
$\mathcal{F}_{X}$ has a non-constant holomorphic first integral $g_{1}$ on X,
such that $g_{1}|_{S}=g$.
###### Proof.
It is easily seen that $S\subset X$ which implies that
$\mathcal{F}_{X}|_{S}=\mathcal{F}_{S}$. Let us extend $g$ to $X$. Fix $p\in
X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})$. It is possible to find a
small neighborhood $W_{p}\subset X$ of $p$ and a holomorphic coordinate chart
$\varphi:W_{p}\rightarrow\triangle$, where $\triangle\subset\mathbb{C}^{n}$ is
a polydisc, such that:
1. (i)
$\varphi(S\cap W_{p})=\\{z_{3}=\ldots=z_{n}=0\\}\cap\triangle$.
2. (ii)
$\varphi_{*}(\mathcal{F}_{X})$ is given by $dz_{n}|_{\triangle}=0$.
Let $\pi_{n}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{2}$ be the projection
defined by $\pi_{n}(z_{1},\ldots,z_{n})=(z_{1},z_{2})$ and set
$\tilde{g}_{p}:=g\circ\varphi^{-1}\circ\pi_{n}|_{\triangle}$. We obtain that
$\tilde{g}$ is a holomorphic function defined in $\triangle$ and is a first
integral of $\varphi_{*}(\mathcal{F}_{X})$. Let
$g_{p}=\tilde{g}_{p}\circ\varphi$. Notice that, if $W_{p}\cap
W_{q}\neq\emptyset$, $p$ and $q$ being regular points for $\mathcal{F}_{X}$,
then we have $g_{p}|_{W_{p}\cap W_{q}}=g_{q}|_{W_{p}\cap W_{q}}$. This follows
easily form the identity principle for holomorphic functions. In particular,
$g$ can be extended to
$W=\bigcup_{p\in X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})}W_{p},$
which is a neighborhood of $X_{reg}\backslash\textsf{Sing}(\mathcal{F}_{X})$.
Call $g_{W}$ this extension.
Since $cod_{X_{reg}}\textsf{Sing}(\mathcal{F}_{X})\geq 2$, by a theorem of
Levi (cf. [15]), $g_{W}$ can be extended to $X_{reg}$, as
$cod_{X_{reg}}(\textsf{Sing}(X))\geq 2$ this allows us to extend $g_{W}$ to
$g_{1}$ as holomorphic first integral for $\mathcal{F}_{X}$, in whole $X$. ∎
### 5.5 End of the proof of Theorem 1 in dimension $n\geq 3$
Since $\mathcal{F}_{X}$ has a non-constant holomorphic first integral on $X$,
Lemma 2.4 imply that $\mathcal{W}$ has a non-constant holomorphic first
integral.
Acknowledgments.– The author is greatly indebted to Alcides Lins Neto for
suggesting the problem and for many stimulating conversations. This is part of
the author’s Ph.D. thesis, written at IMPA. I also want to thank the referee
for his suggestions.
## References
* [1] D. Burns, X. Gong: Singular Levi-flat real analytic hypersurfaces, Amer. J. Math. 121, $(1999)$, no. 1, pp. 23-53.
* [2] M. Brunella: Singular Levi-flat hypersurfaces and codimension one foliations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, pp. 661-672.
* [3] V. Cavalier, D. Lehmann: Introduction à l’étude globale des tissus sur une surface holomorphe. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, pp. 1095-1133.
* [4] V. Cavalier, D. Lehmann: Global structure of holomorphic webs on surfaces. Geometry and topology of caustics–CAUSTICS ’06, pp. 35-44, Banach Center Publ., 82, Polish Acad. Sci. Inst. Math., Warsaw, 2008.
* [5] D. Cerveau, A. Lins Neto: Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation. Amer. J. Math. 133, $(2011)$, no. 3, pp. 677-716.
* [6] A. Fernández-Pérez: On normal forms of singular Levi-flat real analytic hypersurfaces. Bull. Braz. Math. Soc. (N.S.) 42, (2011), no. 1, pp. 75-85.
* [7] ——–: Singular Levi-flat hypersurfaces. An approach through holomorphic foliations. Ph.D Thesis IMPA, Brazil 2010.
* [8] R. Gunning: Introduction to holomorphic functions of several variables. Vol. II. Local theory. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990.
* [9] J. Lebl: Singularities and complexity in CR geometry. Ph.D. Thesis, University of California at San Diego, Spring 2007.
* [10] F. Loray: Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. ; http://hal.archives-ouvertures.fr/ccsd-00016434
* [11] J.F. Mattei, R. Moussu: Holonomie et intégrales premières, Ann. Ec. Norm. Sup. 13, $(1980)$, pp. 469–523.
* [12] J.V. Pereira, L. Pirio: “An invitation to web geometry”. From Abel’s addition theorem to the algebraization of codimension one webs. Publicações Matemáticas do IMPA, $27^{\circ}$ Cóloquio Brasileiro de Matemática, Instituto Nacional de Matemática Pura e Aplicada IMPA. Rio de Janeiro, 2009.
* [13] A. Seidenberg: Reduction of singularities of the differential equation $A\,dy=B\,dx$. Amer. J. Math. 90, $(1968)$, pp. 248–269.
* [14] J.N.A Yartey: Number of singularities of a generic web on the complex projective plane. J. Dyn. Control Syst. 11 $(2005)$, no. 2, pp. 281-296.
* [15] Y.T. Siu: Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics, Vol. 8. Marcel Dekker, Inc., New York, 1974.
Arturo Fernández-Pérez
Departamento de Matemática, UFMG
Av. Antônio Carlos, 6627 C.P. 702,
30123-970 - Belo Horizonte - MG, BRAZIL.
E-mail: afernan@impa.br
|
arxiv-papers
| 2011-09-08T01:03:48 |
2024-09-04T02:49:21.982760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arturo Fern\\'andez-P\\'erez",
"submitter": "Arturo Fernandez",
"url": "https://arxiv.org/abs/1109.1608"
}
|
1109.1636
|
# Non-monotonic quantum to classical transition in multiparticle interference
Young-Sik Ra Department of Physics, Pohang University of Science and
Technology (POSTECH), Pohang, 790-784, Korea Malte C. Tichy Physikalisches
Institut der Albert-Ludwigs-Universität, Hermann-Herder-Str. 3, D-79104
Freiburg Hyang-Tag Lim Department of Physics, Pohang University of Science
and Technology (POSTECH), Pohang, 790-784, Korea Osung Kwon Department of
Physics, Pohang University of Science and Technology (POSTECH), Pohang,
790-784, Korea Florian Mintert Physikalisches Institut der Albert-Ludwigs-
Universität, Hermann-Herder-Str. 3, D-79104 Freiburg Freiburg Institute for
Advanced Studies, Albert-Ludwigs-Universität, Albertstrasse 19, D-79104
Freiburg Andreas Buchleitner Physikalisches Institut der Albert-Ludwigs-
Universität, Hermann-Herder-Str. 3, D-79104 Freiburg Yoon-Ho Kim Department
of Physics, Pohang University of Science and Technology (POSTECH), Pohang,
790-784, Korea
###### Abstract
We experimentally demonstrate the non-monotonic dependence of genuine many-
particle interference signals on the particles’ mutual distinguishability. Our
theoretical analysis shows that such non-monotonicity is a generic feature of
the quantum to classical transition in multiparticle correlation functions of
more than two particles.
###### pacs:
42.50.-p 42.50.Dv 42.50.Ex 42.65.Lm 05.30.Jp,
A quintessential ingredient of quantum physics is the superposition principle.
It becomes manifest in the (self-) interference of single particles, as
observed for systems ranging from photons ref:grangier to fullerene molecules
arndt05 . These phenomena rely on the _coherence_ of the single-particle wave-
function, which guaranties that the different pathways a single particle can
take to a detector – e.g. through the left or through the right slit in a
double-slit experiment – remain indistinguishable.
Interaction with the environment, however, may convey which-path information
to the environment, and then inevitably leads to decoherence cohevol02 .
Thereby, it jeopardizes the ideal interference pattern and induces the
quantum-to-classical transition arndt91 : _probabilities_ instead of
amplitudes need to be summed to obtain event probabilities. The stronger the
decoherence, the weaker is the interference signal englert96 . Expectation
values of observables therefore depend _monotonically_ on the strength of
decoherence, and a _monotonic_ transition between quantum and classical
expectation values takes place.
The superposition principle also applies to _many_ -particle wave-functions,
with entanglement ref:schrodinger and many-particle interference
Peruzzo:2011dq as immediate consequences. For the observation of the latter,
the particles’ mutual indistinguishability is necessary. Given that, one
observes, e.g., the Hong-Ou-Mandel (HOM) effect ref:hong ; ref:pittman : When
a single photon is incident on each of the two input modes of a balanced beam
splitter, the two-particle Feynman-paths of “both photons reflected” and “both
photons transmitted” interfere destructively, leading to the strict
suppression of the event with one particle per output mode.
Much as in the case of single-particle interference discussed above, the
_transition_ between _distinguishable_ and _indistinguishable_ particles is
monotonic also in the HOM setting: the interference-induced suppression of the
balanced output event fades away monotonically with increasing particle
distinguishability ref:hong ; al:2009vn , since particle-distinguishability is
tantamount of the distinguishability of many-particle paths. Therefore,
conveying information on the interfering particles’ identity provides which-
path information in the space of many-particle paths. Monotonic
distinguishability-dependence of bunching events (where all particles are
detected in one output mode) was also observed for four ref:ou1 and six
Niu:2009pr photons, while events other than bunching were not considered yet
for more than two photons.
In our present contribution, we focus on many-particle interference effects in
multi-port output events distinct from bunching. In contrast to the hitherto
established scenario sketched above, we will see that the distinguishibility-
induced suppression of multi-particle interference signals is, in general, a
non-monotonic function of the particles’ distinguishability, provided that
more than two particles are brought to interference ref:malte . This is
explained by a hierarchy of different orders of many-particle interferences
which dominate the total interference signal at different stages of the
distinguishability transition. Only when reduced indistinguishability defines
two unambiguous alternatives does one enter the monotonic realm of the quantum
to classical transition.
Figure 1: (a) HOM type interferometer. The path delay $x$ controls the
distinguishability between the photons from mode $a$ and $b$ which then
scatter on the 50:50 beam splitter (BS). (b) Joint detection of (m,n) photons
at the output modes $c$ and $d$, which is realized by concatenating fiber beam
splitters (FBSs, 50:50) and single-photon counting modules (SPCMs).
Let us first describe our experimental setup shown in Fig. 1(a). Single-photon
or two-photon Fock-states are injected into each input mode ($a$,$b$), and the
particle distributions at the output modes $c$ and $d$ of the beam splitter
(BS, 50:50) are measured while changing the path delay $x$ of the mode $b$,
thereby varying the distinguishability of the photons in the input modes. The
single- and two-photon Fock states are generated via the non-collinear
frequency-degenerate spontaneous parametric down-conversion (SPDC) process at
a 2 mm-thick $\beta$-barium-borate (BBO) crystal pumped by a femtosecond laser
pulse (central wavelength: 390 nm, average power: 120 mW) which is focused
onto the BBO by a lens of 300 mm focal length. The generated photons are
centered at 780 nm and filtered by interference filters of 4 nm full width at
half maximum (FWHM). The photons are coupled into two single-mode fibers
located at a distance of 430 mm from the BBO, where each fiber corresponds to
an input mode ($a$, $b$). Two-fold counts by simultaneous single-photon
detection between the two fibers were recorded at a rate of 13 kHz, by
combination of single-photon counting modules (SPCMs, Perkin-Elmer SPCM-AQ4C)
and a coincidence circuit with 8 ns coincidence window. At the fibers’ output,
half-wave plates and quarter-wave plates adjust the photons’ polarizations,
and a motorized stage connected to the fiber tip at the mode $b$ changes $x$.
The quantum state of the input modes is proportional to
$\sum_{j=0}^{\infty}\eta^{j}|j,j\rangle_{a,b}$, where $|\eta|^{2}$ is the
probability for photon pair generation, and coincidence detection of two or
four photons in the output modes projects the generated state onto the initial
state $|1,1\rangle_{ab}$ or $|2,2\rangle_{ab}$, respectively. The photon
distribution at the output modes is measured by combining 50:50 fiber beam
splitters (FBSs, 2$\times$2 single mode coupler) and SPCMs, as depicted in
Fig. 1(b), such that $m$ and $n$ particles in the first and second output
mode, respectively, are detected simultaneously, to define a $(m,n)$-event.
The distinguishability of the photons in modes $a$ and $b$ depends on $x$.
When $x$ = 0 $\mu$m, the photons are perfectly indistinguishable, and exhibit
ideal interference. When $x$ is much larger than the single-photon coherence
length $l_{c}=$134 $\mu$m, the particles can be treated as distinguishable, so
many-particle interference does not occur.
In a first step, we study the effect of the distinguishability transition of
two photons. While changing $x$, we measured the probabilities of the two-
photon events (1,1) and (2,0). The resulting data in Fig. 2(a) show typical
two-photon HOM-type interferences ref:hong ; ref:yhkim . As expected ref:hong
, the event probabilities are complementary and depend monotonically on the
photons’ distinguishability.
Figure 2: Event probabilities (in absolute coincidence counts) vs. path delay
$x$. (a) HOM-type interference of two-photons, with a clear enhancement of the
(2,0) event (two photons in one output mode; open circles), and the
corresponding suppression of the (1,1) event (one photon in each output mode;
filled circles). Errors are below 36 counts per bin, and omitted for clarity.
Four photon interference with two photons per input mode is monitored in (b)
by the (4,0) (integration over $14100\ {\mathrm{s}}$), and in (c) by the (2,2)
(integration over $9600\ {\mathrm{s}}$) event probabilities, respectively. The
red lines are the event probabilities as predicted by (5) – within the
statistical error in perfect agreement with the experimental data.
Next, the four-photon events, (4,0) and (2,2), are recorded, and we monitor
the resulting four-photon distinguishability transition. A clear enhancement
of (4,0)-events for indistinguishable particles is observed in Fig. 2(b) and,
indeed, this enhancement fades away monotonically as the particles are tuned
to become distinguishable by increasing $x$. The event (2,2), however, in Fig.
2(c), exhibits minima symmetrically displaced from the origin $x=0$, and the
distinguishability transition, as induced by continuous variation of $x$ from
$0$ to $x\gg\ell_{c}$, is not any longer unambiguously reflected by a
monotonic $x$-dependence of the many-particle interference signal. This is
indicative of an intricate interplay of many-particle interference and
distinguishability transition, which we will elucidate in the following.
In our setup sketched in Fig. 1(a), the path delay $x$ controls the
(in-)distinguishability between photons injected into the modes $a$ and $b$,
respectively, by controlling their arrival times at the BS. A photon with
arrival time $t_{i}$ is described as
$\displaystyle\mathcal{A}_{t_{i}}^{\dagger}|0\rangle=|1_{t_{i}}\rangle=\int_{-\infty}^{\infty}\mbox{d}\omega\frac{1}{\sqrt{\pi}\Delta\omega}e^{-\frac{(\omega-\omega_{0})^{2}}{2\Delta\omega^{2}}}e^{i\omega
t_{i}}\mathcal{A}_{\omega}^{\dagger}|0\rangle,$ (1)
where $\omega_{0}$ is the central frequency, $\Delta\omega$ is the frequency
width, and $\mathcal{A}^{\dagger}_{\omega}$ creates a photon of frequency
$\omega$. The distinguishability of two photons with different arrival times
$t_{1}$ and $t_{2}$ is quantified by
$\left|{\alpha}\right|^{2}=\mathrm{exp}\left({-\Delta\omega^{2}{(t_{2}-t_{1})}^{2}/2}\right)\,,\,t_{2}-t_{1}=\frac{x}{c}\,,$
(2)
with $\alpha=\langle{1_{t_{1}}}|{1_{t_{2}}}\rangle$. The photons are strictly
indistinguishable for $\left|{\alpha}\right|^{2}=1$, and fully distinguishable
for $\left|{\alpha}\right|^{2}=0$. Consequently, the $t_{2}$ photon can be
decomposed – with respect to its support on the time axis – by orthogonal
projection onto one strictly indistinguishable component
$\alpha|1_{t_{1}}\rangle$, and onto one (orthogonal) fully distinguishable
component $\sqrt{1-\alpha^{2}}|\widetilde{1}_{t_{1}}\rangle$ 111Since the
phase of $\alpha$ is not observable in our setting, we assume
$\alpha\in\mathbbm{R}$ in the following, without loss of generality.. A two-
photon state with one photon in each input modes of arrival time $t_{1}$ and
$t_{2}$ thus reads
$\displaystyle a_{t_{1}}^{\dagger}b_{t_{2}}^{\dagger}|0\rangle$
$\displaystyle=$
$\displaystyle\alpha|1,1\rangle_{ab}+\sqrt{1-\alpha^{2}}|1,\widetilde{1}\rangle_{ab}$
(3)
after substitution of the specific input modes $a$, $b$ for $\mathcal{A}$.
Note that we have omitted the subscript $t_{1}$ for simplifying the notation.
Herein, the components $|1,1\rangle_{ab}$ and $|1,\widetilde{1}\rangle_{ab}$
are orthogonal, with strictly indistinguishable particles in
$|1,1\rangle_{ab}$, and fully distinguishable particles in
$|1,\widetilde{1}\rangle_{ab}$. The continuous parametrization of the
distinguishability transition by $x$ is now replaced by continuous
parametrization by $\left|{\alpha}\right|^{2}\in[0;1]$. Only the
indistinguishable component in (3) defines indistinguishable two-particle
paths which can interfere, and the resulting interference signal therefore has
to fade away monotonically with decreasing $\left|{\alpha}\right|^{2}$ – this
is just the HOM scenario, as reproduced by our experimental results in Fig.
2(a).
Something qualitatively new happens when more than one particle is injected in
each input mode: The orthogonal decomposition analogous to (3) gives rise to
cross terms between the distinguishable and indistinguishable components.
Specifically, for two photons injected into each mode, we obtain
$\displaystyle\frac{1}{2}\left(a_{t_{1}}^{\dagger}\right)^{2}\left(b_{t_{2}}^{\dagger}\right)^{2}|0\rangle$
$\displaystyle=$
$\displaystyle\alpha^{2}|2,2\rangle_{ab}+\sqrt{2}\alpha\sqrt{1-\alpha^{2}}|2,1\widetilde{1}\rangle_{ab}$
(4) $\displaystyle+(1-\alpha^{2})|2,\widetilde{2}\rangle_{ab}\,,$
with the cross term $\propto|2,1\widetilde{1}\rangle_{ab}$ mutually orthogonal
to $|2,2\rangle_{ab}$ and $|2,\widetilde{2}\rangle_{ab}$. In precise analogy
to the HOM scenario, the $|2,2\rangle_{ab}$ component defines
indistinguishable four-particle paths that give rise to an interference signal
with amplitude $\alpha^{2}$, fading away with increasing amplitude
$(1-\alpha^{2})$ of the (distinguishable) $|2,\widetilde{2}\rangle_{ab}$
component. In contrast, the novel term $\propto|2,1\widetilde{1}\rangle_{ab}$
represents relative timing such that one of the two particles in mode $b$ is
strictly indistinguishable from those in mode $a$, while the other one is
fully distinguishable. Therefore, this term gives rise to a three-particle
interference signal through coherent superposition of indistinguishable three-
particle paths, with amplitude $\sqrt{2}\alpha\sqrt{1-\alpha^{2}}$. The
probability to detect $m$ particles in output mode $c$ and $n$ particles in
output mode $d$, given $N=m+n$ particles on input, is then generally given by
$\displaystyle P^{(N;m,n)}(x)=\sum_{\mathrm{type}}p_{\mathrm{type}}^{(N;m,n)}\
W_{\mathrm{type}}^{(N)}(x)\ ,$ (5)
where we need to sum over the various distinguishibility types of the
contributions as they emerge in (4): strictly indistinguishable (indis),
partially distinguishable (inter), and fully distinguishable (dist). The
associated detection probabilities $p_{\mathrm{type}}^{(N;m,n)}$ are
determined by the geometry of the experimental set-up, computed by mapping
input on output modes via
$\displaystyle
a^{\dagger}\rightarrow\frac{1}{\sqrt{2}}\left(c^{\dagger}+d^{\dagger}\right),\
b^{\dagger}\rightarrow\frac{1}{\sqrt{2}}\left(c^{\dagger}-d^{\dagger}\right)\,,$
(6)
and listed in Table 1 for the specific events experimentally probed in Fig. 2.
Figure 3: Weights of the contributions of strictly indistinguishable
($W^{(N)}_{\mathrm{indis}}$), of partially distinguishable
($W^{(N)}_{\mathrm{inter}}$), and of fully distinguishable
($W^{(N)}_{\mathrm{dist}}$) particles to the many-particle wave-function, as a
function of the path delay $x$ which continuously parametrizes the particles’
distinguishability. (a) For $N=2$, partial distinguishability does not arise,
and $W^{(N)}_{\mathrm{indis}}$ and $W^{(N)}_{\mathrm{dist}}$ are monotonic
functions of $x$, leading to a monotonic distinguishability transition in
$P^{(2;m,n)}(x)$. (b) In contrast, for $N=4$, $W^{(N)}_{\mathrm{inter}}$
contributes non-monotonically to the event probability, and induces the non-
monotonic behavior observed in Fig. 2(c).
$p^{(2;m,n)}$ | indis | dist
---|---|---
$~{}(2,0)~{}$ | 1/2 | 1/4
$~{}(1,1)~{}$ | 0 | 1/2
$p^{(4;m,n)}$ | indis | inter | dist
---|---|---|---
$~{}(4,0)~{}$ | 3/8 | 3/16 | 1/16
$~{}(2,2)~{}$ | 1/4 | 1/8 | 3/8
Table 1: Detection probabilities $p^{(N;m,n)}$ of $m$ ($n$) particles in mode
$c$ ($d$) as derived from the mutually orthogonal amplitudes of the strictly
indistinguishable (indis), partially distinguishable (inter) and fully
distinguishable (dist) contributions in (3,4), for $N=2$ (left panel) and
$N=4$ (right panel).
The weights $W_{\mathrm{type}}^{(4)}(x)$ in (5) are simply given by squaring
the amplitudes of the respective components in the orthogonal decomposition
(4), where we baptize the cross term $\propto|2,1\widetilde{1}\rangle_{ab}$ by
the label “inter”. Their functional dependence on $x$ (derived from their
dependence on $\alpha$ via (2)) is shown in Fig. 3: while the weight of all
particles being either strictly indistinguishable,
$W_{\mathrm{indis}}^{(4)}(x)$, or fully distinguishable,
$W_{\mathrm{dist}}^{(4)}(x)$, always depends monotonically on $x$, the weight
$W_{\mathrm{inter}}^{(4)}(x)$ representing cross terms between distinguishable
and indistinguishable components exhibits a non-monotonic $x$-dependence!
Consequently, in the $N=4$ particle case studied in our experiment, three-
particle interference contributions kick in while four-particle interference
contributions fade away, thus giving rise to the non-monotonicity observed in
Fig. 2(c). The continuous lines in Fig. 2(b,c) are derived from the
$W_{\mathrm{type}}^{(4)}(x)$ in Fig. 3, together with the detection
probabilities from Table 1 inserted in (5), and fit the experimental data
perfectly well, without adjustable parameters. Note that bunching signals, as
the $(4,0)$ event observed here, always exhibit a monotonic transition, as a
consequence of the bosonic enhancement tichy10 of the associated, unique
many-particle path, which induces the strict hierarchy
$p^{(4;m,n)}_{\mathrm{indis}}>p^{(4;m,n)}_{\mathrm{inter}}>p^{(4;m,n)}_{\mathrm{dist}}$
in Table 1.
In general, for $N/2$ particles per input mode there are $N/2-1$ cross terms
of distinguishable and indistinguishable components (represented by the term
$\propto W_{\mathrm{inter}}^{(N)}(x)$ in (5)), which all depend non-
monotonically on $x$, and this non-monotonicity is generically inherited by
the derived event probabilities $P^{(N;m,n)}(x)$ (provided that it not be
counterbalanced by the actual values of the detection probabilities
$p_{\mathrm{indis/inter/dist}}^{(N;m,n)}$). Fig. 4 shows an example for $N=6$.
Figure 4: (a) Event probabilities $P^{(N;m,n)}(x)$, for $N=m+n=6$, and
$m=3,\ldots,6$, for each three photons injected into modes $a$ and $b$ (see
Fig. 1(a)). (b) shows the associated weights of strictly indistinguishable,
partially and fully distinguishable contributions
$W^{(N)}_{\mathrm{indis/inter/dist}}$, with a notation strictly analogous to
that introduced in (4). Much as in Fig. 3(b), the partially distinguishable
contributions’ weights are non-monotonic in $x$.
Let us finally discuss how this generic non-monotonicity of many-particle
interference signals in correlation functions such as $P^{(N;m,n)}(x)$ with
respect to $x$ matches our intuition of the monotonicity of the quantum to
classical transition under decoherence: To begin with, the latter is generally
observed in the interference of indistinguishable single-particle paths at one
detector, while genuine many-particle interference manifests in multiparticle
correlation functions not accessible in single-particle experiments. On that
level, the distinguishability transition – possibly caused by environment
interaction, i.e. coupling to additional degrees of freedom – reduces the
order of many-particle interferences, e.g. from six- to five- and four-
particle, or from four- to three-particle interference, in the examples
considered in Figs. 2-4. As a consequence of the non-monotonic dependence of
the different orders of the many-particle interference contributions on the
particles’ mutual distinguishability, the transition from fully multi-particle
multi-path to single-particle coherence is, in general, non-monotonic and,
yet, fully consistent with our understanding of single particle decoherence
and the quantum to classical transition. However, we speculate that different
types of system-environment interaction (e.g. two vs. three- or many-particle
interactions) may allow to distinguish different orders of many-particle
interference, and that, reciprocally, the dependence of the various
interference terms in (5) on the environment coupling parameters may serve as
an analytic tool to distinguish distinct environment coupling mechanisms.
Financial support by the National Research Foundation of Korea (2009-0070668
and 2009-0084473), through a DAAD/GenKo grant 50739824, and by the German
National Academic Foundation (M.C.T.) is gratefully acknowledged.
## References
* (1) P. Grangier, G. Roger, and A. Aspect, Europhys. Lett. 1, 173 (1986).
* (2) M. Arndt, K. Hornberger, and A. Zeilinger, Physics World 18, 3 (2005).
* (3) A. Buchleitner and K. Hornberger (eds.), Coherent Evolution in Noisy Environments (Springer Verlag, Berlin, 2002).
* (4) M. Arndt, A. Buchleitner, R.N. Mantegna, and H. Walther, Phys. Rev. Lett. 67, 2435 (1991).
* (5) B.G. Englert, Phys. Rev. Lett. 77, 2154 (1996).
* (6) E. Schrödinger, Die Naturwissenschaften 48, 52 (1935).
* (7) A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, Nat. Commun. 2, 224 (2011).
* (8) C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
* (9) T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 77, 1917 (1996).
* (10) L. Chen, C.-F. Li, M. Gong, F.-W. Sun, and G.-C. Guo, Europhys. Lett. 85, 14001 (2009).
* (11) Z. Y. Ou, J.-K. Rhee, and L. J. Wang, Phys. Rev. Lett. 83, 959 (1999).
* (12) X.-L. Niu, Y.-X. Gong, B.-H. Liu, Y.-F. Huang, G.-C. Guo, and Z. Y. Ou, Optics Letters 34, 1297 (2009).
* (13) M. C. Tichy, H.-T. Lim, Y.-S. Ra, F. Mintert, Y.-H. Kim, and A. Buchleitner, Phys. Rev. A 83, 062111 (2011).
* (14) Y.-H. Kim, J. Opt. Soc. Am. B 20, 1959 (2003).
* (15) Since the phase of $\alpha$ is not observable in our setting, we assume $\alpha\in\mathbbm{R}$ in the following, without loss of generality.
* (16) M.C. Tichy, M. Tiersch, F. de Melo, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. 104, 220405 (2010).
|
arxiv-papers
| 2011-09-08T06:53:20 |
2024-09-04T02:49:21.992233
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Young-Sik Ra, Malte C. Tichy, Hyang-Tag Lim, Osung Kwon, Florian\n Mintert, Andreas Buchleitner, Yoon-Ho Kim",
"submitter": "Young-Sik Ra",
"url": "https://arxiv.org/abs/1109.1636"
}
|
1109.1857
|
# Some Remarks about Interpolating Sequences in Reproducing Kernel Hilbert
Spaces
Mrinal Raghupathi Mrinal Raghupathi, Department of Mathematics
United States Naval Academy
Annapolis, MD, 21403 raghupat@usna.edu and Brett D. Wick Brett D. Wick,
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA USA 30332-0160 wick@math.gatech.edu
###### Abstract.
In this paper we study two separate problems on interpolation. We first give
some new equivalences of Stout’s Theorem on necessary and sufficient
conditions for a sequence of points to be an interpolating sequence on a
finite open Riemann surface. We next turn our attention to the question of
interpolation for reproducing kernel Hilbert spaces on the polydisc and
provide a collection of equivalent statements about when it is possible to
interpolation in the Schur-Agler class of the associated reproducing kernel
Hilbert space.
###### Key words and phrases:
Interpolating Sequences, Schur-Agler Class, Riemann Surfaces
###### 2000 Mathematics Subject Classification:
Primary
Research supported in part by a National Science Foundation DMS Grant #
1001098 and # 0955432
## 1\. Introduction and Statement of Main Results
Recall that a sequence $Z=\\{z_{j}\\}\subset\mathbb{D}$ is called an
$H^{\infty}$-interpolating sequence if for every
$a=\\{a_{j}\\}\in\ell^{\infty}$ there exists a function $f\in H^{\infty}$ such
that
$f(z_{j})=a_{j}\quad\forall j.$
Similarly, for the sequence $Z$ let $\ell^{2}(\mu_{Z})$ be the space of all
sequences $a=\\{a_{j}\\}$ such that
$\sum_{j=1}^{\infty}\left|a_{j}\right|^{2}(1-\left|z_{j}\right|^{2}):=\left\|a\right\|_{\ell^{2}(\mu_{Z})}^{2}<\infty.$
Then the sequence $Z=\\{z_{j}\\}$ is called an $H^{2}$-interpolating sequence
if for every $a=\\{a_{j}\\}\in\ell^{2}(\mu_{Z})$ there exists a function $f\in
H^{2}$ such that
$f(z_{j})=a_{j}\quad\forall j.$
As is well known, these sequences turn out to be one in the same and are
characterized by a separation condition on the points in $Z$ and that the
points must generate a Carleson measure for the space $H^{2}$. The following
theorem gives a precise statement of this.
###### Theorem 1.1 (Carleson, [Car2], Shapiro, Shields [ShSh2]).
The following are equivalent:
* (a)
The sequence $Z$ is $H^{2}$-interpolating;
* (b)
The sequence $Z$ is $H^{\infty}$-interpolating;
* (c)
The sequence $Z$ is separated in the pseudo-hyperbolic metric and generates a
$H^{2}$-Carleson measure. In particular, $\sum_{z_{j}\in
Z}(1-\left|z_{j}\right|^{2})\delta_{z_{j}}$ is a $H^{2}$ Carleson measure and
$\inf_{j\neq
k}\left|\frac{z_{j}-z_{k}}{1-\overline{z_{k}}z_{j}}\right|\geq\delta>0;$
* (d)
The sequence $Z$ is strongly separated, namely there exists a constant
$\delta>0$ such that
$\inf_{j}\left|\prod_{j\neq
k}\frac{z_{j}-z_{k}}{1-\overline{z_{k}}z_{j}}\right|\geq\delta>0.$
Since the results of Carleson, [Car2], and Shapiro-Shields, [ShSh2] the
question of characterizing the interpolating sequences for other spaces of
analytic functions has been intensively studied. See any of the papers
[MR2137874, MaSu, Seip] for various generalizations of this question.
In this paper we study the problem of interpolating sequences in two settings.
First, we consider the case of finite Riemann surfaces and obtain a new
equivalences and a different proof of a theorem of Stout. We then go on to
consider a multivariable example: the Schur Agler class. In both cases we will
make heavy use of results on Pick interpolation.
### Interpolation on Riemann Surfaces
Let $\Gamma$ be a Fuchsian group acting on the unit disk. We will assume that
$\Gamma$ finitely-generated. The group $\Gamma$ acts on $H^{\infty}$ by
composition and the associated fixed-point algebra is denoted
$H^{\infty}_{\Gamma}$. It is known that every finite open Riemann surface can
be viewed as the quotient space of the disk by the action of such a group. The
group $\Gamma$ is finitely generated and acts without fixed points on the
disk. A major advantage of viewing the problem in terms of fixed points is
that the algebra $H^{\infty}_{\Gamma}\subseteq H^{\infty}$ and this allows us
to bootstrap results about Riemann surfaces to the classical setting of the
open unit disk. There is also a reproducing kernel Hilbert space
$H^{2}_{\Gamma}$ associated with the group action. We denote by $K^{\Gamma}$
the reproducing kernel for the Hilbert space $H^{2}_{\Gamma}$. This is just
the set of fixed points in $H^{2}$. In [R] it is shown that
$H^{\infty}_{\Gamma}$ is the multiplier algebra for $H^{2}_{\Gamma}$.
Given a sequence of non-zero vectors $\\{x_{n}\\}$ in a Hilbert space $H$ we
define the associated Gramian as the matrix $[\left\langle
x_{n},x_{m}\right\rangle]_{m,n=1}^{\infty}$. The normalized Gramian is defined
as the Gramian of the sequence $\tilde{x_{n}}$, where
$\tilde{x_{n}}=\frac{x_{n}}{\left\|x_{n}\right\|}$.
Our first main result of this paper is the following theorem.
###### Theorem 1.2.
Let $Z=(z_{n})\subseteq\mathbb{D}$ be a sequence of points in
$H^{\infty}_{\Gamma}$ such that no two points lie on the same orbit of
$\Gamma$, where $\Gamma$ is the group of deck transformation associated to a
finite Riemann surface. Let $Z_{n}=Z\setminus\\{z_{n}\\}$. The following are
equivalent:
1. (1)
The sequence $\\{z_{n}\\}$ is interpolating for $H^{\infty}_{\Gamma}$;
2. (2)
The sequence $\\{z_{n}\\}$ is interpolating for $H^{2}_{\Gamma}$;
3. (3)
The sequence $\\{z_{n}\\}$ is $H^{2}_{\Gamma}$-separated and
$\sum_{n=1}^{\infty}K^{\Gamma}(z_{i},z_{i})^{-1}\delta_{z_{i}}$ is a Carleson
measure;
4. (4)
The Gramian
$G=\left[\frac{K^{\Gamma}(z_{i},z_{j})}{\sqrt{K^{\Gamma}(z_{i},z_{i})K^{\Gamma}(z_{j},z_{j})}}\right]$
is bounded below;
5. (5)
There is a constant $\delta>0$ such that $\inf_{n\geq
1}d_{H^{\infty}_{\Gamma}}(z_{n},Z_{n})\geq\delta$.
A similar result was obtained by Stout [S]. However, there are two differences
between the results obtained there and our results. First, we use the
interpolation theorem from [RW] as an essential ingredient in our proof. This
modern approach appears in the work of Marshall and Sundberg on interpolating
sequences for the Dirichlet space. Second, our proof applies to the case of a
subalgebra of $H^{\infty}$ that is fixed by the action of a finitely-generated
discrete group, a more general setting than the case of a finite Riemann
surface.
### Interpolation in the Schur-Agler Class
We now turn to the case where the domain is $\mathbb{D}^{d}$. Here the algebra
in question is the set of functions in the Schur-Agler class. As motivation
for our results we describe the important theorem of Agler and McCarthy that
characterizes the interpolating sequences for $H^{\infty}(\mathbb{D}^{2})$.
Recall that $H^{\infty}(\mathbb{D}^{2})$ is the multiplier algebra for the
space $H^{2}(\mathbb{D}^{2})$, and that this is a reproducing kernel Hilbert
space with kernel given by
$k_{z}(w)=\frac{1}{1-\overline{z_{1}}w_{1}}\frac{1}{1-\overline{z_{2}}w_{2}}$
for $z,w\in\mathbb{D}^{2}$.
A sequence of points $\\{\lambda_{j}\\}\subset\mathbb{D}^{2}$ is called an
$H^{\infty}(\mathbb{D}^{2})$-interpolating sequence if for any sequence of
bounded numbers $\\{w_{i}\\}$ there is a function $f\in
H^{\infty}(\mathbb{D}^{2})$ such that $f(\lambda_{j})=w_{j}$. The sequence of
points is said to be strongly separated if for each integer $i$ there is a
function in $\varphi_{i}\in H^{\infty}(\mathbb{D}^{2})$ of norm at most $M$
such that $\varphi_{i}(\lambda_{i})=1$ and $\varphi_{i}(\lambda_{k})=0$ for
$k\neq i$. The result of Agler and McCarthy then gives a characterization of
the interpolating sequences for $H^{\infty}(\mathbb{D}^{2})$.
###### Theorem 1.3 (Agler and McCarthy, [AgMc]).
Let $\\{\lambda_{j}\\}\in\mathbb{D}^{2}$. The following are equivalent:
* (i)
$\\{\lambda_{j}\\}$ is an interpolating sequence for
$H^{\infty}(\mathbb{D}^{2})$;
* (ii)
The following two conditions hold
* $(a)$
For all admissible kernels $k$, their normalized Gramians are uniformly
bounded above,
$G^{k}\leq MI$
for some $M>0$, item[$(b)$] For all admissible kernels $k$, their normalized
Gramians are uniformly bounded below,
$G^{k}\geq NI$
for some $N>0$;
* (iii)
The sequence $\\{\lambda_{j}\\}$ is strongly separated and condition $(a)$
alone holds;
* (iv)
Condition $(b)$ alone holds.
Here an admissible kernel is one for which the pointwise by $M_{z_{j}}$ is a
contraction on $H(k)$, the reproducing kernel Hilbert space on
$\mathbb{D}^{2}$ with kernel $k$.
We now consider a related question, but for more general products of
reproducing kernel Hilbert spaces. Given $k_{j}$ with $j=1,\ldots,d$
reproducing kernels on $\mathbb{D}$ with the property that
$\frac{1}{k_{j}}(z,w)=1-\left\langle b_{j}(z),b_{j}(w)\right\rangle$
where $b_{j}$ is an analytic map from $\mathbb{D}$ into the open unit ball of
a separable Hilbert space.
The kernel $k_{j}$ is the reproducing kernel for the Hilbert space $H(k_{j})$.
Let $H(k)$ denote the reproducing kernel Hilbert space defined on
$\mathbb{D}^{d}$ with reproducing kernel
$k(z,w)=\prod_{j=1}^{d}k_{j}(z_{j},w_{j})$ for $z,w\in\mathbb{D}^{d}$.
We define $S_{H(k)}(\mathbb{D}^{d})$ to be the set of functions
$m:\mathbb{D}\to\mathbb{C}$ such that
$1-m(z)\overline{m(w)}=\sum_{j=1}^{d}\frac{1}{k_{j}}(z_{j},w_{j})h_{j}(z)\overline{h_{j}(w)}$
for functions $\\{h_{j}\\}$ defined on $\mathbb{D}^{d}$. Note that this is the
Schur-Agler class of multipliers for $H(k)$. In the case where $k_{j}$ is the
Szegö kernel and $d=2$ an application of Ando’s theorem shows that
$H^{\infty}(\mathbb{D}^{2})$ and $S_{H(k)}(\mathbb{D}^{2})$ coincide. In
higher dimensions this is no longer the case.
Let us say that a kernel $k$ is an admissible kernel if we have that
$\frac{1}{k_{j}}(z_{j},w_{j})k(z,w)=(1-\left\langle
b_{j}(z_{j}),b_{j}(w_{j})\right\rangle)k(z,w)\geq 0\text{ for }j=1,\ldots,d.$
Given a sequence of points $\\{\lambda_{j}\\}\in\mathbb{D}^{d}$, then the
normalized Gramian of $k$ is the matrix given by
$G_{ij}^{k}=\frac{k(\lambda_{i},\lambda_{j})}{\sqrt{k(\lambda_{i},\lambda_{i})k(\lambda_{j},\lambda_{j})}}$
A sequence of points $\\{\lambda_{j}\\}\subset\mathbb{D}^{2}$ is called an
$S_{H(k)}(\mathbb{D}^{d})$-interpolating sequence if for any sequence of
bounded numbers $\\{w_{i}\\}$ there is a function $f\in S_{H(k)}$ such that
$f(\lambda_{j})=w_{j}$. The sequence of points is said to be strongly
separated if for each integer $i$ there is a function in $\varphi_{i}\in
S_{H(k)}(\mathbb{D}^{d})$ of norm at most $M$ such that
$\varphi_{i}(\lambda_{i})=1$ and $\varphi_{i}(\lambda_{k})=0$ for $k\neq i$.
Our second main result is the following theorem providing a generalization of
the result of Agler and McCarthy:
###### Theorem 1.4.
Let $\\{\lambda_{j}\\}$ be a sequence of points in $\mathbb{D}^{d}$. The
following are equivalent:
* (i)
$\\{\lambda_{j}\\}$ is an interpolating sequence for
$S_{H(k)}(\mathbb{D}^{d})$;
* (ii)
The following two conditions hold
* $(a)$
For all admissible kernels $k$, their normalized Gramians are uniformly
bounded above,
$G^{k}\leq MI$
for some $M>0$,
* $(b)$
For all admissible kernels $k$, their normalized Gramians are uniformly
bounded below,
$G^{k}\geq NI$
for some $N>0$;
* (iii)
The sequence $\\{\lambda_{j}\\}$ is strongly separated and condition $(a)$
alone holds;
* (iv)
Condition $(b)$ alone holds.
## 2\. Interpolation in Riemann surfaces
Our goal in this section is to prove the analogue of Carleson’s theorem for
Riemann surfaces. We view the Riemann surface as the quotient of the disk by
the action of a Fuchsian group and state our theorems for the corresponding
fixed-point algebra $H^{\infty}_{\Gamma}$.
A central result that we require is a Nevanlinna–Pick type theorem obtained in
[RW]. We briefly recall the parts of that paper that are most relevant to our
work.
Let $C(H^{\infty}_{\Gamma})$ be the set of columns over $H^{\infty}_{\Gamma}$,
similarly, let $R(H^{\infty}_{\Gamma})$ denote the rows. There is a natural
identification between $C(H^{\infty}_{\Gamma})$ and the space of multipliers
$\mathrm{mult}(H^{2}_{\Gamma},H^{2}_{\Gamma}\otimes\ell^{2})$. There is also a
natural identification between $R(H^{\infty}_{\Gamma})$ and
$\mathrm{mult}(H^{2}_{\Gamma}\otimes\ell^{2},H^{2}_{\Gamma})$.
###### Theorem 2.1 ([RW]).
Let $z_{1},\ldots,z_{n}\in\mathbb{D}$, $w_{1},\ldots,w_{n}\in\mathbb{C}$ and
$v_{1},\ldots,v_{n}\in\ell^{2}$. There exists a function $F\in
C(H^{\infty}_{\Gamma})$ such that $\left\|F\right\|\leq C$ and $\left\langle
F(z_{i}),v_{i}\right\rangle=w_{i}$ if and only if the matrix
$[(\alpha^{2}C^{2}\left\langle v_{j},v_{i}\right\rangle-
w_{i}\overline{w_{j}})K^{\Gamma}(z_{i},z_{j})]\geq 0$. The constant $\alpha$
depends on $\Gamma$ but not on the points $z_{1},\ldots,z_{n}$.
A similar argument also establishes the fact that there is a function $F\in
R(H^{\infty}_{\Gamma})$ such that $\left\|F\right\|\leq C$ and
$F(z_{i})=v_{i}$ if and only if the matrix $[(C^{2}\alpha^{2}-\left\langle
v_{j},v_{i}\right\rangle)K^{\Gamma}(z_{i},z_{j})]\geq 0$.
### 2.1. Separation, interpolation, and Carleson measures
In order to state our theorem we need to develop some of the necessary
background on separation of points, interpolating sequences and Carleson
measures. We state our definitions in terms of reproducing kernels and
multiplier algebras. The case we are interested in is the RKHS
$H^{2}_{\Gamma}$ and its multiplier algebra $H^{\infty}_{\Gamma}$. In this
situation there is additional structure that we can exploit.
Let $X$ be a set and let $\\{x_{n}\\}$ be a sequence of points in $X$. Let $H$
be a reproducing kernel Hilbert space of functions on $X$ with kernel $K$ and
let $M(H)$ be its multiplier algebra. We say that $\\{x_{n}\\}$ is an
interpolating sequence for the algebra $M(H)$ if and only if the restriction
map $R:M(H)\to\ell^{\infty}$ given by $R(f)=\\{f(x_{n})\\}$ is surjective.
Given a point $x\in X$ and a set $S\subseteq X$ we define the $M(H)$-distance
from $x$ to $S$ by
$d_{M(H)}(x,S)=\sup\\{\left|f(x)\right|\,:\,f|_{S}=0,\left\|f\right\|_{M(H)}\leq
1\\}$. The sequence $\\{x_{n}\\}$ is called $M(H)$-separated if and only if
there exists a constant $\delta>0$ such that $d_{M(H)}(x_{n},Z_{n})\geq\delta$
for all $n\geq 1$, where $Z_{n}=\\{x_{m}\,:\,m\geq 1\\}\setminus\\{x_{n}\\}$.
If $\\{x_{n}\\}$ is an interpolating sequence, then there exists a constant
$C$ such that for any sequence $w\in\ell^{\infty}$, $R(f)=w$ and
$\left\|f\right\|_{M(H)}\leq C\left\|w\right\|_{\infty}$. Applying this to the
case where $w=e_{j}$ we see that $d_{M(H)}(x_{n},Z_{n})\geq C^{-1}$. Therefore
an interpolating sequence for $M(H)$ is $M(H)$-separated.
Carleson’s theorem states that the converse is true for
$H^{\infty}(\mathbb{D})$, that is, every $H^{\infty}$-separated sequence is an
$H^{\infty}$-interpolating sequence. The modern approach to this problem
relies on the fact that the $H^{\infty}$ is the multiplier algebra of the
Hardy space, and the fact that the Szegö kernel has the complete Pick
property. We will use a similar approach based on Theorem 2.1 and bootstrap
our results to the case of $H^{\infty}$.
There is a related notion of separation in terms of the reproducing kernel of
$H$. In [AgMc2] it is shown that the function
$\rho_{H}(x,y)=\sqrt{1-\frac{\left|K(x,y)\right|^{2}}{K(x,x)K(y,y)}}$ is semi-
metric on the set $X$. A sequence of points is called $H$-separated if and
only if $\inf_{i\not=j}\rho_{H}(x_{i},x_{j})>0$. A sequence is weakly
separated if and only if there is a constant $\delta>0$ and functions
$f_{i,j}\in M(H)$ such that $\left\|f_{i,j}\right\|\leq 1$ with
$f_{i,j}(x_{i})=\delta$ and $f_{i,j}(x_{j})=0$. In general a weakly separated
sequence is $H$-separated, and the converse if true for the case of Riemann
surfaces.
###### Lemma 2.2.
Let $\Gamma$ be a finitely generated discrete group of automorphisms and let
$H^{\infty}_{\Gamma}$ be the corresponding fixed-point algebra. There exists a
constant $C$ such that for any pair of points $z,w\in\mathbb{D}$,
$\rho_{H}(z,w)\geq\delta/C$ if and only if there exists a function $f\in
H^{\infty}_{\Gamma}$ such that $f(w)=\delta$, $f(z)=0$ and
$\left\|f\right\|_{\infty}\leq 1$.
###### Proof.
By the Interpolation Theorem 2.1 there exists a constant $C$ and function
$f\in H^{\infty}_{\Gamma}$ such that $f(w)=\delta$ and $f(z)=0$ with
$\left\|f\right\|_{\infty}\leq 1$ if and only if the matrix
$\begin{bmatrix}C^{2}K^{\Gamma}(z,z)&C^{2}K^{\Gamma}(z,w)\\\
C^{2}K^{\Gamma}(w,z)&(C^{2}-\delta^{2})K^{\Gamma}(w,w)\end{bmatrix}\geq 0,$
where $C$ is a constant that does not depend on the points $z,w$. Since the
diagonal terms of the above matrix are non-negative, the matrix positivity
condition is equivalent to the determinant being non-negative. Computing the
determinant and rearranging we find that $\rho_{H}(z,w)\geq\delta/C$. ∎
###### Corollary 2.3.
A sequence is $H^{2}_{\Gamma}$-separated if and only if the sequence is weakly
separated by $H^{\infty}_{\Gamma}$.
A sequence is called a (universal) interpolating sequence for $H$ if and only
if the map $T:H\to\ell^{2}$ given by
$T(f)=\left\\{\frac{f(x_{n})}{K(x_{n},x_{n})^{1/2}}\right\\}$ is surjective.
In order to state our results we need the notion of a Carleson measure. A
measure $\mu$ on a set $X$ is called a Carleson measure for the Hilbert space
$H$ if and only if there exists a constant $C(\mu)$ such that
$\int_{X}\left|f(x)\right|^{2}\,d\mu\leq C(\mu)\left\|f\right\|_{H}^{2}.$
Given a sequence of points $\\{x_{n}\\}$ we can construct a measure on the set
$X$ by setting $\mu=\sum_{n=1}^{\infty}K(x_{n},x_{n})^{-1}\delta_{x_{n}}$.
When $f\in H$, we see that
$\displaystyle\int_{X}\left|f(x)\right|^{2}\,d\mu$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\left|f(x_{n})\right|^{2}K(x_{n},x_{n})^{-1}$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\left|\left\langle
f,k_{x_{n}}/\left\|k_{x_{n}}\right\|\right\rangle\right|^{2}$
$\displaystyle\leq$ $\displaystyle C(\mu)\left\|f\right\|_{H}^{2}.$
With the last inequality happening if the set of points $\\{x_{n}\\}$ that
generates $\mu$ generates a Carleson measure.
It is helpful to restate the above in terms of sequences in Hilbert space. To
this end let us fix a sequence $x_{n}\in X$, let $k_{x_{n}}$ be the
corresponding reproducing kernel and let
$g_{n}=\frac{k_{x_{n}}}{\left\|k_{x_{n}}\right\|}$. Note that $\\{g_{n}\\}$ is
a unit norm sequence in the Hilbert space $H$. The map $T:H\to\ell^{2}$ given
by
$T(f)=\left\\{\frac{f(x_{n})}{K(x_{n},x_{n})^{1/2}}\right\\}=\\{\left\langle
f,g_{n}\right\rangle\\}$. It is well known that this map is bounded if and
only if $\\{g_{n}\\}$ is a Bessel sequence, i.e., there is a constant $C$ such
that $\sum_{n=1}^{\infty}\left|\left\langle
f,g_{n}\right\rangle\right|^{2}\leq C\left\|f\right\|^{2}$ for all $f\in H$.
This in turn is equivalent to the fact that the measure
$\sum_{n=1}^{\infty}K(x_{n},x_{n})^{-1}\delta_{x_{n}}$ is a Carleson measure
for $H$. In order to make the connection with Pick interpolation later on, we
also point out that the sequence $\\{g_{n}\\}$ is Bessel if and only if the
Gram matrix $G$ whose entries are given by $\left\langle
g_{j},g_{i}\right\rangle$ is bounded, when viewed as an operator on
$\ell^{2}$.
The sequence $\\{x_{n}\\}$ is interpolating for $H$ if and only if the
sequence $\\{g_{n}\\}$ is a Riesz basic sequence, i.e., the sequence
$\\{g_{n}\\}$ is similar to a orthonormal set. In terms of the Gramian this
means that $G$ is both bounded and bounded below.
Before we proceed we need a preliminary lemma that relates the fact that
$\\{g_{n}\\}$ is a Riesz basic sequence to the matrix positivity condition
that appears in Theorem 2.1
###### Lemma 2.4.
Let $C>0$ be a constant. Let $\\{g_{n}\\}$ be a sequence of vectors in a
Hilbert space. The Gramian $[\left\langle g_{j},g_{i}\right\rangle]$ is both
bounded, and bounded below if and only if for all points $(w_{n})_{n\geq
1}\in\mathrm{ball}(\ell^{\infty})$ the matrix
$[(C^{2}-w_{i}\overline{w_{j}})\left\langle g_{j},g_{i}\right\rangle]$ is a
positive matrix.
###### Proof.
Suppose that $[(C^{2}-w_{i}\overline{w_{j}})\left\langle
g_{j},g_{i}\right\rangle]\geq 0$. If $\alpha_{n}$ is a sequence in $\ell^{2}$,
then we get
$C^{2}\sum_{i,j=1}^{\infty}\alpha_{j}\overline{\alpha_{i}}\left\langle
g_{j},g_{i}\right\rangle\geq\sum_{i,j=1}^{\infty}w_{i}\overline{w_{j}}\alpha_{j}\overline{\alpha_{i}}\left\langle
g_{j},g_{i}\right\rangle.$
Choose $w_{i}=\exp(2\pi t_{i}\sqrt{-1})$. This gives,
$C^{2}\sum_{i,j=1}^{\infty}\alpha_{j}\overline{\alpha_{i}}\left\langle
g_{j},g_{i}\right\rangle\geq\sum_{i,j=1}^{\infty}\exp(2\pi(t_{i}-t_{j})\sqrt{-1})\alpha_{j}\overline{\alpha_{i}}\left\langle
g_{j},g_{i}\right\rangle.$
If we integrate both sides from $0$ to $2\pi$ with respect to each of the
variables $t_{1},\ldots,t_{n},\ldots$ then the above equation reduces to
$C^{2}\left\|\sum_{n=1}^{\infty}\alpha_{n}g_{n}\right\|^{2}\geq\sum_{n=1}^{\infty}\left|\alpha_{n}\right|^{2}.$
Now choose $\alpha_{i}=w_{i}\beta_{i}$. We have
$C^{2}\sum_{i,j=1}^{\infty}\exp(2\pi(t_{i}-t_{j})\sqrt{-1})\beta_{j}\overline{\beta_{i}}\left\langle
g_{j},g_{i}\right\rangle\geq\sum_{i,j=1}^{\infty}\beta_{j}\overline{\beta_{i}}\left\langle
g_{j},g_{i}\right\rangle.$
Integrating as before, we obtain
$C^{2}\sum_{n=1}^{\infty}\left|\beta_{n}\right|^{2}\geq\left\|\sum_{n=1}^{\infty}\alpha_{n}g_{n}\right\|^{2}.$
For the converse assume that $B\geq[\left\langle g_{j},g_{i}\right\rangle]\geq
B^{-1}$. Let $D_{w}$ be the matrix with diagonal entries $\\{w_{n}\\}$. Since
$D_{w}$ is a contraction we obtain the equation
$BG\geq D_{w}GD_{w}^{*}.$
Since $G$ is invertible we have
$\left\|G^{-1/2}D_{w}G^{1/2}\right\|\leq\left\|G^{-1/2}\right\|\left\|G^{1/2}\right\|\leq
B$. Therefore,
$B^{2}I-(G^{-1/2}D_{w}G^{1/2})(G^{-1/2}D_{w}G^{1/2})^{*}\geq 0.$
Which gives $B^{2}G\geq D_{w}GD_{w}^{*}$. This is just
$[(B^{2}-w_{i}\overline{w_{j}})\left\langle g_{j},g_{i}\right\rangle]\geq 0$.
∎
### 2.2. Proof of Theorem 1.2
Our goal is to prove Theorem 1.2 (stated again for ease):
###### Theorem 2.5.
Let $Z=(z_{n})\subseteq\mathbb{D}$ be a sequence of points in
$H^{\infty}_{\Gamma}$ such that no two points lie on the same orbit of
$\Gamma$, where $\Gamma$ is the group of deck transformation associated to a
finite Riemann surface. Let $Z_{n}=Z\setminus\\{z_{n}\\}$. The following are
equivalent:
1. (1)
The sequence $\\{z_{n}\\}$ is interpolating for $H^{\infty}_{\Gamma}$;
2. (2)
The sequence $\\{z_{n}\\}$ is interpolating for $H^{2}_{\Gamma}$;
3. (3)
The sequence $\\{z_{n}\\}$ is $H^{2}_{\Gamma}$-separated and
$\sum_{n=1}^{\infty}K^{\Gamma}(z_{i},z_{i})^{-1}\delta_{z_{i}}$ is a Carleson
measure;
4. (4)
The Gramian
$G=\left[\frac{K^{\Gamma}(z_{i},z_{j})}{\sqrt{K^{\Gamma}(z_{i},z_{i})K^{\Gamma}(z_{j},z_{j})}}\right]$
is bounded below;
5. (5)
There is a constant $\delta>0$ such that $\inf_{n\geq
1}d_{H^{\infty}_{\Gamma}}(z_{n},Z_{n})\geq\delta$.
The strategy that we will follow is to prove that (1) and (2) are equivalent
and that (4) and (1) are equivalent. We will then show that (5) implies (1).
Finally, we establish that (2) implies (3) and that (3) implies (5). We now
prove the first of our claims that lead to the proof of Theorem 1.2.
We fix notation as follows: Let $\\{z_{n}\\}\subseteq\mathbb{D}$ be a sequence
of points and let $K^{\Gamma}_{z_{n}}$ be the reproducing kernel for the space
$H^{2}_{\Gamma}$ at the point $z_{n}$. The corresponding normalized kernel
function will be denoted $g_{n}$. The Gram matrix will be denoted
$G=[\left\langle g_{j},g_{i}\right\rangle]$.
###### Proposition 2.6.
(1) $\Leftrightarrow$ (2) A sequence of points $\\{z_{n}\\}$ is interpolating
for $H^{\infty}_{\Gamma}$ if and only if it is interpolating for
$H^{2}_{\Gamma}$.
###### Proof.
Suppose that $\\{z_{n}\\}$ is a sequence of points in $\mathbb{D}$. Let
$g_{n}$ be the normalized kernel function for $H^{2}_{\Gamma}$ at the point
$z_{n}$. The restriction map $R:H^{\infty}_{\Gamma}\to\ell^{\infty}$ is
surjective if and only if there is a constant $M$ such that for every sequence
$w=(w_{n})\in\mathrm{ball}(\ell^{\infty})$ there is a function $f\in
H^{\infty}_{\Gamma}$ of norm at most $M$ such that $R(f)=w$. By the
Nevanlinna–Pick type Theorem 2.1 this is equivalent to the matrix
$[(C^{2}M^{2}-w_{i}\overline{w_{j}})\left\langle g_{j},g_{i}\right\rangle]\geq
0$ for all choices of $w$. Using Lemma 2.4 we see that this is equivalent to
the Gramian $G$ being bounded and bounded below which in turn is equivalent to
the sequence $\\{g_{n}\\}$ being a Riesz basic sequence. By our comments
earlier the sequence $g_{n}$ is a Riesz basic sequence if and only if
$T:H^{2}_{\Gamma}\to\ell^{2}$ given by $T(f)=\\{\left\langle
f,g_{n}\right\rangle\\}$ is bounded and surjective, i.e., $\\{z_{n}\\}$ is
interpolating for $H^{2}_{\Gamma}$. ∎
###### Proposition 2.7.
(5) $\Rightarrow$ (1) If there is a constant $\delta$ such that
$d_{H^{\infty}_{\Gamma}}(z_{n},Z_{n})\geq\delta>0$, then $\\{z_{n}\\}$ is an
interpolating sequence for $H^{\infty}_{\Gamma}$.
###### Proof.
Given a set $Z\subseteq\mathbb{D}$, let $\Gamma
Z:=\\{\gamma(z)\,:\,\gamma\in\Gamma,z\in Z\\}$. Suppose that
$d_{H^{\infty}_{\Gamma}}(z_{n},Z_{n})\geq\delta>0$. Then, by definition, there
exist functions $f_{n}\in H^{\infty}_{\Gamma}$ such that
$\left\|f_{n}\right\|_{\infty}\leq 1$, $\left|f_{n}(z_{n})\right|\geq\delta$
and $f_{n}|_{Z_{n}}=0$. It follows that $f_{n}|_{\Gamma Z_{n}}=0$. The proof
of [S]*Theorem 6.3 shows that the sequence $\Gamma Z$ is an interpolating
sequence for $H^{\infty}$.
Now given a sequence $\\{w_{n}\\}\in\ell^{\infty}$, let
$\tilde{w}_{n,\gamma}=w_{n}$ for $\gamma\in\Gamma$ and $n\geq 1$. Since
$\Gamma Z$ is interpolating for $H^{\infty}$, there exists a function
$\tilde{f}\in H^{\infty}$ such that
$\tilde{f}(\gamma(z_{n}))=\tilde{w}_{n,\gamma}=w_{n}$.
Next we invoke a result of Earle and Marden [EM]*Theorem page 274. Their
result shows that there is a polynomial $p$ such that the map
$(\Phi
g)(z)=\frac{\sum_{\gamma\in\Gamma}p(\gamma(z))g(\gamma(z))\gamma^{\prime}(z)^{2}}{\sum_{\gamma\in\Gamma}p(\gamma(z))\gamma^{\prime}(z)^{2}}$
defines a bounded projection from $H^{\infty}$ onto $H^{\infty}_{\Gamma}$. If
$g\in H^{\infty}$ and $g(\gamma(\zeta))=c$ for some constant $c$, then $(\Phi
g)(\zeta)=c$. It follows that the function $f=\Phi\tilde{f}$ is in
$H^{\infty}_{\Gamma}$ and that $f(z_{n})=w_{n}$. Hence, $z_{n}$ is an
interpolating sequence for $H^{\infty}_{\Gamma}$. ∎
###### Proposition 2.8.
(4) $\Leftrightarrow$ (1) The Gram matrix is bounded below if and only if the
sequence is interpolating for $H^{\infty}_{\Gamma}$.
###### Proof.
If $G\geq C^{-2}>0$, then by Theorem 2.1, there exists a function $F\in
R(H^{\infty}_{\Gamma})$ such that $\left\|F\right\|\leq C$ and
$F(z_{n})=e_{n}$. If we write $F=(f_{1},\ldots,)$, then
$f_{m}(z_{n})=\delta_{m,n}$ with $\left\|f_{m}\right\|\leq C$. Let
$\phi_{n}=f_{n}^{2}$. Given a sequence $w=\\{w_{n}\\}\in\ell^{\infty}$ let
$f=\sum_{n=1}^{\infty}w_{n}f_{n}^{2}$. We have,
$\displaystyle\left|f(z)\right|$
$\displaystyle\leq\left|\sum_{n=1}^{\infty}w_{n}f_{n}(z)^{2}\right|\leq\sum_{n=1}^{\infty}\left|w_{n}\right|\left|f_{n}(z)\right|^{2}$
$\displaystyle\leq\left(\sup_{n\geq
1}\left|w_{n}\right|\right)\left\|F(z)\right\|^{2}\leq\left\|w\right\|_{\ell^{\infty}}\left\|F\right\|^{2}\leq\left\|w\right\|_{\ell^{\infty}}C^{2}.$
This proves that the sequence is interpolating for $H^{\infty}_{\Gamma}$.
If the sequence $z_{n}$ is interpolating for $H^{\infty}_{\Gamma}$, then for
any choice of sequence $(w_{n})\in\ell^{\infty}$ such that
$\left|w_{n}\right|\leq 1$, there exists a function $f\in
H^{\infty}_{\Gamma}$, with $\left\|f\right\|_{\infty}\leq C$ such that
$f(z_{n})=w_{n}$. Hence, the matrix
$[(C^{2}-w_{i}\overline{w_{j}})\left\langle g_{i},g_{j}\right\rangle]\geq 0$
for all $(w_{n})\in\mathrm{ball}(\ell^{\infty})$. From Lemma 2.4 we see that
the Gramian is bounded below. ∎
###### Proposition 2.9.
(2) $\Rightarrow$ (3) If $\\{z_{n}\\}$ is an interpolating sequence for
$H^{2}_{\Gamma}$, then $\\{z_{n}\\}$ is $H^{2}_{\Gamma}$-separated and
$\sum_{n=1}^{\infty}K^{\Gamma}(z_{n},z_{n})^{-1}\delta_{z_{n}}$ is a Carleson
measure.
###### Proof.
This result is true for any RKHS and the proof can be found in [Seip]. ∎
###### Proposition 2.10.
(3) $\Rightarrow$ (5) If the sequence $\\{z_{n}\\}$ is
$H^{2}_{\Gamma}$-separated and
$\sum_{n=1}^{\infty}K^{\Gamma}(z_{n},z_{n})^{-1}\delta_{z_{n}}$ is a Carleson
measure, then the sequence $\\{z_{n}\\}$ is $H^{\infty}_{\Gamma}$-separated.
###### Proof.
Fix one of the indices $m$. Since the sequence is $H^{2}_{\Gamma}$ separated,
by Lemma 2.2 there exist functions $f_{n}$ such that $\left\|f_{n}\right\|\leq
1$ with $f_{n}(z_{n})=0$ and
$f_{n}(z_{m})\geq\rho_{H^{2}_{\Gamma}}(z_{n},z_{m})$.
We now consider the sequence of products $\phi_{n}=f_{1}\cdots f_{n}$. This
sequence has a weak-* limit in the unit ball of $H^{\infty}_{\Gamma}$. Denote
this limit by $\phi$.
The claim is that $\phi(z_{m})>\delta^{\prime}$ and $\phi(z_{n})=0$, where
$\delta^{\prime}$ is a constant that does not depend on $j$.
To see this we note that the infinite product that defines $\phi(z_{m})$
converges to a non-zero value if and only if the series
$\sum_{n\not=m}1-\left|f_{n}(z_{m})\right|^{2}<+\infty$.
Using the Carleson condition we get that
$\sum_{n=1}^{\infty}K^{\Gamma}(z_{n},z_{n})\left|K^{\Gamma}_{z_{m}}(z_{n})\right|^{2}\leq
C\left\|K^{\Gamma}_{z_{m}}\right\|^{2}$, where the constant $C$ is independent
of $m$. Rewriting this we get
$\sum_{n=1}^{\infty}\dfrac{\left|K^{\Gamma}(z_{m},z_{n})\right|^{2}}{K^{\Gamma}(z_{n},z_{n})K^{\Gamma}(z_{m},z_{m})}\leq
C.$
Now we invoke the fact that
$f_{n}(z_{m})\geq\rho_{H^{2}_{\Gamma}}(z_{n},z_{m})$ from which we get that
the sum $\sum_{n\not=m}1-\left|f_{n}(z_{m})\right|^{2}\leq C$. ∎
Combining all these Propositions then gives the Proof of Theorem 1.2.
### 2.3. Applications to the Feichtinger conjecture
In this section we make some observations that are relevant to the Kadison-
Singer problem. This has been a significant problem in operator algebras for
the past 50 years. We refrain from stating the problem in its original form,
and instead focus on an equivalent statement: the Feichtinger conjecture.
The Feichtinger conjecture asks whether every bounded frame $\\{f_{n}\\}$ can
be written as the union of finitely many Riesz basic sequences. In [CCLV] it
is shown the term bounded frame can be replaced by bounded Bessel sequence.
The term bounded here means that $\inf_{n\geq 1}\left\|f_{n}\right\|>0$.
Perhaps, bounded below is a better term. In recent years there has been
interest in this problem from the perspective of function theory. The frames
of interest are sequences of normalized reproducing kernels. Given a kernel
function $K$ on a set $X$, the normalized reproducing kernel at $x$ is the
function $g_{x}=\frac{k_{x}}{K(x,x)^{1/2}}$. Given a sequence of points in $X$
we obtain a sequence of unit norm vectors $g_{x_{n}}$.
###### Theorem 2.11.
Let $\\{z_{n}\\}$ be a sequence of points in the unit disk. The Bessel
sequence $\\{g_{z_{n}}\\}$ of reproducing kernels for $H^{2}_{\Gamma}$ can be
written as a union of finitely many Riesz basic sequences. The Feichtinger
conjecture is true for such Bessel sequences.
###### Proof.
Let $\\{z_{n}\\}$ be a sequence of points in the unit disk. Let $K^{\Gamma}$
be the reproducing kernel for the space $H^{2}_{\Gamma}$. The sequence
$\\{g_{z_{n}}\\}$ is a sequence of unit norm vectors in $H^{2}_{\Gamma}$. The
condition that the sequence $\\{g_{z_{n}}\\}$ be a Bessel sequence is
equivalent to the Carleson condition on the points $\\{z_{n}\\}$. A result of
[M], a proof of which can be found in [AgMc2] and [Seip], shows that a Bessel
sequence can be written as a union of finitely many $H$-separated sequences.
By Theorem 1.2 an $H^{2}_{\Gamma}$-separated Bessel sequence of normalized
reproducing kernels is an interpolating sequence for $H^{2}_{\Gamma}$. From
Theorem 1.2 we see that the Bessel sequence of reproducing kernels for
$H^{2}_{\Gamma}$ can be written as a union of finitely many Riesz basic
sequences. ∎
## 3\. Interpolation for Products of Kernels
In this section we prove a generalization of a theorem of Agler-McCarthy
[AgMc] on interpolating sequences in several variables. Our results depend on
the Pick Interpolation theorem due to Tomerlin [T]. Our goal is to give a
proof of Theorem 1.4.
We proceed in much the same way as in [AgMc] by first defining related
conditions that will help us in studying the equivalences between the various
interpolation problems. Condition $(a)$ from Theorem 1.4 is equivalent to the
following: There exists a constant $M$ and positive semi-definite infinite
matrices $\Gamma^{j}$, $j=1,\ldots,d$, such that
($a^{\prime}$)
$M\delta_{ij}-1=\sum_{l=1}^{d}\Gamma_{ij}^{l}\frac{1}{k_{l}}(\lambda_{i}^{l},\lambda_{j}^{l})$
While Condition $(b)$ is equivalent to the following: There exists a constant
$N$ and positive semi-definite infinite matrices $\Delta^{j}$, $j=1,\ldots,d$,
such that
($b^{\prime}$)
$1-N\delta_{ij}=\sum_{l=1}^{d}\Delta_{ij}^{l}\frac{1}{k_{l}}(\lambda_{i}^{l},\lambda_{j}^{l})$
### 3.1. Proof that $(a)\Leftrightarrow(a^{\prime})$ and
$(b)\Leftrightarrow(b^{\prime})$
Note that a kernel $K$ defined on a subset $Y\subseteq\mathbb{D}^{d}$ can
always be extended to a weak kernel $\tilde{K}$ on $\mathbb{D}^{d}$ by setting
$\tilde{K}(z,w)=K(z,w)$ for $z,w\in Y$ and $\tilde{K}(z,w)=0$ otherwise.
We need to prove that if $K$ is a kernel on $\Lambda$ such that $(MI-J)\cdot
K\geq 0$, then $MI-J$ is a sum of the above form. The proof of this follows
from a basic Hilbert space argument.
If $A,B\in M_{n}$, then the Schur product of $A$ and $B$ is the matrix $A\cdot
B=[a_{i,j}b_{i,j}]$. It is a well-known fact that the Schur product of two
positive matrices is positive. Let $M_{n}^{h}$ denote the set of $n\times n$
Hermitian matrices. The space $M_{n}^{h}$ is a real Hilbert space in the inner
product $\left\langle A,B\right\rangle=\operatorname{trace}(AB)=\left\langle
A\cdot Be,e\right\rangle$ where $e$ is the vector in $\mathbb{R}^{n}$ all of
whose entries are 1.
If $C$ is a wedge in a Hilbert space $H$, then the dual wedge $C^{\prime}$ is
defined as the collection of all elements $h$ of $H$ such that $\left\langle
h,x\right\rangle\geq 0$ for all $x\in C$. The following observation appears in
[P].
###### Proposition 3.1.
Let $C\subseteq M_{n}^{h}$ be a set of matrices such that for every positive
matrix $P$, and $X\in C$, $P\cdot X\in C$. Then,
$C^{\prime}=\\{H\in M_{n}^{h}\,:\,H\cdot X\geq 0\text{ for all }X\in C\\}.$
###### Proof.
If $H\cdot X\geq 0$, then $\left\langle H,X\right\rangle=\left\langle H\cdot
Xe,e\right\rangle\geq 0$. On the other hand, assume that $H\in C^{\prime}$ and
let $v\in\mathbb{C}^{n}$. If $X=[x_{i,j}]$, then the matrix
$v^{*}Xv=[v_{i}x_{i,j}\overline{v_{j}}]=(vv^{*})\cdot X\in C$, since $vv^{*}$
is positive. Since $H\in C^{\prime}$ we have that $\left\langle H\cdot
Xv,v\right\rangle=\left\langle H,v^{*}Xv\right\rangle\geq 0$. ∎
With this proposition in hand we can prove our claim. We are assuming that $K$
is kernel function on $\Lambda$ and that $(MI-J)\cdot K\geq 0$.
Let $R_{l}$ be the matrix
$\left(\frac{1}{k_{l}}(\lambda_{i}^{l},\lambda_{j}^{l})\right)_{i,j=1}^{n}$.
Note that $R_{l}$ is self-adjoint. Let $\mathcal{R}_{l}$ be the set of
matrices of the form $P\cdot R_{l}$ where $P\geq 0$. Note that this collection
is a closed wedge that satisfies the hypothesis of Proposition 3.1. If $K$ is
an admissible kernel, then $K\cdot R_{l}\geq 0$ and so $K\cdot P\cdot
R_{l}\geq 0$ for all $P$. Hence, $K\in\mathcal{R}_{l}^{\prime}$ for
$l=1,\ldots,d$.
Let $\mathcal{K}$ be the collection of positive matrices $K$ such that $K\cdot
R_{l}\geq 0$ for all $l$. We have just shown that
$\mathcal{K}=\mathcal{R}_{1}^{\prime}\cap\cdots\cap\mathcal{R}_{d}^{\prime}$.
Let $K$ be an admissible kernel such that $(MI-J)\cdot K\geq 0$. Note that any
positive matrix $P$ such that $(MI-J)\cdot P\geq 0$ can be extended to an
admissible kernel. This means that the matrix $MI-J$ is in
$\mathcal{K}^{\prime}$.
If $C_{1},C_{2}$ and $C$ are closed wedges, then $(C_{1}\cap
C_{2})^{\prime}=C_{1}^{\prime}+C_{2}^{\prime}$ and $C^{\prime\prime}=C$.
Applying this result we get
$\mathcal{K}^{\prime}=(\mathcal{R}_{1}^{\prime}\cap\cdots\cap\mathcal{R}_{d}^{\prime})^{\prime}=\mathcal{R}_{1}^{\prime\prime}+\cdots+\mathcal{R}_{d}^{\prime\prime}=\mathcal{R}_{1}+\cdots+\mathcal{R}_{d}$.
Hence, there exists matrices $\Gamma_{l}\geq 0$ such that
$MI-J=\sum_{l=1}^{d}\Gamma_{l}\cdot R_{l}$.
### 3.2. Equivalence of Conditions $(a^{\prime})$ and $(b^{\prime})$ to
vector-valued interpolation problems
We next show that conditions $(a^{\prime})$ and $(b^{\prime})$ are equivalent
to certain vector-valued interpolation problems. The general idea is to follow
the proof in [AgMc], but to use related results by Tomerlin [T] that are
directly applicable to our setting.
First, some notation. Let $E$ and $E_{*}$ denote separable Hilbert spaces and
let $\mathcal{L}(E,E_{*})$ denote the space of bounded linear operators from
$E$ to $E_{*}$. Let $\\{e_{i}\\}$ denote the standard basis in
$\ell^{2}(\mathbb{N})$.
Finally, let $S_{H(k)}(\mathbb{D}^{d};\mathcal{L}(E,E_{*}))$ denote the set of
functions $M$ such that there exist functions $H_{j}$ on $\mathbb{D}^{d}$ and
auxiliary Hilbert spaces $E_{j}$ with values in $\mathcal{L}(E_{j},E_{*})$
such that
$I_{E_{*}}-M(z)M(w)^{*}=\sum_{j=1}^{d}\frac{1}{k_{j}}(z_{j},w_{j})H_{j}(z)H_{j}(w)^{*}.$
###### Theorem 3.2 (Tomerlin [T]).
Let $z_{1},\ldots,z_{n}$ be points in $\mathbb{D}^{d}$, let
$x_{1},\ldots,x_{n}\in\mathcal{L}(\mathcal{E}_{*},\mathcal{H}_{n})$, let
$y_{1},\ldots,y_{n}\in\mathcal{L}(\mathcal{E},\mathcal{H}_{n})$. Then there
exists an element $W\in
S_{H(k)}(\mathbb{D}^{d};\mathcal{L}(\mathcal{E},\mathcal{E}_{*}))$ such that
$x_{i}W(z_{i})=y_{i}$ if and only if there exist block matrices
$[\Gamma_{i,j}^{l}]$ such that
$x_{i}x_{j}^{*}-y_{i}y_{j}^{*}=\sum_{l=1}^{d}\Gamma^{l}_{i,j}\frac{1}{k_{l}}(z_{i},z_{j})$
With this notation and result, we can now state an alternate equivalence
between condition $(b^{\prime})$.
###### Lemma 3.3.
Let $\\{\lambda_{j}\\}$ be a sequence of points in $\mathbb{D}^{d}$. The
following are equivalent:
* $(b^{\prime})$
There exists a constant $N$ and positive semi-definite infinite matrices
$\Delta^{j}$, $j=1,\ldots,d$, such that
$1-N\delta_{ij}=\sum_{l=1}^{d}\Delta_{ij}^{l}\frac{1}{k_{l}}(\lambda_{i}^{l},\lambda_{j}^{l});$
* $(b^{\prime\prime})$
There exists a function $\Phi\in
S_{H(k)}(\mathbb{D}^{d};\mathcal{L}(\mathbb{C},\ell^{2}(\mathbb{N})))$ of norm
at most $\sqrt{N}$ such that
$\Phi(\lambda_{i})=e_{i}.$
Similarly, we have the following lemma giving an equivalent condition for
$(a^{\prime})$.
###### Lemma 3.4.
Let $\\{\lambda_{j}\\}$ be a sequence of points in $\mathbb{D}^{d}$. The
following are equivalent:
* $(a^{\prime})$
There exists a constant $M$ and positive semi-definite infinite matrices
$\Gamma^{j}$, $j=1,\ldots,d$, such that
$M\delta_{ij}-1=\sum_{l=1}^{d}\Gamma_{ij}^{l}\frac{1}{k_{l}}(\lambda_{i}^{l},\lambda_{j}^{l});$
* $(a^{\prime\prime})$
There exists a function $\Psi\in
S_{H(k)}(\mathbb{D}^{d};\mathcal{L}(\ell^{2}(\mathbb{N}),\mathbb{C}))$ of norm
at most $\sqrt{M}$ such that
$\Psi(\lambda_{i})e_{i}=1.$
###### Proof of Lemma 3.3.
Suppose that $(b^{\prime})$ is true. Consider the interpolation problem with
$x_{i}=\sqrt{N}\in\mathbb{C}$ and $y_{i}=e_{i}$ viewed as a map from
$\ell^{2}(\mathbb{N})$ to $\mathbb{C}$. Then
$x_{i}x_{j}^{*}-y_{i}y_{j}^{*}=NJ-I$. From the interpolation Theorem 3.2 we
see that there exits an element $\tilde{\Psi}\in
S_{H(k)}(\mathbb{D}^{d};\mathcal{L}(\ell^{2}(\mathbb{N}),\mathbb{C}))$ such
that $\sqrt{N}\tilde{\Psi}(\lambda_{i})=y_{i}=e_{i}$. Hence,
$\Psi=\sqrt{N}\tilde{\Psi}$ has norm at most $\sqrt{N}$ and has the property
that $\Psi(\lambda_{i})=e_{i}$.
The converse follows from the fact that a multiplier $\Psi$ of norm at most
$\sqrt{N}$ has the property that
$N-\frac{1}{N}\Psi(\lambda)\Psi(\mu)^{*}=\sum_{l=1}^{d}\frac{1}{k_{l}}(\lambda,\mu)\Gamma^{l}(\lambda,\mu)$
for some positive semidefinite functions $\Gamma^{1},\ldots,\Gamma^{d}$. When
restricted to the points $\lambda_{i}$ we see that $NJ-I$ is a sum of the
appropriate form. ∎
The proof of Lemma 3.4 is similar to the above. We have seen that condition
$(a)$ is equivalent to the condition $(a^{\prime})$ which is equivalent to
$(a^{\prime\prime})$. A similar equivalence is true for $(b)$, $(b^{\prime})$
and $(b^{\prime\prime})$.
Before we prove Theorem 1.4 recall that strong separation of a sequence
$\\{\lambda_{n}\\}$ by $S_{H(k)}$ if there is a constant $M>0$ and functions
$f_{n}\in S_{H(k)}$ such that $\left\|f_{n}\right\|_{S_{H(K)}}\leq M$,
$f_{n}(\lambda_{n})=1$, and $f_{n}(\lambda_{m})=0$ for $n\not=m$.
### Proof of Theorem 1.4
We are now ready to prove the second main theorem (stated again for ease on
the reader)
###### Theorem 3.5.
Let $\\{\lambda_{j}\\}$ be a sequence of points in $\mathbb{D}^{d}$. The
following are equivalent:
* (i)
$\\{\lambda_{j}\\}$ is an interpolating sequence for
$S_{H(k)}(\mathbb{D}^{d})$;
* (ii)
The following two conditions hold
* $(a)$
For all admissible kernels $k$, their normalized Gramians are uniformly
bounded above,
$G^{k}\leq MI$
for some $M>0$,
* $(b)$
For all admissible kernels $k$, their normalized Gramians are uniformly
bounded below,
$G^{k}\geq NI$
for some $N>0$;
* (iii)
The sequence $\\{\lambda_{j}\\}$ is strongly separated and condition $(a)$
alone holds;
* (iv)
Condition $(b)$ alone holds.
###### Proof.
Let $\lambda_{i}$ be an interpolating sequence and suppose that $k$ is an
admissible kernel. Let $k_{j}$ denote the normalized kernel function at the
point $\lambda_{j}$. Let $w_{i}$ be a sequence of points such that
$\left|w_{i}\right|\leq 1$ for all $i$.
Using the interpolation Theorem 3.2 we see that there exists a function $f$
such that $f/\sqrt{M}\in S_{H(k)}$ and $f(\lambda_{i})=w_{i}$ if and only if
the matrix $(M-w_{i}\overline{w_{j}})\cdot\left\langle
k_{j},k_{i}\right\rangle\geq 0$ for all admissible kernels $k$. This statement
is equivalent to the fact that
$M\left\|\sum_{i=1}^{\infty}\alpha_{i}k_{i}\right\|^{2}\geq\left\|\sum_{i=1}^{\infty}\alpha_{i}w_{i}k_{i}\right\|^{2}$
for all sequences $\\{\alpha_{i}\\}\in\ell^{2}$.
It follows from the argument in [AgMc]*Lemma 2.1 that both
$(M\delta_{i,j}-J)\cdot K$ and $(J-M\delta_{i,j})\cdot K$ are positive.
It is also clear that (i) and (ii) are equivalent to the conditions (iii) and
(iv).
We now come to the equivalence of (iii) and (iv). The proof that (iii) and
(iv) are equivalent is essentially that given by Agler-McCarthy [AgMc]. If
there exists $\Phi$ such that $\Phi/\sqrt{M}\in
S_{H(k)}(\mathbb{D}^{d},\mathcal{L}(\mathbb{C},\ell^{2}))$ such that
$\Phi(\lambda_{i})^{*}e_{i}=1$, then writing $\Phi=(\phi_{1},\ldots,)^{t}$ we
see that $\overline{\phi_{i}(\lambda_{i})}=1$. Since we have assumed strong
separation, there exist $f_{i}$ and a constant $C$ such that
$f_{i}(\lambda_{j})=\delta_{i,j}$ and $\left\|f_{i}\right\|\leq C$. Therefore
$\Psi=(\phi_{1}f_{1},\ldots)$ has the property that $\left\|\Psi\right\|\leq
C\sqrt{M}$ and $\Psi(\lambda_{i})=e_{i}$.
Conversely if $\Psi=(\psi_{1},\ldots)$ has the property that
$\Psi(\lambda_{i})=e_{i}$ then $\psi_{j}(\lambda_{i})=\delta_{i,j}$. Therefore
the functions $\psi_{i}$ strongly separate $\lambda_{i}$. Setting
$\Phi=\Psi^{t}$ we see that
$\Phi(\lambda_{i})^{*}e_{i}=\overline{\phi_{i}(\lambda_{i})}=1$. ∎
## References
|
arxiv-papers
| 2011-09-08T23:33:36 |
2024-09-04T02:49:22.013452
|
{
"license": "Public Domain",
"authors": "Mrinal Raghupathi and Brett D. Wick",
"submitter": "Mrinal Raghupathi",
"url": "https://arxiv.org/abs/1109.1857"
}
|
1109.2064
|
# Thermal States in Conformal QFT. II
Paolo Camassa, Roberto Longo, Yoh Tanimoto, Mihály Weiner111Permanent address:
Budapest University of Technology and Economics Department of Analyses, Pf.
91, 1521 Budapest, Hungary
Università di Roma “Tor Vergata”, Dipartimento di Matematica
Via della Ricerca Scientifica, 1 - 00133 Roma, Italy
###### Abstract
We continue the analysis of the set of locally normal KMS states w.r.t. the
translation group for a local conformal net ${\mathcal{A}}$ of von Neumann
algebras on $\mathbb{R}$. In the first part we have proved the uniqueness of
KMS state on every completely rational net. In this second part, we exhibit
several (non-rational) conformal nets which admit continuously many primary
KMS states. We give a complete classification of the KMS states on the
$U(1)$-current net and on the Virasoro net ${\rm Vir}_{1}$ with the central
charge $c=1$, whilst for the Virasoro net ${\rm Vir}_{c}$ with $c>1$ we
exhibit a (possibly incomplete) list of continuously many primary KMS states.
To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem
within the locally normal system framework: if there is an inclusion of split
nets ${\mathcal{A}}\subset{\mathcal{B}}$ and ${\mathcal{A}}$ is the fixed
point of ${\mathcal{B}}$ w.r.t. a compact gauge group, then any locally
normal, primary KMS state on ${\mathcal{A}}$ extends to a locally normal,
primary state on ${\mathcal{B}}$, KMS w.r.t. a perturbed translation.
Concerning the non-local case, we show that the free Fermi model admits a
unique KMS state.
Dedicated to Rudolf Haag on the occasion of his 90th birthday
Research supported in part by the ERC Advanced Grant 227458 OACFT “Operator
Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network
“Noncommutative Geometry” MRTN-CT-2006-0031962.
Email: camassa@mat.uniroma2.it, longo@mat.uniroma2.it,
tanimoto@mat.uniroma2.it, mweiner@renyi.hu
## 1 Introduction
We continue here our study of the thermal state structure in Conformal Quantum
Field Theory, namely we study the set of locally normal KMS states on a local
conformal net of von Neumann algebras on the real line with respect to the
translation automorphism group.
As is known, local conformal nets may be divided in two classes [28] that
reflect the sector (equivalence class of representations on the circle)
structure. For a local conformal net ${\mathcal{A}}$, to be completely
rational (this condition is characterized intrinsically by the finiteness of
the global index [22]) is equivalent to the requirement that ${\mathcal{A}}$
has only finitely many inequivalent irreducible sectors and all of them have
finite index. If ${\mathcal{A}}$ is not completely rational then either
${\mathcal{A}}$ has uncountably many inequivalent irreducible sectors or
${\mathcal{A}}$ has at least one irreducible sector with infinite index.
Faithful KMS states of ${\mathcal{A}}$ w.r.t. translations are locally normal
on the real line (assuming the general split property) and are associated with
locally normal GNS representations of the restriction of the net
${\mathcal{A}}$ to the real line. One may wonder whether the structure of
these representations, i.e. of the KMS states, also strikingly depends on the
rational/non-rational alternative.
In the first part of our work [7] we have indeed shown the general result
that, if ${\mathcal{A}}$ is a completely rational local conformal net, then
there exists only one locally normal KMS state with respect to translations on
${\mathcal{A}}$ at any fixed inverse temperature $\beta>0$. This state is the
geometric KMS state $\varphi_{\rm geo}$ which is canonically constructed for
any (rational or non-rational) local conformal (diffeomorphism covariant) net.
In this paper we examine the situation when ${\mathcal{A}}$ is not completely
rational. In contrast to the completely rational case, we shall see that there
are non-rational nets with continuously many KMS states.
We shall focus our attention on two important models. The first one is the
free field, i.e. the net generated by the $U(1)$-current. In this model we
manage to classify all KMS states. We shall show that the primary (locally
normal) KMS states of the $U(1)$-current net are in one-to-one correspondence
with real numbers $q\in{\mathbb{R}}$; as we shall see, each state
$\varphi^{q}$ is uniquely and explicitly determined by its value on the
current. The geometric KMS state is $\varphi_{\rm geo}=\varphi^{0}$ and any
other primary KMS state is obtained by composition of the geometric one with
the automorphisms $\gamma_{q}$ of the net (see Section 4.2):
$\varphi^{q}=\varphi_{\rm geo}\circ\gamma_{q}.$
The second model we study is the Virasoro net ${\rm Vir}_{c}$, the net
generated by the stress-energy tensor with a given central charge $c$. This
net is fundamental and is contained in any local conformal net [21]. If $c$ is
in the discrete series, thus $c<1$, the net ${\rm Vir}_{c}$ is completely
rational, so there exists a unique KMS state by the first part of our work
[7].
In the case $c=1$ we are able to classify all the KMS states. The primary
(locally normal) KMS states of the ${\rm Vir}_{1}$ net w.r.t. translations are
in one-to-one correspondence with positive real numbers
$\left|q\right|\in{\mathbb{R}}^{+}$; each state $\varphi^{\left|q\right|}$ is
uniquely determined by its value on the stress-energy tensor $T$:
$\varphi^{\left|q\right|}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx.$
The geometric KMS state corresponds to $q=0$, because it is the restriction of
the geometric KMS state on the $U(1)$-current net, and the corresponding value
of the ‘energy density’ $\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}$ is the
lowest in the set of the KMS states. We construct these KMS states by
composing the geometric state with automorphisms on the larger $U(1)$-current
net.
We mention that, as a tool here, we adapt the Araki-Haag-Kastler-Takesaki
theorem to locally normal systems with the help of split property. We show
that, if we have an inclusion of split nets with a conditional expectation,
then any extremal invariant state on the smaller net extends to the larger
net. The original theorem will be discussed in detail, since we need an
extension of a KMS state on the fixed point subnet to the whole net.
Furthermore, we warn the reader that the original proof of the theorem appears
to be incomplete (see Appendix A), yet we are able to give a complete proof
for the case of split nets (Corollary 3.11), which suffices for our purpose.
Then we consider the case $c>1$. In this case we produce a continuous family
which is probably exhaustive. While we leave open the problem of the
completeness of this family, we mention that the formulae on polynomials of
fields should be useful. There is a set of primary (locally normal) KMS states
of the ${\rm Vir}_{c}$ net with $c>1$ w.r.t. translations in one-to-one
correspondence with positive real numbers $\left|q\right|\in{\mathbb{R}}^{+}$;
each state $\varphi^{\left|q\right|}$ can be evaluated on the stress-energy
tensor
$\varphi^{\left|q\right|}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx$
and the geometric KMS state corresponds to
$q=\frac{1}{\beta}\sqrt{\frac{\pi\left(c-1\right)}{6}}$ and energy density
$\frac{\pi c}{12\beta^{2}}$. It is even possible to evaluate
$\varphi^{\left|q\right|}$ on polynomials of the stress-energy tensor and
these values are already determined by the value above on $T(f)$, hence by the
number $|q|$. This should give an important information for the complete
classification.
We shall also consider a non-local rational model. We will see that there is
only one KMS state at each temperature in the free Fermi model. This model
contains the Virasoro net ${\rm Vir}_{c}$ with $c=\frac{1}{2}$, which is
completely rational [21]. Then by a direct application of the results in Part
I, we obtain the existence and the uniqueness of KMS state in this case.
We end this introduction by pointing out that our results are relevant for the
construction of Boundary Quantum Field Theory nets on the interior of the
Lorentz hyperboloid. As shown in particular in [27], one gets such a net from
any translation KMS state on a conformal net on the real line, so our results
directly apply.
## 2 Preliminaries
Here we collect basic notions and technical devices regarding nets of
observables and thermal states. Although our main result in this paper is the
classification of KMS states on certain conformal nets on $S^{1}$, we need to
adapt standard results on $C^{*}$-dynamical systems to our locally normal
systems. Since these materials can be stated for more general nets of von
Neumann algebras, we first formulate the problems without referring to the
circle.
### 2.1 Net of von Neumann algebras on a directed set
#### 2.1.1 Axioms and further properties
Let ${\mathcal{I}}$ be a directed set. We always assume that there is a
countable subset $\\{I_{i}\\}_{i\in{\mathbb{N}}}\subset{\mathcal{I}}$ with
$I_{i}\prec I_{i+1}$ of indices such that for any index $I$ there is some $i$
such that $I\prec I_{i}$. A net (of von Neumann algebras) ${\mathcal{A}}$ on
${\mathcal{I}}$ assigns a von Neumann algebra ${\mathcal{A}}(I)$ to each
element $I$ of ${\mathcal{I}}$ and satisfies the following conditions:
* •
(Isotony) If $I\prec J$ then ${\mathcal{A}}(I)\subset{\mathcal{A}}(J)$.
* •
(Covariance) There is a strongly-continuous unitary representation $U$ of
${\mathbb{R}}$ and an order-preserving action of ${\mathbb{R}}$ on
${\mathcal{I}}$ such that
$U(t){\mathcal{A}}(I)U(t)^{*}={\mathcal{A}}(t\cdot I),$
and for any index $I$ and for any compact set $C\Subset{\mathbb{R}}$, there is
another index $I_{C}$ such that $t\cdot I\prec I_{C}$ for $t\in C$.
Since the net ${\mathcal{A}}$ is directed, it is natural to consider the norm-
closed union of $\\{{\mathcal{A}}(I)\\}_{I\in{\mathcal{I}}}$. We simply denote
$\mathfrak{A}=\overline{\bigcup_{I\in{\mathcal{I}}}{\mathcal{A}}(I)}^{\|\cdot\|}$
and call it the quasilocal algebra. Each algebra ${\mathcal{A}}(I)$ is
referred to as a local algebra. If each local algebra is a factor, then we
call ${\mathcal{A}}$ a net of factors. The adjoint action ${\hbox{\rm
Ad\,}}U(t)$ naturally extends to an automorphism of the quasilocal algebra
$\mathfrak{A}$. We denote by $\tau_{t}$ this action of ${\mathbb{R}}$ and call
it translation (note that in this article $\tau_{t}$ is a one-parameter family
of automorphisms, although in Part I [7, Section 2.3], where we assumed
diffeomorphism covariance, we denoted it by ${\hbox{\rm Ad\,}}U(\tau_{t})$ to
unify the notation).
An automorphism of the net ${\mathcal{A}}$ (not just of $\mathfrak{A}$) is a
family $\\{\gamma_{I}\\}$ of automorphisms of local algebras
$\\{{\mathcal{A}}(I)\\}$ such that if $I\prec J$ then
$\gamma_{J}|_{{\mathcal{A}}(I)}=\gamma_{I}$. Such an automorphism extends by
norm continuity to an automorphism of the quasilocal algebra $\mathfrak{A}$
which preserves all the local algebras. Conversely, any automorphism of
$\mathfrak{A}$ which preserves each local algebra can be described as an
automorphism of the net ${\mathcal{A}}$.
A net ${\mathcal{A}}$ is said to be asymptotically $\gamma$-abelian if there
is an automorphism $\gamma$ of the quasilocal $C^{*}$-algebra $\mathfrak{A}$
implemented by a unitary operator $U(\gamma)$ such that
* •
$\gamma$ is normal on each local algebra ${\mathcal{A}}(I)$ and maps it into
another local algebra ${\mathcal{A}}(\gamma\cdot I)$, where we consider that
the automorphism acts also on the set ${\mathcal{I}}$ of indices by a little
abuse of notation.
* •
for any pair of indices $I,J$ there is a sufficiently large $n$ such that
${\mathcal{A}}(I)$ and ${\mathcal{A}}(\gamma^{n}\cdot
J)=U(\gamma)^{n}{\mathcal{A}}(J)(U(\gamma)^{*})^{n}$ commute,
* •
$\gamma$ and $\tau_{t}$ commute.
It is also possible (and in many cases more natural) to consider a one-
parameter group $\\{\gamma_{s}\\}$ of automorphisms for the notion of
asymptotic $\gamma$-abelianness (and weakly $\gamma$-clustering, see below).
In that case, we assume that $\\{\gamma_{s}\\}$ is implemented by a strongly-
continuous family $\\{U(\gamma_{s})\\}$ and the corresponding conditions above
can be naturally translated.
We say that a net ${\mathcal{A}}$ is split if, for the countable set
$\\{I_{i}\\}$ in the definition of the net, there are type I factors
$\\{{\mathcal{F}}_{i}\\}$ such that
${\mathcal{A}}(I_{i})\subset{\mathcal{F}}_{i}\subset{\mathcal{A}}(I_{i+1})$.
Note that in this case the argument in the appendix of [22] applies.
#### 2.1.2 Examples of nets
The definition of nets looks quite general, but we have principally two types
of examples in mind.
The first comes from the nets on the circle $S^{1}$ which we have studied in
Part I. For the readers’ convenience, we recall the axioms. A conformal net
${\mathcal{A}}$ on $S^{1}$ is a map from the family of intervals
${\mathcal{I}}$ of $S^{1}$ to the family of von Neumann algebras on
${\mathcal{H}}$ such that:
1. (1)
Isotony. If $I_{1}\subset I_{2}$, then
${\mathcal{A}}(I_{1})\subset{\mathcal{A}}(I_{2})$.
2. (2)
Locality. If $I_{1}\cap I_{2}=\emptyset$, then
$[{\mathcal{A}}(I_{1}),{\mathcal{A}}(I_{2})]=0$.
3. (3)
Möbius covariance. There exists a strongly continuous unitary representation
$U$ of the Möbius group ${\rm PSL}(2,{\mathbb{R}})$ such that for any interval
$I$ it holds that
$U(g){\mathcal{A}}(I)U(g)^{*}={\mathcal{A}}(gI),\mbox{ for }g\in{\rm
PSL}(2,{\mathbb{R}}).$
4. (4)
Positivity of energy. The generator of the one-parameter subgroup of rotations
in the representation $U$ is positive.
5. (5)
Existence of vacuum. There is a unique (up to a phase) unit vector $\Omega$ in
${\mathcal{H}}$ which is invariant under the action of $U$, and cyclic for
$\bigvee_{I\in\mathcal{I}}{\mathcal{A}}(I)$.
6. (6)
Conformal covariance. The representation $U$ extends to a projective unitary
representation of ${\rm Diff}(S^{1})$ such that for any interval $I$ and
$x\in{\mathcal{A}}(I)$ it holds that
$\displaystyle U(g){\mathcal{A}}(I)U(g)^{*}={\mathcal{A}}(gI),\mbox{ for
}g\in{\rm Diff}(S^{1}),$ $\displaystyle U(g)xU(g)^{*}=x,\mbox{ if }{\rm
supp}(g)\subset I^{\prime}.$
Strictly speaking, a net is a pair $({\mathcal{A}},U)$ of a family of von
Neumann algebras ${\mathcal{A}}$ and a group representation $U$, yet for
simplicity we denote it simply by ${\mathcal{A}}$.
We identify $S^{1}$ and the one-point compactification
${\mathbb{R}}\cup\\{\infty\\}$ by the Cayley transform. If ${\mathcal{A}}$ is
a conformal net on $S^{1}$, we consider the restriction
${\mathcal{A}}|_{\mathbb{R}}$ with the family of all finite intervals in
${\mathbb{R}}$ as the index set. The translations in the present setting are
the ordinary translations. If we take a finite translation as $\gamma$, this
system is asymptotically $\gamma$-abelian. To consider split property, we can
take the sequence of intervals $I_{n}=(-n,n)$. It is known [17] that each
local algebra of a conformal net is a (type $\mathrm{I\\!I\\!I}_{1}$)
_factor_. This property is exploited when we extend a KMS state on a smaller
net to a larger net.
The second type is a net of observables on Minkowski space ${\mathbb{R}}^{d}$
(see [18] for a general account). In this case the index set is the family of
bounded open sets in ${\mathbb{R}}^{d}$. The group of translations in some
fixed timelike direction plays the role of ”translations”, while a fixed
spacelike translation plays the role of $\gamma$. The net satisfies asymptotic
$\gamma$-abelianness.
In both cases, it is natural to consider the continuous group $\gamma_{s}$ of
(space-)translations for the notion of $\gamma$-abelianness.
### 2.2 States on a net
For a $C^{*}$-algebra $\mathfrak{A}$ and a one-parameter automorphism group
$\\{\tau_{t}\\}$, it is possible to consider KMS states on $\mathfrak{A}$ with
respect to $\tau$. Since our local algebras are von Neumann algebras, it is
natural to consider locally normal objects. Let $\varphi$ be a state on the
quasilocal algebra $\mathfrak{A}$. It is said to be locally normal if each
restriction of $\varphi$ to a local algebra ${\mathcal{A}}(I)$ is normal. A
$\beta$-KMS state $\varphi$ on $\mathfrak{A}$ with respect to $\tau$ is a
state with the following properties: for any $x,y\in\mathfrak{A}$, there is an
analytic function $f$ in the interior of $D_{\beta}:=\\{0\leq\Im
z\leq\beta\\}$ where $\Im$ means the imaginary part, continuous on
$D_{\beta}$, such that
$f(t)=\varphi(x\tau_{t}(y)),f(t+i\beta)=\varphi(\tau_{t}(y)x).$ (1)
The parameter $\frac{1}{\beta}$ is called the temperature. In Part I we
considered only the case $\beta=1$ since our main subject were the conformal
nets, in which case the phase structure is uniform with respect to $\beta$.
Furthermore, we studied completely rational models and proved that they admit
only one KMS state at each temperature. Also in this Part II the main examples
are conformal, but these models admit continuously many different KMS states
and it should be useful to give concrete formulae which involve also the
temperature.
A KMS state $\varphi$ is said to be primary if the GNS representation of
$\mathfrak{A}$ with respect to $\varphi$ is factorial, i.e.,
$\pi_{\varphi}(\mathfrak{A})^{\prime\prime}$ is a factor. Any KMS states can
be decomposed into primary states [39, Theorem 4.5] in many practical
situation, for example if the net is split or if each local algebra is a
factor. Hence, to classify KMS states of a given system, it is enough to
consider the primary ones.
If the net ${\mathcal{A}}$ comes from a conformal net on $S^{1}$, namely if we
assume the diffeomorphism covariance, we saw in Part I that there is at least
one KMS state, the geometric state $\varphi_{\rm geo}$ [7, Section 2.8]. It is
easy to obtain a formula for $\varphi_{\rm geo}$ with general temperature
$\frac{1}{\beta}$. We exhibit it for later use: let
$\omega:=\langle\Omega,\cdot\Omega\rangle$ be the vacuum state, then
$\varphi_{\rm geo}:=\omega\circ{\rm Exp}_{\beta}$, where, for any
$I\Subset{\mathbb{R}}$, ${\rm Exp}_{\beta}|_{{\mathcal{A}}(I)}={\hbox{\rm
Ad\,}}U(g_{\beta,I})|_{{\mathcal{A}}(I)}$ and $g_{\beta,I}$ is a
diffeomorphism of ${\mathbb{R}}$ with compact support such that for $t\in I$
it holds that $g_{\beta,I}(t)=e^{\frac{2\pi t}{\beta}}$.
If $\varphi$ is $\gamma$-invariant (invariant under an automorphism $\gamma$
or a one-parameter group $\\{\gamma_{s}\\}$) and cannot be written as a linear
combination of different locally normal $\gamma$-invariant states, then it is
said to be extremal $\gamma$-invariant.
We denote the GNS representation of $\mathfrak{A}$ with respect to $\varphi$
by $\pi_{\varphi}$, the Hilbert space by ${\mathcal{H}}_{\varphi}$ and the
vector which implements the state $\varphi$ by $\Omega_{\varphi}$. If
$\varphi$ is invariant under the action of an automorphism $\tau_{t}$
(respectively $\gamma$, $\gamma_{s}$), we denote by $U_{\varphi}(t)$ (resp.
$U_{\varphi}(\gamma)$, $U_{\varphi}(\gamma_{s})$) the canonical unitary
operator which implements $\tau_{t}$ (resp. $\gamma$, $\gamma_{s}$) and leaves
$\Omega_{\varphi}$ invariant. If $\varphi$ is locally normal, the GNS
representation $\pi_{\varphi}$ is locally normal as well, namely the
restriction of $\pi_{\varphi}$ to each ${\mathcal{A}}(I)$ is normal. Indeed,
let us denote the restriction $\varphi_{i}:=\varphi|_{{\mathcal{A}}(I_{i})}$.
The representation $\pi_{\varphi_{i}}$ is normal on ${\mathcal{A}}(I_{i})$.
The Hilbert space is the increasing union of ${\mathcal{H}}_{\varphi_{i}}$ and
the restriction of $\pi_{\varphi}$ to ${\mathcal{A}}(I_{i})$ on
${\mathcal{H}}_{\varphi_{j}}$ ($i\leq j$) is $\pi_{\varphi_{j}}$, hence is
normal. Then $\pi_{\varphi}|_{{\mathcal{A}}(I_{i})}$ is normal.
Furthermore, the map $t\mapsto U_{\varphi}(t)$ is weakly (and hence strongly)
continuous, since the one-parameter automorphism $\tau_{t}$ is weakly (or even
*-strongly) continuous and $U_{\varphi}(t)$ is defined as the closure of the
map
$\pi_{\varphi}(x)\Omega_{\varphi}\longmapsto\pi_{\varphi}(\tau_{t}(x))\Omega_{\varphi}.$
Thus the weak continuity of $t\mapsto U_{\varphi}(t)$ follows from the local
normality of $\pi_{\varphi}$ and boundedness of $U_{\varphi}(t)$, which
follows from the invariance of $\varphi$. By the same reasoning, if there is a
one-parameter family $\gamma_{s}$, the GNS implementation
$U_{\varphi}(\gamma_{s})$ is weakly continuous.
If for any locally normal $\gamma$-invariant state $\varphi$ the algebra
$E_{0}\pi_{\varphi}(\mathfrak{A})E_{0}$ is abelian, where $E_{0}$ is the
projection onto the space of $U_{\varphi}(\gamma)$-invariant (resp.
$\\{U_{\varphi}(\gamma_{s})\\}$) vectors, then the net ${\mathcal{A}}$ is said
to be $\gamma$-abelian.
A locally normal state $\varphi$ on $\mathfrak{A}$ is said to be weakly
$\gamma$-clustering if it is $\gamma$-invariant and
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\varphi(\gamma^{n}(x)y)=\varphi(x)\varphi(y).$
for any pair of $x,y\in\mathfrak{A}$. For a one parameter group
$\\{\gamma_{s}\\}$, we define $\gamma$-clustering by
$\lim_{N\to\infty}\frac{1}{N}\int_{0}^{N}\varphi(\gamma_{s}(x)y)ds=\varphi(x)\varphi(y).$
At the end of this subsection, we remark that, in our principal examples
coming from conformal nets on $S^{1}$, KMS states are automatically locally
normal by the following general result [39, Theorem 1].
###### Theorem 2.1 (Takesaki-Winnink).
Let ${\mathcal{A}}$ be a net such that ${\mathcal{A}}(I_{i})$ are
$\sigma$-finite properly infinite von Neumann algebras. Then any KMS-state on
${\mathcal{A}}$ is locally normal.
If ${\mathcal{A}}$ is a conformal net on $S^{1}$ defined on a separable
Hilbert space, then each local algebra ${\mathcal{A}}(I)$ is a type
$\mathrm{{I\\!I\\!I}_{1}}$ factor, in particular it is properly infinite, and
obviously $\sigma$-finite, hence Theorem 2.1 applies.
### 2.3 Subnets and group actions
Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be two nets with the same index set
${\mathcal{I}}$ acting on the same Hilbert space. If for each index $I$ it
holds ${\mathcal{A}}(I)\subset{\mathcal{B}}(I)$, then we say that
${\mathcal{A}}$ is a subnet of ${\mathcal{B}}$ and write simply
${\mathcal{A}}\subset{\mathcal{B}}$. We always assume that each inclusion of
algebras has a normal conditional expectation
$E_{I}:{\mathcal{B}}(I)\to{\mathcal{A}}(I)$ such that
* •
(Compatibility) For $I\prec J$ it holds that
$E_{J}|_{{\mathcal{B}}(I)}=E_{I}$.
* •
(Covariance) $\tau_{t}\circ E_{I}=E_{t\cdot I}\circ\tau_{t}$, and
See [26] for a general theory on nets with a conditional expectation.
Principal examples come again from nets of observables on $S^{1}$. As remarked
in Part I [7, Section 2.2], if we have an inclusion of nets on $S^{1}$ there
is always a compatible and covariant family of expectations.
Another case has a direct relation with one of our main results. Let
${\mathcal{A}}$ be a net on ${\mathcal{I}}$ and assume that there is a family
of *-strongly continuous actions $\alpha_{I,g}$ of a compact Lie group $G$ on
${\mathcal{A}}(I)$ such that if $I\subset J$ then
$\alpha_{J,g}|_{{\mathcal{A}}(I)}=\alpha_{I,g}$ and
$\tau_{t}\circ\alpha_{I,g}=\alpha_{t\cdot I,g}\circ\tau_{t}$. By the first
condition (compatibility of $\alpha$) we can extend $\alpha$ to an
automorphism of the quasilocal $C^{*}$-algebra $\mathfrak{A}$, and by the
second condition (covariance of $\alpha$) $\alpha$ and $\tau$ commute. Then
for each index $I$ we can consider the fixed point subalgebra
${\mathcal{A}}(I)^{G}=:{\mathcal{A}}^{G}(I)$. Then ${\mathcal{A}}^{G}$ is
again a net on ${\mathcal{I}}$. Furthermore, since the group is compact, there
is a unique normalized invariant mean $dg$ on $G$. Then it is easy to see that
the map $E(x):=\int_{G}\alpha_{g}(x)dg$ is a locally normal conditional
expectation ${\mathcal{A}}\to{\mathcal{A}}^{G}$. The group $G$ is referred to
as the gauge group of the inclusion ${\mathcal{A}}^{G}\subset{\mathcal{A}}$.
The *-strong continuity of the group action is valid, for example, when the
group action is implemented by weakly (hence strongly) continuous unitary
representation of $G$. In fact, if $g_{n}\to g$, then $U_{g_{n}}\to U_{g}$
strongly, hence $\alpha_{g_{n}}(x)={\hbox{\rm Ad\,}}U_{g_{n}}(x)\to{\hbox{\rm
Ad\,}}U_{g}(x)$ and $\alpha_{g_{n}}(x^{*})={\hbox{\rm
Ad\,}}U_{g_{n}}(x^{*})\to{\hbox{\rm Ad\,}}U_{g}(x^{*})$ strongly since
$\\{U_{g_{n}}\\}$ is bounded. This is the case, as are our principal examples,
when the net is defined in the vacuum representation (see [7, Section 2.1])
and the vacuum state is invariant under the action of $G$.
If the net ${\mathcal{A}}$ is asymptotically $\gamma$-abelian, then we always
assume that $\gamma$ commutes with $\alpha_{g}$.
### 2.4 $C^{*}$-dynamical systems
A pair of a $C^{*}$-algebra $\mathfrak{A}$ and a pointwise norm-continuous
one-parameter automorphism group $\alpha_{t}$ is called a $C^{*}$-dynamical
system. The requirement of pointwise norm-continuity is strong enough to allow
extensive general results. Although our main objects are not $C^{*}$-dynamical
systems, we recall here a standard result.
All notions defined for nets, namely compact (gauge) group action, asymptotic
$\gamma$-abelianness, $\gamma$-abelianness, weakly $\gamma$-clustering of
states and inclusion of systems, and corresponding results in Section 3,
except Corollary 3.6, have variations for $C^{*}$-dynamical system ([20], see
also [3]). Among them, important is the theorem of Araki-Haag-Kastler-Takesaki
[1]: for a $C^{*}$-dynamical system with the fixed point subalgebra with
respect to a gauge group, any KMS state on the smaller algebra extends to a
KMS state with respect to a slightly different one-parameter group. In fact,
to obtain the full extension, it is necessary to assume that the state
$\varphi$ is faithful and that there is the net structure. A detailed
discussion is collected in Appendix A.
### 2.5 Regularization
To classify the KMS states on ${\rm Vir}_{1}$, we need to extend a KMS state
on ${\rm Vir}_{1}$ to ${\mathcal{A}}_{SU(2)_{1}}$ (explained below). Since
${\rm Vir}_{1}$ is the fixed point subnet of ${\mathcal{A}}_{SU(2)_{1}}$ with
respect to the action of $SU(2)$ [35], one would like to apply Theorem A.1.
The trouble is, however, that the theorem applies only to $C^{*}$-dynamical
systems where the actions of the translation group and the gauge group are
pointwise continuous in norm. The pointwise norm-continuity seems essential in
the proof and it is not straightforward to modify it for locally normal
systems; we instead aim to reduce our cases to $C^{*}$-dynamical systems.
More precisely, we assume that the net ${\mathcal{A}}$ has a locally
*-strongly continuous action $\tau$ of translations (covariance, in subsection
2.1.1) and $\alpha$ of a gauge group $G$ (subsection 2.3), has an automorphism
$\gamma$ (subsection 2.1.1) and they commute, then we construct a
$C^{*}$-dynamical system $({\mathfrak{A}_{\mathrm{r}}},\tau)$ with the regular
subalgebra ${\mathfrak{A}_{\mathrm{r}}}$ *-strongly dense in $\mathfrak{A}$.
###### Proposition 2.2.
For any net ${\mathcal{A}}$ with locally *-strongly continuous action
$\tau\times\alpha$ of ${\mathbb{R}}\times G$ and an automorphism $\gamma$
commuting with $\tau\times\alpha$, the set ${\mathfrak{A}_{\mathrm{r}}}$ of
elements of the quasilocal algebra $\mathfrak{A}$ on which $\tau\times\alpha$
act pointwise continuously in norm is a ($\tau\times\alpha,\gamma$)-globally
invariant *-strongly dense $C^{*}$-subalgebra. Any local element
$x\in{\mathcal{A}}(I)$ can be approximated *-strongly by a bounded sequence
from ${\mathfrak{A}_{\mathrm{r}}}\cap{\mathcal{A}}(I_{C})$ for some
$I_{C}\succ I$.
If we consider a continuous action $\gamma_{s}$, then we can take
${\mathfrak{A}_{\mathrm{r}}}$ such that ${\mathfrak{A}_{\mathrm{r}}}$ is
$\\{\gamma_{s}\\}$-invariant and the action of $\gamma$ is pointwise
continuous in norm.
###### Proof.
Let ${\mathfrak{A}_{\mathrm{r}}}$ be the set of elements of $\mathfrak{A}$ on
which ${\mathbb{R}}\times G$ acts pointwise continuously in norm;
${\mathfrak{A}_{\mathrm{r}}}$ is clearly a $\ast$-algebra and is norm-closed,
hence is a $C^{*}$-subalgebra of $\mathfrak{A}$. Global invariance follows
since $\tau$, $\gamma$ and $\alpha$ commute.
Let $x$ be an element of some local algebra ${\mathcal{A}}(I)$. We consider
the smearing of $x$ with a smooth function $f$ on ${\mathbb{R}}\times G$ with
compact support
$x_{f}:=\int_{{\mathbb{R}}\times G}f(t,g)\alpha_{g}(\tau_{t}(x))dtdg.$
By the definition of net and the compactness of the support of $f$, the
integrand belongs to another local algebra ${\mathcal{A}}(I_{C})$ and the
actions $\alpha$ and $\tau$ are normal on ${\mathcal{A}}(I_{C})$, hence the
weak integral can be defined. Smoothness of the actions on $x_{f}$ is easily
seen from the smoothness of $f$, thus $x_{f}\in{\mathfrak{A}_{\mathrm{r}}}$.
Take a sequence of functions approximating the Dirac distribution, i.e. a
sequence of $f_{n}$ with $\int_{{\mathbb{R}}\times G}f_{n}(x,g)dxdg=1$ and
whose supports shrink to the unit element in the group ${\mathbb{R}}\times G$,
then $x_{f_{n}}$ converges *-strongly to $x$, since group actions $\alpha$ and
$\tau$ are *-strongly continuous by assumption. Thus, any element $x$ in a
local algebra ${\mathcal{A}}(I)$ can be approximated by a bounded sequence of
smeared elements in a slightly larger local algebra ${\mathcal{A}}(I_{C})$. As
any element in $\mathfrak{A}$ can be approximated in norm (and a fortiori
*-strongly) by local elements, ${\mathfrak{A}_{\mathrm{r}}}$ is *-strongly
dense in $\mathfrak{A}$.
Moreover, as the actions are norm continuous on ${\mathfrak{A}_{\mathrm{r}}}$,
if $x\in{\mathfrak{A}_{\mathrm{r}}}$ then $x_{f_{n}}$ converges in norm to
$x$. This means that the norm closure of the linear space generated by the
smeared elements $\\{x_{f}\\}$ is an algebra and coincides with
${\mathfrak{A}_{\mathrm{r}}}$.
For a continuous action $\gamma_{s}$, it is enough to consider a smearing on
${\mathbb{R}}\times G\times{\mathbb{R}}$ with respect to the action of
$\tau\times\alpha\times\gamma$. ∎
###### Remark 2.3.
If ${\mathcal{A}}$ is the fixed point subnet of ${\mathcal{B}}$ in the sense
of Section 2.3 ($\mathfrak{A}=\mathfrak{B}^{G}$), then
${\mathfrak{A}_{\mathrm{r}}}={\mathfrak{B}_{\mathrm{r}}}^{G}$. Indeed, from
${\mathfrak{A}_{\mathrm{r}}}\subset{\mathfrak{B}_{\mathrm{r}}}\subset\mathfrak{B}$
it follows that
${\mathfrak{A}_{\mathrm{r}}}\subset{\mathfrak{B}_{\mathrm{r}}}^{G}\subset\mathfrak{B}^{G}=\mathfrak{A}$,
since the elements of $\mathfrak{A}$ are $G$-invariant; on the other side,
from ${\mathfrak{B}_{\mathrm{r}}}^{G}\subset\mathfrak{A}$ it follows that
${\mathfrak{B}_{\mathrm{r}}}^{G}\subset{\mathfrak{A}_{\mathrm{r}}}$, since the
elements of ${\mathfrak{B}_{\mathrm{r}}}^{G}$ are regular. Thus we obtain an
inclusion of $C^{*}$-dynamical systems
${\mathfrak{A}_{\mathrm{r}}}\subset{\mathfrak{B}_{\mathrm{r}}}$.
###### Lemma 2.4.
If a state $\varphi$ on the net ${\mathcal{A}}$ is weakly $\gamma$-clustering,
then the restriction of $\varphi$ to the regular system
$({\mathfrak{A}_{\mathrm{r}}},\tau)$ is again weakly $\gamma$-clustering.
###### Proof.
The definition of weakly $\gamma$-clustering of a smaller algebra
${\mathfrak{A}_{\mathrm{r}}}$ refers only to elements in
${\mathfrak{A}_{\mathrm{r}}}$, hence it is weaker than the counterpart for
$\mathfrak{A}$. ∎
###### Lemma 2.5.
Let $\varphi$ be a locally normal state on $\mathfrak{A}$ which is a KMS state
on ${\mathfrak{A}_{\mathrm{r}}}$. Then $\varphi$ is a KMS state on
$\mathfrak{A}$.
###### Proof.
We only have to confirm the KMS condition for $\mathfrak{A}$. Let
$x,y\in\mathfrak{A}$ and take bounded sequences $\\{x_{n}\\},\\{y_{n}\\}$ from
${\mathfrak{A}_{\mathrm{r}}}$ which approximate $x,y$ *-strongly. Since
$\varphi$ is a KMS state on ${\mathfrak{A}_{\mathrm{r}}}$, there is an
analytic function $f_{n}$ such that
$\displaystyle f_{n}(t)=\varphi(x_{n}\tau_{t}(y_{n})),$ $\displaystyle
f_{n}(t+i)=\varphi(\tau_{t}(y_{n})x_{n}).$
In terms of GNS representation with respect to $\varphi$, these functions can
be written as
$\displaystyle\varphi(x_{n}\tau_{t}(y_{n}))=\langle\pi_{\varphi}(x_{n}^{*})\Omega_{\varphi},U_{\varphi}(t)\pi_{\varphi}(y_{n})\Omega_{\varphi}\rangle,$
$\displaystyle\varphi(\tau_{t}(y_{n})x_{n})=\langle
U_{\varphi}(t)\pi_{\varphi}(y_{n}^{*})\Omega_{\varphi},\pi_{\xi}(x_{n})\Omega_{\varphi}\rangle.$
Note that $\pi_{\varphi}(x_{n})$ (respectively $\pi_{\varphi}(y_{n})$) is
*-strongly convergent to $\pi_{\varphi}(x)$ (resp. $\pi_{\varphi}(y)$) since
the sequence $\\{x_{n}\\}$ (resp. $\\{y_{n}\\}$) is bounded. Let us denote a
common bound of norms by $M$. We can estimate the difference as follows:
$\displaystyle\left|\varphi(x\tau_{t}(y))-\varphi(x_{n}\tau_{t}(y_{n}))\right|$
$\displaystyle=$
$\displaystyle\left|\langle\pi_{\varphi}(x^{*})\Omega_{\varphi},U_{\varphi}(t)\pi_{\varphi}(y)\Omega_{\varphi}\rangle-\langle\pi_{\varphi}(x_{n}^{*})\Omega_{\varphi},U_{\varphi}(t)\pi_{\varphi}(y_{n})\Omega_{\varphi}\rangle\right|$
$\displaystyle\leq$ $\displaystyle
M\left\|\pi_{\varphi}(x^{*})-\pi_{\varphi}(x_{n}^{*})\Omega_{\varphi}\right\|+M\left\|\pi_{\varphi}(y)-\pi_{\varphi}(y_{n})\Omega_{\varphi}\right\|$
and this converges to $0$ uniformly with respect to $t$. Analogously we see
that $\varphi(\tau_{t}(y_{n})x_{n})$ converges to
$\varphi(\tau_{t}(y_{n})x_{n})$ uniformly. Then by the three-line theorem
(which can be applied because $f_{n}$ are bounded: see [3, Prop. 5.3.7])
$f_{n}(z)$ is uniformly convergent on the strip $0\leq\Im z\leq 1$ and the
limit $f$ is an analytic function. Obviously $f$ connects
$\varphi(x\tau_{t}(y))$ and $\varphi(t_{t}(y)x)$, hence $\varphi$ satisfies
the KMS condition for $\mathfrak{A}$. ∎
###### Lemma 2.6.
Let $\varphi$ be a locally normal state on $\mathfrak{A}$ which is a KMS state
on ${\mathfrak{A}_{\mathrm{r}}}$. If each local algebra ${\mathcal{A}}(I)$ is
a factor, then $\varphi$ is faithful on $\mathfrak{A}$.
###### Proof.
By Lemma 2.5, $\varphi$ is a KMS state on $\mathfrak{A}$. The GNS
representation $\pi_{\varphi}$ is locally normal and hence locally faithful
since each local algebra is a factor, then it is faithful also on the norm
closure $\mathfrak{A}$. On the other hand, [3, Corollary 5.3.9] (which applies
also to locally normal systems) tells us that the GNS vector
$\Omega_{\varphi}$ is separating for
$\pi_{\varphi}(\mathfrak{A})^{\prime\prime}$, thus
$\varphi(\cdot)=\langle\Omega_{\varphi},\pi_{\varphi}(\cdot)\Omega_{\varphi}\rangle$
is faithful. ∎
## 3 Extension results
### 3.1 Extension of clustering states
In this section we provide variations of standard results on $C^{*}$-dynamical
systems. Parts of the proofs of Lemma 3.7 and Proposition 3.8 are adaptations
of [20] for the locally normal case, as we shall see. In particular, when we
consider the one-parameter group $\\{\gamma_{s}\\}$, we need local normality
to assure the weak-continuity of the GNS implementation
$\\{U_{\varphi}(\gamma_{s})\\}$. For some propositions we need the split
property in connection with local normality.
###### Remark 3.1.
If we treat one-parameter group $\\{\gamma_{s}\\}$, in the following
propositions (except for Proposition 3.5, where the corresponding modification
shall be explicitly indicated) it is enough to take just the von Neumann
algebra
$(\pi_{\varphi}(\mathfrak{A})\cup\\{U_{\varphi}(\gamma_{s})\\})^{\prime\prime}$
and to consider invariance under $\\{\gamma_{s}\\}$ or
$\\{U_{\varphi}(\gamma_{s})\\}$ and the corresponding notion of
$\gamma$-clustering property of states. Since $\\{U_{\varphi}(\gamma_{s})\\}$
is weakly continuous, we can utilize the mean ergodic theorem in this case as
well.
The following proposition is known (see e.g. [3, 20]).
###### Proposition 3.2.
A state $\varphi$ is extremal $\gamma$-invariant if and only if
$(\pi_{\varphi}(\mathfrak{A})\cup\\{U_{\varphi}(\gamma)\\})^{\prime\prime}=B({\mathcal{H}}_{\varphi})$.
Note that any finite convex decomposition of a locally normal state consists
of locally normal states, because a state dominated by a normal state is
normal, too.
The following is essential to our argument of extension for locally normal
systems.
###### Theorem 3.3 ([14], A 86).
Let ${\mathcal{H}}=\int^{\oplus}_{X}{\mathcal{H}}_{\lambda}d\mu(\lambda)$ be a
direct integral Hilbert space,
$T_{i}=\int^{\oplus}_{X}T_{i,\lambda}\mu(\lambda)$ be a sequence of
decomposable operators, ${\mathcal{M}}$ be the von Neumann algebra generated
by $\\{T_{i}\\}$, and ${\mathcal{M}}_{\lambda}$ be the von Neumann algebra
generated by $\\{T_{i,\lambda}\\}$. Then the algebra ${\mathcal{Z}}$ of
diagonalizable operators is maximally commutative in ${\mathcal{M}}^{\prime}$
if and only if ${\mathcal{M}}_{\lambda}=B({\mathcal{H}}_{\lambda})$ for almost
all $\lambda$.
Since we assume the split property of the net ${\mathcal{A}}$, there is a
sequence of indices $I_{i}$ and type I factors ${\mathcal{F}}_{i}$. Let
$K_{i}$ be the ideal of compact operators of ${\mathcal{F}}_{i}$, and
$\mathfrak{K}$ be the $C^{*}$-algebra generated by $\\{K_{i}\\}$. With a
slight modification about the index set, the following applies to our
situation.
###### Theorem 3.4 ([22], Proposition 56).
Let $\pi$ be a locally normal representation of a split net ${\mathcal{A}}$ on
a separable Hilbert space and denote by $\pi_{\mathfrak{K}}$ the restriction
to the algebra $\mathfrak{K}$. If we have a disintegration
$\pi_{\mathfrak{K}}=\int^{\oplus}_{X}\pi_{\lambda}d\mu(\lambda),$
then $\pi_{\lambda}$ extends to a locally normal representation
$\widetilde{\pi}_{\lambda}$ of $\mathfrak{A}$ for almost all $\lambda$.
We need further a variation of a standard result. The next Proposition would
follow from a general decomposition of an invariant state into extremal
invariant states and [39, Corollary 5.3] which affirms that any decomposition
is locally normal. In the present article we take another way through
decomposition of representation.
###### Proposition 3.5.
Let $\varphi$ be a locally normal $\gamma$-invariant state of the
$C^{*}$-algebra $\mathfrak{A}$ and $\pi_{\varphi}$ be the corresponding GNS
representation, then $\varphi$ decomposes into an integral of locally normal
extremal $\gamma$-invariant states.
###### Proof.
We take a separable subalgebra $\mathfrak{K}$ as above analogously as in [22].
We fix a maximally abelian subalgebra $\mathfrak{m}$ in the commutant
$(\pi_{\mathfrak{K}}(\mathfrak{K})\cup\\{U_{\varphi}(\gamma)\\})^{\prime}$.
Since $\mathfrak{K}$ is separable, we can apply [13, Section II.3.1 Corollary
1] to obtain a measurable space $X$, a standard measure $\mu$ on $X$, a field
of Hilbert spaces ${\mathcal{H}}_{\lambda}$ and a field of representations
$\pi_{\lambda}$ such that the original restricted representation
$\pi_{\mathfrak{K}}$ is unitarily equivalent to the integral representation:
$\pi_{\mathfrak{K}}=\int^{\oplus}_{X}\pi_{\lambda}d\mu(\lambda)$
and $\mathfrak{m}=L^{\infty}(X,\mu)$. Now, by Theorem 3.4 (note that the
representation space ${\mathcal{H}}_{\varphi}$ of the GNS representation with
respect to a locally normal state $\varphi$ is separable since we assume that
the original net ${\mathcal{A}}$ is represented on a separable Hilbert space
${\mathcal{H}}$), we may assume that $\pi_{\lambda}$ is locally normal for
almost all $\lambda$, hence it extends to a locally normal representation
$\widetilde{\pi}_{\lambda}$ and the original representation $\pi_{\varphi}$
decomposes into
$\pi_{\varphi}=\int^{\oplus}_{X}\widetilde{\pi}_{\lambda}d\mu(\lambda).$
Furthermore, the GNS vector $\Omega_{\varphi}$ decomposes into a direct
integral
$\Omega_{\varphi}=\int^{\oplus}_{X}\Omega_{\lambda}d\mu(\lambda).$
The representative $U_{\varphi}(\gamma)$ decomposes into direct integrals as
well, since $\mathfrak{m}$ commutes with $U_{\varphi}(\gamma)$:
$U_{\varphi}(\gamma)=\int^{\oplus}_{X}U_{\lambda}(\gamma)d\mu(\lambda).$
From this it holds that $\Omega_{\lambda}$ is invariant under
$U_{\lambda}(\gamma)$, thus the state
$\varphi_{\lambda}(\cdot):=\langle\Omega_{\lambda},\pi_{\lambda}(\cdot)\Omega_{\lambda}\rangle$
is invariant under the action of $\gamma$, for almost all $\lambda$. By the
definition of the direct integral it holds that
$\varphi=\int^{\oplus}_{X}\varphi_{\lambda}d\mu(\lambda).$
It is obvious that $\varphi_{\lambda}$ is locally normal.
It remains to show that each $\varphi_{\lambda}$ is extremal
$\gamma$-invariant. By assumption, $\mathfrak{m}$ is maximally commutative in
the commutant of
$(\pi_{\mathfrak{K}}(\mathfrak{K})\cup\\{U_{\varphi}(\gamma)\\})^{\prime\prime}$.
This von Neumann algebra is generated by a countable dense subset
$\\{\pi_{\mathfrak{K}}(x_{i})\\}$ and a representative $U_{\varphi}(\gamma)$.
Then, by Theorem 3.3, this is equivalent to the condition that
$(\\{\pi_{\lambda}(x_{i})\\}\cup\\{U_{\lambda}(\gamma)\\})^{\prime\prime}=B({\mathcal{H}}_{\lambda})$,
namely $\varphi_{\lambda}$ is extremal $\gamma$-invariant.
If we consider a continuous family $\\{\gamma_{s}\\}$, we only have to take a
countable family of operators
$\\{\pi_{\mathfrak{K}}(x_{i})\\}\cup\\{U_{\varphi}(\gamma_{s})\\}_{s\in{\mathbb{Q}}}$.
∎
###### Corollary 3.6.
Let ${\mathcal{A}}\subset{\mathcal{B}}$ be an inclusion of split nets with a
locally normal conditional expectation which commutes with $\gamma$. If
$\varphi$ is an extremal $\gamma$-invariant state on $\mathfrak{A}$, then
$\varphi$ extends to an extremal $\gamma$-invariant state on the quasilocal
algebra $\mathfrak{B}$ of the net ${\mathcal{B}}$.
###### Proof.
The composition $\varphi\circ E$ is a $\gamma$-invariant state on
$\mathfrak{B}$. By Proposition 3.5, $\varphi\circ E$ can be written as an
integral of extremal $\gamma$-invariant states:
$\varphi\circ E=\int^{\oplus}_{X}\psi_{\lambda}d\mu(\lambda).$
By assumption, the restriction of $\varphi\circ E$ to $\mathfrak{A}$ is equal
to $\varphi$, which is extremal $\gamma$-invariant, hence the restriction
$\psi_{\lambda}|_{\mathcal{A}}$ coincides with $\varphi$ for almost all
$\lambda$. Hence, each of $\psi_{\lambda}$ is an extremal $\gamma$-invariant
extension of $\varphi$. ∎
###### Lemma 3.7.
If the net ${\mathcal{A}}$ is asymptotically $\gamma$-abelian, then it is
$\gamma$-abelian.
###### Proof.
Let $\varphi$ be a locally normal $\gamma$-invariant state on $\mathfrak{A}$.
The action of $\gamma$ is canonically unitarily implemented by
$U_{\varphi}(\gamma)$. Let $E_{0}$ be the projection onto the space of
$U_{\varphi}(\gamma)$-invariant vectors in ${\mathcal{H}}_{\varphi}$ and
$\Psi_{1},\Psi_{2}\in E_{0}{\mathcal{H}}_{\varphi}$. Let us put
$\psi(x)=\langle\Psi_{1},\pi_{\varphi}(x)\Psi_{2}\rangle$.
By the assumption of asymptotically $\gamma$-abelianness, it is easy to see
that
$\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\psi(\gamma^{n}(x)y)=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\psi(y\gamma^{n}(x)).$
On the other hand, by the mean ergodic theorem we have
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\psi(\gamma^{n}(x)y)$
$\displaystyle=$
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\langle\Psi_{1},U_{\varphi}(\gamma)^{n}\pi_{\varphi}(x)(U_{\varphi}(\gamma)^{*})^{n}\pi_{\varphi}(y)\Psi_{2}\rangle$
$\displaystyle=$
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\langle\Psi_{1},\pi_{\varphi}(x)(U_{\varphi}(\gamma)^{*})^{n}\pi_{\varphi}(y)\Psi_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi_{1},\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)\Psi_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi_{1},E_{0}\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)E_{0}\Psi_{2}\rangle.$
Similarly we have
$\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\psi(y\gamma^{n}(x))=\langle\Psi_{1},E_{0}\pi_{\varphi}(y)E_{0}\pi_{\varphi}(x)E_{0}\Psi_{2}\rangle$.
Together with the above equality we see that
$\langle\Psi_{1},E_{0}\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)E_{0}\Psi_{2}\rangle=\langle\Psi_{1},E_{0}\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)E_{0}\Psi_{2}\rangle$,
which means that $E_{0}\pi_{\varphi}(x)E_{0}$ and $E_{0}\pi_{\varphi}(y)E_{0}$
commute. ∎
###### Proposition 3.8.
If $\varphi$ is a locally normal $\gamma$-invariant state on the
asymptotically $\gamma$-abelian net ${\mathcal{A}}$, then the following are
equivalent:
* (a)
in the GNS representation $\pi_{\varphi}$, the space of invariant vectors
under $U_{\varphi}(\gamma)$ is one dimensional.
* (b)
$\varphi$ is weakly $\gamma$-clustering.
* (c)
$\varphi$ is extremal $\gamma$-invariant.
###### Proof.
First we show the equivalence (a)$\Leftrightarrow$(b). By the asymptotic
$\gamma$-abelianness we have
$\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi(\gamma^{n}(x)y)=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi(y\gamma^{n}(x)),$
and it holds by the mean ergodic theorem that
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi(\gamma^{n}(x)y)$
$\displaystyle=$
$\displaystyle\langle\Omega_{\varphi},E_{0}\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)E_{0}\Omega_{\varphi}\rangle,$
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi(y\gamma^{n}(x))$
$\displaystyle=$
$\displaystyle\langle\Omega_{\varphi},E_{0}\pi_{\varphi}(y)E_{0}\pi_{\varphi}(x)E_{0}\Omega_{\varphi}\rangle.$
Now if $E_{0}$ is one dimensional, then it holds that
$\langle\Omega_{\varphi},E_{0}\pi_{\varphi}(y)E_{0}\pi_{\varphi}(x)E_{0}\Omega_{\varphi}\rangle=\langle\Omega_{\varphi},\pi_{\varphi}(y)\Omega_{\varphi}\rangle\langle\Omega_{\varphi},\pi_{\varphi}(x)\Omega_{\varphi}\rangle=\langle\Omega_{\varphi},E_{0}\pi_{\varphi}(x)E_{0}\pi_{\varphi}(y)E_{0}\Omega_{\varphi}\rangle,$
and this is weakly $\gamma$-clustering.
Conversely, if ${\mathcal{A}}$ is weakly $\gamma$-clustering, the above
equality holds and it implies that $E_{0}$ is one dimensional, since
$\Omega_{\varphi}$ is cyclic for $\pi_{\varphi}(\mathfrak{A})$.
Next we see the implication (a)$\Rightarrow$(c). Let us take a projection $P$
in the commutant
$(\pi_{\varphi}(\mathfrak{A})\cup\\{U_{\varphi}(\gamma)\\})^{\prime}$. Since
$P$ commutes with $U_{\varphi}(\gamma)$, $P\Omega_{\varphi}$ is again an
invariant vector. By assumption the space of invariant vector is one
dimensional, it holds that $P\Omega_{\varphi}=\Omega_{\varphi}$ or that
$P\Omega_{\varphi}=0$. We may assume that $P\Omega_{\varphi}=\Omega_{\varphi}$
(otherwise consider ${\mathbbm{1}}-P$). By the cyclicity of $\Omega_{\varphi}$
for $\pi_{\varphi}(\mathfrak{A})$, it is separating for
$\pi_{\varphi}(\mathfrak{A})^{\prime}$, thus $P={\mathbbm{1}}$.
Finally, we prove the implication (c)$\Rightarrow$(a). By Lemma 3.7, the
algebra $E_{0}\pi_{\varphi}(\mathfrak{A})E_{0}$ is abelian, but by assumption
(c), $\pi_{\varphi}(\mathfrak{A})\cup\\{U_{\varphi}(\gamma)\\}$ act
irreducibly and $U_{\varphi}(\gamma)$ acts trivially on $E_{0}$. Hence
$E_{0}\pi_{\varphi}(\mathfrak{A})E_{0}$ acts irreducibly on $E_{0}$. This is
possible only if $E_{0}$ is one dimensional. ∎
### 3.2 Extension of KMS states
In this section we partly follow the steps in [1]. We give an overview of the
proof of Theorem II.4 of [1] in Appendix A, where some notations are
introduced.
Let ${\mathcal{A}}\subset{\mathcal{B}}$ be an inclusion of asymptotically
$\gamma$-abelian split nets of factors, and suppose that ${\mathcal{A}}$ is
the fixed point subnet of a locally normal action $\alpha$ by a separable
compact group $G$ which commutes with $\gamma$ and $\tau$. We take a weakly
$\gamma$-clustering primary $\tau$-KMS state $\varphi$ on ${\mathcal{A}}$ and
fix a $\gamma$-clustering extension $\psi$ to ${\mathcal{B}}$ (whose existence
is assured by Corollary 3.6).
###### Lemma 3.9.
There is a one-parameter group $\varepsilon_{t}\in Z(G_{\psi},G)$ such that
the restriction of $\psi$ to ${\mathcal{B}}^{G_{\psi}}$ is a faithful KMS
state with respect to
$\tau^{\prime}_{t}:=\tau_{t}\circ\alpha_{\varepsilon_{t}}$.
###### Proof.
We consider the inclusion of $C^{*}$-algebras
${\mathfrak{A}_{\mathrm{r}}}={\mathfrak{B}_{\mathrm{r}}}^{G}\subset{\mathfrak{B}_{\mathrm{r}}}$
and the restriction of $\tau$. This is an inclusion of $C^{*}$-systems. The
restriction of $\psi$ to the regular subalgebra ${\mathfrak{B}_{\mathrm{r}}}$
is still $\gamma$-clustering by Lemma 2.4. We claim that the restriction of
$\psi$ (hence of $\varphi$) to ${\mathfrak{B}_{\mathrm{r}}}^{G}$ (see Remark
2.3) is still a primary KMS state. Indeed, the GNS representation of
$\varphi|_{{\mathfrak{B}_{\mathrm{r}}}^{G}}$ can be identified with a subspace
of the representation $\pi_{\varphi}$ of ${\mathcal{A}}$. By the local
normality, this subspace for ${\mathfrak{B}_{\mathrm{r}}}^{G}$ (which
coincides with ${\mathfrak{A}_{\mathrm{r}}}$) includes the subspace generated
by ${\mathcal{A}}(I)$ for each fixed index set. The whole representation space
of $\pi_{\varphi}$ is the closed union of such subspaces, hence these spaces
coincide. Furthermore, by the local normality,
$\pi_{\varphi}({\mathfrak{B}_{\mathrm{r}}}^{G})^{\prime\prime}$ contains
$\pi_{\varphi}({\mathcal{A}}(I))^{\prime\prime}$ for each $I$. Hence the von
Neumann algebras generated by $\pi_{\varphi}({\mathcal{A}})$ and
$\pi_{\varphi}({\mathfrak{B}_{\mathrm{r}}}^{G})$ coincide and
$\varphi|_{{\mathfrak{B}_{\mathrm{r}}}^{G}}$ is primary.
Now we can apply Lemma A.2 to obtain a one-parameter group $\varepsilon_{t}\in
Z(G_{\psi},G)$ such that $\psi$ restricted to
${\mathfrak{B}_{\mathrm{r}}}^{G_{\psi}}$ is a KMS state with respect to
$\tau^{\prime}_{t}$. Then by Lemmas 2.5, 2.6 we see that $\psi$ is a KMS state
on the net ${\mathcal{B}}^{G_{\psi}}$ and it is faithful. ∎
###### Theorem 3.10.
Let ${\mathcal{A}}\subset{\mathcal{B}}$ be an inclusion of asymptotically
$\gamma$-abelian split nets of factors, and suppose that ${\mathcal{A}}$ is
the fixed point subnet of a locally normal action $\alpha$ by a separable
compact group $G$ which commutes with $\gamma$ and $\tau$. Then, for any
weakly $\gamma$-clustering extension $\psi$ to ${\mathcal{B}}$ of a primary
$\tau$-KMS state $\varphi$ on ${\mathcal{A}}$ (such an extension always exists
by Corollary 3.6), there is a one-parameter subgroup $(\varepsilon\circ\zeta)$
in $G$ such that $\psi$ is a primary $\widetilde{\tau}$-KMS state where
$\widetilde{\tau}_{t}=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$.
The state $\psi$ is automatically faithful.
###### Proof.
The restriction of $\psi$ to ${\mathfrak{B}_{\mathrm{r}}}$ is primary as we
saw in Lemma 3.9.
This time we consider the inclusion
${\mathfrak{B}_{\mathrm{r}}}^{G_{\psi}}\subset{\mathfrak{B}_{\mathrm{r}}}$.
Any locally normal representation of the regularized algebra extends to a
representation of the net on the same Hilbert space, hence it is faithful on
the quasilocal algebra since we treat a net of factors. Then we can apply
Lemma A.3 together with Theorem A.5 to see that there is a one-parameter
subgroup $\zeta_{t}\in G_{\psi}$ such that $\psi|_{\mathfrak{B}_{\mathrm{r}}}$
is a $\widetilde{\tau}$-KMS state where
$\widetilde{\tau}_{t}=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$ and
$\varepsilon_{t}$ is taken from Lemma 3.9. By Lemma 2.5 $\psi$ is a KMS state
on the net ${\mathcal{B}}$. Again by local normality, the primarity of
$\psi|_{\mathfrak{B}_{\mathrm{r}}}$ and $\psi$ are equivalent. The
faithfulness is proved as in Proposition 3.9. ∎
We have the following corollary. Note that the gauge group of a split net is
separable because the underlying Hilbert space is automatically separable.
###### Corollary 3.11.
Let ${\mathcal{A}}\subset{\mathcal{B}}$ be an inclusion of split conformal
nets on the real line ${\mathbb{R}}$, and suppose that ${\mathcal{A}}$ is the
fixed point subnet of a locally normal action $\alpha$ by a compact group $G$
which commutes with translations $\tau$. Then, for every weakly
$\tau$-clustering primary $\tau$-KMS state $\varphi$ on ${\mathcal{A}}$, there
exists a weakly $\tau$-clustering extension $\psi$ to ${\mathcal{B}}$. For any
such an extension $\psi$ there is a one-parameter subgroup
$(\varepsilon\circ\zeta)$ in $G$ such that $\psi$ is a primary
$\widetilde{\tau}$-KMS state where
$\widetilde{\tau}_{t}=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$.
The state $\psi$ is automatically faithful.
## 4 The $U(1)$-current model
From now on, we discuss concrete examples from one-dimensional Conformal Field
Theory. We recall some constructions regarding the $U(1)$-current and discuss
its KMS states for two reasons: being a free field model, it is simple enough
to allow a complete classification of the KMS states, showing an example of
non completely rational model with multiple KMS states; it is useful in the
classification of states for the Virasoro nets, whose restrictions to
${\mathbb{R}}$ are translation-covariant subnets of the $U(1)$-current net.
### 4.1 The $U(1)$-current model
The $U(1)$-current model is the chiral component of the derivative of massless
scalar free field in the 2-dimensional Minkowski space time. See [5, 24] for
detail.
In the ${\mathbb{R}}$ picture, the space
${C_{c}^{\infty}({\mathbb{R}},{\mathbb{R}})}$ can be completed to a complex
Hilbert space (the one-particle space) with the complex scalar product
$(f,g):=\int_{p>0}2p\overline{\widehat{f}(p)}\widehat{g}(p)$, where
$\widehat{f}$ is the Fourier transform of $f$, and the imaginary unit is given
by $\widehat{\textit{I}f}(p):=-i\,\textrm{sgn}(p)\widehat{f}(p)$. The
imaginary part of the scalar product is a symplectic form
$\sigma(f,g):=\int_{{\mathbb{R}}}fg^{\prime}dx$. The $U(1)$-current algebra
${\mathcal{A}}_{U(1)}$ is the Weyl algebra constructed on this symplectic
space, generated by Weyl operators $W\left(f\right)=e^{iJ\left(f\right)}$
acting on the corresponding Fock space (if $f$ is a real function, $J(f)$ is
essentially self-adjoint on the finite particle-number subspace). The net
structure is given by ${\mathcal{A}}_{U(1)}(I):=\\{W(f):{\rm supp}(f)\subset
I\\}^{\prime\prime}$. This defines a conformal net on $S^{1}$ in the sense of
Part I. The current operators satisfy $\left[J(f),J(g)\right]=i\sigma(f,g)$
and the Weyl operators satisfy
$W\left(f\right)W\left(g\right)=W\left(f+g\right)\exp\left(-\frac{i}{2}\sigma(f,g)\right).$
Let us briefly discuss the split property of the $U(1)$-current net. A
sufficient condition for the split property for a conformal net on $S^{1}$ is
the trace class condition, namely the condition that the operator
$e^{-sL_{0}}$, where $L_{0}$ is the generator of the rotation automorphism, is
a trace class operator for each $s>0$ [12, 4]. The Fock space is spanned by
the vectors of the following form $J(e_{-n_{1}})J(e_{-n_{2}})\cdots
J(e_{-n_{k}})\Omega$, where $e_{n}(\theta)=e^{i2\pi n\theta}$, $0\leq
n_{1}\leq n_{2}\leq\cdots\leq n_{k}$, $k\in{\mathbb{N}}$, and all these
vectors are linearly independent and eigenvectors of $L_{0}$ with eigenvalue
$\sum_{i=1}^{k}n_{i}$. Hence the dimension of the eigenspace with eigenvalue
$N$ is $p(N)$, the partition number of $N$. There is an asymptotic estimate of
the partition function [19]: $p(n)\sim\frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}$.
Hence with some constants $C_{s},D_{s}$, we have
$\mathrm{Tr}(e^{-sL_{0}})=\sum_{n=0}^{\infty}p(n)e^{-sn}\leq\sum_{n=0}^{\infty}C_{s}e^{-D_{s}n},$
which is finite for a fixed $s>0$. Namely we have the trace class condition,
and the split property.
The Sugawara construction $T:=\frac{1}{2}:J^{2}:$, using normal ordering,
gives the stress-energy tensor, satisfying the commutation relations:
$\left[T(f),T(g)\right]=iT(\left[f,g\right])+i\frac{c}{12}\int_{{\mathbb{R}}}f^{\prime\prime\prime}g\,dx$
(2)
with $c=1$ and $\left[f,g\right]=fg^{\prime}-gf^{\prime}$. This is the
relation of $\mathrm{Vect}(S^{1})$, which is the Lie algebra of ${\rm
Diff}(S^{1})$. This (projective) representation $T$ of $\mathrm{Vect}(S^{1})$
integrates to a (projective) representation $U$ of ${\rm Diff}(S^{1})$.
Furthermore, $T$ and $J$ satisfy the following commutation relations
$\left[T(f),J(g)\right]=iJ(fg^{\prime}).$ (3)
Accordingly, $U$ acts on $J$ covariantly: if $\gamma$ is a diffeomorphism of
${\mathbb{R}}$, then $U(\gamma)J(f)U(\gamma)^{*}=J(f\circ\gamma^{-1})$ (see
[6, 33] for details).
### 4.2 KMS states of the $U(1)$-current model
We give here the complete classification of the KMS states of the
$U(1)$-current model, first appeared in [40, Theorem 3.4.11].
###### Proposition 4.1.
There is a one-parameter group $q\mapsto\gamma_{q}$ of automorphisms of
${\mathcal{A}}_{U(1)}|_{{\mathbb{R}}}$ commuting with translations, locally
unitarily implementable, such that
$\gamma_{q}\left(W\left(f\right)\right)=e^{iq\int_{{\mathbb{R}}}fdx}W\left(f\right).$
(4)
###### Proof.
For any $I\Subset{\mathbb{R}},$ let $s_{I}$ be a function in
${C_{c}^{\infty}({\mathbb{R}},{\mathbb{R}})}$ such that $\forall x\in I$
$s_{I}\left(x\right)=x$; then $\sigma(s_{I},f):=\int_{{\mathbb{R}}}fdx$ if
${\rm supp}f\subset I$ and therefore
${\hbox{\rm
Ad\,}}W\left(qs_{I}\right)\,W\left(f\right)=e^{-i\sigma(qs_{I},f)}W\left(f\right)=e^{iq\int_{{\mathbb{R}}}fdx}W\left(f\right).$
Set $\gamma_{q}|_{{\mathcal{A}}\left(I\right)}={\hbox{\rm
Ad\,}}W\left(qs_{I}\right)$, this is a well-defined automorphism, since
${\hbox{\rm
Ad\,}}W\left(qs_{I}\right)|_{{\mathcal{A}}\left(I\right)}={\hbox{\rm
Ad\,}}W\left(qs_{J}\right)|_{{\mathcal{A}}\left(I\right)}$ when $I\subset J$,
which can be extended to the norm closure $\mathfrak{A}_{U(1)}$ satisfying (4)
and commuting with translations because so is the integral. ∎
###### Lemma 4.2.
A state $\varphi$ is a primary KMS state of the $U(1)$-current model if and
only if so is $\varphi\circ\gamma_{q}$ for one value (and hence all) of
$q\in{\mathbb{R}}$.
###### Proof.
By a direct application of the KMS condition and the fact that $\gamma_{q}$ is
an automorphism commuting with translations. ∎
###### Theorem 4.3.
The primary (locally normal) KMS states of the $U(1)$-current model at inverse
temperature $\beta$ are in one-to-one correspondence with real numbers
$q\in{\mathbb{R}}$; each state $\varphi^{q}$ is uniquely determined by its
value on the Weyl operators
$\varphi^{q}\left(W\left(f\right)\right)=e^{iq\int f\,dx}\cdot
e^{-\frac{1}{4}\left\|f\right\|_{S_{\beta}}^{2}}$ (5)
where $\left\|f\right\|_{S_{\beta}}^{2}=\left(f,S_{\beta}f\right)$ and the
operator $S_{\beta}$ is defined by
$\widehat{S_{\beta}f}$$\left(p\right)$:=$\coth\frac{\beta
p}{2}\widehat{f}\left(p\right)$. The geometric KMS state is $\varphi_{\rm
geo}=\varphi^{0}$ and any other primary KMS state is obtained by composition
of the geometric one with the automorphisms (4):
$\varphi^{q}=\varphi_{\rm geo}\circ\gamma_{q}.$
###### Proof.
The algebra of the $U(1)$-current model is a Weyl CCR algebra, for which the
general structure of KMS states w.r.t. a Bogoliubov automorphism is
essentially known: see e.g. [36, Theorem 4.1] or [3, Example 5.3.2]. It is
however easier to do an explicit and straightforward calculation for the
present case.
Let $\varphi$ be a KMS state and $f,g\in
C_{c}^{\infty}\left({\mathbb{R}},{\mathbb{R}}\right)$. Recall that a product
of Weyl operators is again a (scalar multiple of) Weyl operator, so that the
quasilocal $C^{*}$-algebra is linearly generated by Weyl operators. Hence the
state $\varphi$ is uniquely determined by its values on $\\{W(f)\\}$.
Furthermore, under the KMS condition, the function $t\mapsto
F(t)=\varphi\left(W\left(f\right)W\left(g_{t}\right)\right)$, where
$g_{t}\left(x\right):=g\left(x-t\right)$, has analytic continuation in the
interior of $D_{\beta}:=\\{0\leq\Im z\leq\beta\\}$, continuous on $D_{\beta}$,
satisfying
$F(t+i\beta)=e^{-i\sigma(g_{t},f)}F(t)=e^{i\sigma(f,g_{t})}F(t).$ (6)
We search for a solution $F_{0}$ of the form $F_{0}(z)=\exp{K(z)}$, where $K$
is analytic in the interior of $D_{\beta}$ and has to satisfy the logarithm of
(6), $K(t+i\beta)=i\sigma(f,g_{t})+K(t)$. The Fourier transform of $t\mapsto
i\sigma\left(f,g_{t}\right)$ is $p\mapsto
p\overline{\widehat{f}\left(p\right)}\widehat{g}\left(p\right)$, thus we have
a simple equation for the Fourier transform w.r.t. $t$: $\exp(-\beta
p)\widehat{K}\left(p\right)=\widehat{K}\left(p\right)+p\overline{\widehat{f}\left(p\right)}\widehat{g}\left(p\right)$,
from which
$\widehat{K}\left(p\right)=\frac{p\overline{\widehat{f}\left(p\right)}\widehat{g}\left(p\right)}{\exp(-\beta
p)-1}$. It can be explicitly checked that $F_{0}$ is a solution of (6); any
other solution, divided by the never vanishing function $F_{0}$, has to be
constant (w.r.t. $t$) by analyticity. The general solution can therefore be
written as $F(t)=c(f,g_{t})\cdot F_{0}(t)$, with $c(f,g_{t})$ independent of
$t$.
To obtain (5), notice that
$\varphi\left(W\left(f+g_{t}\right)\right)=F\left(t\right)e^{\frac{i}{2}\sigma\left(f,g_{t}\right)}=c(f,g_{t})\cdot\exp\left[K\left(t\right)+\frac{i}{2}\sigma\left(f,g_{t}\right)\right],$
and $K\left(t\right)+\frac{i}{2}\sigma\left(f_{t},g\right)$ is the Fourier
antitransform of
$p\overline{\widehat{f}\left(p\right)}\left(\frac{1}{e^{-\beta
p}-1}+\frac{1}{2}\right)\widehat{g}\left(p\right)=-\frac{1}{2}p\overline{\widehat{f}\left(p\right)}\coth\frac{\beta
p}{2}\widehat{g}\left(p\right)=-\frac{1}{2}p\overline{\widehat{f}\left(p\right)}\widehat{S_{\beta}g}\left(p\right)$
which is given by
$-\frac{1}{2}\int
e^{itp}p\overline{\widehat{f}\left(p\right)}\widehat{S_{\beta}g}\left(p\right)dp=-\frac{1}{2}\left(f,S_{\beta}g_{t}\right)=-\frac{1}{4}\left(\left\|f+g_{t}\right\|_{S_{\beta}}^{2}-\left\|f\right\|_{S_{\beta}}^{2}-\left\|g_{t}\right\|_{S_{\beta}}^{2}\right),$
since $(f,S_{\beta}g_{t})$ is a real form. Note that
$\left\|g_{t}\right\|_{S_{\beta}}^{2}$ is independent of $t$. We finally have
the general solution in the form
$\varphi\left(W\left(f+g_{t}\right)\right)=c(f,g_{t})\cdot
e^{\frac{1}{4}\left(\|f\|^{2}_{S_{\beta}}+\|g_{t}\|^{2}_{S_{\beta}}\right)}\cdot
e^{-\frac{1}{4}\left\|f+g_{t}\right\|_{S_{\beta}}^{2}}.$
Note that factors $\varphi(W(f+g_{t}))$ and $e^{-\frac{1}{4}\|f+g_{t}\|^{2}}$
depend only on the sum $f+g_{t}$, hence so does the remaining factor: we
define $c(f+g_{t}):=c(f,g_{t})\cdot
e^{\frac{1}{4}\left(\|f\|^{2}_{S_{\beta}}+\|g_{t}\|^{2}_{S_{\beta}}\right)}$.
Since $c(f,g_{t})$ and $\|g_{t}\|_{S_{\beta}}$ are independent of $t$, so is
$c(f+g_{t})$. As $\overline{\varphi(W(f))}=\varphi(W(-f))$,
$\overline{c(f)}=c(-f)$. Now we have
$\varphi(W(f+g_{t}))=c(f+g_{t})\cdot
e^{-\frac{1}{4}\|f+g_{t}\|^{2}_{S_{\beta}}},$
and we only have to determine $c(f+g_{t})$.
Concerning the continuity, we notice that
$\left\|f\right\|_{S_{\beta}}\geq\left\|f\right\|$, because $\coth p\geq 1$
for any $p\in{\mathbb{R}}_{+}$; the map $f\mapsto W(f)$ is weakly continuous
when $C_{c}^{\infty}\left({\mathbb{R}},{\mathbb{R}}\right)$ is given the
topology of the (one-particle space) norm $\left\|\cdot\right\|$ and a
fortiori of the norm $\left\|\cdot\right\|_{S_{\beta}}$; being $\varphi$ a KMS
state and locally normal, $f\mapsto\varphi(W(f))$ is continuous w.r.t. both
norms and $f\mapsto
c(f)=\varphi(W(f))\cdot\exp({-\frac{1}{4}\left\|f\right\|_{S_{\beta}}^{2}})$
is continuous w.r.t. the norm $\left\|\cdot\right\|_{S_{\beta}}$; finally,
both $\lambda\mapsto\lambda f$ and $t\mapsto f_{t}$ are continuous w.r.t. the
$\left\|\cdot\right\|_{S_{\beta}}$ norm, thus in particular $\lambda\mapsto
c(\lambda f)$ (and trivially the constant function $t\mapsto c(f+g_{t})$) is
continuous.
If we require $\varphi$ to be primary, it satisfies the clustering property:
for $t\rightarrow\infty$
$\varphi(W(f+g_{t}))=\varphi(W(f)W(g_{t}))\exp\left(\frac{i}{2}\sigma(f,g_{t})\right)\rightarrow\varphi(W(f))\varphi(W(g))$
and thus
$c(f+g)=c(f)\cdot c(g),$ (7)
because both $\sigma(f,g_{t})$ and $(f,S_{\beta}g_{t})$ go to $0$. It follows
that $c(0)=1$, $c(-f)=c(f)^{-1}=\overline{c(f)}$ and $|c(f)|=1$. As
${\mathbb{R}}\ni\lambda\mapsto c(\lambda f)$ is a continuous curve in
$\\{z\in{\mathbb{C}}:|z|=1\\}$, there is a unique functional
$\rho:C_{c}^{\infty}\left({\mathbb{R}},{\mathbb{R}}\right)\rightarrow{\mathbb{R}}$
s.t. $c(f)=\exp(i\rho(f))$, $\rho(0)=0$ and $\lambda\mapsto\rho(\lambda f)$ is
continuous.
Clearly, (7) implies $\rho(f+g)-\rho(f)-\rho(g)\in 2\pi{\mathbb{Z}}$; by
continuity of $\lambda\mapsto\rho(\lambda f+\lambda g)-\rho(\lambda
f)-\rho(\lambda g)$ and $\rho(0)=0$, we get $\rho(f+g)=\rho(f)+\rho(g)$.
Similarly, from [29, Proposition 6.1.2] we know that $\rho$ has the same
continuity property of $c$, i.e. w.r.t. the $\left\|\cdot\right\|_{S_{\beta}}$
norm; $c(f_{t})=c(f)$ implies $\rho(f_{t})-\rho(f)\in 2\pi{\mathbb{Z}}$, but
this difference vanishes because $t\mapsto\rho(f_{t})$ is continuous.
Therefore, $\rho$ is a real, translation invariant and linear functional.
According to [32], any translation invariant linear functional (even without
requiring continuity) $\rho$ on
$C_{c}^{\infty}\left({\mathbb{R}},{\mathbb{R}}\right)$ is of the form
$\rho(f)=q\int f(x)dx$. So, if $\varphi$ is a primary KMS state, it has to be
of the form (5). Conversely, Lemma 4.2 implies that all these states are KMS.
These are regular states (i.e. $\lambda\mapsto\varphi(W(\lambda f))$ is a
$C^{\infty}$ function $\forall f$) and the one point and two points functions
are given by
$\displaystyle\varphi^{q}\left(J\left(f\right)\right)$ $\displaystyle=$
$\displaystyle q\int f\,dx$ (8)
$\displaystyle\varphi^{q}\left(J\left(f\right)J\left(g\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Re\left(f,S_{\beta}g\right)+\frac{i}{2}\sigma\left(f,g\right)+q^{2}\int
f\,dx\int g\,dx,$ (9)
where $\Re$ means the real part. The geometric KMS state has to coincide with
one of those: it is $\varphi^{0}$. This can be proved by noticing that, if
${\rm supp}f\subset I$ ,
$\varphi_{\rm geo}\left(W\left(\lambda f\right)\right)=\left(\Omega,{\hbox{\rm
Ad\,}}{U\left(\gamma_{I,\beta}\right)W\left(\lambda
f\right)}\Omega\right)=\left(\Omega,W\left(\lambda
f\circ\gamma_{I,\beta}^{-1}\right)\Omega\right)=e^{-\frac{1}{4}\lambda^{2}\left\|f\circ\gamma_{I,\beta}^{-1}\right\|^{2}}$
where the exponent is a quadratic form in $f$, therefore the state is regular
and taking the derivative w.r.t. $\lambda$ we get $\varphi_{\rm
geo}\left(J\left(f\right)\right)=0$, which implies $q=0$ by comparison with
(8). ∎
###### Remark 4.4.
The gauge automorphism $\gamma_{z}$ defined by the map
$J\left(f\right)\mapsto-J\left(f\right)$ acts as a change in the sign of $q$:
$\varphi^{q}\circ\gamma_{z}=\varphi^{-q}$.
The ’energy density’ of a state can be read from the expectation value of the
stress-energy tensor as the constant $c$ in the formula
$\varphi\left(T\left(f\right)\right)=c\int f\,dx$. Beside its physical
interpretation, this formula is also useful to classify the states on the
Virasoro net (see Sections 5.2 and 5.3). In order to evaluate
$\varphi^{q}\left(T(f)\right)$, we need two technical lemmas.
In the following,
$D_{\mathrm{fin}}:=\mathrm{span}\left\\{\psi=J(f_{1})...J(f_{n})\Omega:n\in{\mathbb{N}},f_{1},\ldots,f_{n}\in
C^{\infty}(S^{1},{\mathbb{R}})\right\\}$ is the space of finite number of
particles and $D_{\infty}:=\cap_{n\in{\mathbb{N}}}D(L_{0}^{n})$ is the common
domain of the powers of $L_{0}$; $D(L_{0}^{n})\supset D_{\infty}$ and
$D_{\mathrm{fin}}$ are all dense in the vacuum Hilbert space, contain the
space of finite energy vectors and are cores for $L_{0}^{n}$ (the following
Lemma implies also that $D_{\infty}\supset D_{\mathrm{fin}}$).
###### Lemma 4.5 (Energy bounds).
Let $P_{n}\left(J,T,L_{0}\right)$ be a (noncommutative) polynomial in $L_{0}$
and some $J(f_{i})$ and $T(f_{j})$ of total degree $n$, with $f_{i},f_{j}\in
C^{\infty}(S^{1},{\mathbb{R}})$, then $\forall\psi\in D_{\infty}$
$\left\|P_{n}\left(J,T,L_{0}\right)\psi\right\|\leq
r_{n}\left\|({\mathbbm{1}}+L_{0})^{n}\psi\right\|,$ (10)
with an appropriate $r_{n}$ (depending on $\left\\{f_{k}\right\\}$ and on $n$
but not on $\psi$).
###### Proof.
The operators $J(f)$ and $T(f)$ satisfy similar bounds
$\left\|J(f)\psi\right\|\leq
c_{f}\left\|({\mathbbm{1}}+L_{0})\psi\right\|\;\;\;\;\;\;\;\;\left\|T(f)\psi\right\|\leq
c_{f}\left\|({\mathbbm{1}}+L_{0})\psi\right\|$ (11)
for any $\psi\in D_{\mathrm{fin}}$ with $c_{f}$ independent of $\psi$ [6,
ineqalities (2.21) and (2.23)], and similar commutation relations on
$D_{\mathrm{fin}}$: $[L_{0},J(f)]=iJ(\partial_{\theta}f)$,
$[L_{0},T(f)]=iT(\partial_{\theta}f)$. Since $D_{\mathrm{fin}}$ is a core for
$L_{0}$, $\forall\psi\in D(L_{0})$, using a sequence $\psi_{n}\in
D_{\mathrm{fin}}$, s.t. $\psi_{n}\rightarrow\psi$ and
$L_{0}\psi_{n}\rightarrow L_{0}\psi$, and the closedness of $J(f)$ and $T(f)$,
the bounds (11) hold on $D(L_{0})\supset D_{\infty}$; hence the commutators
hold also on $D_{\infty}$, using $\forall\psi\in D_{\infty}$ a sequence
$\psi_{n}\in D_{\mathrm{fin}}$, s.t. $\psi_{n}\rightarrow\psi$ and
$L_{0}^{2}\psi_{n}\rightarrow L_{0}^{2}\psi$ (from which
$L_{0}\psi_{n}\rightarrow L_{0}\psi$). One sees also that $D_{\infty}$ is
invariant under $J(f)$ and $T(f)$.
We can generalize the inequalities (11), which are equivalent to (10) for
$n=1$, to any $n$. Indeed, induction and commutation relations show that on
$D_{\infty}$
$({\mathbbm{1}}+L_{0})^{n}J(f)=\sum_{0\leq k\leq
n}\left(\begin{array}[]{c}n\\\ k\\\
\end{array}\right)i^{k}J(\partial_{\theta}^{k}f)({\mathbbm{1}}+L_{0})^{n-k}.$
(12)
Then, we use induction in the degree of the polynomial to prove (10). Suppose
(10) holds for degree $n$. Any polynomial of degree $n+1$ is a linear
combination of polynomials of degree $n$ multiplied from the right by $J(f)$
or $T(f)$ or $L_{0}$.
First, let us consider $J(f)$. $\|P_{n}\left(J,T,L_{0}\right)J(f)\psi\|\leq
r_{n}\|(1+L_{0})^{n}J(f)\psi\|$ by induction hypothesis and, applying (12)
(notice that ${\mathbbm{1}}+L_{0}\geq L_{0},{\mathbbm{1}}$), the last norm is
smaller than $\sum_{0\leq k\leq
n}c_{k}\|J(\partial_{\theta}^{k}f_{n})({\mathbbm{1}}+L_{0})^{n-k}\psi\|$ where
each term is estimated, using (11), by constants times
$\|({\mathbbm{1}}+L_{0})^{n+1}\psi\|$.
Secondly, we consider $T(f)$. With $T$ in place of $J$, equation (12) still
holds and the same argument as above applies.
Finally, $\|P_{n}\left(J,T,L_{0}\right)L_{0}\psi\|\leq
r_{n}\|(1+L_{0})^{n}L_{0}\psi\|\leq r_{n}\|(1+L_{0})^{n+1}\psi\|$ and thus
(10) holds for degree $n+1$.
∎
###### Lemma 4.6.
$D_{\infty}$ is invariant for the Weyl operator $W(f)=e^{iJ(f)}$, $\forall
f\in C^{\infty}(S^{1})$, and the unitary $U(g)$, $\forall g\in{\rm
Diff}(S^{1})$, implementing the conformal symmetry.
###### Proof.
The subspace $D_{\mathrm{fin}}$ is included in $D_{\infty}$ by (10), and it is
invariant under $L_{0}$, as $[L_{0},J(f)]=iJ(\partial_{\theta}f)$. Using also
the commutator $[J(f),J(g)^{k}]=ik\sigma(f,g)J(g)^{k-1}$ (easy consequence of
$[J(f),J(g)]=i\sigma(f,g)$), we compute $\forall\psi\in D_{\mathrm{fin}}$
$\left[L_{0},J(f)^{n}\right]\psi=\left(inJ(f)^{n-1}J(\partial_{\theta}f)-\frac{n(n-1)}{2}J(f)^{n-2}\sigma(\partial_{\theta}f,f)\right)\psi.$
(13)
We apply it to the expansion of Weyl operators
$W(f)=\sum_{k}\frac{i^{k}}{k!}J(f)^{k}$, which is absolutely convergent on
$D_{\mathrm{fin}}$ (it is well known that finite particle vectors are analytic
for the free field, see e.g. the proof of [34, Theorem X.41], with the
estimate $\left\|J(f)^{k}\psi\right\|\leq
2^{k/2}\sqrt{(n+k)!}\|f\|^{k}\|\psi\|$, where $n$ is the number of particles
of $\psi$). By the closedness of $L_{0}$ and the absolute convergence of
$L_{0}\sum_{k}\frac{i^{k}}{k!}J(f)^{k}\psi$, thanks to (13), we conclude that
$W(f)D_{\mathrm{fin}}$ is in the domain of $L_{0}$. We then easily compute,
using the convergent series, the commutation relations
$W(f)^{\ast}L_{0}W(f)=L_{0}-J(\partial_{\theta}f)+\frac{1}{2}\sigma(\partial_{\theta}f,f)$
and their powers
$W(f)^{\ast}L_{0}^{n}W(f)\psi=\left(L_{0}-J(\partial_{\theta}f)+\frac{1}{2}\sigma(\partial_{\theta}f,f)\right)^{n}\psi.$
(14)
Finally, (10) applied to the r.h.s., which is a polynomial of degree $n$ in
$J(\partial_{\theta}f)$ and $L_{0}$, gives
$\|L_{0}^{n}W(f)\psi\|\leq r\|({\mathbbm{1}}+L_{0})^{n}\psi\|$ (15)
$\forall\psi\in D_{\mathrm{fin}}$. As $D_{\mathrm{fin}}$ contains the space of
finite energy vectors (the vectors of $D_{\mathrm{fin}}$ where
$f_{1},\cdots,f_{n}$ are trigonometric polynomials), it is dense in
$D_{\infty}$ and is a core for $L_{0}^{n}$; any $\psi\in D_{\infty}$ is the
limit of a sequence $\\{\psi_{i}:i\in{\mathbb{N}}\\}$ such that
$({\mathbbm{1}}+L_{0})^{n}\psi_{i}$ is convergent, thus, by (15) and the
closedness of $L_{0}^{n}W(f)$, $W(f)D_{\infty}$ is in the domain of
$L_{0}^{n}$. We have proved that $W(f)D_{\infty}\subset D_{\infty}$; the same
is true for $W(f)^{-1}=W(-f)$, thus $W(f)D_{\infty}=D_{\infty}$.
A similar argument apply to $U(g)$. First one consider the case where $g=\exp
T(f)$ is contained in a one-parameter group. We replace (14) with the known
transformation property of the stress-energy tensor ($L_{0}=T(1)$, where $1$
has to be understood as the generator of rotations, the constant vector field
on the circle; in the real line picture, it would be the smooth vector field
$x\mapsto 1+x^{2}$) [16]:
$U(g)L_{0}^{n}U(g)^{\ast}=\left(T(g_{\ast}1)+r_{g}{\mathbbm{1}}\right)^{n}$
(16)
and then apply (10). For a general diffeomorphism $g$, it is possible to write
$g$ as a finite product of diffeomorphisms contained in one-parameter groups,
since ${\rm Diff}(S^{1})$ is algebraically simple [15, 31] and the subgroup
generated by one-parameter groups is normal, hence ${\rm Diff}(S^{1})$ itself.
Thus we obtained the claimed invariance for any element $g$. ∎
###### Theorem 4.7.
For any primary KMS state $\varphi^{q}$ (cf. (5)) the map
$t\mapsto\varphi^{q}\left(e^{itT(f)}\right)$ is $C^{\infty}$, $\forall f\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$, and the expectation value of the
stress-energy tensor is given by
$\varphi^{q}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx.$ (17)
Moreover, in the GNS representation
$(\pi_{\varphi^{q}},{\mathcal{H}}_{\varphi^{q}},\Omega_{\varphi^{q}})$,
$\Omega_{\varphi^{q}}$ is in the domain of any (non commutative) polynomial of
the stress-energy tensors
$\pi_{\varphi^{q}}(T(f_{k})):=-i\frac{d}{dt}\pi_{\varphi^{q}}(e^{iT(f_{k})})$,
with $f_{k}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$, $k=1,\dots,n$.
###### Proof.
Fix $f\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$ with ${\rm supp}f\subset
I\Subset{\mathbb{R}}$.
We first consider the case $q=0$. According to the proof of Proposition 2.4
and Theorem 2.5 of Part I [7], the GNS representation of $\varphi_{\rm geo}$
is the triple $\left(\pi_{\varphi_{\rm geo}}={\rm
Exp}_{\beta},{\mathcal{H}}_{\Omega},\Omega\right)$ and there is a
$g_{\beta,I}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$ s.t. ${\rm
Exp}_{\beta}|_{{\mathcal{A}}\left(I\right)}={\hbox{\rm
Ad\,}}U\left(g_{\beta,I}\right)$. It follows that the one parameter group
$t\mapsto\pi_{\varphi_{\rm geo}}\left(e^{itT(f)}\right)={\hbox{\rm
Ad\,}}U\left(g_{\beta,I}\right)\left(e^{itT(f)}\right)$ has a generator
${\hbox{\rm Ad\,}}U\left(g_{\beta,I}\right)\left(T(f)\right)$ which can be
computed: indeed, [16, Proposition 3.1] proves that, in general diffeomorphism
covariant nets, if $g\in{\rm Diff}(S^{1})$ fixes the point $\infty$,
${\hbox{\rm
Ad\,}}U\left(g\right)T\left(f\right)=T\left(g_{\ast}f\right)+r^{{\mathbb{R}}}\left(g,f\right)$,
with $g_{\ast}f\left(x\right)=g^{\prime}\cdot
f\left(g^{-1}\left(x\right)\right)$ and
$r^{{\mathbb{R}}}\left(g,f\right)=\frac{c}{12\pi}\int\sqrt{g^{\prime}\left(x\right)}\frac{d^{2}}{dx^{2}}\frac{f\left(x\right)}{\sqrt{g^{\prime}\left(x\right)}}dx$
with the central charge $c$ set equal to $1$ for the $U(1)$ case. Therefore,
with $g_{\beta,I}$ in place of $g$, recalling that
$g_{\beta,I}(t)=e^{\frac{2\pi t}{\beta}}$ on the support of $f$, we get
$\pi_{\varphi_{\rm geo}}\left(T(f)\right)={\hbox{\rm
Ad\,}}U\left(g_{I}\right)T\left(f\right)=T\left({g_{I}}_{\ast}f\right)+\frac{\pi
c}{12\beta^{2}}\int f\,dx.$ (18)
The vacuum vector $\Omega$ is in the domain of the operator (18) and any
product of such operators; from $\left(\Omega,T(h)\Omega\right)=0$ for any
$h\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$, we easily compute (17). The
case $q=0$ is proved.
We now consider the general case for $q$. In this case the GNS representation
is $\left(\pi_{\varphi^{q}}={\rm
Exp}\circ\gamma_{q},{\mathcal{H}}_{\Omega},\Omega\right)$ with
$\gamma_{q}|_{{\mathcal{A}}\left(I\right)}={\hbox{\rm
Ad\,}}W\left(qs_{I}\right)$ defined in Proposition 4.1. The one parameter
group $t\mapsto{\hbox{\rm Ad\,}}U\left(g_{I}\right)\circ{\hbox{\rm
Ad\,}}W\left(qs_{I}\right)\left(e^{itT(f)}\right)$ has a self-adjoint
generator ${\hbox{\rm Ad\,}}U\left(g_{I}\right)\circ{\hbox{\rm
Ad\,}}W\left(qs_{I}\right)\left(T(f)\right)$, which has to be computed.
According to Lemma 4.6, for any $\psi\in D_{\mathrm{fin}}\subset D_{\infty}$
with a finite number of particles, ${\hbox{\rm
Ad\,}}W\left(qs_{I}\right)\left(T(f)\right)\psi$ is well-defined because
$D_{\infty}$ is in the domain of $T(f)$. Using, as for equation (13),
$[J(f),J(g)^{k}]=ik\sigma(f,g)J(g)^{k-1}$ and $[T(f),J(g)]=iJ(fg^{\prime})$,
we compute $\forall\psi\in D_{\mathrm{fin}}$ a generalization of (13):
$\left[T(f),J(g)^{n}\right]\psi=\left(inJ(g)^{n-1}J(fg^{\prime})-\frac{n(n-1)}{2}J(g)^{n-2}\sigma(fg^{\prime},g)\right)\psi.$
We use a similar argument to that following equation (13). The expansion of
Weyl operators $W(g)=\sum_{k}\frac{i^{k}}{k!}J(g)^{k}$ is absolutely
convergent on $D_{\mathrm{fin}}$; using the absolute convergence of
$L_{0}\sum_{k}\frac{i^{k}}{k!}J(g)^{k}\psi$ and the estimate
$\left\|T(f)\psi\right\|\leq c_{f}\left\|({\mathbbm{1}}+L_{0})\psi\right\|$,
we conclude that also $T(f)\sum_{k}\frac{i^{k}}{k!}J(g)^{k}\psi$ is absolutely
convergent and therefore, by the closedness of $T(f)$, $W(g)D_{\mathrm{fin}}$
is in the domain of $T(f)$. The convergent series lets us compute (cf. (14))
$W(g)^{\ast}T(f)W(g)\psi=\left(T(f)-J(fg^{\prime})+\frac{1}{2}\sigma(fg^{\prime},g)\right)\psi$.
In the particular case in which $g=-qs_{I}$, and thus $fg^{\prime}=-qf$
(recall that ${\rm supp}f\subset I$), we obtain
${\hbox{\rm
Ad\,}}W\left(qs_{I}\right)\left(T(f)\right)=T(f)+qJ(f)+\frac{q^{2}}{2}\int
fdx$
on the dense set $D_{\mathrm{fin}}$ and also on $D_{\infty}$, where both sides
are defined. We can apply ${\hbox{\rm Ad\,}}U\left(g_{I}\right)$ to this
operator, as $D_{\infty}$ is invariant for $U\left(g_{I}\right)$, and taking
into account its action on $J(f)$ and $T(f)$, we get
$\pi_{\varphi^{q}}\left(T(f)\right)=T\left({g_{I}}_{\ast}f\right)+\frac{\pi}{12\beta^{2}}\int
fdx+qJ\left(f\circ g_{I}^{-1}\right)+\frac{q^{2}}{2}\int fdx.$ (19)
$\Omega$ is in the domain of the operator (19) and any power of such
operators; as before, using also $\left(\Omega,J(h)\Omega\right)=0$ for any
$h\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$, we easily compute (17). ∎
We finally observe that the thermal completion (defined in Part I [7]), in the
case of the $U(1)$-current model, does not give any new net.
###### Theorem 4.8.
The thermal completion of the $U(1)$-current net w.r.t. any of its primary
(locally normal) KMS states is unitarily equivalent to the original net.
###### Proof.
In the case of the geometric KMS state, this is the content of Theorem 2.5 in
Part I [7]. The general case follows from the fact that any other primary KMS
state of the $U(1)$-current model is obtained by composition of the geometric
one with an automorphism, so that the local algebras
$\hat{{\mathcal{A}}}_{\varphi^{q}}(e^{2\pi t},e^{2\pi
s}):={\mathcal{A}}_{\varphi^{q}}(t,\infty)\cap{\mathcal{A}}_{\varphi^{q}}(s,\infty)^{\prime}$
do not depend on the value of $q$. ∎
## 5 The case of Virasoro nets
### 5.1 The geometric KMS state of ${\rm Vir}_{c}$
The Virasoro nets ${\rm Vir}{}_{c}$ with $c<1$ are completely rational [21,
Cor. 3.4], so our results in Part I [7] apply and thus they have a unique KMS
state: the geometric state $\varphi_{\rm geo}$. This is not the case for
$c\geq 1$. Before going to the classification of the KMS states of ${\rm
Vir}_{1}$ and a (possibly incomplete) list of KMS states for the Virasoro net
with central charge $c>1$, we characterize the geometric state for any $c$
[40, Theorem 3.6.2].
###### Theorem 5.1.
The (primary locally normal) geometric KMS states of the ${\rm Vir}_{c}$ net
w.r.t. translations assume the following value on the stress-energy tensor
$\varphi_{\rm geo}\left(T\left(f\right)\right)=\left(\frac{\pi
c}{12\beta^{2}}\right)\int f\,dx.$ (20)
###### Proof.
The evaluation of the state on the stress-energy tensor (20) follows from (18)
using the same argument of the proof of Theorem 4.7. ∎
### 5.2 KMS states of the Virasoro net ${\rm Vir}_{1}$
Recall [7, Section 2.3] that the Virasoro net ${\rm Vir}_{1}$ is defined as
the net generated by the representatives of diffeomorphisms. In fact, it holds
that ${\rm Vir}_{1}(I)=\\{e^{iT(f)}:{\rm supp}(f)\subset I\\}^{\prime\prime}$,
since the latter contains the representatives of one-parameter
diffeomorphisms, which form a normal subgroup of ${\rm Diff}(I)$ (the group of
diffeomorphisms with support in $I$), then this turns out to be the full group
because ${\rm Diff}(I)$ is algebraically simple [15, 31]. The net ${\rm
Vir}_{1}$ is realized as a subnet of the $U(1)$-current; we have seen that
$e^{-sL_{0}}$ is trace class, hence ${\rm Vir}_{1}$ is split as well.
The primary (locally normal) KMS states of the $U(1)$-current, restricted to
the Virasoro net, give primary (locally normal) KMS states. They are still
primary because primarity for KMS states is equivalent to extremality in the
set of $\tau$-invariant states [3, Theorem 5.3.32], and this is in turn
equivalent to the clustering property (Proposition 3.8) for asymptotically
abelian nets; clustering property is obviously preserved under restriction. We
denote these states $\varphi^{\left|q\right|}$. We know their values on the
stress-energy tensor (17). Notice that the two different states $\varphi^{q}$
and $\varphi^{-q}$ coincide when restricted to ${\rm Vir}_{1}$. We have thus a
family of primary (locally normal) KMS states classified by a positive number
$\left|q\right|\in{\mathbb{R}}^{+}$. We will show that these exhaust the KMS
states on ${\rm Vir}_{1}$.
An important observation for this purpose is that the $U(1)$-current net and
${\rm Vir}_{1}$ can be viewed as subnets of an even larger net. Namely, let
${\mathcal{B}}:={\mathcal{A}}_{SU(2)_{1}}$ be the net generated by the vacuum
representation of the loop group $LSU(2)$ at level $1$ [17], or by the
$SU(2)$-chiral current at level $1$ [35], on which the compact group $SU(2)$
acts as inner symmetry (an automorphism of the net which preserves the vacuum
state). This net satisfies the trace class condition by an analogous estimate
as for $U(1)$-current net in Section 4.1, hence it is split. It has been shown
[35] that the Virasoro net ${\rm Vir}_{1}$ can be realized as the fixed point
subnet of ${\mathcal{B}}$ with respect to this inner symmetry. Moreover, as
shown in [8], all the subnets of ${\mathcal{B}}$ are classified as fixed
points w.r.t. the actions of closed subgroups of $SU(2)$ (conjugate subgroups
give rise to isomorphic fixed points); in particular, let
${\mathcal{A}}_{U(1)}$ be the $U(1)$-current net, it is the fixed point
${\mathcal{B}}^{H}$ of the net ${\mathcal{B}}$ w.r.t. the action of the
subgroup $H\simeq S^{1}$ of rotations around a fixed axis. Therefore, we have
the double inclusion
${\rm
Vir}_{1}={\mathcal{B}}^{SU(2)}\subset{\mathcal{A}}_{U(1)}={\mathcal{B}}^{H}\subset{\mathcal{A}}_{SU(2)_{1}}=:{\mathcal{B}},$
and a complete classification of the KMS states of the intermediate net
${\mathcal{A}}_{U(1)}$. As we are not able to directly extend a $\tau$-KMS
state on ${\rm Vir}_{1}$ to a $\tau$-KMS state on ${\mathcal{A}}_{U(1)}$, we
use an auxiliary extension to ${\mathcal{B}}$ exploiting the existence of the
gauge group $SU(2)$ and Corollary 3.11.
###### Theorem 5.2.
The primary (locally normal) KMS states of the ${\rm Vir}_{1}$ net w.r.t.
translations are in one-to-one correspondence with positive real numbers
$\left|q\right|\in{\mathbb{R}}^{+}$; each state $\varphi^{\left|q\right|}$ can
be evaluated on the stress-energy tensor and it gives
$\varphi^{\left|q\right|}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx.$ (21)
###### Proof.
For any $q\in{\mathbb{R}}$, the restriction of the KMS state $\varphi^{q}$ to
the ${\rm Vir}_{1}$ subnet gives a KMS state. The evaluation of the state on
the stress-energy tensor (21), depending only on $|q|$, follows again from
(18) using the same argument of the proof of Theorem 4.7.
We have to prove that any primary KMS state of ${\rm Vir}_{1}$ arises in this
way. Let $\varphi$ be a primary KMS state of ${\rm
Vir}_{1}={\mathcal{B}}^{SU(2)}$. By applying Corollary 3.11, we obtain a
locally normal primary (i.e. extremal) $\tau$-invariant extension
$\tilde{\varphi}$ on ${\mathcal{B}}$, which is a KMS state w.r.t. the one
parameter group
$t\mapsto\tilde{\tau_{t}}=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$,
with a suitable one parameter group $t\mapsto\varepsilon_{t}\circ\zeta_{t}\in
SU(2)$. The image of $t\mapsto\varepsilon_{t}\circ\zeta_{t}\in SU(2)$ is a
closed subgroup $H\simeq S^{1}$ since $SU(2)$ has rank $1$ and any one-
parameter subgroup forms a maximal torus, therefore, if we consider the subnet
${\mathcal{A}}={\mathcal{B}}^{H}$, it is $\tilde{\tau}$ invariant and, as
$\tilde{\tau_{t}}|_{{\mathcal{A}}}=\tau_{t}|_{{\mathcal{A}}}$, the state
$\tilde{\varphi}$ is a primary KMS state of ${\mathcal{A}}$ w.r.t. $\tau$. It
then follows that the KMS state $\varphi$ of ${\rm Vir}_{1}$ is the
restriction of a KMS state $\tilde{\varphi}|_{\mathcal{A}}$ of
${\mathcal{A}}$, isomorphic to the $U(1)$-current net ${\mathcal{A}}_{U(1)}$.
∎
###### Remark 5.3.
The geometric KMS state corresponds to $q=0$, because it is the restriction of
the geometric KMS state on the $U(1)$-current net, and the corresponding value
of the ‘energy density’ $\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}$ is the
lowest in the set of the KMS states.
###### Remark 5.4.
In contrast to the case of the $U(1)$-current net (Theorem 4.3), here the
different primary KMS states are not obtained through composition of the
geometric one with automorphisms of the net.
By contradiction, suppose that there were an automorphism $\alpha$ of the net
such that $\varphi^{\left|q\right|}=\varphi\circ\alpha$ with $q\neq 0$. The
KMS condition for $\varphi\circ\alpha$ w.r.t. the one parameter group
$t\mapsto\tau_{t}$ is equivalent to the KMS condition for $\varphi$ w.r.t. the
one parameter group $t\mapsto\alpha\circ\tau_{t}\circ\alpha^{-1}$ and, by the
uniqueness of the modular group, $\tau_{t}$ has to coincide with
$\alpha\circ\tau_{t}\circ\alpha^{-1}$, i.e. the automorphism of the net
commutes with translations. By Proposition 4.2 of Part I [7], $\alpha$ cannot
preserve the vacuum state and, by Lemma 4.5 of Part I, there is a continuous
family of pairwise non unitarily equivalent automorphisms of
${\mathcal{A}}|_{\mathbb{R}}$ commuting with translations. By Proposition 4.6
of Part I, there is a continuous family of automorphic sectors of
${\mathcal{A}}$, which contradicts the fact, proved in [9], that ${\rm
Vir}_{1}$ can have at most countable sectors with finite statistical
dimension.
Recall that in Part I the thermal completion net played a crucial role. Let
${\mathcal{A}}_{\varphi}(t,s):=\pi_{\varphi}({\mathcal{A}}(t,s))$ and
${\mathcal{A}}^{d}_{\varphi}(t,s):={\mathcal{A}}_{\varphi}(t,\infty)\cap{\mathcal{A}}_{\varphi}(s,\infty)^{\prime}$.
Putting ${\mathcal{A}}\equiv{\rm Vir}_{1}$ and
$\varphi\equiv\varphi^{\left|q\right|}$ with $q\neq 0$, we have examples for
which
${\mathcal{A}}_{\varphi}(t,s)\neq{\mathcal{A}}^{d}_{\varphi}(t,s).$
Indeed, if the inclusion
${\mathcal{A}}_{\varphi}(t,s)\subset{\mathcal{A}}^{d}_{\varphi}(t,s)$ were an
equality, as ${\mathcal{A}}={\rm Vir}_{1}$ has the split property, Theorem 3.1
of Part I tells that $\varphi$ would have to be $\varphi_{\rm
geo}\circ\alpha$. The observation in the previous paragraph would give a
contradiction222This shows that the formula (10) in [38] is incorrect and does
not hold in general..
### 5.3 KMS states of the Virasoro net ${\rm Vir}_{c}$ with $c>1$
Here we show a (possibly incomplete) list of KMS states of the net ${\rm
Vir}_{c}$ with $c>1$.
The restriction of ${\rm Vir}_{1}$ to the real line ${\mathbb{R}}$ can be
embedded as a subnet of the restriction to ${\mathbb{R}}$ of the
$U(1)$-current net. One can simply define a new stress-energy tensor [6,
equation (4.6)], with $k\in{\mathbb{R}}$ and $f\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$
$\widetilde{T}\left(f\right):=T\left(f\right)+kJ\left(f^{\prime}\right)$
and, using the commutation relations (2), calculate that
$\left[\widetilde{T}\left(f\right),\widetilde{T}\left(g\right)\right]=i\widetilde{T}\left(\left[f,g\right]\right)+i\frac{1+k^{2}}{12}\int_{{\mathbb{R}}}f^{\prime\prime\prime}g\,dx.$
It follows that the net generated by $\widetilde{T}\left(f\right)$ as ${\rm
Vir}_{c}\left(I\right):=\left\\{e^{i\widetilde{T}\left(f\right)}:{\rm
supp}f\subset I\right\\}^{\prime\prime}$ with $I\Subset{\mathbb{R}}$, is the
restriction to ${\mathbb{R}}$ of the Virasoro net with $c=1+k^{2}>1$ [6]. We
observe that ${\rm Vir}_{c}(I)\subset{\mathcal{A}}_{U(1)}(I)$ for
$I\Subset{\mathbb{R}}$. Indeed, we know the locality of $J$ and $T$, hence if
${\rm supp}(f)\subset I$, then $e^{i\widetilde{T}}(f)$ commutes with $W(g)$
with ${\rm supp}(g)\subset I^{\prime}$ by the Trotter formula. By the Haag
duality it holds that $e^{i\widetilde{T}(f)}\in{\mathcal{A}}_{U(1)}(I)$.
The primary (locally normal) KMS states of the $U(1)$-current, restricted
again to this Virasoro net, give primary locally normal KMS states, noticing
that $\varphi^{q}\left(J\left(f^{\prime}\right)\right)=q\int
f^{\prime}\,dx=0$:
$\varphi^{\left|q\right|}\left(\widetilde{T}\left(f\right)\right)=\varphi^{\left|q\right|}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx;$
as in the $c=1$ case, the restrictions of $\varphi^{q}$ and $\varphi^{-q}$ are
equal. We have thus the following
###### Theorem 5.5.
There is a set of primary (locally normal) KMS states of the ${\rm Vir}_{c}$
net with $c>1$ w.r.t. translations in one-to-one correspondence with positive
real numbers $\left|q\right|\in{\mathbb{R}}^{+}$; each state
$\varphi^{\left|q\right|}$ can be evaluated on polynomials of stress-energy
tensor $T(f)$ and on a single $T(f)$ it gives:
$\varphi^{\left|q\right|}\left(T\left(f\right)\right)=\left(\frac{\pi}{12\beta^{2}}+\frac{q^{2}}{2}\right)\int
f\,dx.$ (22)
The geometric KMS state corresponds to
$q=\frac{1}{\beta}\sqrt{\frac{\pi\left(c-1\right)}{6}}$ and energy density
$\frac{\pi c}{12\beta^{2}}$.
###### Proof.
As in the case of ${\rm Vir}_{1}$, the restriction of a primary KMS state of
the $U(1)$-current net is a primary KMS state and $\varphi^{q}=\varphi^{p}$ if
and only if $q=\pm p$.
The last statement on the geometric KMS state follows by comparison of (22)
with (20).∎
###### Remark 5.6.
Unlike the ${\rm Vir}_{1}$ case, here the geometric KMS state does not
correspond either to $q=0$ or the lowest possible value
$\frac{\pi}{12\beta^{2}}$ of the energy density.
#### An argument toward classification
We give here an argument that could be useful in the classification of KMS
states on Virasoro nets.
Let $\varphi$ be a primary (locally normal) KMS state on the ${\rm Vir}_{c}$
net w.r.t. translations and suppose that $\varphi\left(\left(T(f_{1})\cdots
T(f_{n})\right)^{\ast}\left(T(f_{1})\cdots T(f_{n})\right)\right)<\infty$,
$f_{1},\cdots,f_{n}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$. This is the
case for all the known KMS states, listed above, although we cannot prove it
for a general KMS state. As the state is locally normal, the GNS
representation $\pi_{\varphi}$ is locally normal (thus a unitary equivalence
of type III factors) and can be extended to the stress-energy tensors $T(f)$
($f\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$), which are unbounded
operators affiliated to local von Neumann algebras. The above hypothesis is
equivalent to the requirement that the GNS vector $\Omega_{\varphi}$ is in the
domain of any (noncommutative) polynomial of the represented stress-energy
tensors $\pi_{\varphi}(T(f))$. We show that the values of the state on
polynomials of the stress-energy tensor $\varphi(T(f_{1})\cdots T(f_{n}))$ are
uniquely determined by the value of the state on a single stress-energy tensor
$\varphi\left(T(f)\right)$, for $f\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$. This fact seems to determine
uniquely the KMS state $\varphi$, as the net is in some sense generated by
such polynomials, however this is not a rigorous statement.
First of all, one can generalize the KMS condition in order to treat unbounded
operators: it is shown in [40, Prop. 3.5.2] that equations (1) hold with $x,y$
possibly unbounded operators affiliated to a local algebra, such that
$\Omega_{\varphi}$ is in the domain of
$\pi_{\varphi}(x),\pi_{\varphi}(x^{\ast}),\pi_{\varphi}(y)$ and
$\pi_{\varphi}(y^{\ast})$. Then we show, by induction in $n$, that
$\varphi\left(T(f)\right)$, together with the KMS conditions, uniquely
determines the values $\varphi\left(T(f_{1})...T(f_{n})\right)$. It is obvious
for $n=1$. It is supposed that $\Omega_{\varphi}$ is in the domain of the
polynomials of $T(f)$, the value of
$\varphi\left(\left[T(f_{1})...T(f_{n-1}),T(f_{n})\right]\right)$ can be
computed from the values of $\varphi$ on polynomials of degree $n-1$, using
the commutation relations (2) which hold on $\Omega_{\varphi}$. According to
the KMS condition, there is a function
$F(t)=\varphi\left(T(f_{1})...T(f_{n-1})\tau_{t}T(f_{n})\right)$, continuous
and bounded in $D_{\beta}:=\\{0\leq\Im z\leq\beta\\}$ and analytic in its
interior, such that
$F(t+i\beta)-F(t)=\varphi\left(\left[T(f_{1})...T(f_{n-1}),\tau_{t}T(f_{n})\right]\right)$.
If $G$ has the same properties, then $F-G$ is continuous in $D_{\beta}$,
analytic in its interior and $(F-G)(t+i\beta)-(F-G)(t)=0$, thus $F-G$ can be
continued to an analytic bounded functions on ${\mathbb{C}}$, which has to be
a constant. As $\varphi$ is primary, the clustering property implies that the
constant is $0$:
$\lim_{t\rightarrow\infty}F(t)=\varphi\left(T(f_{1})...T(f_{n-1})\right)\varphi\left(T(f_{n})\right)=\lim_{t\rightarrow\infty}G(t)$.
Thus $F$ is uniquely determined and, in particular,
$\varphi\left(T(f_{1})...T(f_{n-1})T(f_{n})\right)$.
If the above argument can be made rigorous, one would get that a KMS state on
the Virasoro net is uniquely determined by the value of the ’energy density’,
the constant $k$ appearing in $\varphi(T(f))=k\int f(x)dx$ (this in the only
possible expression with translation invariance). In order to prove that the
list in (22) is complete, it would be enough to prove that the set of possible
energy density has $\frac{\pi}{12\beta^{2}}$ as greatest lower bound.
## 6 The free fermion model
In this section we consider the free fermion net and the KMS states on its
quasilocal $C^{*}$-algebra. For an algebraic treatment of this model, see [2,
30]. In contrast to the $U(1)$-current model, the free boson model, it turns
out to admit a unique KMS state (for each temperature). The model is not
local, but rather graded local. It is still possible to define a (fermionic)
net [11].
The free fermion field $\psi$ defined on $S^{1}$ satisfies the following
Canonical Anticommutation Relation (CAR):
$\\{\psi(z),\psi(w)\\}=2\pi i\cdot\delta(z-w),$
and the Hermitian condition $\psi(z)^{*}=z\psi(z)$, or, if we consider the
smeared field, we have
$\\{\psi(f),\psi(g)\\}=\oint_{S^{1}}\frac{dz}{2\pi iz}f(z)g(z).$
We put the Neveu-Schwarz boundary condition: $\psi(ze^{2\pi i})=\psi(z)$. Then
it is possible to expand $\psi(z)$ in terms of Fourier modes as follows.
$\psi(z)=\sum_{r\in{\mathbb{Z}}+\frac{1}{2}}b_{r}z^{-r-\frac{1}{2}}.$
The Fourier components satisfy the commutation relation
$\\{b_{s},b_{r}\\}=\delta_{s,-r}{\mathbbm{1}}$,
$s,r\in{\mathbb{Z}}+\frac{1}{2}$.
There is a faithful *-representation of this algebra which contains the lowest
weight vector $\Omega$, i.e., $b_{s}\Omega=0$ for $s>0$ (we omit the symbol
for the representation since it is faithful). This representation is Möbius
covariant [2, Appendix A]. Let $U$ be the unitary representation $U$ of
$SL(2,{\mathbb{R}})\cong SU(1,1)$ which makes $\psi$ covariant. It holds that
$U(g)\Omega=\Omega$.
Let $P$ be the orthogonal projection onto the space generated by even
polynomials of $\\{b_{s}\\}$. It commutes with $U(g)$ and the unitary operator
$\Gamma=2P-{\mathbbm{1}}$ defines an inner symmetry (an automorphism which
preserves the vacuum state $\langle\Omega,\cdot\Omega\rangle$).
For an interval $I$, we put ${\mathcal{A}}(I):=\\{\psi(f):{\rm supp}(f)\subset
I\\}^{\prime\prime}$. Then ${\mathcal{A}}$ is a Möbius covariant fermi net in
the sense of [11], and graded locality is implemented by $Z$, where
$Z:=\frac{{\mathbbm{1}}-i\Gamma}{{\mathbbm{1}}-i}$. As a consequence, we have
twisted Haag duality: It holds that
${\mathcal{A}}(I^{\prime})=Z{\mathcal{A}}(I)^{\prime}Z^{*}$. In addition, we
have Bisognano-Wichmann property: $\Delta^{it}=U(\Lambda(-2\pi t))$, where
$\Delta^{it}$ is the modular group of ${\mathcal{A}}({\mathbb{R}}_{+})$ with
respect to $\Omega$ under the identification of $S^{1}$ and
${\mathbb{R}}\cup\\{\infty\\}$, and $\Lambda$ is the unique one-parameter
group of $SL(2,{\mathbb{R}})$ which projects to the dilation subgroup in
$PSL(2,{\mathbb{R}})$ under the quotient by
$\\{{\mathbbm{1}},-{\mathbbm{1}}\\}$ [12].
With $\\{b_{s}\\}$ we can construct a representation of the Virasoro algebra
with $c=\frac{1}{2}$ as follows (see [30]):
$L_{n}:=\frac{1}{2}\sum_{s>\frac{n}{2}}\left(s-\frac{n}{2}\right)b_{-s}b_{n+s},\mbox{
for }n\geq 0,$
and $L_{-n}=L_{n}^{*}$. For a smooth function $f$ on $S^{1}$, we can define
the smeared stress-energy tensor $T(f):=\sum_{n}f_{n}L_{n}$, where
$f_{n}=\oint_{S^{1}}\frac{dz}{2\pi i}z^{-n-1}f(z)$. The two fields $\psi$ and
$T$ are relatively local, namely if $f$ and $g$ have disjoint supports, then
$[\psi(f),T(g)]=0$ ($\psi(f)$ is a bounded operator and this holds on a core
of $T(g)$).
By the twisted Haag duality, we have $e^{iT(g)}\in{\mathcal{A}}(I)$ if ${\rm
supp}(g)\subset I$ (since $\psi(f)$ is bounded for a smooth function $f$,
there is no problem of domains). Let us define ${\rm
Vir}_{\frac{1}{2}}(I):=\\{e^{iT(g)}:{\rm supp}(g)\subset I\\}$. This Virasoro
net ${\rm Vir}_{\frac{1}{2}}$ has been studied in [21] and it has been shown
that ${\rm Vir}_{\frac{1}{2}}$ admits a unique nonlocal, relatively local
extension with index $2$. Hence the fermi net ${\mathcal{A}}$ is the
extension. Furthermore, by the relative locality, ${\mathcal{A}}$ is
diffeomorphism covariant by an analogous argument as in [10, Theorem 3.7].
We consider the restricted net ${\mathcal{A}}|_{\mathbb{R}}$ on ${\mathbb{R}}$
as in Section 2.1.1, the quasilocal $C^{*}$-algebra $\mathfrak{A}$ and
translation.
###### Theorem 6.1.
The free fermion net ${\mathcal{A}}$ admits one and only KMS state at each
temperature.
###### Proof.
By the diffeomorphism covariance and Bisognano-Wichmann property, we can
construct the geometric KMS state as in Part I [7, Section 2.8] (locality is
not necessary). On the other hand, ${\rm Vir}_{\frac{1}{2}}$ is completely
rational [21], hence it admits a unique KMS state. In this case, we have
proved without using locality [7, Theorem 4.11] that also the finite index
extension ${\mathcal{A}}$ admits only the geometric KMS state. ∎
#### Acknowledgement
We would like to thank the referee for pointing out imprecise statements in
Appendix A.
## Appendix Appendix A On the full extension of a KMS state
In this Appendix we discuss the theorem of Araki-Haag-Kastler-Takesaki [1].
Let $\mathfrak{B}$ be a $C^{*}$-algebra, $G$ a compact group acting on
$\mathfrak{B}$ and $\mathfrak{A}=\mathfrak{B}^{G}$ the fixed point with
respect to the action of $G$. We take a KMS state $\varphi$ on ${\mathcal{A}}$
and a weakly $\gamma$-clustering extension $\psi$. If one looks at the
statement carefully, it splits into two parts. The first part (Theorem II.4)
claims that there is a distinguished subgroup $N_{\psi}$ (depending on $\psi$)
of $G$ such that $\psi$ is a KMS state on $\mathfrak{B}^{N_{\psi}}$ with
respect to an appropriate one-parameter automorphism group $\widetilde{\tau}$.
Then the second part (Remark II.4) says that $N_{\psi}$ is trivial when $\psi$
is faithful on $\mathfrak{A}$, so $\psi$ is a KMS state on the whole algebra
$\mathfrak{B}$. We believe that the first part is correct, but the proof of
the second part is missing in the paper and we provide a counterexample at the
end of this Appendix. Hence the extension to the full algebra $\mathfrak{B}$
is not clear in general333The same statement of full extension is found, for
example, in [3, Theorem 5.4.25]. But we think that at least the argument is
flawed. We will give later a counterexample to the argument in [3]. Here we
prove this complete extension with an additional assumption, which can be
applied to the case of nets.
Let $G_{\psi}$ be the group of the stabilizers of $\psi$:
$G_{\psi}:=\\{g\in G:\psi(\alpha_{g}(a))=\psi(a)\mbox{ for all
}a\in\mathfrak{B}\\}.$
The actions $\alpha$ of $G$, $\tau$ of ${\mathbb{R}}$ and $\rho$ of
${\mathbb{Z}}$ are assumed to be norm-continuous. We always assume that $G$ is
compact, the action of $\gamma$ on $\mathfrak{B}$ is asymptotically abelian.
We precisely cite (the relevant part of)[1, Theorem II.4] (Note that we
changed the notation. In the original literature they use
${\mathcal{A}},{\mathcal{F}}$ for algebras, $\alpha$ for the time-translation,
$\gamma$ for the compact group action, $\tau$ for the space-translation and
$\varphi$ for the state).
###### Theorem A.1 (Araki-Haag-Kastler-Takesaki).
Assume that $G$ is separable. Let $\psi$ be a weakly $\gamma$-clustering state
of $\mathfrak{B}$, whose restriction to $\mathfrak{A}$ is an extremal
$(\tau_{t},\beta)$-KMS state. Then there exists a closed normal subgroup
$N_{\psi}$ of $G_{\psi}$, a continuous one-parameter subgroup
$\varepsilon_{t}$ of $Z(G_{\psi},G)$ and a continuous one-parameter subgroup
$\zeta_{t}$ of $G_{\psi}$ such that the restriction of $\psi$ to the fixed
point algebra under $N_{\psi}$
$\mathfrak{B}^{N_{\psi}}=\\{a\in\mathfrak{B}:\alpha_{g}(a)=a\mbox{ for all
}g\in N_{\psi}\\}$
is a $(\widetilde{\tau}_{t},\beta)$-KMS state where
$\widetilde{\tau}_{t}=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$.
We recall that the proof of this Theorem is further split into two parts ([1,
Theorem II.2, Section II.5 and Section II.6]).
###### Lemma A.2.
Under the hypothesis of Theorem A.1, there is a one-parameter subgroup
${\mathbb{R}}\ni t\longmapsto\varepsilon_{t}\in Z(G_{\psi},G)$
such that the restriction of $\psi$ to $\mathfrak{B}^{G_{\psi}}$ is an
$(\tau^{\prime}_{t},\beta)$-KMS state where
$\tau^{\prime}_{t}:=\tau_{t}\circ\alpha_{\varepsilon_{t}}$.
###### Lemma A.3.
Under the hypothesis of Theorem A.1, there is a continuous one-parameter
subgroup $\zeta_{t}$ of $G_{\psi}$ such that the restriction of $\varphi$ to
$\mathfrak{B}^{N_{\psi}}$ is an $(\widetilde{\tau}_{t},\beta)$-KMS state where
$\widetilde{\tau}_{t}:=\tau_{t}\circ\alpha_{\varepsilon_{t}\circ\zeta_{t}}$.
We think both of Lemmas are correct, hence the only task is to show that
$N_{\psi}$ trivially acts on $\mathfrak{B}$ under certain conditions. Then let
us recall how $N_{\psi}$ is defined.
Consider the space of functions
$C_{\psi}(G_{\psi}):=\\{f_{a,b}^{\psi}\in
C(G_{\psi}):f_{a,b}^{\psi}(g)=\psi(a\alpha_{g}(b)),a,b\in\mathfrak{B}\\}.$
It has been shown that the norm closure $\overline{C_{\psi}(G_{\psi})}$ is a
Banach subalgebra of $C(G_{\psi})$ [1, Lemma II.3], thus the intersection
$\overline{C_{\psi}(G_{\psi})}\cap\overline{C_{\psi}(G_{\psi})}^{*}$ is a
$C^{*}$-subalgebra of $C(G_{\psi})$. It is easy to see that this intersection
is globally invariant under left and right translation by $G_{\psi}$ since by
definition $\psi$ is invariant under $G_{\psi}$, hence there is a closed
normal subgroup $N_{\psi}$ such that
$\overline{C_{\psi}(G_{\psi})}\cap\overline{C_{\psi}(G_{\psi})}^{*}\cong
C(G_{\psi}/N_{\psi})$, where the isomorphism intertwines the natural actions
of $G_{\psi}$ [1, Lemma A.1]. Explicitly, $N_{\psi}$ is defined as follows:
$N_{\psi}:=\\{g\in G_{\psi}:f(g\cdot)=f(\cdot)\mbox{ for all
}f\in\overline{C_{\psi}(G_{\psi})}\cap\overline{C_{\psi}(G_{\psi})}^{*}\\}$
On the other hand, we can define another normal subgroup $N^{\prime}_{\psi}$
of $G_{\psi}$:
$N_{\psi}^{\prime}:=\\{g\in
G_{\psi}:f_{a,b}^{\psi}(g\cdot)=f_{a,b}^{\psi}(\cdot)\mbox{ for all
}a,b\in\mathfrak{B}\\}.$
It is easy to see, by uniform approximation, that
$N_{\psi}^{\prime}:=\\{g\in G_{\psi}:f(g\cdot)=f(\cdot)\mbox{ for all
}f\in\overline{C_{\psi}(G_{\psi})}\\}.$
Hence, $N_{\psi}^{\prime}\subset N_{\psi}$. Under a general assumption,
$N^{\prime}_{\psi}$ has a simple interpretation.
###### Lemma A.4.
Suppose that the GNS representation $\pi_{\psi}$ of $\mathfrak{B}$ is
faithful. Then it holds that $N^{\prime}_{\psi}=\\{g\in
G_{\psi}:\alpha_{g}(a)=a\mbox{ for all }a\in\mathfrak{B}\\}$, namely
$N^{\prime}_{\psi}$ is the subgroup of the elements acting trivially on
$\mathfrak{B}$.
###### Proof.
We show that $N^{\prime}_{\psi}\subset\\{g\in G_{\psi}:\alpha_{g}(a)=a\mbox{
for all }a\in\mathfrak{B}\\}$, since the other inclusion is obvious. In the
GNS representation, the defining equation of $N^{\prime}_{\psi}$ is equivalent
to
$\langle\pi_{\psi}(a^{*})\Omega_{\psi},U_{\psi}(g)\pi_{\psi}(b)\Omega_{\psi}\rangle=\langle\pi_{\psi}(a^{*})\Omega_{\psi},\pi_{\psi}(b)\Omega_{\psi}\rangle,\mbox{
for all }a,b\in\mathfrak{B},$
which implies that $U_{\psi}(g)={\mathbbm{1}}$ and ${\hbox{\rm
Ad\,}}U_{\psi}(g)={\rm id}$. In particular, we have
$\pi_{\psi}(\alpha_{g}(a))=\pi_{\psi}(a)$ for all $a\in\mathfrak{B}$ and, by
the assumed faithfulness of $\pi_{\psi}$, we obtain $\alpha_{g}(a)=a$. ∎
###### Theorem A.5.
If the GNS representation of $\mathfrak{B}^{G_{\psi}}$ with respect to (the
restriction of) $\psi$ is faithful and if $\pi_{\psi}$ is faithful on
$\mathfrak{B}$, then $N_{\psi}$ acts trivially on $\mathfrak{B}$.
###### Proof.
We only have to show that $N_{\psi}=N^{\prime}_{\psi}$ by Lemma A.4 and the
latter hypothesis. Under the former assumption, we show that the intersection
$\overline{C_{\psi}(G_{\psi})}\cap\overline{C_{\psi}(G_{\psi})}^{*}$ is equal
to $\overline{C_{\psi}(G_{\psi})}$, then the Theorem follows from the
definitions of $N_{\psi}$ and $N^{\prime}_{\psi}$.
We remark that the assumption implies that $\psi$ is faithful on
$\mathfrak{B}$. Indeed, first the assumption that the GNS representation of
$\psi$ restricted to $\mathfrak{B}^{G_{\psi}}$ is faithful implies that $\psi$
is faithful on $\mathfrak{B}^{G_{\psi}}$, since the GNS vector of a KMS state
is separating [3, Corollary 5.3.9]. Now let $x\in\mathfrak{B}$ such that
$\psi(x^{*}x)=0$. Then, by the definition of $G_{\psi}$, $\psi$ is invariant
under $G_{\psi}$, thus we have
$0=\int_{G_{\psi}}\psi(\alpha_{g}(x^{*}x))dg=\psi\left(\int_{G_{\psi}}\alpha_{g}(x^{*}x)dg\right).$
But $\int_{G_{\psi}}\alpha_{g}(x^{*}x)dg$ is positive and belongs to
$\mathfrak{B}^{G_{\psi}}$, hence must be zero by the faithfulness of $\psi$ on
$\mathfrak{B}^{G_{\psi}}$. This is possible only if $x^{*}x=0$ by the
continuity of $\alpha$.
As recalled in Appendix B, for $f_{a,b}^{\psi}\in C_{\psi}(G_{\psi})$, one can
take its Fourier component $f_{a,b_{\chi}}^{\psi}$ and the original function
$f_{a,b}^{\psi}$ is uniformly approximated by its components. Hence it is
enough to consider irreducible representations. If $f_{a,b}^{\psi}$ contains
$\chi$-component for some $a,b\in\mathfrak{B}$, then this in particular means
that $b_{\chi}\neq 0$. By the faithfulness of $\psi$ on $\mathfrak{B}$ proved
above, one sees that $\psi(b_{\chi}b_{\chi}^{*})\neq 0$. Since $b_{\chi}^{*}$
belongs to the irreducible representation $\overline{\chi}$, one concludes
that the conjugate representation $\overline{\chi}$ is contained in
${\mathcal{H}}_{\psi}$. Then any function in
$\overline{C_{\overline{\chi}}(G_{\psi})}$ (see Appendix B) belongs to
$\overline{C_{\psi}(G_{\psi})}$.
Summing up, the adjoint of each component of $f_{a,b}^{\psi}\in
C_{\psi}(G_{\psi})$ belongs again to $C_{\psi}(G_{\psi})$ and each function in
$\overline{C_{\psi}(G_{\psi})}$ is recovered from its components. This
completes the proof of self-adjointness of $\overline{C_{\psi}(G_{\psi})}$. ∎
The hypothesis of the Theorem are satisfied not only in our case of conformal
nets, as we see in Section 3.2, but also in a wide class of models of
statistical mechanics where local algebras are finite dimensional factors
$M_{n}({\mathbb{C}})$.
#### On the proofs of full-extension in the literature
As noted before, Theorem A.5 without the assumption of faithfulness of
$\pi_{\psi}$ is claimed in [1] without proof. In [3, Theorem 5.4.25] the
theorem of full extension (i.e. $N_{\psi}$ acts trivially) is stated with the
assumption of faithfulness of $\varphi=\psi|_{\mathfrak{A}}$ on
$\mathfrak{A}$. But we think that the proof is not complete. The argument in
[3] goes as follows. At the first step, they assume that $\psi$ is faithful on
$\mathfrak{A}$ and show that $\psi$ is a KMS state on
$\mathfrak{B}^{G_{\psi}}$. At the second step, they say that one can assume
that $\psi$ is invariant under $G$ and the rest follows. The point is that, in
the first extension, the faithfulness of $\psi$ on $\mathfrak{B}^{G_{\psi}}$
is not automatic. The symmetry of the spectrum of $\pi_{\psi}$ is essential in
the second extension and the faithfulness is used for it. Here we provide an
example which shows that this faithfulness does not hold in general. We do not
know whether the theorem holds without these assumptions. The same
construction gives a counterexample to [1, Remark II.4].
We take an auxiliary system $(\mathfrak{B},\mathfrak{A},\tau,\alpha,\gamma)$,
where $\mathfrak{A}=\mathfrak{B}^{G}$ and take a KMS state $\varphi$ on
$\mathfrak{A}$ with respect to $\tau$ and a $\gamma$-clustering extension
$\psi$. Suppose for simplicity that $\psi$ is faithful and has the whole group
as the stabilizer: $G_{\psi}:=\\{g\in G:\psi\circ\alpha_{g}=\psi\\}=G$. We
have many such examples: one can take just the geometric KMS state on the
regularized quasilocal algebra of a conformal net with a compact group action
and the inclusion of the fixed point subnet.
Consider now the field system
$(\widehat{\mathfrak{B}},\widehat{\mathfrak{A}},\widehat{\tau},\widehat{\alpha},\widehat{\gamma})$
where $\widehat{\mathfrak{B}}:=\mathfrak{B}\oplus\mathfrak{B}$,
$\widehat{G}:=(G\times G)\rtimes{\mathbb{Z}}_{2}$ with ${\mathbb{Z}}_{2}$
acting on $G\times G$ as the flip,
$\widehat{\tau}_{t}:=\tau_{t}\oplus\tau_{t}$, the action $\widehat{\alpha}$ of
$(G\times G)\rtimes{\mathbb{Z}}_{2}$ on $\mathfrak{B}\oplus\mathfrak{B}$ being
the action $\alpha$ of each copy of $G$ on each copy of $\mathfrak{B}$ and the
action of ${\mathbb{Z}}_{2}$ as the flip. The fixed point
$\mathfrak{B}^{\widehat{G}}$ is the diagonal algebra
$\widehat{\mathfrak{A}}\subset\mathfrak{A}\oplus\mathfrak{A}$, which is
isomorphic to $\mathfrak{A}$. The system
$(\widehat{\mathfrak{B}},\widehat{\tau})$ is asymptotically abelian as so is
$(\mathfrak{B},\tau)$.
Let $\pi_{i}:\widehat{\mathfrak{B}}\to\mathfrak{B}$ be the projections on a
component and $\psi_{i}:=\psi\circ\pi_{i}.$ The two states $\psi_{i}$ are the
two $\gamma$-clustering extensions of $\varphi$ on $\widehat{\mathfrak{B}}$
(other extensions are convex combinations of $\psi_{1}$ and $\psi_{2}$ and are
KMS states w.r.t. $\widehat{\tau}$). The stabilizer is in both cases
$\widehat{G}_{\psi_{i}}=(G\times G)$, a normal subgroup of $\widehat{G}$,
while the flip exchanges the two states: $\psi_{i}\circ\alpha_{z}=\psi_{zi}$
for $z\in{\mathbb{Z}}_{2}$. The intermediate algebra is
$\widehat{\mathfrak{B}}^{\widehat{G}_{\psi_{i}}}=\mathfrak{A}\oplus\mathfrak{A}$
and $\psi_{i}$ is obviously not faithful on it; the faithfulness was assumed
implicitly in the second step of the proof in [3].
Let $\pi_{\psi}:\mathfrak{B}\to B({\mathcal{H}}_{\psi})$ be the GNS
representation of $\psi$ (faithful as $\psi$ is faithful), then
$\pi_{\psi_{i}}=\pi_{\psi}\circ\pi_{i}:\widehat{\mathfrak{B}}\to
B({\mathcal{H}}_{\psi})$ is not faithful, although it is true that
$\pi_{\psi_{i}}(\widehat{\mathfrak{A}})^{\prime\prime}=\pi_{\psi_{i}}(\widehat{\mathfrak{B}}^{G_{\psi_{i}}})^{\prime\prime}$.
As $\pi_{\psi_{i}}$ is not faithful, although $\psi_{i}$ is faithful on
$\widehat{\mathfrak{A}}$, we cannot deduce that
$N^{\prime}_{\psi_{i}}:=\\{g\in\widehat{G}:\psi_{i}(a\widehat{\alpha}_{g}(b))=\psi_{i}(ab)\mbox{
for all }a,b\in\widehat{\mathfrak{B}}\\}$ is trivial nor that it acts
trivially. Indeed, $N^{\prime}_{\psi_{1}}=(1,G,1)$ and
$N^{\prime}_{\psi_{2}}=(G,1,1)$; this is in contradiction with [1, Remark
II.4], since we show that $N^{\prime}_{\psi_{i}}=N_{\psi_{i}}$, also in this
case. By proceeding as in the third paragraph of the proof of Theorem A.5, let
$b_{\chi}$ belong to the irreducible representation $\chi$ and
$\psi_{1}(b_{\chi}^{*}b_{\chi})\neq 0$. Then, as $b_{\chi}$ is of the form
$b_{\chi}=b_{1}\oplus b_{2}$,
$\psi_{1}(b_{\chi}b_{\chi}^{*})=\psi(b_{1}b_{1}^{*})\neq 0$ by the
faithfulness of $\psi$, which implies that $\overline{\chi}$ is contained in
${\mathcal{H}}_{\psi}$; the rest follows as in Theorem A.5. One sees that
$N_{\psi_{i}}$ is a normal subgroup of $\widehat{G}_{\psi_{i}}$, $\psi_{i}$
and $\pi_{\psi_{i}}$ are faithful neither on $\widehat{\mathfrak{B}}$ nor on
$\widehat{\mathfrak{B}}^{N_{\psi_{1}}}=\mathfrak{B}\oplus\mathfrak{A}$.
Moreover, the one parameter group $\widehat{\tau}$ w.r.t. which $\psi_{i}$ is
KMS is not uniquely defined, as $\psi_{1}$ is not faithful and is KMS w.r.t.
$\widehat{\tau}\circ\alpha_{(1,g_{t},1)}$ for any $t\mapsto g_{t}\in G$.
## Appendix Appendix B Noncommutative harmonic analysis
Here we briefly summarize elementary methods to treat actions of a compact
group $G$ on a $C^{*}$-algebra. For the classical facts from the
representation theory of compact groups, we refer to the standard textbooks,
for example, [23]. The classical Peter-Weyl theorem says that any irreducible
representation of $G$ is finite dimensional. To a finite-dimensional
representation one can associate a character $\chi$ in the space $C(G)$ of
continuous functions on $G$. On this space $G$ acts by left and right
translations. This becomes a pre-Hilbert space by the inner product induced by
the Haar measure and its completion is denoted by $L^{2}(G)$. The action by
translation is referred to as the left or right regular representation. Again
the Peter-Weyl theorem states that the left or right regular representation
contains any irreducible representation and the multiplicity is equal to its
dimension. If a function $f$ belongs to an irreducible representation $\chi$
of dimension $n$ of the left (or right) regular representation, then the
images of $f$ under right and left translation of $G\times G$ span the whole
$n^{2}$ dimensional space. Here we call this subspace $C_{\chi}(G)$. Two
characters $\chi,\chi^{\prime}$ are orthogonal iff the corresponding
representations are disjoint. Any unitary representation $U$ can be written as
the direct sum of irreducible representations. The decomposition into classes
of inequivalent representations is canonical: for a character $\chi$
associated to an irreducible representation, the map
$\xi\mapsto\xi_{\chi}=\int_{G}\overline{\chi(g)}U(g)\xi dg.$
is the projection from the representation space onto the direct sum of
irreducible subrepresentations of $U$ equivalent to the one corresponding to
$\chi$. It holds that $\xi=\sum_{\chi}\xi_{\chi}$ and
$\xi_{\chi}\perp\xi_{\chi^{\prime}}$ if $\chi$ and $\chi^{\prime}$ are
inequivalent. The above formula is an extension of the Fourier decomposition.
An action $\alpha$ of $G$ on a $C^{*}$-algebra $\mathfrak{B}$ is an infinite
dimensional representation of $G$ on a Banach space. It is still possible to
define the Fourier components: for $a\in\mathfrak{B}$, we put
$a_{\chi}:=\int_{G}\overline{\chi(g)}\alpha_{g}(a)dg.$
In general the sum $\sum_{\chi}a_{\chi}$ is not necessarily norm-convergent.
Now let us assume that there is a $G$-invariant state $\psi$. Then in the GNS
representation $({\mathcal{H}}_{\psi},\pi_{\psi},\Omega_{\psi})$ there is a
unitary representation $U_{\psi}$ which implements the action $\alpha$. The
components defined for $U_{\psi}$ and $\alpha$ are compatible: we have
$\pi_{\psi}(a_{\chi})\Omega_{\psi}=\int_{G}\overline{\chi(g)}\pi_{\psi}(\alpha_{g}(a))\Omega_{\psi}=\int_{G}\overline{\chi(g)}U_{\psi}(g)\pi_{\psi}(a)\Omega_{\psi}=(\pi_{\psi}(a)\Omega_{\psi})_{\chi}.$
From the orthogonality in the representation $U_{\psi}$, one sees that if
$\chi$ and $\chi^{\prime}$ correspond to two disjoint representations, then
$\psi((a_{\chi})^{*}\alpha_{g}(b_{\chi^{\prime}}))=\langle\pi_{\psi}(a_{\chi})\Omega_{\psi},U_{\psi}(g)\pi_{\psi}(b_{\chi^{\prime}}))\Omega_{\psi}\rangle=0$.
It is immediate to see that the function
$g\mapsto\psi(a\alpha_{g}(b_{\chi}))=\psi((a_{\overline{\chi}})\alpha_{g}(b))$
belongs to $C_{\chi}(G)$. The decomposition of the vector
$\pi_{\psi}(b)\Omega_{\psi}=\sum_{\chi}\pi_{\psi}(b_{\chi})\Omega_{\psi}$
converges in norm, hence for the function
$f_{a,b}^{\psi}(g):=\psi(a\alpha_{g}(b))$, the decomposition
$f_{a,b}^{\psi}=\sum_{\chi}f_{a,b}^{\chi}$ where
$f_{a,b}^{\chi}(g)=\langle\pi_{\psi}(a^{*})\Omega_{\psi},U_{\psi}(g)\pi_{\psi}(b_{\chi})\Omega_{\psi}\rangle$
converges uniformly in the norm of $C(G)$.
## References
* [1] H. Araki, D. Kastler, M. Takesaki and R. Haag: Extension of KMS states and chemical potential. Comm. Math. Phys. 53 (1977), no. 2, 97-134.
* [2] J. Böckenhauer: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177 (1996), 265–304.
* [3] O. Bratteli and D. Robinson: Operator algebras and quantum statistical mechanics. Vol. 2, Springer-Verlag, Berlin, 1997.
* [4] D. Buchholz, C. D’Antoni and R. Longo: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270 (2007), 267–293.
* [5] D. Buchholz, G. Mack and I. Todorov: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proc. Suppl.) 5B (1988), 20–56.
* [6] D. Buchholz and H. Schulz-Mirbach: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2 (1990), 105–125.
* [7] P. Camassa, R. Longo, Y. Tanimoto and M. Weiner: Thermal states in conformal QFT. I. Commun. Math. Phys. 309 (2011), 703–735.
* [8] S. Carpi: Classification of subsystems for the Haag-Kastler nets generated by $c=1$ chiral current algebras. Lett. Math. Phys. 47 (1999), 353–364.
* [9] S. Carpi: The Virasoro algebra and sectors with infinite statistical dimension. Ann. Henri Poincaré 4 (2003), 601–611.
* [10] S. Carpi: On the representation theory of Virasoro nets. Comm. Math. Phys. 244 (2004), 261–284.
* [11] S. Carpi, Y. Kawahigashi and R. Longo: Structure and classification of superconformal nets. Ann. Henri Poincaré. 9 (2008), 1069–1121.
* [12] C. D’Antoni, R. Longo and F. Radulescu: Conformal nets, maximal temperature and models from free probability. J. Op. Theory 45 (2001), 195–208.
* [13] J. Dixmier. _Von Neumann algebras_. North-Holland, 1981.
* [14] J. Dixmier. _$C^{*}$ -algebras_. North-Holland, 1982.
* [15] D. B. A. Epstein: Commutators of $C^{\infty}$-diffeomorphisms. Appendix to: “A curious remark concerning the geometric transfer map” by John N. Mather [Comment. Math. Helv. 59 (1984), no. 1, 86–110] Comment. Math. Helv. 59 (1984), 111–122.
* [16] C. J. Fewster and S. Hollands: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17 (2005), 577–612.
* [17] J. Fröhlich and F. Gabbiani: Operator algebras and conformal field theory. Commun. Math. Phys. 155 (1993), no. 3, 569–640.
* [18] R. Haag: Local quantum physics. Fields, particles, algebras. Springer-Verlag, Berlin, 1996.
* [19] G. H. Hardy: Asymptotic theory of partitions. In Ramanujan: Twelve lectures on subjects suggested by his life and work, 3rd ed., 113–131. Chelsea, New York, 1999.
* [20] D. Kastler: Equilibrium states of matter and operator algebras, Symposia Mathematica, Vol. XX. Academic Press, London-New York, 1976.
* [21] Y. Kawahigashi and R. Longo: Classification of local conformal nets. Case $c<1$ Ann. of Math. 160 (2004), 493–522.
* [22] Y. Kawahigashi, R. Longo and M. Müger: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219 (2001), 631–669.
* [23] A. Knapp: Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press, 2001.
* [24] R. Longo: Real Hilbert subspaces, modular theory, ${\rm SL}(2,{{\mathbb{R}}})$ and CFT. In Von Neumann algebas in Sibiu: Conference Proceedings, 33–91. Theta, Bucharest, 2008.
* [25] R. Longo and C. Peligrad: Noncommutative topological dynamics and compact actions on $C^{\ast}$-algebras. J. Funct. Anal. 58 (1984), no. 2, 157-174.
* [26] R. Longo and K.-H. Rehren: Nets of subfactors. Rev. Math. Phys. 7 (1995), no. 4, 567–597.
* [27] R. Longo and K.-H. Rehren: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys., to appear.
* [28] R. Longo and F. Xu: Topological sectors and a dichotomy in conformal field theory. Comm. Math. Phys. 251 (2004), no. 2, 321-364.
* [29] J. Manuceau: Étude de quelques automorphismes de la $C^{*}$-algèbre du champ de bosons libres. Ann. Inst. H. Poincaré Sect. A (N.S.) 8 (1968), 117–138.
* [30] G. Mack and V. Schomerus: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys. 134 (1990), 139–196.
* [31] J. N. Mather: Commutators of diffeomorphisms. Comment. Math. Helv. 49 (1974), 512–528.
* [32] G. H. Meisters: Translation-invariant linear forms and a formula for the Dirac measure. J. Funct. Anal. 8 (1971), 173–188.
* [33] J. T. Ottesen: Infinite-dimensional groups and algebras in quantum physics. Springer-Verlag, Berlin, 1995.
* [34] M. Reed and B. Simons: _Methods of modern mathematical physics._ Vol. 2: Fourier analysis, self-adjointness. Academic Press, 1975.
* [35] K.-H. Rehren: A new view of the Virasoro algebra. Lett. Math. Phys. 30 (1994), 125–130.
* [36] F. Rocca, M. Sirigue and D. Testard: On a class of equilibrium states under the Kubo-Martin-Schwinger condition II. Bosons. Commun. Math. Phys. 19 (1970), 119–141.
* [37] K. Schmüdgen: Strongly commuting selfadjoint operators and commutants of unbounded operator algebras. Proc. Amer. Math. Soc. 102 (1988), 365–372.
* [38] B. Schroer and H.-W. Wiesbrock: Looking beyond the thermal horizon: hidden symmetries in chiral models. Rev. Math. Phys. 12 (2000), no. 3, 461-473.
* [39] M. Takesaki and M. Winnink: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30 (1973), 129–152.
* [40] Y. Wang: Locally normal KMS states of diffeomorphism covariant nets w.r.t. translation subgroups. Ph.D. thesis Univ. of Roma ”Tor Vergata”(2008).
|
arxiv-papers
| 2011-09-09T16:38:49 |
2024-09-04T02:49:22.028017
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Paolo Camassa, Roberto Longo, Yoh Tanimoto, Mihaly Weiner",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1109.2064"
}
|
1109.2268
|
# Efficient Collection of Single Photons Emitted from a Trapped Ion into a
Single Mode Fiber for Scalable Quantum Information Processing
Taehyun Kim, Peter Maunz, and Jungsang Kim Fitzpatrick Institute for
Photonics, Electrical and Computer Engineering Department, Duke University,
Durham, North Carolina 27708, USA
###### Abstract
Interference and coincidence detection of two photons emitted by two remote
ions can lead to an entangled state which is a critical resource for scalable
quantum information processing. Currently, the success probabilities of
experimental realizations of this protocol are mainly limited by low coupling
efficiency of a photon emitted by an ion into a single mode fiber. Here, we
consider two strategies to enhance the collection probability of a photon
emitted from a trapped Yb+ ion, using analytic methods that can be easily
applied to other types of ion or neutral atoms. Our analysis shows that we can
achieve fiber coupling efficiency of over 30 % with an optical cavity made of
a flat fiber tip and a spherical mirror. We also investigate ways to increase
the fiber coupling efficiency using high numerical aperture optics, and show
that collection probability of over 15 % is possible with proper control of
aberration.
###### pacs:
32.80.Qk,42.50.Pq,42.15.Eq
## I INTRODUCTION
Entanglement is a unique feature of quantum physics and it has found various
applications ranging from the experimental verification of the foundation of
quantum physics Einstein _et al._ (1935) to many quantum information
protocols such as quantum teleportation Bennett _et al._ (1993), quantum
cryptography Ekert (1991), quantum computation Gottesman and Chuang (1999);
Raussendorf and Briegel (2001), and long distance quantum communication
Briegel _et al._ (1998). Most of these applications require a large number of
entangled states distributed over long distances, which can be generated using
one of two approaches. In the first approach, one can locally generate a pair
of entangled photons Kim _et al._ (2006), distribute them to remote sites,
and map these flying qubits into stationary (memory) qubits Lloyd _et al._
(2001). To efficiently transfer a flying qubit into a stationary qubit, the
spectral and spatial modes of the flying qubit have to be matched to the time-
inverted emission properties of the memory qubit, which is the most
significant challenge in this approach. In the second approach, one first
generates flying qubits that are entangled with a local stationary qubit at
each site, sends these photons to a common location, and then interferes these
two photons using a 50:50 beam splitter Duan _et al._ (2001); Simon and
Irvine (2003). With 50 % chance, these photon pairs are projected into one of
two distinguishable Bell states, and once one of these two Bell states is
detected, the two stationary qubits at the remote sites are also projected to
a maximally entangled Bell state Pan _et al._ (1998). The advantage of this
approach is that the spectral properties of the two photons are naturally
matched if one uses the same type of memory qubits. In order to achieve
interference of the two photons with high visibility, the photons have to be
indistinguishable. Therefore, we need to collect emitted photons into a single
spatial mode, such as a single mode fiber. Several experiments demonstrated
establishment of entanglement between remote trapped ions using this approach
Moehring _et al._ (2007); Matsukevich _et al._ (2008); Olmschenk _et al._
(2009). While these experiments realized high fidelity entangled states, the
success probability of the probabilistic entanglement process is very small.
The success probability is mainly limited by the small photon collection
probability of about 0.4 % given by the small numerical aperture (NA) of the
collection optics and an insufficient overlap between the spatial mode of the
single mode fiber and the emission pattern of an ion. Because this protocol
requires one to collect photons from both sites and observe a coincidence, the
overall success probability scales quadratically with the single fiber
coupling efficiency, and was limited to only about $2\cdot 10^{-8}$ (see Table
2 in Sec. IV) Olmschenk _et al._ (2009). To apply this system for large scale
quantum information processing or long distance quantum communication, it is
critical to dramatically enhance the collection probability of photons into a
single mode fiber.
In this paper, we show systematic approaches to increase the fiber coupling
efficiency of a single photon while preserving ion-photon entanglement. We
will analyze the design of an optimized optical cavity that can be integrated
with a surface ion trap, and evaluate various ways to achieve high NA
collection optics compatible with surface traps to enhance light collection
into a single mode fiber.
Figure 1: Ion-photon entanglement schemes in ${}^{171}{\rm Yb}^{+}$. (a)
Simplified level scheme of ${}^{171}{\rm Yb}^{+}$ with dipole-allowed
transitions. Zeeman levels are denoted by $|F,m_{F}\rangle$, where $F$ is the
total angular momentum including nuclear spin and $m_{F}$ is its projection
onto the quantization axis. ${}^{2}S_{1/2}|1,0\rangle$ and
${}^{2}S_{1/2}|0,0\rangle$ are first-order magnetic field-insensitive
hyperfine levels (clock states). (b) When the ion decays from
${}^{2}P_{1/2}|1,0\rangle$ to one of the Zeeman levels in ${}^{2}S_{1/2}|1,\pm
1\rangle$, the polarization of the emitted photon gets entangled with two
Zeeman levels Matsukevich _et al._ (2008). (c) The frequency of the emitted
photon and the atomic clock states are entangled after a decay from
${}^{2}P_{1/2}|1,-1\rangle$ Moehring _et al._ (2007). (d) Due to the
selection rules, excitation of ${}^{171}{\rm Yb}^{+}$ prepared in a
superposition of the clock states in ${}^{2}S_{1/2}$ generates a coherent
superposition of the excited states (two ${}^{2}P_{1/2}$ states with
$m_{F}=0$). Once excited state decays back to the original state, the
frequency of the emitted photon is entangled with clock states Olmschenk _et
al._ (2009). In ${}^{171}{\rm Yb}^{+}$ system, ${}^{2}S_{1/2}|0,0\rangle$ is
generally used as an initial state due to its high preparation fidelity.
Therefore (c) and (d) scheme require additional microwave or optical Raman
transitions denoted by thick arrows.
Our analysis mainly focuses on ${}^{171}{\rm Yb}^{+}$ whose simplified level
scheme is shown in Figure 1 (a) Olmschenk _et al._ (2007), but the
conclusions can be applied to other atomic systems. There are several ways to
entangle a photon’s degree of freedom with internal electronic states of a
trapped ${}^{171}{\rm Yb}^{+}$ ion. If Zeeman levels are denoted by
$|F,m_{F}\rangle$, where $F$ is the total angular momentum and $m_{F}$ is its
projection onto the quantization axis, one can excite the Yb ion to
${}^{2}P_{1/2}|1,0\rangle$ and collect the photon from the $\sigma^{\pm}$
transition as shown in Figure 1 (b). The polarization of the photon will then
be entangled with the two ground state Zeeman levels ${}^{2}S_{1/2}|1,\pm
1\rangle$ Matsukevich _et al._ (2008). Alternatively, one can collect a
photon generated by $\sigma$-transitions from ${}^{2}P_{1/2}|1,-1\rangle$ as
shown in Figure 1 (c) or by $\pi$-transitions from a superposition of
${}^{2}P_{1/2}|1,0\rangle$ and ${}^{2}P_{1/2}|0,0\rangle$ to the atomic clock
states composed of ${}^{2}S_{1/2}|1,0\rangle$ and ${}^{2}S_{1/2}|0,0\rangle$
as shown in Figure 1 (d). The frequency of the photon is then entangled with
the clock state qubits. The spatial emission pattern of photons from
$\sigma^{\pm}$ and $\pi$-transitions and their spectral properties together
with the ability to filter out photons from unwanted transitions will dictate
strategies that can be adopted to enhance the photon collection efficiency.
For remote entanglement generation, we will analyze the approach using an
optical cavity in Section II, and an approach using high NA optics (either a
reflector or a lens) in Section III. Section IV estimates the entanglement
generation rate we can achieve based on these approaches, and conclusions are
presented in Section V.
## II Cavity enhanced photon collection
A small high finesse cavity can be used to collect light from a single atom by
induced emission into the cavity mode. Although efficient light collection
using an optical cavity has been demonstrated for neutral atoms McKeever _et
al._ (2004); Hijlkema _et al._ (2007), additional challenges have to be met
for trapped ions: readout transitions of trapped ions are typically in the
ultraviolet (UV) and the ion trap has to fit in a small cavity with UV-
dielectric coatings. In addition, trapped ions have to be shielded from the
charges which can accumulate on dielectric surfaces Harlander _et al._
(2010).
The collection probability of a single photon from an excited atom into a
single mode fiber is dictated by several factors. First, the excitation has to
leave the coupled atom-cavity system via the cavity decay channel rather than
spontaneous emission, characterized by the photon extraction probability
$P_{cavity}$. While some photons leave the cavity through the desired output
channel, other photons are either lost due to scattering and absorption in the
mirrors or transmitted through the undesired mirror(s). We define $r_{t}$ as
the fraction of the desired transmission over the total cavity decay. Finally,
we consider overlap between the mode of the cavity and that of the single mode
fiber, which will be called mode matching factor $\epsilon$. The overall
probability $P_{fiber}$ of extracting the photon through the single mode fiber
is then given by
$P_{fiber}=\epsilon\cdot r_{t}\cdot P_{cavity}.$ (1)
In this section, we will investigate how to maximize each factor within the
physical constraints. We will mainly consider a cavity geometry shown in
Figure 2 Kim and Kim (2009) and optimize system parameters to maximize the
collection efficiency under the given constraints. This discussion will also
identify components that deserve further investigation to improve the
performance of the system in the future, and provide a metric to compare
different cavity systems.
Figure 2: Geometry of an optical cavity integrated with surface trap. Optical
cavity is composed of a small spherical mirror and a fiber in a ferrule whose
tip is coated with high reflecting dielectric layers. $l$ is cavity length,
$w_{0}$ is beam waist at the fiber tip, and $r_{ion}$ is beam radius at the
trapping position of the ion. Surface trap can be patterned with Au on the
fiber/ferrule surface whose diameter is $>$2 mm.
When an atomic system is coupled to a cavity mode, a single photon can be
generated either by fast excitation with resonant $\pi$-pulse followed by
spontaneous emission Moehring _et al._ (2007); Matsukevich _et al._ (2008);
Olmschenk _et al._ (2009); Bochmann _et al._ (2008), or by stimulated Raman
adiabatic passage (STIRAP) process McKeever _et al._ (2004); Hijlkema _et
al._ (2007). In this section, we will also compare these two methods in terms
of collection probability $P_{fiber}$ and repetition rate.
### II.1 Fast excitation with resonant $\pi$-pulse: Design principles
In this subsection, we consider the case where the ion is excited by an
ultrafast pulse whose duration is short compared to any time constants of the
system, such that the initial state is an excited state of the ion. When an
excited ion is placed inside an optical cavity, the spontaneous emission rate
into the cavity mode can be enhanced by the Purcell factor, $2C+1$ Mundt _et
al._ (2002), where $C=g^{2}/2\kappa\gamma$ is the cooperativity, $g$ is the
coherent ion-field coupling constant, $\kappa$ is the cavity-field decay rate,
and $2\gamma$ is the spontaneous emission rate.
The parameter $g^{2}$ is inversely proportional to the cavity mode volume and
$\kappa=\pi\cdot FSR/\mathcal{F}$, where $FSR$ and $\mathcal{F}$ are the free
spectral range and the finesse of the cavity, respectively. Therefore, we need
a high finesse cavity with small mode volume to achieve large cooperativity.
Small mode volume cavities can be realized by many different approaches such
as microdisk Gayral _et al._ (1999), microsphere Braginsky _et al._ (1989),
microtoroid Armani _et al._ (2003), and photonic crystals Painter _et al._
(1999). Integration with ion traps requires a cavity with empty mode volume,
so we will consider only traditional open cavities formed by two (curved)
mirrors. Cavity mode shape between two spherical mirrors is Gaussian and the
mode volume is $\pi w_{0}^{2}l/4$, where $w_{0}$ is the beam waist of the
Gaussian mode function and $l$ is length of the cavity. The cooperativity is
reduced to
$C=3\mathcal{F}\cdot\text{Br}\cdot\lambda^{2}/(\pi^{3}r_{ion}^{2})$, where Br
is a branching ratio, $\lambda$ is wavelength of transition, and $r_{ion}$ is
the radius of the mode function at ion location. In this system, we need high
$\mathcal{F}$ and small $r_{ion}$ to maximize $C$.
Using these cavity parameters, one can express the standard formula for
$P_{cavity}$ Cui and Raymer (2005) as a function of the cavity length ($l$)
and the beam radius at ion location ($r_{ion}$):
$\displaystyle P_{cavity}$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa}{\kappa+\gamma}\right)\left(\frac{2C}{2C+1}\right)$
(2) $\displaystyle=$
$\displaystyle\left(\frac{1}{1+l/l_{c}}\right)\left(\frac{1}{1+r_{ion}^{2}/r_{c}^{2}}\right),$
where $l_{c}$ is a characteristic cavity length defined as $l_{c}=\pi
c/(2\gamma\mathcal{F})$ and $r_{c}$ is a characteristic beam radius defined as
$r_{c}=\sqrt{6\lambda^{2}\mathcal{F}\cdot\text{Br}/\pi^{3}}$. Eq. (2) implies
that if either $l$ or $r_{ion}$ is much larger than the corresponding
characteristic lengths, the overall $P_{cavity}$ will decrease rapidly. The
characteristic cavity length (beam radius) decreases (increases) as the
finesse of the cavity increases. This indicates that for high finesse cavities
one should use a short cavity but the beam radius can be larger, and the
design shifts to longer cavities with smaller beam radius as the finesse of
the cavity decreases.
Cavity finesse $\mathcal{F}$ is defined by
$2\pi\sqrt{1-\mathcal{L}}/\mathcal{L}$, where
$\mathcal{L}=T_{f}+T_{e}+L_{f}+L_{e}$ is the total loss of the cavity, $T_{f}$
and $L_{f}$ ($T_{e}$ and $L_{e}$) are transmission and passive loss through
dielectric coating on the fiber tip (the other end mirror) in Figure 2,
respectively. Passive loss is due to scattering loss mainly determined by
surface roughness Trupke _et al._ (2005) and absorption in the dielectric
coating. We can define $\mathcal{F}_{0}\equiv
2\pi\sqrt{1-(L_{f}+L_{e})}/(L_{f}+L_{e})$, to denote the maximum achievable
finesse limited by the uncontrollable passive loss. Then the actual finesse
can be approximated to $\mathcal{F}\simeq\mathcal{F}_{0}(1-r_{t})$ where
$r_{t}=T_{f}/\mathcal{L}$. Assuming the mode matching factor $\epsilon$ is
constant, one can see that $P_{fiber}$ defined in Eq. (1) is maximized when
$r_{t}=r_{t0}\equiv\frac{1}{1+1/\sqrt{(1+2C_{0})(1+l/l_{c0})}}$ (3)
to a value
$P_{fiber,max}=\frac{C_{0}}{1+C_{0}+l/2l_{c0}+\sqrt{(1+2C_{0})(1+l/l_{c0})}},$
(4)
where $C_{0}\equiv
3\mathcal{F}_{0}\cdot\text{Br}\cdot\lambda^{2}/(\pi^{3}r_{ion}^{2})$ and
$l_{c0}\equiv\pi c/(2\gamma\mathcal{F}_{0})$ are the cooperativity and
characteristic length with maximum possible finesse $\mathcal{F}_{0}$,
respectively. From Eq. (4), one can see that $C_{0}$ should be maximized while
$l$ should be minimized in order to maximize the fiber coupling efficiency.
Finally, fiber collection efficiency is further reduced by mismatch between
cavity mode ($E$) and fiber mode ($G_{fiber}$) at the fiber tip characterized
by the mode matching factor
$\epsilon=\frac{|\int_{A}da\vec{E}^{*}\cdot\vec{G}_{fiber}|^{2}}{\int_{A}da|\vec{E}|^{2}\int_{A}da|\vec{G}_{fiber}|^{2}},$
(5)
where $G_{fiber}$ and $E$ are the transverse Gaussian mode shapes of fiber and
cavity, respectively. The optimal $w_{0}$ of the fiber cavity should be
determined to maximize the coupling $P_{fiber}$ to an ion at height $h_{ion}$,
which may not necessarily match with the fiber mode. If one uses additional
optics to couple the cavity mode into a fiber mode rather than integrate the
fiber mode directly underneath the flat cavity mirror, the mode matching
factor given by Eq. (5) can be improved substantially, approaching a value of
1.
### II.2 Cavity design example
The optimum design of the cavity-ion trap system is often constrained by the
ability to achieve high performance components. In our system, the dominating
constraint is the finesse $\mathcal{F}_{0}$ and the radius of curvature (RoC)
for the curved cavity mirror one can achieve, limited by the coating quality
and polishing capability. Compared to near-infrared wavelengths used in
neutral atom experiments McKeever _et al._ (2004); Hijlkema _et al._ (2007),
the passive loss of cavity mirror coating is significantly higher in the
ultraviolet range relevant for most trapped ions. Scattering loss from a rough
surface is given by $1-\exp[-(4\pi\sigma/\lambda)^{2}]$ Trupke _et al._
(2005) where $\sigma$ is RMS surface roughness. For example, we observed
$\approx$ 150 parts per million (ppm) loss at 369.5 nm on a super-polished
substrate whose RMS surface roughness is measured to be $<$0.1 nm,
corresponding to estimated scattering loss of $<$12 ppm. We can attribute most
of the loss in this case to absorption loss in the coating material in the UV
wavelength range. Since the RMS surface roughness of the fiber tip and the
mirror substrate with small RoC is generally larger than that of a super-
polished mirror substrate with large RoC, it is reasonable to assume that the
uncontrolled loss in our mirrors is $L_{f}+L_{e}\approx$1500 ppm,
corresponding to $\mathcal{F}_{0}\approx 4200$.
Next, we have to pick the radius of the cavity mode $r_{ion}$ at the ion
location and the cavity length $l$. In order to maximize $C_{0}$, one should
find a minimum value of $r_{ion}$. Since the ion cannot be trapped at the beam
waist in a fiber tip cavity shown in Fig. (2), the optimal waist $w_{0}$ is
determined by the distance between the ion and the waist which is located on
the flat fiber tip. When this distance is given by $h_{ion}$, $r_{ion}$ can be
minimized if $w_{0}=\sqrt{h_{ion}\cdot\lambda/\pi}$. In a surface trap, the
minimum ion height from the surface is constrained by considerations such as
the exposed dielectric material Harlander _et al._ (2010), anomalous heating
Labaziewicz _et al._ (2008) and the scattering of laser light from the
surface, and $h_{ion}$ = 50 $\mu$m is a good compromise. In this case, the
optimal waist is $w_{0}$ = 2.4 $\mu$m, corresponding to $r_{ion}$ = 3.4
$\mu$m.
For a cavity created between a flat mirror and a concave mirror, the beam
waist of the Gaussian mode is located at the flat mirror surface. The length
of the cavity $l$ is related to the Rayleigh range $z_{R}=\pi
w_{0}^{2}/\lambda$ of the Gaussian mode as $z_{R}^{2}=({\rm RoC}-l)l$. To trap
the ion inside the cavity volume, we require $l>h_{ion}=z_{R}$ and therefore
$l$ should be chosen to be very close to the RoC. A practical mirror one can
polish using conventional super-polishing technique has a RoC $\geq$ 5 mm, so
we choose $l\approx{\rm RoC}-z_{R}^{2}/{\rm RoC}\approx$ 5 mm. The cavity
length must be aligned to be about 0.5 $\mu$m shorter than the RoC of the
mirror. Laser ablation can create a smooth curved surface with RoC between 40
$\mu$m and 2 mm with 0.2 nm RMS roughness Hunger _et al._ (2010), but the
aperture diameter of a mirror fabricated in this way is limited to
$<100~{}\mu$m which is too small for near-concentric cavity geometry.
Using these parameters, we obtain $C_{0}\approx 4.8$ and $l_{c0}\approx$ 1.8
mm. The transmission of the out coupling mirror is determined by setting
$r_{t}\approx 86\%$ according to Eq. (3), corresponding to $T_{f}\approx$
9,500 ppm. This choice determines critical cavity parameters as
$\mathcal{F}=570$, $C=$ 0.65, $l_{c}$=13 mm and $r_{c}=$ 3.9 $\mu$m. Measured
beam waist of a typical UV fiber at 369.5 nm is about 1.5 $\mu$m and in Figure
3 the change of the $P_{fiber}$ as a function of the cavity length $l$ for an
ion height $h_{ion}$ = 50 $\mu$m is shown. The mode matching efficiency
$\epsilon$ is about 82 % if $w_{0}$ is optimized for maximum $P_{cavity}$
($w_{0}$ = 2.4 $\mu$m), and $P_{cavity}$ is reduced when $\epsilon$ is 100 %.
Therefore, there is an optimal length that maximizes $P_{fiber}$, near which
one can achieve more than 30 % of coupling efficiency compared to 0.4 %
Olmschenk _et al._ (2009).
If additional optics is used to couple the cavity mode into a single mode
fiber (instead of using fiber tip as the flat mirror of the cavity), we can
achieve $\epsilon\approx$1 and a maximum fiber coupling efficiency of over 35
%. A comparison with a cavity constructed using two concave mirrors is
presented in Appendix A.
Figure 3: Probability of collecting photons with a fiber cavity into the mode
of a single mode fiber as a function of cavity length ($l$). As the cavity
length approaches the RoC of the spherical mirror, the beam waist at the fiber
tip shrinks. $T_{f}$ = 9,500 ppm and $T_{e}+L_{f}+L_{e}$= 1500 ppm are
assumed. The radius $r_{ion}$ is smallest for a cavity length of 4.9995 mm
while best mode overlap between cavity and fiber is achieved for a length of
4.9999 mm.
In summary, we expect to reach $(g,\kappa,\gamma)/2\pi\approx(18,26,10)$ MHz
using a fiber tip cavity. With pulsed excitation, one can achieve more than 30
% coupling efficiency into the single mode fiber if the minimum available RoC
is about 5 mm, and the average rate of single photon emission process is on
the order of $\text{max}(1/g,1/\kappa)$.
### II.3 Excitation by stimulated Raman adiabatic passage
In an atomic system where the initial state $|i\rangle$ and the final states
$|f_{1}\rangle$, $|f_{2}\rangle$ are different as shown in Fig. 1 (b) and can
be connected by a resonantly enhanced Raman transition, the STIRAP process can
be used to generate a single photon in the cavity mode McKeever _et al._
(2004); Hijlkema _et al._ (2007).
For this process, a laser beam is resonant with one branch of the Raman
transition ($|i\rangle\Leftrightarrow|e\rangle$) with Rabi frequency
$\Omega_{P}$, and the cavity vacuum stimulates the emission of the photon on
the other branch ($|f\rangle\Leftrightarrow|e\rangle$) with vacuum Rabi
frequency $2g$. As $\Omega_{P}$ is slowly increased from zero, the ion
adiabatically follows the dark state of the Hamiltonian starting from
$|i\rangle$ to $|f\rangle$. Because the excited state $|e\rangle$ is not
populated, spontaneous emission does not occur. The photon collection
probability $P_{cavity}$ is $2C/(2C+1)$. If used together with the fiber
cavity geometry discussed above, $P_{cavity}\approx 0.56$ and
$P_{fiber}\approx 0.40$ can be reached.
If one is interested in an ideal single photon source, STIRAP is a better
choice because of its larger photon collection efficiency per trial.
Furthermore, in contrast to the fast excitation, the pulse shape of the
emitted photon can be controlled by $\Omega_{P}(t)$. For the protocol to
generate entangled ion pair based on entanglement swapping between a photon
pair, successful generation rate of the photon pair is the relevant figure of
merit, and is proportional to $P_{fiber}^{2}\cdot R_{rep}$, where $R_{rep}$ is
repetition rate of excitation process. Therefore in addition to the
availability of the necessary atomic states, the optimal excitation method
depends on the details of the experimental setup, such as the time required to
prepare the initial state as well as cavity QED parameters.
To estimate the time required to complete a single STIRAP photon generation,
we need to add the time necessary to prepare the initial state $|i\rangle$ as
well as the time for the adiabatic evolution and/or emission. Adiabaticity
requires that $\dot{\Omega}_{P}\ll g^{2}$, and if $\Omega_{P}$ increases
linearly over time $T$, $T\gg 1/g$. In the strong coupling regime
($g\gg\kappa,\gamma$), slow ramping of $\Omega_{P}$ guarantees that the
quantum state of the system will be adiabatically transferred from
$|i,n=0\rangle$ to $|f,n=1\rangle$ without spontaneous emission where $n$ is
the number of photons in the cavity mode. The single photon then leaves the
cavity with a rate of $\kappa$. The Rabi rate $\Omega_{P}$ should remain at
its maximum value until the single photon leaves the cavity. Therefore with
strong coupling, the total photon generation time includes
$\mathcal{O}(1/\kappa)$ as well as time necessary for adiabatic process ($\gg
1/g$). In the weak coupling regime, the cavity decay process will occur much
faster than the adiabatic transfer, and therefore the minimum time is mainly
limited by $T\gg 1/g$. Because Yb ion-cavity system lies in weak coupling
regime, STIRAP process will require much more time to extract a photon
compared to fast excitation case. In the rest of this paper, we will mainly
consider fast excitation to compare the cavity result with high NA optics in
the following section.
## III Collection of photons with high numerical aperture optics
Figure 4: Definitions of coordinate systems and symbols with respect to
spherical mirror. (a) Ion is located at the origin of coordinate system, and
polar (cylindrical) coordinates are used to describe the optical field before
(after) reflection. $\theta$ is the polar angle with respect to
$\hat{z}$-axis, and $\phi$ is azimuthal angle from $\hat{x}$-axis. (b)
$\theta_{max}$ is the maximum half opening angle corresponding to the mirror
aperture. $h$ is a distance to the analysis plane from the center of sphere.
Surface ion trap can be fabricated on the substrate area surronding mirror.
As an alternative method to collect light from a single trapped ion, we
consider the use of high NA collection optics whose geometric aberrations are
well-corrected to achieve diffraction limited coupling performance into a
single mode fiber. Reflective optics is frequently used as an alternative to
refractive optics to realize a large NA Maiwald _et al._ (2009); Noek _et
al._ (2010); Shu _et al._ (2009, 2010); Sondermann _et al._ (2007); Luo _et
al._ (2009); Lindlein _et al._ (2007). A parabolic mirror can perfectly
collimate light emitted from a point source, and has been studied extensively
Sondermann _et al._ (2007); Luo _et al._ (2009); Lindlein _et al._ (2007).
In this section we will investigate the properties of light reflected by a
spherical mirror and numerically calculate the collection probability into a
single mode fiber. Although the analysis is presented for reflective optics,
the general conclusions apply to other approaches if similar NA and aberration
control can be realized.
### III.1 Dipole emission and Gaussian mode
Distinguishing light from $\pi$ and $\sigma$ transitions is essential for
achieving high fidelities in remote entanglement protocols. Therefore, the
modes of photons from $\pi$ and $\sigma$ transitions have to be orthogonal
after collection. For large numerical aperture, this is only possible if we
choose $\vec{\it{B}}$ parallel to the optical axis $\hat{z}$. The optical
fields of photons emitted by the ion via the three possible transitions are
given by Luo _et al._ (2009)
$\displaystyle\vec{E}_{\Delta l=1,\Delta m=0}$ $\displaystyle=$
$\displaystyle\frac{ie^{ikr}}{r}\sqrt{\frac{3}{8\pi}}\sin\theta\hat{\theta},~{}~{}$
$\displaystyle\vec{E}_{\Delta l=1,\Delta m=\pm 1}$ $\displaystyle=$
$\displaystyle\frac{ie^{ikr}}{r}e^{\pm
i\phi}\sqrt{\frac{3}{16\pi}}(\pm\cos\theta\hat{\theta}+i\hat{\phi}).$ (6)
Once it is reflected by a curved surface, spherical waves from an ion become
paraxial waves, and the reflected optical field will be described in
cylindrical coordinates shown in Figure 4 (a). If the curved surface is
parabolic and the trapped ion is located at the focus of paraboloid, all the
reflected rays propagate parallel to the $\hat{z}$-axis and the traveling
wavefronts are flat.
For a spherical mirror, most of the reflected rays propagate with radially
varying phase delay and non-zero angle with respect to the $\hat{z}$-axis due
to spherical aberration, even if the ion is trapped at the nominal focus
(midpoint between the center of sphere and the vertex of the spherical
surface). In this case, rays cross each other and the optical field
distribution in $\rho$-coordinate changes as it propagates along the
$\hat{z}$-direction [Figure 4 (b)]. The probability of collecting a photon
into a single mode fiber can be estimated by numerically calculating the mode
matching factor [Eq. (5)] between the reflected optical field distribution
($\vec{R}_{l,m}$) incident on the fiber and the Gaussian fiber mode
($\vec{G}_{fiber}$) by setting $\vec{E}=\vec{R}_{l,m}$.
Luo et al. Luo _et al._ (2009) analyzed the case of a parabolic mirror and
calculated the overlap of the reflected mode with a Gaussian mode. They showed
that the collection probability of a photon emitted by the $\pi$-transition
($\Delta l=1,\Delta m=0$) vanishes due to the rotational cylindrical symmetry
of polarization, while the collection efficiency for $\sigma$-transition
($\Delta l=1,\Delta m=\pm 1$) approaches 50 % as the mirror aperture
($\rho_{0}$) becomes much larger than the focal length ($f$) of the parabolic
mirror. Provided the axis of single mode fiber is collinear with the
quantization axis, a single mode fiber can be used as transition-selective
filter which maps photons generated by $\sigma^{+}$ and
$\sigma^{-}$-transitions into two orthogonal circular polarizations
($(\hat{x}\mp i\hat{y})/\sqrt{2}$ ) respectively, and rejects photons
generated by the $\pi$ transition. For the frequency qubit shown in Fig. 1
(d), it would be interesting to collect photons from $\pi$-transitions and
reject $\sigma$ transition photons Olmschenk _et al._ (2009). We can
implement this orthogonal filter by adding an additional $\phi$-dependent
phase ($e^{i\phi}$) to the optical field after it is reflected by the mirror
[Eq. (33) and (35) of Ref. Luo _et al._ (2009)]. A phase that depends on the
azimuthal angle can be added using computer generated holograms Heckenberg
_et al._ (1992); Arlt _et al._ (1998), diffractive optics Khonina _et al._
(2000); Kennedy _et al._ (2002), phase plates Beijersbergen _et al._ (1994);
Sueda _et al._ (2004), or spatial light modulators Leach _et al._ (2004);
Ren _et al._ (2010). Therefore we assume that whenever we intend to collect
light from $\pi$-transitions, an additional phase factor $e^{i\phi}$ is added
to the (near-) collimated field.
Figure 5: Maximum collection probability of photons emitted by a
$\sigma$-transition ($\Delta l=1,\Delta m=\pm 1$) and a $\pi$-transition
($\Delta l=1,\Delta m=0$). For the $\pi$-transition calculation, we assume
that phase $e^{i\phi}$ is added to the collimated field. We also plotted the
amount of light collimated by parabolic mirror before being collected by
single mode fiber. $\rho_{0}$ is the radius of mirror aperture and $f$ is the
focal length of the parabolic mirror. Inset: Spherical probability
distribution of photons emitted by different transitions with respect to the
quantization axis. Distance from the origin is proportional to the probability
of photons propagating in the corresponding angle.
The collection probabilities of light from a single ion onto a large area
detector (referred to as “collimated case”) and into a single mode fiber
(“fiber coupled case”) are shown in Figure 5 as a function of the mirror
aperture radius ($\rho_{0}$) for both $\sigma$\- and $\pi$-transitions. When
$\rho_{0}$ is smaller than $f$, the fiber coupling efficiency of
$\sigma$-transition photons is much larger than that of $\pi$-transition
photons due to the dipole emission pattern [inset in Figure 5]. When
$\rho_{0}$ is much larger than $f$, spatial overlap with fiber mode becomes
less sensitive to the details of the emission pattern but the spatially
varying polarization distribution limits the overall coupling efficiencies of
both transitions near 50 %.
Figure 6: Matching collimated ray distribution to Gaussian beam with finite
wavefront radius of curvature. For clarity, figures are not drawn to scale.
The integration of an ion trap with large aperture parabolic mirrors has been
considered Maiwald _et al._ (2009), In this work, we will concentrate our
analysis on the collection probability one can achieve with a spherical
mirror, which can be integrated with a surface trap, as shown schematically in
Figure 4 (b) Noek _et al._ (2010). As spherical mirrors feature substantial
aberrations at large NA, this analysis also provides strategies for
effectively managing aberrations present in most practical collection optics.
When the ion is trapped at the nominal focus of the spherical mirror, the rays
collimated by a spherical mirror are not perfectly parallel to the optical
axis but converge initially and eventually diverge after crossing the optical
axis (Figure 6). The resulting far-field wavefronts are close to spherical
with residual optical path difference (OPD) as shown in Fig. 6. To estimate
the probability of coupling this mode to an optical fiber, we calculate the
mode overlap to a Gaussian mode with spherical wavefronts:
$\vec{G}(\rho,\phi,z)=\frac{1}{w}\sqrt{\frac{2}{\pi}}\exp\left(i\psi\right)\exp\left(-\frac{\rho^{2}}{w^{2}(z)}\right)\exp\left(\frac{ik\rho^{2}}{2R(z)}\right)(\alpha\hat{x}+\beta\hat{y}),$
(7)
where $R(z)$ is the radius of curvarture (RoC) of the wavefront, $w(z)$ is the
beam radius at the measurement plane, and $\psi=\arctan(-z/2z_{R})$ is Gouy
phase. The residual OPD between the wavefront of collimated light field and
the spherical wavefronts of the Gaussian mode limits the maximum coupling
efficiency one can achieve with this method. The coupling efficiency can be
recovered to values similar to the ideal parabolic case if the residual OPD is
reduced to below the diffraction limit.
### III.2 Design examples
The residual OPD (expressed in units of wavelength $\lambda$ = 369.5 nm) of
the rays reflecting off a micromirror with RoC $R$ = 160 $\mu$m, measured at a
location $h$ = 50 mm compared to a best-matching Gaussian mode is plotted in
Figure 7. The ion is assumed to be located at the focal point of the spherical
mirror. The two horizontal gray dashed lines show the limits of Rayleigh
criterion representing the diffraction limit (OPD $\leq\pm\lambda/4$), below
which residual OPD does not lead to reduced coupling efficiency Barakat
(1965). The black dashed line shows the residual OPD for a spherical
micromirror with opening angle ($\theta_{max}$) of 32∘, where 10.5 % of
$\sigma$-transition light falls within the aperture and is collimated by the
mirror. In this case, the OPD remains within the Rayleigh criterion, and a
numerical calculation estimates that 6.2 % of photons from
$\sigma$-transitions can be matched to a single mode fiber. In contrast, the
light field generated by the $\pi$-transition has little overlap with a
Gaussian mode for $\theta_{max}=32^{\circ}$ due to the emission pattern.
Figure 7: Example of residual OPD expressed in units of wavelength
($\lambda=369.5$ nm). The RoC of matching wavefront is chosen to maximize the
mode overlap rather than minimizing the maximum OPD. According to Rayleigh
wavefront criteria Barakat (1965), maximum OPD should be less than $\lambda/4$
to avoid the effect of non-zero OPD and this range is marked by two dashed
lines. The dark dashed line and dotted line shows the residual OPD (measured
at a location $h$ = 50 mm) for the rays reflected by a spherical mirror ($R$ =
160 $\mu$m) with $\theta_{max}$ = 32∘ and $\theta_{max}$ = 48∘, respectively.
In the inset, thickness variation of phase plate is plotted, designed to
compensate OPD shown by the dotted line. 500$\mu m$-thick fused silica
substrate is assumed in this design. With this phase correcting plate placed
at $h$ = 50 mm, residual OPD at $h$ = 50 mm is dramatically suppressed. Solid
line is OPD of phase corrected wavefront measured at $h$ = 100 mm, which shows
that OPD barely changes for additional propagation once OPD is compensated.
When $\theta_{max}$ is extended to $48^{\circ}$ to increase the amount of
collimated light from $\pi$-transition to 7.3 %, the maximum OPD becomes
larger than $\lambda$ due to spherical aberrations (dotted line in Figure 7)
and the maximum coupling efficiency is still limited to 0.4 %. The residual
OPD must be compensated in order to recover the fiber coupling efficiency. If
the distribution of the OPD is known, one can utilize a phase plate
Beijersbergen _et al._ (1994); Sueda _et al._ (2004) or a spatial light
modulator Leach _et al._ (2004); Ren _et al._ (2010) to introduce
compensating OPD to cancel out the residual OPD. Inset of Figure 7 shows an
example of a phase plate which can be fabricated on a 500 $\mu$m-thick fused
silica substrate to compensate the residual OPD for the $\pi$-transition case.
Using this phase corrector plate, the coupling efficiency into the fiber mode
can be enhanced to 3.0 %. We used $h$ = 50 mm as a typical location to place
phase correcting optics, but the general conclusion is independent of where
the mode matching is calculated. We also explored optimization of ion height
along the optical axis, and concluded that no substantial improvement can be
achieved.
Figure 8: Fiber coupling efficiency of a photon emitted by $\sigma$\- and
$\pi$-transition using a spherical mirror with a phase corrector plate when
$R=160\mu m$. We also plotted fiber coupling efficiency by parabolic mirror
for comparison. This plot shows that most of aberration of spherical mirror
can be corrected by phase plate when $\theta_{max}<50^{\circ}$.
Figure 8 shows the probability of coupling emitted photons into a single mode
fiber using a spherical mirror and an optimized phase corrector plate for each
value of $\theta_{max}$. With optimum phase correction, the collection
probability as a function of the opening angle $\theta_{max}$ of a spherical
mirror grows similar to that of a parabolic mirror until $\theta_{max}$
reaches $\sim 50^{\circ}$ where matching of slightly diverging rays to a
Gaussian mode starts to fail. Beyond this limit, we need to correct the angle
of collimated rays in addition to the phase which is beyond the scope of this
paper. Likewise, $\pi$-transition coupling efficiency could be improved to
over 3.0 % from 0.4 %, approaching the parabolic mirror limit if all
aberrations are properly compensated.
Use of a microfabricated mirror integrated under a surface trap has been
proposed for efficient collection of emitted photons for qubit state detection
Noek _et al._ (2010). In that approach, however, the aberrations of the
microfabricated mirror cannot be controlled well enough to produce a flat
wavefront. One important advantage of using a microfabricated mirror compared
to a conventional macroscopic mirror Shu _et al._ (2010) is that the
geometric aberrations scale as the dimensions of the optics are reduced Brady
and Hagen (2009). The residual OPD is reduced while the wavelength ($\lambda$)
remains the same eventually achieving diffraction limit. Table 1 summarizes
photon coupling efficiency using a single spherical mirror with the same large
NA ($\theta_{max}$ = 48∘) without any additional aberration correction. Last
column for RoC = 16 $\mu$m shows that there will be a significant improvement
if the dimensions of the mirror and the position of the ion are shrunk by a
factor of 10. For such micro optics, the detailed shape of the reflector has
little impact on the collection efficiency. On the other hand, if the
dimensions of mirror are increased by a factor of 10, the collection
probability will vanish unless there is proper phase compensating optics
(first column). Therefore the microfabricated mirror with RoC $\ll$ 100 $\mu$m
can increase the fiber collection efficiency drastically. However due to
technical limitations of the ion height, it is impractical to use mirrors with
RoC much smaller than $\sim$ 100 $\mu$m at present day.
Mirror RoC ($\mu$m) | 1600 | 160 | 16
---|---|---|---
Transition | $\sigma$ | $\pi$ | $\sigma$ | $\pi$ | $\sigma$ | $\pi$
Collimated light by mirror (%) | 21.2 | 7.3 | 21.2 | 7.3 | 21.2 | 7.3
Fiber coupling probability (%) | $<$0.1 | $<$0.1 | 6.2 | 0.4 | 15.8 | 2.8
Table 1: Estimation of fiber coupling efficiency of $\sigma$ and $\pi$
transition light by ideal spherical mirrors of different dimensions.
$\theta_{max}$ is set to 48∘. In our simulation, we added an ideal hyperbolic
lens in the beam path, which can transform the matched Gaussian mode into UV
fiber mode ($w_{0}\sim$ 1.54 $\mu$m) and performed full ray tracing from the
ion position to the fiber plane to evaluate the mode overlap integral.
### III.3 Alternative approaches
The analysis in this section shows that efficient coupling of light into a
single mode fiber can be achieved provided one can correct the aberration of a
large NA collection optics to conform to the wavefront of a Gaussian beam, to
below the diffraction limit. Practical means to realize such high NA optics
with well-corrected aberration include (1) design of multi-element refractive
lenses, (2) fabrication of paraboloidal mirrors (using e.g. diamond turning
technology), (3) spherical mirror with corrector plates to compensate for the
residual aberration, and (4) Fresnel reflectors or diffractive optical
components.
Fresnel lenses have been investigated as a way to collect light with large NA
without spherical aberration Cruz-Cabrera _et al._ (2007); Streed _et al._
(2011) because it adds the necessary phase shift to a continuous wavefront.
Using the similar design principle, we can consider integrating Fresnel
reflector with an ion trap in place of the micro mirror shown in Figure 4 (b).
One critical consideration for Fresnel diffractive optical elements is the
diffraction efficiency that depends on the grating periods Cruz-Cabrera _et
al._ (2007). The Fresnel reflector deserves more careful analysis for use in
aberratioin-free fiber coupling applications.
## IV Application to remote entanglement generation
Based on our estimation of photon coupling efficiency into a single mode
fiber, we can calculate the expected entangled ion pair generation rate
considering practical experimental constraints. Table 2 compares the expected
entanglement generation rate of various approaches with existing experimental
results. Matsukevich et al. Matsukevich _et al._ (2008) generated entangled
ion pairs by using polarization qubit [Figure 1 (b)]. Because state
preparation step requires only optical pumping into ${}^{2}S_{1/2}|0,0\rangle$
without any qubit rotation Olmschenk _et al._ (2007), the same procedure can
be repeated at 520 kHz repetition rate, and the entangled pair generation rate
is analyzed in the first column of Table 2. When the remote entanglement is
established by coincident detection of a photon pair, the ionic qubit is
encoded in the Zeeman substates ${}^{2}S_{1/2}|1,\pm 1\rangle$ which are
susceptible to magnetic field fluctuation and therefore these two states
should be transferred to atomic clock states ${}^{2}S_{1/2}|\pm 1,0\rangle$ by
microwave transitions or Raman transitions. The photonic polarization qubit is
also susceptible to decoherence from birefringence in the vacuum window,
downstream optics and single mode fibers. Therefore this method requires
careful control of potential decoherence mechanisms to achieve high fidelity.
Olmschenk et al. Olmschenk _et al._ (2009) used the first-order magnetic
field-insensitive hyperfine clock state qubit accompanied by $\pi$ transition
as shown in Figure 1 (d), and collected emitted photons from a direction
orthogonal to the quantization axis to maximize collection of photons
generated by $\pi$ transition [inset of Figure 5]. To generate ion-photon
entangled state, the ion should be prepared in a superposition state before
excitation, which requires rotation of clock states using microwave transition
and limits the overall repetition rate to 75 kHz. The analysis of this
experiment is listed in the second column of Table 2. The limitation of this
collection geometry is the increased overlap between different polarizations
from $\sigma$ and $\pi$ transitions, which reduces fidelity as NA is increased
Olmschenk _et al._ (2009).
Numerical aperture | 0.23 | 0.6 | cavity
---|---|---|---
Optic axis vs. B-field | $\parallel$ | $\perp$ | $\parallel$ | $\parallel$ | $\parallel$ | $\parallel$
Type of atomic transition | $\sigma$ | $\pi$ | $\sigma$ | $\pi$ | $\sigma$ | $\sigma$
Photonic qubit | Pol. | Freq. | Pol. | Freq. | Freq. | Pol.
Detector efficiency | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.3
Decay via right transition | 2/3 | 1/3 | 2/3 | 1/3 | 2/3 | n.a.
Collected fraction | 0.0198 | 0.0198 | 0.136 | 0.028 | 0.136 | 0.337
Mode overlap | 0.82 | 0.82 | 0.85 | 0.32 | 0.85 | 0.95
Misc. efficiency | 0.26 | 0.25 | 0.63 | 0.62 | 0.63 | 0.63
Single Photon collection efficiency (%) | 0.0554 | 0.0269 | 1.46 | 0.0547 | 1.46 | 6.04
Bell state identification | 0.25 | 0.25 | 0.5 | 0.5 | 0.5 | 0.5
Coincidence efficiency | $7.7\times 10^{-8}$ | $1.8\times 10^{-8}$ | $1.1\times 10^{-4}$ | $1.5\times 10^{-7}$ | $1.1\times 10^{-4}$ | $1.8\times 10^{-3}$
Repetition rate (kHz) | 520 | 75 | 500 | 75 | 75 | 500
Entanglement rate (Hz) | 0.04 | 0.0014 | 53 | 0.011 | 8 | 913
Table 2: Comparison of entanglement generation rate based on different setup.
Theoretically maximum two types of Bell states encoded in a photon pair can be
unambiguously identified from the others, and we include this identification
efficiency on the 5th row. When atom decays to the ground state, some of the
decay channels are undesired and we included the fraction of desired decay
channel in 7th row. Collected fraction corresponds to the fraction of
collimated light when the photons are collected with reflective or refractive
optics and $r_{t}\cdot P_{cavity}$ for the case of the cavity system.
Experimental condition of each column is described in the main text.
From the third to the fifth column in Table 2 shows the expected entangled ion
pair generation rates when the numerical aperture is increased to 0.6 using
either reflective optics or refractive optics discussed in Section III. We
considered several practical improvements that can be made over the existing
experiments. Although previous experiments Moehring _et al._ (2007);
Matsukevich _et al._ (2008); Olmschenk _et al._ (2009) detected only the
singlet state out of four Bell states, one of triplet states can also be
distinguished from the other two. Therefore we assume that probability of Bell
state measurement is 1/2 rather than 1/4. In previous experiments, various
losses in photon collection efficiency caused by non-ideal optical components
further reduced the overall efficiency by a factor of $\sim$0.25. We estimate
that we can use better optical components to increase this efficiency to more
than 60 % based on realistic analysis. Third column analyzes experimental
setup similar to the first column Matsukevich _et al._ (2008) when the
numerical aperture is increased to 0.6. This setup can achieve the highest
expected generation rate among the three cases, but requires decoherence
control similar to the first column setup. Forth column lists values we can
achieve using similar setup to the second column Olmschenk _et al._ (2009),
but we assume that optic axis of collection optics is parallel to the
quantization axis compared to orthogonal setup used in the second column. As
was discussed in section III, if a phase plate is used in the parallel
configuration, only the $\pi$ transition can be coupled into a single mode
fiber and the fidelity will remain high even with very large numerical
aperture. However, NA of 0.6 is not large enough to show significant
improvement due to non-ideal emission pattern of $\pi$-transition in this
collection direction. Fifth column estimates the generation rate which can be
achieved with a scheme in Figure 1 (c) Moehring _et al._ (2007). This
approach has advantages of high fidelity due to frequency qubit and $\sigma$
transition which allows high collection efficiency. However, to achieve near
unity excitation probability, the initial state should be prepared to either a
superposition of the clock states or one of ${}^{2}S_{1/2}|1,\pm 1\rangle$,
which limits the repetition rate around 75 kHz.
The last column shows the entanglement generation rate which can be achieved
by an optical cavity discussed in Section II. In schemes using optical
cavities, the FSR of cavity should be matched to either 12.6 GHz or 14.7 GHz
to collect photons encoding frequency qubits. These constraints lead to
minimum cavity length of $\sim$10 mm, precision control of RoC of the
spherical mirror down to $\mu$m level, and an upper limit on the cavity
finesse of $\sim$ 700 to cover 19.6 MHz linewidth of the ${}^{2}P_{1/2}$
state. Therefore frequency qubit is not compatible with optical cavity
schemes. However, we can achieve highest entanglement generation rate compared
to other approaches using polarization qubit or time-bin qubit. In this case,
the remaining challenge will be the minimization of decoherence similar to all
the other schemes involving the polarization qubit.
Based on the above analysis, collection of photons generated by
$\sigma$-transition along quantization axis is the optimal choice for both the
large NA system and the cavity system. Cavity system prefers polarization
qubit as the photonic qubit of choice, but it can achieve the highest
entanglement generation rate. For large NA system, frequency qubit has lower
generation rate than polarization qubit mainly due to slow initial state
preparation. In principle, this limitation can be overcome with fast microwave
transition or Raman transition. A custom lens system designed for optimal
chamber configuration can achieve a NA of more than 0.6 with diffraction-
limited performance. Even though the maximum collection probability with free
space optics is smaller, free space optics has the advantage that it can be
realized with off-the-shelf technology today. Therefore, $\sigma$-transition-
based frequency qubit system with large numerical aperture lens can be a good
compromise between high fidelity and high throughput.
## V Conclusions
In this paper, we discussed various ways to enhance collection of photons
emitted by a single atomic ion into a single-mode optical fiber. Based on this
analysis, we conclude that the generation rate of entangled ion pairs through
entanglement swapping of photons can be dramatically enhanced beyond the
current values using either a high NA collection optics or a carefully
designed optical cavity. Such enhancement is expected to enable the
realization of scalable quantum multi-processor and quantum repeaters based on
nodes of trapped ions connected via photonic network.
This research was funded by Office of the Director of National Intelligence
and Intelligence Advanced Research Projects Activity through Army Research
Office.
## Appendix A Cavity with two concave mirrors
In this Appendix, we estimate the maximum collection efficiency one can
achieve with a cavity made of two curved mirrors with an ion trapped in the
center. Compared to the fiber cavity, ion can be trapped at the beam waist of
the cavity mode, and we can estimate the lower bound and the upper bound of
$P_{cavity}$. In the near-concentric limit, we estimate the optimal $w_{0}$ to
be $0<w_{0}<\sqrt{\lambda/\pi}\sqrt[4]{\text{RoC}\cdot\lambda/2}$, which
corresponds to $1>1/(1+(w_{0}/r_{c})^{2})>0.87$. Combined with
$1/(1+\text{2RoC}/l_{c})\approx 0.53$ which is possible with symmetric
geometry, we can estimate that $0.53>P_{cavity}>0.46$ which is comparable to
$P_{cavity}\approx$ 0.41 of the fiber cavity. Mode matching has to be
performed to couple these photons into a single mode fiber, and it is not easy
to integrate this geometry with surface traps.
## References
* Einstein _et al._ (1935) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 0777 (1935).
* Bennett _et al._ (1993) C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wooters, Phys. Rev. Lett. 70, 1895 (1993).
* Ekert (1991) A. Ekert, Phys. Rev. Lett. 67, 661 (1991).
* Gottesman and Chuang (1999) D. Gottesman and I. Chuang, Nature 402, 390 (1999).
* Raussendorf and Briegel (2001) R. Raussendorf and H. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
* Briegel _et al._ (1998) H. Briegel, W. Dur, J. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).
* Kim _et al._ (2006) T. Kim, M. Fiorentino, and F. Wong, Phys. Rev. A 73 (2006).
* Lloyd _et al._ (2001) S. Lloyd, M. Shahriar, J. Shapiro, and P. Hemmer, Phys. Rev. Lett. 87, 167903 (2001).
* Duan _et al._ (2001) L. Duan, M. Lukin, J. Cirac, and P. Zoller, Nature 414, 413 (2001).
* Simon and Irvine (2003) C. Simon and W. Irvine, Phys. Rev. Lett. 91 (2003).
* Pan _et al._ (1998) J. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998).
* Moehring _et al._ (2007) D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe, Nature 449, 68 (2007).
* Matsukevich _et al._ (2008) D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Phys. Rev. Lett. 100 (2008).
* Olmschenk _et al._ (2009) S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L. M. Duan, and C. Monroe, Science 323, 486 (2009).
* Olmschenk _et al._ (2007) S. Olmschenk, K. C. Younge, D. L. Moehring, D. N. Matsukevich, P. Maunz, and C. Monroe, Phys. Rev. A 76 (2007).
* McKeever _et al._ (2004) J. McKeever, A. Boca, A. Boozer, R. Miller, J. Buck, A. Kuzmich, and H. Kimble, Science 303, 1992 (2004).
* Hijlkema _et al._ (2007) M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, Nature Physics 3, 253 (2007).
* Harlander _et al._ (2010) M. Harlander, M. Brownnutt, W. Haensel, and R. Blatt, New J. Phys. 12 (2010).
* Kim and Kim (2009) J. Kim and C. Kim, Quant. Inf. Comput. 9, 181 (2009).
* Bochmann _et al._ (2008) J. Bochmann, M. Muecke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, Phys. Rev. Lett. 101 (2008).
* Mundt _et al._ (2002) A. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 89 (2002).
* Gayral _et al._ (1999) B. Gayral, J. Gerard, A. Lemaitre, C. Dupuis, L. Manin, and J. Pelouard, Appl. Phys. Lett. 75, 1908 (1999).
* Braginsky _et al._ (1989) V. Braginsky, M. Gorodetsky, and V. Ilchenko, Phys. Rev. A 137, 393 (1989).
* Armani _et al._ (2003) D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, Nature 421, 925 (2003).
* Painter _et al._ (1999) O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, Science 284, 1819 (1999).
* Cui and Raymer (2005) G. Cui and M. Raymer, Opt. Express 13, 9660 (2005).
* Trupke _et al._ (2005) M. Trupke, E. Hinds, S. Eriksson, E. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87 (2005).
* Labaziewicz _et al._ (2008) J. Labaziewicz, Y. Ge, P. Antohi, D. Leibrandt, K. R. Brown, and I. L. Chuang, Phys. Rev. Lett. 100 (2008).
* Hunger _et al._ (2010) D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Haensch, and J. Reichel, New J. Phys. 12 (2010).
* Maiwald _et al._ (2009) R. Maiwald, D. Leibfried, J. Britton, J. C. Bergquist, G. Leuchs, and D. J. Wineland, Nature Physics 5, 551 (2009).
* Noek _et al._ (2010) R. Noek, C. Knoernschild, J. Migacz, T. Kim, P. Maunz, T. Merrill, H. Hayden, C. S. Pai, and J. Kim, Opt. Lett. 35, 2460 (2010).
* Shu _et al._ (2009) G. Shu, M. R. Dietrich, N. Kurz, and B. B. Blinov, J. Phys. B 42 (2009).
* Shu _et al._ (2010) G. Shu, N. Kurz, M. R. Dietrich, and B. B. Blinov, Phys. Rev. A 81 (2010).
* Sondermann _et al._ (2007) M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, Appl. Phys. B 89, 489 (2007).
* Luo _et al._ (2009) L. Luo, D. Hayes, T. A. Manning, D. N. Matsukevich, P. Maunz, S. Olmschenk, J. D. Sterk, and C. Monroe, Fortschr. Phys. 57, 1133 (2009).
* Lindlein _et al._ (2007) N. Lindlein, R. Maiwald, H. Konermann, M. Sondermann, U. Peschel, and G. Leuchs, Laser Phys. 17, 927 (2007).
* Heckenberg _et al._ (1992) N. Heckenberg, R. McDuff, C. Smith, H. Rubinszteindunlop, and M. Wegener, Opt. Quant. Electron. 24, S951 (1992).
* Arlt _et al._ (1998) J. Arlt, K. Dholakia, L. Allen, and M. Padgett, J. Mod. Opt. 45, 1231 (1998).
* Khonina _et al._ (2000) S. Khonina, V. Kotlyar, R. Skidanov, V. Soifer, P. Laakkonen, and J. Turunen, Opt. Comm. 175, 301 (2000).
* Kennedy _et al._ (2002) S. Kennedy, M. Szabo, H. Teslow, J. Porterfield, and E. Abraham, Phys. Rev. A 66 (2002).
* Beijersbergen _et al._ (1994) M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Comm. 112, 321 (1994).
* Sueda _et al._ (2004) K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, Opt. Express 12, 3548 (2004).
* Leach _et al._ (2004) J. Leach, E. Yao, and M. Padgett, New J. Phys. 6 (2004).
* Ren _et al._ (2010) Y.-X. Ren, M. Li, K. Huang, J.-G. Wu, H.-F. Gao, Z.-Q. Wang, and Y.-M. Li, Appl. Opt. 49, 1838 (2010).
* Barakat (1965) R. Barakat, J. Opt. Soc. Am. 55, 572 (1965).
* Brady and Hagen (2009) D. J. Brady and N. Hagen, Opt. Express 17, 10659 (2009).
* Cruz-Cabrera _et al._ (2007) A. A. Cruz-Cabrera, S. A. Kemme, J. R. Wendt, D. Kielpinski, E. W. Streed, T. R. Carter, and S. Samora, Proc. of the SPIE 6482, 48209 (2007).
* Streed _et al._ (2011) E. W. Streed, B. G. Norton, A. Jechow, T. J. Weinhold, and D. Kielpinski, Phys. Rev. Lett. 106 (2011).
|
arxiv-papers
| 2011-09-11T00:02:37 |
2024-09-04T02:49:22.053328
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Taehyun Kim, Peter Maunz, Jungsang Kim",
"submitter": "Jungsang Kim",
"url": "https://arxiv.org/abs/1109.2268"
}
|
1109.2300
|
# A characterization of optimal entanglement witnesses
Xiaofei Qi Department of Mathematics, Shanxi University , Taiyuan 030006, P.
R. of China; qixf1980@126.com and Jinchuan Hou Department of Mathematics
Taiyuan University of Technology
Taiyuan 030024, P. R. of China houjinchuan@tyut.edu.cn;
jinchuanhou@yahoo.com.cn
###### Abstract.
In this paper, we present a characterization of optimal entanglement witnesses
in terms of positive maps and then provide a general method of checking
optimality of entanglement witnesses. Applying it, we obtain new
indecomposable optimal witnesses which have no spanning property. These also
provide new examples which support a recent conjecture saying that the so-
called structural physical approximations to optimal positive maps (optimal
entanglement witnesses) give entanglement breaking maps (separable states).
PACS. 03.65.Ud, 03.65.Db, 03.67.-a
Key words and phrases. Quantum states, entanglement, positive linear maps
This work is partially supported by Research Fund for the Doctoral Program of
Higher Education of China (20101402110012), A grant from International
Cooperation Program in Sciences and Technology of Shanxi (2011081039),
Tianyuan Funds of China (11026161) and Natural Science Foundation of China
(11171249, 11101250).
## 1\. Introduction
Let $H$ be a separable complex Hilbert space. Recall that a quantum state on
$H$ is a density operator $\rho\in{\mathcal{B}}(H)$ (the von Neumann algebra
of all bounded linear operators) which is positive and has trace 1. Denote by
${\mathcal{S}}(H)$ the set of all states on $H$. If $H$ and $K$ are finite
dimensional, a state in the bipartite composition system
$\rho\in{\mathcal{S}}(H\otimes K)$ is said to be separable if $\rho$ can be
written as $\rho=\sum_{i=1}^{k}p_{i}\rho_{i}\otimes\sigma_{i},$ where
$\rho_{i}$ and $\sigma_{i}$ are states on $H$ and $K$ respectively, and
$p_{i}$ are positive numbers with $\sum_{i=1}^{k}p_{i}=1$. For the case that
at least one of $H$ and $K$ is of infinite dimension, a state $\rho$ acting on
$H\otimes K$ is called separable if it can be approximated in the trace norm
by the states of the form
$\sigma=\sum_{i=1}^{n}p_{i}\rho_{i}\otimes\sigma_{i},$ where $\rho_{i}$ and
$\sigma_{i}$ are states on $H$ and $K$ respectively, and $p_{i}$ are positive
numbers with $\sum_{i=1}^{n}p_{i}=1$. Otherwise, $\rho$ is said to be
inseparable or entangled (ref. [2, 17]).
Entanglement is a basic physical resource to realize various quantum
information and quantum communication tasks such as quantum cryptography,
teleportation, dense coding and key distribution [17]. It is very important
but also difficult to determine whether or not a state in a composite system
is separable. A most general approach to characterize quantum entanglement is
based on the notion of entanglement witnesses (see [8]). A Hermitian (i.e.,
self-adjoint) operator $W$ acting on $H\otimes K$ is an entanglement witness
(briefly, EW) if $W$ is not positive and ${\rm Tr}(W\sigma)\geq 0$ holds for
all separable states $\sigma$. Thus, if $W$ is an EW, then there exists an
entangled state $\rho$ such that ${\rm Tr}(W\rho)<0$ (that is, the
entanglement of $\rho$ can be detected by $W$). It was shown that, a state is
entangled if and only if it is detected by some entanglement witness [8].
However, constructing entanglement witnesses is a hard task. There was a
considerable effort in constructing and analyzing the structure of
entanglement witnesses for both finite and infinite dimensional systems [3,
19, 5, 13, 11]. However, complete characterization and classification of EWs
is far from satisfactory.
Due to the Choi-Jamiołkowski isomorphism [4, 14], a self-adjoint operator
$W\in{\mathcal{B}}(H\otimes K)$ with $\dim H\otimes K<\infty$ is an EW if and
only if there exists a positive linear map which is not completely positive
(NCP) $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ and a maximally
entangled state $P^{+}\in{\mathcal{B}}(H\otimes H)$ such that
$W=W_{\Phi}=(I_{n}\otimes\Phi)P^{+}$. Recall that a maximally entangled state
is a pure state $P^{+}=|\psi^{+}\rangle\langle\psi^{+}|$ with
$|\psi^{+}\rangle=\frac{1}{\sqrt{n}}(|11\rangle+|22\rangle+\cdots|nn\rangle)$,
where $n=\dim H$, $\\{|i\rangle\\}_{i=1}^{n}$ is an orthonormal basis of $H$.
Thus, up to a multiple by positive scalar, $W_{\Phi}$ can be written as the
matrix $W_{\Phi}=(\Phi(E_{ij}))$, where $E_{ij}=|i\rangle\langle j|$. For a
positive linear map $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$, we
always denote $W_{\Phi}$ the Choi-Jamiołkowski matrix of $\Phi$ with respect
to a given basis of $H$, that is $W_{\Phi}=(\Phi(E_{ij}))$, and we say that
$W_{\Phi}$ is the witness associated to the positive map $\Phi$. Conversely,
for an entanglement witness $W$, we denote $\Phi_{W}$ for the associated
positive map so that $W=W_{\Phi_{W}}$.
For any entanglement witness $W$, let
$\mathcal{D}_{W}=\\{\rho:\rho\in{\mathcal{S}}(H\otimes K),{\rm
Tr}(W\rho)<0\\},$
that is, ${\mathcal{D}}_{W}$ is the set of all entangled states that detected
by $W$. For entanglement witnesses $W_{1},W_{2}$, we say that $W_{1}$ is finer
than $W_{2}$ if $\mathcal{D}_{W_{2}}\subset\mathcal{D}_{W_{1}}$, denoted by
$W_{2}\prec W_{1}$. While, an entanglement witness $W$ is optimal if there
exists no other witness finer than it. Obviously, a state $\rho$ is entangled
if and only if there is some optimal EW such that ${\rm Tr}(W\rho)<0$. In
[16], Lewenstein, Kraus, Cirac and Horodecki proved that: (1) $W$ is an
optimal entanglement witness if and only if $W-Q$ is no longer an entanglement
witness for arbitrary positive operator $Q$; (2) $W$ is optimal if
${\mathcal{P}}_{W}=\\{|e,f\rangle\in H\otimes K:\langle e,f|W|e,f\rangle=0\\}$
spans the whole $H\otimes K$ (in this case, we say that $W$ has spanning
property). For the infinite dimensional version of these results, see [12]. To
the best of the author’s knowledge, the above criterion (2) is the only method
we have known by now that is practical of checking optimality of witnesses. In
fact, almost all known optimal EWs are checked by using of the criterion (2)
(Ref. [6, 7] and the references therein). However, the criterion is only a
sufficient condition. There are known optimal witnesses that have no spanning
property. For example, consider the Choi map $\phi$ from $M_{3}$ into $M_{3}$
defined by
$None$ $\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\
a_{31}&a_{32}&a_{33}\end{array}\right)\mapsto\left(\begin{array}[]{ccc}a_{11}+a_{33}&-a_{12}&-a_{13}\\\
-a_{21}&a_{22}+a_{11}&-a_{23}\\\
-a_{31}&-a_{32}&a_{33}+a_{22}\end{array}\right).$
It is well known that the associated entanglement witness $W_{\phi}$ is
optimal (by proving that $\phi$ is an extremal point of the convex set of all
completely positive linear maps on $M_{3}$) and
span$({\mathcal{P}}_{W_{\phi}})\not={\mathbb{C}}^{3}\otimes{\mathbb{C}}^{3}$.
Thus a natural question rises: Are there any other practical methods to detect
the optimality of entanglement witnesses?
The purpose of this paper is to give a necessary and sufficient condition for
an EW to be optimal in terms of positive maps. Based on this result, we give a
general approach of how to check that an EW is optimal or not. This approach
is practical. Applying it we show that the entanglement witnesses arising from
the positive maps in [18] are indecomposable optimal witnesses. Moreover,
these optimal EWs give new examples supporting a recent conjecture posed in
[15] saying that the so-called structural physical approximations (SPA) to
optimal positive maps (optimal EWs) give entanglement breaking maps (separable
states).
Recall that an entanglement witness $W$ is called decomposable if
$W=Q_{1}+Q_{2}^{\Gamma}$ for some operators $Q_{1},Q_{2}\geq 0$, where
$Q_{2}^{\Gamma}$ stands for any one of $Q_{2}^{T_{1}}$ and $Q_{2}^{T_{2}}$,
the partial transpose of $Q_{2}$ with respect to the subsystem $H$ and $K$,
respectively; a positive map $\Delta$ is said to be decomposable if it is the
sum of a completely positive map $\Delta_{1}$ and the composition of a
completely positive map $\Delta_{2}$ and the transpose $\bf T$, i.e.,
$\Delta=\Delta_{1}+\Delta_{2}\circ{\bf T}$. Hence $W_{\Phi}$ is decomposable
if and only if $\Phi$ is decomposable. A completely positive map $\Lambda$ is
called entanglement breaking (EB) if its partial action $\Lambda\otimes I$
sends every state to a separable state.
Throughout this paper, $H$ and $K$ are complex Hilbert spaces, and
$\langle\cdot|\cdot\rangle$ stands for the inner product in both of them.
${\mathcal{B}}(H,K)$ (${\mathcal{B}}(H)$ when $K=H$) is the Banach space of
all (bounded linear) operators from $H$ into $K$. $A\in{\mathcal{B}}(H)$ is
self-adjoint if $A=A^{\dagger}$ ($A^{\dagger}$ stands for the adjoint operator
of $A$); and $A$ is positive, denoted by $A\geq 0$, if
$\langle\psi|A|\psi\rangle\geq 0$ for all $|\psi\rangle\in H$. For any
positive integer $n$, $H^{(n)}$ denotes the direct sum of $n$ copies of $H$. A
linear map $\Phi$ from ${\mathcal{B}}(H)$ into ${\mathcal{B}}(K)$ is positive
if $A\in{\mathcal{B}}(H)$ is positive implies that $\Phi(A)$ is positive;
$\Phi$ is $k$-positive if $\Phi_{k}=\Phi\otimes I_{k}:{\mathcal{B}}(H)\otimes
M_{k}\rightarrow{\mathcal{B}}(K)\otimes M_{k}$ defined by
$\Phi_{k}((A_{ij})_{k\times k})=(\Phi(A_{ij}))_{k\times k}$ is positive;
$\Phi$ is completely positive (CP) if $\Phi_{k}$ is positive for all positive
integers $k$. By Choi’s well-known result, if $H$ and $K$ are finite
dimensional, then $\Phi$ is completely positive if and only if
$W_{\Phi}=(\Phi(E_{ij}))$ is a positive (semi-definite) matrix. A linear map
$\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ is called an elementary
operator if there are two finite sequences
$\\{A_{i}\\}^{n}_{i=1}\subset{\mathcal{B}}(H,K)$ and
$\\{B_{i}\\}^{n}_{i=1}\subset{\mathcal{B}}(K,H)$ such that
$\Phi(X)=\sum_{i=1}^{n}A_{i}XB_{i}$ for all $X\in{\mathcal{B}}(H)$. Note that
every linear map from ${\mathcal{B}}(H)$ into ${\mathcal{B}}(K)$ is an
elementary operator if $H$ and $K$ are finite dimensional.
## 2\. A characterization of the optimality of entanglement witnesses
In this section we first give a characterization of optimality of EWs in terms
of positive elementary operators. Then, by using of the result, we develop a
general approach how to check the optimality of entanglement witnesses.
Before stating the main results in this section, let us recall some notions
and give a lemma from [9].
Let $l$, $k\in\mathbb{N}$ (the set of all natural numbers), and let
$A_{1},\cdots,A_{k}$, and $C_{1},\cdots,C_{l}\in{\mathcal{B}}(H$, $K$). If,
for each $|\psi\rangle\in H$, there exists an $l\times k$ complex matrix
$(\alpha_{ij}(|\psi\rangle))$ (depending on $|\psi\rangle$) such that
$C_{i}|\psi\rangle=\sum_{j=1}^{k}\alpha_{ij}(|\psi\rangle)A_{j}|\psi\rangle,\qquad
i=1,2,\cdots,l,$
we say that $(C_{1},\cdots,C_{l})$ is a locally linear combination of
$(A_{1},\cdots,A_{k})$, $(\alpha_{ij}(|\psi\rangle))$ is called a local
coefficient matrix at $|\psi\rangle$. Furthermore, if a local coefficient
matrix $(\alpha_{ij}(|\psi\rangle))$ can be chosen for every $|\psi\rangle\in
H$ so that its operator norm $\|(\alpha_{ij}(|\psi\rangle))\|\leq 1$, we say
that $(C_{1},\cdots,C_{l})$ is a contractive locally linear combination of
$(A_{1},\cdots,A_{k})$; if there is a matrix $(\alpha_{ij})$ such that
$C_{i}=\sum_{j=1}^{k}\alpha_{ij}A_{j}$ for all $i$, we say that
$(C_{1},\cdots,C_{l})$ is a linear combination of $(A_{1},\cdots,A_{k})$ with
coefficient matrix $(\alpha_{ij})$. Sometimes we also write
$\\{A_{i}\\}_{i=1}^{k}$ for $(A_{1},\cdots,A_{k})$.
The following characterization of positive elementary operators was obtained
in [9], also, see [10].
Lemma 2.1. Let $H$ and $K$ be complex Hilbert spaces of any dimension,
$\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a linear map defined by
$\Phi(X)=\sum_{i=1}^{k}C_{i}XC_{i}^{\dagger}-\sum_{j=1}^{l}D_{j}XD_{j}^{\dagger}$
for all $X$. Then $\Phi$ is positive if and only if $(D_{1},\cdots,D_{l})$ is
a contractive locally linear combination of $(C_{1},\cdots,C_{k})$.
Furthermore, $\Phi$ is completely positive if and only if
$(D_{1},\cdots,D_{l})$ is a linear combination of $(C_{1},\cdots,C_{k})$ with
a contractive coefficient matrix, and in turn, if and only if there exist
$E_{1},E_{2},\ldots,E_{r}$ in ${\rm span}\\{C_{1},\cdots,C_{k}\\}$ such that
$\Phi=\sum_{i=1}^{r}E_{i}(\cdot)E_{i}^{\dagger}.$
Since every linear map between matrix algebras is an elementary operator, and
every hermitian preserving linear map is of the form
$\Phi(\cdot)=\sum_{i=1}^{k}C_{i}(\cdots)C_{i}^{\dagger}-\sum_{j=1}^{l}D_{j}(\cdot)D_{j}^{\dagger}$,
Lemma 2.1 gives a characterization of positive maps from ${\mathcal{B}}(H)$
into ${\mathcal{B}}(K)$ in the case that both $H$ and $K$ are finite
dimensional.
The following is the main result of this section.
Theorem 2.2. Let $H$ and $K$ be finite dimensional complex Hilbert spaces. Let
$\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a positive linear map.
Then $W_{\Phi}$ is an optimal entanglement witness if and only if, for any
$C\in{\mathcal{B}}(H,K)$, the map $X\mapsto\Phi(X)-CXC^{\dagger}$ is not a
positive map.
Proof. Fix an orthonormal basis of $H$ and $K$, respectively. Assume that
$W_{\Phi}$ is optimal, by [16], $W_{\Phi}-D$ is not an entanglement witness
for any nonzero $D\geq 0$. Take any $C\in{\mathcal{B}}(H,K)$ and consider the
map $A\mapsto\Phi(A)-CAC^{\dagger}$. Since the map $A\mapsto CAC^{\dagger}$ is
completely positive, the corresponding Choi-Jamiołkowski matrix $W_{C}$ is
positive. The optimality of $W_{\Phi}$ implies that $W_{\Phi}-W_{C}$ is not an
EW, and so the map $A\mapsto\Phi(A)-CAC^{\dagger}$ is not positive.
On the other hand, if the map $X\mapsto\Phi(X)-CXC^{\dagger}$ is not a
positive map for any $C\in{\mathcal{B}}(H,K)$, we will show that $W_{\Phi}$ is
optimal. By the Choi-Jamiołkowski isomorphism, any positive operator $D$
corresponds to a completely positive linear map $\Phi_{D}$. Thus there exist
non-zero operators $E_{1},\ldots,E_{k}$ such that
$\Phi_{D}(X)=\sum_{i=1}^{k}E_{i}XE_{i}^{\dagger}$ for all $X$. By the
assumption, $X\mapsto\Phi(X)-E_{1}XE_{1}^{\dagger}$ is not a positive map, and
hence the map $X\mapsto\Phi(X)-\sum_{i=1}^{k}E_{i}XE_{i}^{\dagger}$ is not
positive either. So $W_{\Phi}-D$ is not an EW for any positive operator
$D\in{\mathcal{B}}(H,K)$, which implies that $W_{\Phi}$ is optimal. $\Box$
Corollary 2.3. Let $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a
positive linear map defined by
$\Phi(X)=\sum_{i=1}^{k}A_{i}XA_{i}^{\dagger}-\sum_{j=1}^{l}B_{j}XB_{j}^{\dagger}$
for all $X$. Then $W_{\Phi}$ is optimal if and only if, for any operator
$C\in{\mathcal{B}}(H,K)$ which is a contractive locally linear combination of
$\\{A_{i}\\}_{i=1}^{k}$, the map $\Psi$ defined by
$\Psi(X)=\Phi(X)-CXC^{\dagger}$ is not positive.
Proof. The “only if” part is clear by Theorem 2.2. For the “if” part, assume
that $C\in{\mathcal{B}}(H,K)$ is not a contractive locally linear combination
of $\\{A_{i}\\}_{i=1}^{k}$. By Lemma 2.1, the map $\Psi$ defined by
$\Psi(X)=\Phi(X)-CXC^{\dagger}$ is not positive. So, together with the
hypotheses, we see that the map $X\mapsto\Phi(X)-CXC^{\dagger}$ is not a
positive map for any $C$. Then, by Theorem 2.2, $W_{\Phi}$ is an optimal EW.
$\Box$
By Corollary 2.3, we provide a method of checking the optimality of an
entanglement witness.
A general approach of checking optimality. Assume that $\dim H=n$ and $\dim
K=m$. Identify $H$ and $K$ as ${\mathbb{C}}^{n}$ and ${\mathbb{C}}^{m}$,
respectively. If $W$ is an EW of the system $H\otimes K$, then there exists
some NCP positive linear map $\Phi:M_{n}\rightarrow M_{m}$ such that
$W=W_{\Phi}=(\Phi(E_{ij}))$. By Lemma 2.1, $\Phi$ has the form
$\Phi(X)=\sum_{i=1}^{k}C_{i}XC_{i}^{\dagger}-\sum_{j=1}^{l}D_{j}XD_{j}^{\dagger}$
for all $X\in M_{n}$, where $(D_{1},\cdots,D_{l})$ is a contractive locally
linear combination of $(C_{1},\cdots,C_{k})$. To check the optimality of $W$,
let $C\in{\mathcal{B}}({\mathbb{C}}^{n},{\mathbb{C}}^{m})$ be any operator
such that $C$ is a contractive locally linear combination of
$(C_{1},\cdots,C_{k})$. If $X\mapsto\Phi(X)-CXC^{\dagger}$ is positive, then,
by Lemma 2.1, for any $|x\rangle\in{\mathbb{C}}^{n}$, there exist scalars
$\\{\alpha_{ij}(|x\rangle)\\}_{i=1;j=1}^{k;l}$ and
$\\{\gamma_{i}(|x\rangle)\\}_{i=1}^{k}$ such that
$D_{j}|x\rangle=\sum_{i=1}^{k}\alpha_{ij}(|x\rangle)A_{i}|x\rangle$,
$C|x\rangle=\sum_{i=1}^{k}\gamma_{i}(|x\rangle)A_{i}|x\rangle$, and the matrix
$F(|x\rangle)=\left(\begin{array}[]{cccc}\alpha_{11}(|x\rangle)&\alpha_{12}(|x\rangle)&\cdots&\alpha_{1k}(|x\rangle)\\\
\alpha_{21}(|x\rangle)&\alpha_{22}(|x\rangle)&\cdots&\alpha_{2k}(|x\rangle)\\\
\vdots&\vdots&\ddots&\vdots\\\
\alpha_{l1}(|x\rangle)&\alpha_{l2}(|x\rangle)&\cdots&\alpha_{lk}(|x\rangle)\\\
\gamma_{1}(|x\rangle)&\gamma_{2}(|x\rangle)&\cdots&\gamma_{k}(|x\rangle)\end{array}\right)$
is contractive. Thus, $W=W_{\Phi}$ is optimal if and only if, for any operator
$C$, there exists some vector $|x\rangle$ such that
$\|F(|x\rangle)F(|x\rangle)^{\dagger}\|>1$ for all possible choice of
coefficient matrices $F(|x\rangle)$.
This approach is very useful especially for those entanglement witnesses $W$
with span$({\mathcal{P}}_{W})\not=H\otimes K$. In the next section, we will
use this method to show that $W_{\Phi^{(n,k)}}$ ($n\geq 3$ and
$k=1,2,\ldots,n-1$) are indecomposable optimal entanglement witnesses if
$k\not=\frac{n}{2}$, where $\Phi^{(n,k)}$s are NCP positive maps constructed
in [18].
## 3\. Optimality of some indecomposable entanglement witnesses
The following kind of NCP positive linear maps
$\Phi^{(n,k)}:M_{n}({\mathbb{C}})\rightarrow M_{n}({\mathbb{C}})$ are
constructed in [18],
$None$
$\Phi^{(n,k)}(A)=(n-1)\sum_{i=1}^{n}E_{ii}AE_{ii}+\sum_{i=1}^{n}E_{i,\pi^{k}(i)}AE_{\pi^{k}(i),i}-A$
for every $A\in M_{n}$, where $n\geq 3$ and $k=1,2,\cdots,n-1$, $E_{ij}$ are
the matrix units as usual and $\pi^{1}=\pi$ is a permutation of
$\\{1,2,\cdots,n\\}$ defined by $\pi(i)=(i+1)\ {\rm mod}\ n$,
$\pi^{k}(i)=(i+k)\ {\rm mod}\ n$ ($k>1$), $i=1,2,\cdots,n$. That is,
$\Phi^{(n,k)}$ maps $n\times n$ matrix $(a_{ij})$ to ${\rm
diag}((n-1)a_{11}+a_{k+1,k+1},(n-1)a_{22}+a_{k+2,k+2},\cdots,(n-1)a_{nn}+a_{kk})-(a_{ij}).$
Moreover, it was shown in [18] that $\Phi^{(n,k)}$ is indecomposable whenever
either $n$ is odd or $n$ is even but $k\neq\frac{n}{2}$. For the case $n=3$
and $k=2$, one gets the Choi map $\phi=\Phi^{(3,2)}$ defined by Eq.(0.1). The
purpose of this section is to show, by using of the approach provided in the
previous section, that all $W_{\Phi^{(n,k)}}$s are indecomposable optimal
entanglement witnesses whenever $k\not=\frac{n}{2}$; while in the case $n$ is
even, $W_{\Phi^{(n,\frac{n}{2})}}$ is decomposable and not optimal.
The following lemma is obvious but useful to our purpose.
Lemma 3.1. Assume that $F=\left(\begin{array}[]{cc}1&b\\\
\bar{b}&a\end{array}\right)\in M_{2}({\mathbb{C}})$ is positive semi-definite.
If $b\not=0$, then $\|F\|>1$.
Theorem 3.2. For $n\geq 3$, $k=1,2,\ldots,n-1$, let
$\Phi^{(n,k)}:M_{n}({\mathbb{C}})\rightarrow M_{n}({\mathbb{C}})$ be the
positive linear maps defined by Eq.(3.1). Then
(1) the entanglement witness $W_{\Phi^{(n,k)}}$ is indecomposable and optimal
whenever $k\not=\frac{n}{2}$;
(2) the entanglement witness $W_{\Phi^{(n,\frac{n}{2})}}$ is decomposable and
not optimal, in this case $n\geq 4$ is an even integer.
Proof. We first prove the assertion (1). We give the details of proof for the
maps $\Phi=\Phi^{(n,1)}$. Other $\Phi^{(n,k)}$s are dealt with similarly
whenever $k\not=\frac{n}{2}$ in the case that $n$ is even.
It is clear that $W_{\Phi}$ is indecomposable as $\Phi$ is indecomposable by
[18]. In the sequel we show that $W_{\Phi}$ is also optimal by using of the
approach presented in Section 2.
Take any $C\in M_{n}$ and let
$\begin{array}[]{rl}\Psi_{C}(A)=&\Phi(A)-CAC^{\dagger}\\\
=&(n-1)\sum_{i=1}^{n}E_{ii}AE_{ii}^{\dagger}+\sum_{i=1}^{n}E_{i,i+1}AE_{i,i+1}^{\dagger}-A-CAC^{\dagger}\end{array}$
for all $A\in M_{n}$. By Corollary 2.3, we only need to consider the case that
$C$ is a contractive locally linear combination of
$\\{\sqrt{n-1}E_{11},\sqrt{n-1}E_{22},\cdots,\sqrt{n-1}E_{nn},E_{12},E_{23},\cdots,E_{n1}\\}.$
Since $\Phi$ is positive, by Theorem 2.1, for any
$|x\rangle=(x_{1},x_{2},\ldots,x_{n})^{T}$, there exist scalars
$\alpha_{1},\alpha_{2},\cdots,\alpha_{n},\beta_{1},\beta_{2},\cdots,\beta_{n}$
(depending on $|x\rangle$) with
$\sum_{i=1}^{n}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})\leq 1$ such that
$None$
$|x\rangle=\sum_{i=1}^{n}\alpha_{i}(\sqrt{n-1}E_{ii})|x\rangle+\sum_{i=1}^{n}\beta_{i}E_{i,i+1}|x\rangle.$
Consider the case that $x_{i}\not=0$ for all $i$. By Eq.(3.2), we get
$x_{i}=\sqrt{n-1}\alpha_{i}x_{i}+\beta_{i}x_{i+1}$, and so
$None$ $\beta_{i}=(1-\sqrt{n-1}\alpha_{i})\frac{x_{i}}{x_{i+1}}\ \ {\rm for\ \
all}\ \ i=1,2,\cdots,n.$
Write
$|x\rangle=(|x_{1}|e^{i\theta_{1}},|x_{2}|e^{i\theta_{2}},\cdots,|x_{n}|e^{i\theta_{n}})^{T}$
and let $r_{i}=|\frac{x_{i}}{x_{i+1}}|^{2}$ for all $i=1,2,\cdots,n$. Define a
function
$\begin{array}[]{rl}f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})=&\sum_{i=1}^{n}|\alpha_{i}|^{2}+\sum_{i=1}^{n}|\beta_{i}|^{2}\\\
=&\sum_{i=1}^{n}|\alpha_{i}|^{2}+\sum_{i=1}^{n}|1-\sqrt{n-1}\alpha_{i}|^{2}r_{i}.\end{array}$
For every $j$, write $\alpha_{j}=a_{j}+ib_{j}$ with $a_{j}$ and $b_{j}$ real.
Then the above equation reduces to
$None$
$\begin{array}[]{rl}f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})=&\sum_{i=1}^{n}a_{i}^{2}+\sum_{i=1}^{n}b_{i}^{2}+\sum_{i=1}^{n}(n-1)a_{i}^{2}r_{i}\\\
&+\sum_{i=1}^{n}(n-1)b_{i}^{2}r_{i}+\sum_{i=1}^{n}r_{i}-2\sqrt{n-1}\sum_{i=1}^{n}a_{i}r_{i},\end{array}$
where $\prod_{i=1}^{n}r_{i}=1$ with $r_{i}>0$ for $i=1,2,\cdots,n$.
Now, for the given matrix $C=(c_{ij})\in M_{n}$, by the assumption, there
exist some $\\{\delta_{i},\gamma_{i}\\}_{i=1}^{n}$ (depending on $|x\rangle$)
such that
$C|x\rangle=\sum_{i=1}^{n}\delta_{i}(\sqrt{n-1}E_{ii})|x\rangle+\sum_{i=1}^{n}\gamma_{i}E_{i,i+1}|x\rangle.$
It follows that
$\sum_{j=1}^{n}c_{ij}x_{j}=\sqrt{n-1}\delta_{i}x_{i}+\gamma_{i}x_{i+1}$
for each $i$, which implies that
$None$
$\gamma_{i}=\sum_{i\not=j}c_{ij}\frac{x_{j}}{x_{i+1}}+(c_{ii}-\sqrt{n-1}\delta_{i})\frac{x_{i}}{x_{i+1}},\quad
i=1,2,\cdots,n.$
Let
$F=F_{x}=\left(\begin{array}[]{cccccccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{n}&\beta_{1}&\beta_{2}&\cdots&\beta_{n}\\\
\delta_{1}&\delta_{2}&\cdots&\delta_{n}&\gamma_{1}&\gamma_{2}&\cdots&\gamma_{n}\end{array}\right)$.
Note that
$FF^{\dagger}=\left(\begin{array}[]{cc}\sum_{i=1}^{n}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})&\sum_{i=1}^{n}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})\\\
\sum_{i=1}^{n}(\bar{\alpha_{i}}\delta_{i}+\bar{\beta_{i}}\gamma_{i})&\sum_{i=1}^{n}(|\delta_{i}|^{2}+|\gamma_{i}|^{2})\end{array}\right),$
and $\|F\|>1$ if and only if $\|FF^{\dagger}\|>1$. So, to prove that
$W_{\Phi}$ is optimal, we only need to check that $\|FF^{\dagger}\|>1$ for
some suitable $|x\rangle$ and any choice of the coefficient matrix $F=F_{x}$.
Case 1. ${\rm Tr}(C)\not=0$ or there exists at least one of $c_{ij}$ with
$i\not=j$ such that $c_{ij}\not=0$.
We consider the special case that $r_{i}=1$ for all $i$ in Eq.(3.4). Then it
follows from Eq.(3.4) that
$\begin{array}[]{rl}f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\geq&n\sum_{i=1}^{n}a_{i}^{2}+n-2\sqrt{n-1}\sum_{i=1}^{n}a_{i}\\\
=&\sum_{i=1}^{n}(na_{i}^{2}-2\sqrt{n-1}a_{i}+1).\end{array}$
Let $g(t)=nt^{2}-2\sqrt{n-1}t+1$. It is easily checked that $g$ attains its
minimum $\frac{1}{n}$ at the point $t_{0}=\frac{\sqrt{n-1}}{n}$. So
$f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\geq 1$ and attains its minimum 1
at the point
$(\frac{\sqrt{n-1}}{n},\frac{\sqrt{n-1}}{n},\cdots,\frac{\sqrt{n-1}}{n})$.
Thus the best contractive coefficient matrix is
$None$
$\begin{array}[]{rl}&(\alpha_{1},\alpha_{2},\cdots,\alpha_{n},\beta_{1},\beta_{2},\cdots,\beta_{n})\\\
=&(\frac{\sqrt{n-1}}{n},\frac{\sqrt{n-1}}{n},\cdots,\frac{\sqrt{n-1}}{n},\frac{{1}}{n}\frac{x_{1}}{x_{2}},\frac{{1}}{n}\frac{x_{2}}{x_{3}},\cdots,\frac{{1}}{n}\frac{x_{n}}{x_{1}}),\end{array}$
and $\sum_{i=1}^{n}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})=1$. We may take
$x_{i}=e^{i\theta_{i}}$ for each $i$ as $r_{i}=1$ for each $i$. Thus
$|x\rangle=(e^{i\theta_{1}},e^{i\theta_{2}},\cdots,e^{i\theta_{n}})^{T}$, and,
for such $|x\rangle$, Eq.(3.5) becomes
$None$
$\gamma_{i}=\sum_{i\not=j}c_{ij}e^{i(\theta_{j}-\theta_{i+1})}+(c_{ii}-\sqrt{n-1}\delta_{i})e^{i(\theta_{i}-\theta_{i+1})},\quad
i=1,2,\cdots,n.$
By Eqs.(3.6)-(3.7), one obtains
$\sum_{i=1}^{n}(\alpha_{i}\overline{\delta_{i}}+\beta_{i}\overline{\gamma_{i}})=\frac{1}{n}(\sum_{i=1}^{n}c_{ii}+\sum_{i\not=j}\overline{c_{ij}}e^{i(\theta_{i}-\theta_{j})}).$
As $\sum_{i=1}^{n}c_{ii}={\rm Tr}(C)\not=0$ or $c_{ij}\not=0$ for some $i,j$
with $i\not=j$, we can choose suitable $\theta_{i}$s such that
$\sum_{i=1}^{n}(\alpha_{i}\overline{\delta_{i}}+\beta_{i}\overline{\gamma_{i}})\not=0$.
It follows, by Lemma 3.1, that $\|FF^{\dagger}\|>1$ for any choice of the
coefficient $(\delta_{1},\ldots,\delta_{n},\gamma_{1},\ldots,\gamma_{n})$.
Hence $\Psi_{C}$ is not positive by Lemma 2.1.
Case 2. ${\rm Tr}(C)=0$ and $c_{ij}=0$ for all $1\leq i\not=j\leq n$.
In this case, $C={\rm diag}(c_{11},\ldots,c_{nn})$ with
$\sum_{i=1}^{n}c_{ii}=0$, and, Eq.(3.4) implies
$\begin{array}[]{rl}f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})=&\sum_{i=1}^{n}a_{i}^{2}+\sum_{i=1}^{n}b_{i}^{2}+\sum_{i=1}^{n}(n-1)a_{i}^{2}r_{i}\\\
&+\sum_{i=1}^{n}(n-1)b_{i}^{2}r_{i}+\sum_{i=1}^{n}r_{i}-2\sqrt{n-1}\sum_{i=1}^{n}a_{i}r_{i}\\\
\geq&\sum_{i=1}^{n}(1+(n-1)r_{i})a_{i}^{2}+\sum_{i=1}^{n}r_{i}-2\sqrt{n-1}\sum_{i=1}^{n}a_{i}r_{i}\\\
=&\sum_{i=1}^{n}((1+(n-1)r_{i})a_{i}^{2}+r_{i}-2\sqrt{n-1}a_{i}r_{i}).\end{array}$
Notice that the function $h(t)=(1+(n-1)r)t^{2}-2\sqrt{n-1}rt+r$ achieves its
minimum $\frac{r}{1+(n-1)r}$ at $t_{0}=\frac{\sqrt{n-1}r}{1+(n-1)r}$. Hence
$f(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ attains its minimum
$\sum_{i=1}^{n}\frac{r_{i}}{1+(n-1)r_{i}}$ at the point
$(\alpha_{1},\ldots,\alpha_{n})=(\frac{\sqrt{n-1}r_{1}}{1+(n-1)r_{1}},\ldots,\frac{\sqrt{n-1}r_{n}}{1+(n-1)r_{n}})$.
Together with Eq.(3.3), we see that the coefficient matrix
$None$
$\begin{array}[]{rl}&(\alpha_{1},\alpha_{2},\cdots,\alpha_{n},\beta_{1},\beta_{2},\cdots,\beta_{n})\\\
=&(\frac{\sqrt{n-1}r_{1}}{1+(n-1)r_{1}},\frac{\sqrt{n-1}r_{2}}{1+(n-1)r_{2}},\cdots,\frac{\sqrt{n-1}r_{n}}{1+(n-1)r_{n}},\\\
&\frac{\sqrt{r_{1}}}{1+(n-1)r_{1}}e^{i(\theta_{1}-\theta_{2})},\frac{\sqrt{r_{2}}}{1+(n-1)r_{2}}e^{i(\theta_{2}-\theta_{3})},\cdots,\frac{\sqrt{r_{n}}}{1+(n-1)r_{n}}e^{i(\theta_{n}-\theta_{1})})\end{array}$
attains the minimal norm for
$|x\rangle=(|x_{1}|e^{i\theta_{1}},|x_{2}|e^{i\theta_{2}},\cdots,|x_{n}|e^{i\theta_{n}})^{T}$
with $r_{i}=|\frac{x_{i}}{x_{i+1}}|^{2}$, $i=1,2,\cdots,n$.
For the given $C={\rm diag}(c_{11},\ldots,c_{nn})$, write
$c_{ii}=s_{i}+it_{i}$. Let
$(\delta_{1},\ldots,\delta_{n},\gamma_{1},\ldots,\gamma_{n})$ be the
associated coefficients of $C$ at the above vector $|x\rangle$. Write
$\delta_{j}$ in the form $\delta_{j}=u_{j}+iv_{j}$. Consider the function
$f_{C}(\delta_{1},\delta_{2},\cdots,\delta_{n})=\sum_{i=1}^{n}(|\delta_{i}|^{2}+|\gamma_{i}|^{2})$.
By Eq.(3.5), we have
$None$
$\begin{array}[]{rl}f_{C}(\delta_{1},\delta_{2},\cdots,\delta_{n})=&\sum_{i=1}^{n}u_{i}^{2}+\sum_{i=1}^{n}v_{i}^{2}+\sum_{i=1}^{n}(n-1)u_{i}^{2}r_{i}+\sum_{i=1}^{n}s_{i}^{2}r_{i}\\\
&+\sum_{i=1}^{n}(n-1)(n-1)v_{i}^{2}r_{i}+\sum_{i=1}^{n}t_{i}^{2}r_{i}\\\
&-2\sqrt{n-1}\sum_{i=1}^{n}u_{i}s_{i}r_{i}-2\sqrt{n-1}\sum_{i=1}^{n}v_{i}t_{i}r_{i}\\\
=&\sum_{i=1}^{n}[(1+(n-1)r_{i})u_{i}^{2}-2\sqrt{n-1}u_{i}s_{i}r_{i}+s_{i}^{2}r_{i}]\\\
&+\sum_{i=1}^{n}[(1+(n-1)r_{i})v_{i}^{2}-2\sqrt{n-1}v_{i}t_{i}r_{i}+t_{i}^{2}r_{i}].\end{array}$
Consider the function
$h_{C}(y)=(1+(n-1)r)y^{2}-2\sqrt{n-1}sry+s^{2}r.$
A simple calculation shows that $h_{C}$ attains the minimum
$\frac{rs^{2}}{1+(n-1)r}$ at the point $y_{0}=\frac{\sqrt{n-1}rs}{1+(n-1)r}$.
Thus, by Eq.(3.9), we get
$f_{C}(\delta_{1},\delta_{2},\cdots,\delta_{n})\geq\sum_{i=1}^{n}\frac{\sqrt{n-1}r_{i}|c_{i}|^{2}}{1+(n-1)r_{i}}.$
Moreover, $f_{C}(\delta_{1},\delta_{2},\cdots,\delta_{n})$ achieves its
minimum at
$(\delta_{1},\delta_{2},\cdots,\delta_{n})=(\frac{\sqrt{n-1}r_{1}c_{1}}{1+(n-1)r_{1}},\frac{\sqrt{n-1}r_{2}c_{2}}{1+(n-1)r_{2}},\cdots,\frac{\sqrt{n-1}r_{n}c_{n}}{1+(n-1)r_{n}}),$
and the associated coefficient matrix is
$None$
$\begin{array}[]{rl}&(\delta_{1},\delta_{2},\cdots,\delta_{n},\gamma_{1},\gamma_{2},\cdots,\gamma_{n})\\\
=&(\frac{\sqrt{n-1}r_{1}c_{1}}{1+(n-1)r_{1}},\frac{\sqrt{n-1}r_{2}c_{2}}{1+(n-1)r_{2}},\cdots,\frac{\sqrt{n-1}r_{n}c_{n}}{1+(n-1)r_{n}},\\\
&\frac{\sqrt{r_{1}}c_{1}}{1+(n-1)r_{1}}e^{i(\theta_{1}-\theta_{2})},\frac{\sqrt{r_{2}}c_{2}}{1+(n-1)r_{2}}e^{i(\theta_{2}-\theta_{3})},\cdots,\frac{\sqrt{r_{n}}c_{n}}{1+(n-1)r_{n}}e^{i(\theta_{n}-\theta_{1})}).\end{array}$
By Eq.(3.8) and (3.10), we get
$FF^{\dagger}=\left(\begin{array}[]{cc}\sum_{i=1}^{n}\frac{r_{i}}{1+(n-1)r_{i}}&\sum_{i=1}^{n}\frac{r_{i}\bar{c_{i}}}{1+(n-1)r_{i}}\\\
\sum_{i=1}^{n}\frac{r_{i}c_{i}}{1+(n-1)r_{i}}&\sum_{i=1}^{n}\frac{r_{i}|c_{i}|^{2}}{1+(n-1)r_{i}}\end{array}\right).$
Let
$|y_{(r_{1},r_{2},\cdots,r_{n})}\rangle=FF^{\dagger}\left(\begin{array}[]{c}1\\\
0\end{array}\right)=\begin{pmatrix}\sum_{i=1}^{n}\frac{r_{i}}{1+(n-1)r_{i}}\\\
\sum_{i=1}^{n}\frac{r_{i}c_{i}}{1+(n-1)r_{i}}\end{pmatrix}.$
Then
$None$ $\begin{array}[]{rl}&\||y_{(r_{1},r_{2},\cdots,r_{n})}\rangle\|^{2}\\\
=&(\sum_{i=1}^{n}\frac{r_{i}}{1+(n-1)r_{i}})^{2}+|\sum_{i=1}^{n}\frac{r_{i}c_{i}}{1+(n-1)r_{i}}|^{2}\\\
=&(\sum_{i=1}^{n}\frac{r_{i}}{1+(n-1)r_{i}})^{2}+(\sum_{i=1}^{n}\frac{r_{i}s_{i}}{1+(n-1)r_{i}})^{2}+(\sum_{i=1}^{n}\frac{r_{i}t_{i}}{1+(n-1)r_{i}})^{2}.\end{array}$
Since $\sum_{i=1}^{n}c_{ii}=0$, there exists at least a number $t_{i}$, say
$t_{n}$, such that $t_{1}+t_{2}+\cdots+t_{n-1}=-t_{n}>0$ (or there exists at
least one $s_{i}$, say $s_{n}$, such that $s_{1}+\cdots+s_{n-1}=-s_{n}>0)$.
Assume, without loss of generality, that
$t_{0}=t_{1}+t_{2}+\cdots+t_{n-1}=-t_{n}>0$. Note that
$\prod_{i=1}^{n}r_{i}=1$ with $r_{i}>0$ for each $i$. Let
$r_{i}\rightarrow\infty$ for $i=1,2,\ldots,n-1$, then we have
$r_{n}\rightarrow 0$. Since
$\lim_{r_{i}\rightarrow\infty}\frac{r_{i}}{1+(n-1)r_{i}}=\frac{1}{n-1}\
(i=1,2,\cdots,n-1)\quad{\rm and}\lim_{r_{n}\rightarrow
0}\frac{r_{n}}{1+(n-1)r_{n}}=0,$
for any $\varepsilon>0$, there exists $N$ such that
$0<\frac{1}{n-1}-\frac{r_{i}}{1+(n-1)r_{i}}<\varepsilon$ for $i=1,\ldots,n-1$
and $\frac{r_{n}}{1+(n-1)r_{n}}<\varepsilon$ whenever $r_{i}\geq N$,
$i=1,2,\ldots,n-1$. Thus we can write
$None$
$\frac{r_{i}}{1+(n-1)r_{i}}=\frac{1}{n-1}-\delta_{r_{i}}\varepsilon\quad{\rm
with}\quad i=1,2,\cdots,n-1$
and
$None$ $\frac{r_{n}}{1+(n-1)r_{n}}=\delta_{r_{n}}\varepsilon$
with $0<\delta_{r_{i}}<1$ for each $i$. Combining Eqs.(3.11)-(3.13), one
obtains
$None$ $\begin{array}[]{rl}&\||y_{(r_{1},r_{2},\cdots,r_{n})}\rangle\|^{2}\\\
\geq&[\sum_{i=1}^{n-1}(\frac{1}{n-1}-\delta_{r_{i}}\varepsilon)+\delta_{r_{n}}\varepsilon]^{2}+[\sum_{i=1}^{n-1}(\frac{t_{i}}{n-1}-\delta_{r_{i}}t_{i}\varepsilon)+\delta_{r_{n}}t_{n}\varepsilon]^{2}\\\
=&[1-(\sum_{i=1}^{n-1}\delta_{r_{i}}-\delta_{r_{n}})\varepsilon]^{2}+[\frac{t_{0}}{n-1}-(\sum_{i=1}^{n-1}\delta_{r_{i}}t_{i}+\delta_{r_{n}}t_{0})\varepsilon]^{2}\\\
=&1+\mu^{2}\varepsilon^{2}-2\mu\varepsilon+\frac{t_{0}^{2}}{(n-1)^{2}}+\nu^{2}\varepsilon^{2}-2\frac{t_{0}\nu}{n-1}\varepsilon\\\
=&1+\frac{t_{0}^{2}}{(n-1)^{2}}+(\mu^{2}\varepsilon-2\mu+\nu^{2}\varepsilon-2\frac{t_{0}\nu}{n-1})\varepsilon\\\
\geq&1+\frac{t_{0}^{2}}{(n-1)^{2}}-2(\mu+\frac{t_{0}\nu}{n-1})\varepsilon,\end{array}$
where $\mu=\sum_{i=1}^{n-1}\delta_{r_{i}}-\delta_{r_{n}}$ and
$\nu=\sum_{i=1}^{n-1}\delta_{r_{i}}t_{i}+\delta_{r_{n}}t_{0}$. Note that
$|\mu|\leq n$ and $|\nu|\leq\sum_{i=1}^{n}|t_{i}|\leq\sum_{i=1}^{n}|c_{ii}|$.
Thus, if we take $\varepsilon$ so that
$\varepsilon<\frac{t_{0}^{2}}{2(n-1)(n(n-1)+t_{0}\sum_{i=1}^{n}|c_{ii}|)},$
then, by Eq.(3.14) we see that
$\begin{array}[]{rl}\||y_{(r_{1},r_{2},\cdots,r_{n})}\rangle\|^{2}\geq&1+\frac{t_{0}^{2}}{(n-1)^{2}}-2(\mu+\frac{t_{0}\nu}{n-1})\varepsilon\\\
\geq&1+\frac{t_{0}^{2}}{(n-1)^{2}}-2(n+\frac{t_{0}\sum_{i=1}^{n}|c_{ii}|}{n-1})\varepsilon>1.\end{array}$
It follows that, for those $|x\rangle$ with $r_{i}\geq N$, $i=1,2,\ldots,n-1$,
$\|FF^{\dagger}\|>1$ for any possible choice of the coefficients. Hence
$\Psi_{C}$ is again not positive.
Now, by use of Corollary 2.3, $W_{\Phi}$ is optimal, completing the proof of
the assertion (1).
Next let us prove the assertion (2). Let $n\geq 4$ be an even integer. One can
check that, the entanglement witness $W_{\Phi^{(n,\frac{n}{2})}}$ has the form
$W_{\Phi^{(n,\frac{n}{2})}}=P+Q^{T}$, where
$P=\sum_{i=1}^{n}(n-2)E_{ii}\otimes
E_{ii}-\sum_{i\not=j,|i-j|\not=\frac{n}{2}}E_{ij}\otimes E_{ij}\geq 0,$
and
$Q=\sum_{i=1}^{n}E_{i+\frac{n}{2},i+\frac{n}{2}}\otimes
E_{ii}-\sum_{i=1}^{\frac{n}{2}}(E_{i+\frac{n}{2},i}\otimes
E_{i,i+\frac{n}{2}}+E_{i,i+\frac{n}{2}}\otimes E_{i+\frac{n}{2},i})\geq 0.$
Hence $W_{\Phi^{(n,\frac{n}{2})}}$ is decomposable, and not optimal as
$P\not=0$. As a consequence, we see that the positive map
$\Phi^{(n,\frac{n}{2})}$ is decomposable.
The proof of Theorem 3.2 is finished. $\Box$
Remark 3.3. We remark here that our $W_{\Phi^{(n,n-1)}}$ coincides with
$W_{(n,1)}$ discussed in [1], where the authors asked whether $W_{(n,1)}$ is
optimal. Theorem 3.2 gives an affirmative answer to this problem. It is worth
to noting that it was shown in [1] that $W_{(n,1)}$ has no spanning property,
that is,
span${\mathcal{P}}_{W_{(n,1)}}\not={\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$.
In fact all $W_{\Phi^{(n,k)}}$s have no spanning property. Assume that
$|x\rangle\otimes|y\rangle=(x_{1}y_{1},x_{2}y_{1},\ldots,x_{n}y_{1},x_{1}y_{2},\ldots,x_{n}y_{n})^{T}$
is a product vector from ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$ such that
on which $W_{\Phi^{(n,k)}}$ has zero mean values, where
$|x\rangle=(x_{1},\ldots,x_{n})^{T}$ and $|y\rangle=(y_{1},\ldots,y_{n})^{T}$.
If all $y_{i}$ are nonzero, then it is not too difficult to check that
$|x\rangle\otimes|y\rangle$ has the form of
$\sum_{k=1}^{n}e^{i\theta_{k}}|k\rangle\otimes\sum_{k=1}^{n}e^{-i\theta_{k}}|k\rangle.$
These states span a subspace
$L=\\{(\xi_{11},\xi_{21},\ldots,\xi_{n1},\xi_{12},\xi_{22}\ldots,\xi_{nn})^{T}:\xi_{ij}\in{\mathbb{C}},\xi_{11}=\xi_{22}=\cdots=\xi_{nn}\\}$
of ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$. If $y_{i}=0$ for some $i$, then
one may check that $y_{j}\not=0$ implies that $x_{j}=0$. This forces that
$\xi_{ii}=x_{i}y_{i}=0$ and hence $|x\rangle\otimes|y\rangle\in L$. Therefore,
$L$ is in fact the subspace spanned by all product vectors on which
$W_{\Phi^{(n,k)}}$ has zero mean values. As $\dim L=n^{2}-n+1<n^{2}$,
$W_{\Phi^{(n,k)}}$ has no spanning property.
## 4\. Extending the set of witnesses which support the SPA conjecture
Now let us turn to another topic so-called structural physical approximation
(SPA) [1, 15]. Among the criteria of separability of states, probably the most
powerful one is the positive map criterion: a given state $\rho$ acting on
$M_{n}({\mathbb{C}})\otimes M_{m}({\mathbb{C}})$ is separable if and only if,
for any positive map $\Phi:M_{n}\rightarrow M_{m}$, the operator
$(I_{n}\otimes\Phi)\rho$ is positive. Despite its proven efficiency in
entanglement detection, the positive map criterion of separability above is
not directly applicable in experiments as the NCP positive maps do not
represent physical processes. So it is important to design methods which could
make the experimental detection of entanglement with the aid of positive maps
possible. SPA is one of such methods. Let $D:M_{n}\rightarrow M_{m}$ be the
completely depolarizing channel, i.e., $D(\rho)={\rm Tr}(\rho)I_{m}/m$. $D$ is
certainly an interior of the convex set of positive maps from $M_{n}$ into
$M_{m}$. It is clear $D$ is completely positive. For any positive map
$\Phi:M_{n}\rightarrow M_{m}$, let
$None$ $\tilde{\Phi}[p]=(1-p)D+p\Phi\quad(0\leq p\leq 1).$
From the C-J isomorphism it is clear that there exists a $p_{*}\in(0,1)$ such
that $\Phi[p]$ is completely positive whenever $0\leq p\leq p_{*}$ (if $\Phi$
is trace-preserving, $\tilde{\Phi}[p]$ represents a quantum channel whenever
$0\leq p\leq p_{*}$), and thus, in principle, it can represent some physical
process. The least noisy completely positive map from the class
$\tilde{\Phi}[p]$ ($0\leq p\leq p_{*}$), i.e., $\tilde{\Phi}[p_{*}]$ is called
the structural physical approximation of $\Phi$.
Recall that a completely positive map $\Phi$ is said to be entanglement
breaking if $I\otimes\Phi$ sends all states to separable states.
The following conjecture is posed in [15]. Here, $\Phi$ is said to be optimal
if the corresponding entanglement witness $W_{\Phi}$ is optimal.
Conjecture 4.1. Let $\Phi$ be an optimal (trace-preserving) positive map. Then
its SPA is entanglement breaking map (channel).
Applying C-J isomorphism, one can define SPA of an entanglement witness: Let
$W$ be a normalized EW, i.e., ${\rm Tr}(W)=1$. An operator $\tilde{W}(p)$
defined by
$\tilde{W}(p)=(1-p)\frac{I_{n}\otimes I_{m}}{mn}+pW\quad(0\leq p\leq 1)$
is called structural physical approximation (SPA) of $W$ if $\tilde{W}(p)\geq
0$. The maximal value of such $p$ is given by $p_{*}=1/(1+mn\lambda)$, where
$-\lambda<0$ is the smallest eigenvalue of $W$. Thus Conjecture 4.1
reformulated as
Conjecture 4.1′. Let $W$ be an optimal entanglement witness with ${\rm
Tr}(W)=1$. Then $\tilde{W}(p_{*})$ defines a separable state.
Conjecture 4.1 (4.1′) is supported by several examples (ref. [1, 15]. We shall
show that the normalized optimal indecomposable entanglement witnesses
$W^{(n,k)}=\frac{1}{n(n-1)}W_{\Phi^{(n,k)}}$ ($k\not=\frac{n}{2}$) support the
above conjecture,too.
Proposition 4.2. Let $n\geq 3$ and $1\leq k\leq n-1$ with $k\not=\frac{n}{2}$.
Then $\tilde{W}^{(n,k)}(p_{*})$ is a separable state.
Proof. It is clear that the minimal eigenvalue of $W^{(n,k)}$ is
$-\frac{1}{n(n-1)}$. So $p_{*}=\frac{n-1}{2n-1}$. Thus
$\begin{array}[]{rl}\tilde{W}^{(n,k)}(p_{*})=&(1-p_{*})\frac{I_{n}\otimes
I_{n}}{n^{2}}+p_{*}{W}^{(n,k)}\\\
=&\frac{1}{n(2n-1)}(I_{n^{2}}+\sum_{i=1}^{n}(n-2)E_{ii}\otimes
E_{ii}+\sum_{i=1}^{n}E_{i+(n-k),i+(n-k)}\otimes E_{ii}\\\
&-\sum_{i=1}^{n}E_{i,i+k}\otimes E_{i,i+k}-\sum_{i=1}^{n}E_{i+k,i}\otimes
E_{i+k,i})\\\ =&\frac{1}{n(2n-1)}(\sum_{i\not=j}E_{ii}\otimes
E_{jj}+\sum_{i=1}^{n}E_{i+(n-k),i+(n-k)}\otimes E_{ii}\\\
&+(n-1)\sum_{i=1}^{n}E_{ii}\otimes E_{ii}-\sum_{i=1}^{n}E_{i,i+k}\otimes
E_{i,i+k}-\sum_{i=1}^{n}E_{i+k,i}\otimes E_{i+k,i}).\end{array}$
Since $\sum_{i\not=j,|i-j|\not=k}E_{ii}\otimes
E_{jj}+\sum_{i=1}^{n}E_{i+(n-k),i+(n-k)}\otimes E_{ii}$ is non-normalized
separable density matrix, we only need to prove that
$\sigma=(n-1)\sum_{i=1}^{n}E_{ii}\otimes E_{ii}-\sum_{i=1}^{n}E_{i,i+k}\otimes
E_{i,i+k}-\sum_{i=1}^{n}E_{i+k,i}\otimes E_{i+k,i}$
is separable. However, $\sigma=\sum_{i=1}^{n}\sigma_{i,i+k}$ with
$\sigma_{i,i+k}=E_{ii}\otimes E_{ii}+E_{i+k}\otimes E_{i+k}+E_{i,i}\otimes
E_{i+k,i+k}+E_{i+k,i+k}\otimes E_{ii}-E_{i,i+k}\otimes
E_{i,i+k}-E_{i+k,i}\otimes E_{i+k,i}$. Note that $\sigma_{i,i+k}$ stands for
two-qubit matrix embedded in $M_{n^{2}}$ and has positive partial
transposition. So $\sigma_{i,i+k}$ is separable, which implies that $\sigma$
is also separable. Therefore, $\tilde{W}^{(n,k)}(p_{*})$ is a separable state,
as desired. $\Box$
## 5\. Optimal entanglement witnesses for infinite dimensional systems
Based on the results in Section 3, we can obtain some optimal indecomposable
entanglement witnesses for infinite dimensional systems.
Let $H$ and $K$ be separable infinite dimensional Hilbert spaces, and, for any
positive integer $n\geq 3$, let $\\{|i\rangle\\}_{i=1}^{n}$ and
$\\{|j^{\prime}\rangle\\}_{j=1}^{n}$ be any orthonormal sets of $H$ and $K$,
respectively. For each $k=1,2,\cdots,n-1$, let
$\hat{\Phi}^{(n,k)}:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be defined by
$None$
$\begin{array}[]{rl}\hat{\Phi}^{(n,k)}(A)=&(n-1)\sum_{i=1}^{n}E_{ii}AE_{ii}^{\dagger}+\sum_{i=1}^{n}E_{i,\pi^{k}(i)}AE_{i,\pi^{k}(i)}^{\dagger}\\\
&-(\sum_{i=1}^{n}E_{ii})A(\sum_{i=1}^{n}E_{ii})^{\dagger}\end{array}$
for every $A\in{\mathcal{B}}(H)$, where $\pi(i)=\pi^{1}(i)=(i+1)\ {\rm mod}\
n$, $\pi^{k}(i)=(i+k)\ {\rm mod}\ n$ ($k>1$), $i=1,2,\cdots,n$ and
$E_{ji}=|j^{\prime}\rangle\langle i|$. It is shown in [18] that,
$\hat{\Phi}^{(n,k)}$s are NCP positive linear maps. Moreover,
$\hat{\Phi}^{(n,k)}$ is indecomposable whenever $k\not=\frac{n}{2}$.
Let $P_{n}^{+}=|\psi_{n}\rangle\langle\psi_{n}|$, where
$|\psi_{n}\rangle=|11\rangle+|22\rangle+\cdots+|nn\rangle$, and let
$None$ $\hat{W}_{\hat{\Phi}^{(n,k)}}=(I\otimes\hat{\Phi}^{(n,k)})P^{+}_{n}.$
Then $\hat{W}_{\hat{\Phi}^{(n,k)}}$ is an entanglement witness for the system
living in $H\otimes K$.
Theorem 5.1. Let $H$ and $K$ be infinite dimensional complex Hilbert spaces.
For any positive integers $n\geq 3$ and $k$, let $\hat{\Phi}^{(n,k)}$ and
$\hat{W}_{\hat{\Phi}^{(n,k)}}$ be the positive maps and the entanglement
witnesses defined in Eq.(5.1) and Eq.(5.2), respectively.
(1) $\hat{W}_{\hat{\Phi}^{(n,k)}}$ is indecomposable and optimal whenever
$k\not=\frac{n}{2}$.
(2) $\hat{W}_{\hat{\Phi}^{(n,\frac{n}{2})}}$ is decomposable and not optimal.
Proof. (1) Assume that $k\not=\frac{n}{2}$. Denote by $P$ and $Q$ the $n$-rank
projection with range the subspace spanned by $\\{|i\rangle\\}_{i=1}^{n}$ and
$\\{|j^{\prime}\rangle\\}_{j=1}^{n}$. Then we have
$\hat{\Phi}^{(n,k)}(A)=Q[\hat{\Phi}^{(n,k)}(PAP)]Q$ holds for all
$A\in{\mathcal{B}}(H)$.
By [12], an entanglement witness $W$ for an infinite dimensional system is
optimal if and only if $W-D$ can not be an entanglement witness anymore for
any nonzero operator $D\geq 0$. So, if $\hat{W}_{\hat{\Phi}^{(n,k)}}$ is not
optimal, then there exists a nonzero positive operator
$\hat{D}\in{\mathcal{B}}(H\otimes K)$ such that
$\hat{W}_{\hat{\Phi}^{(n,k)}}-\hat{D}$ is an entanglement witness. Note that
$(P\times Q)\hat{W}_{\hat{\Phi}^{(n,k)}}(P\otimes
Q)=\hat{W}_{\hat{\Phi}^{(n,k)}}$. Then $\hat{W}_{\hat{\Phi}^{(n,k)}}-\hat{D}$
is an entanglement witness implies that, for any separable pure state
$\sigma_{lh}=|l\rangle\langle l|\otimes|h^{\prime}\rangle\langle h^{\prime}|$
with $l,h>n$, we have
$-{\rm Tr}(\hat{D}\sigma_{lh})={\rm
Tr}((\hat{W}_{\hat{\Phi}^{(n,k)}}-\hat{D})\sigma_{lh})\geq 0,$
which forces that
$\langle lh^{\prime}|\hat{D}|lh^{\prime}\rangle=0$
for all $l,h>n$. Since $\hat{D}\geq 0$, we see that
$\hat{D}=(P\otimes Q)\hat{D}(P\otimes Q).$
Observe that $\hat{W}_{\hat{\Phi}^{(n,k)}}|_{(P\otimes Q)(H\otimes
K)}=W_{\Phi^{(n,k)}}$ with $W_{\Phi^{(n,k)}}$ the same as that in Theorem 3.2.
Denote by $D=\hat{D}|_{(P\otimes Q)(H\otimes K)}$. Then $\hat{D}=D\oplus 0$.
Now, for any separable state $\sigma\in{\mathcal{S}}(P(H)\otimes Q(K))$, there
exists a separable state $\sigma^{\prime}\in{\mathcal{S}}(H\otimes K)$ with
$\sigma^{\prime}=(P\otimes Q)\sigma^{\prime}(P\otimes Q)$ such that
$\sigma=\sigma^{\prime}|_{(P\otimes Q)(H\otimes K)}$. Then we have
${\rm Tr}((W_{\Phi^{(n,k)}}-D)\sigma)={\rm
Tr}((\hat{W}_{\hat{\Phi}^{(n,k)}}-\hat{D})\sigma^{\prime})\geq 0,$
which means that $W_{\Phi^{(n,k)}}-D$ is an entanglement witness,
contradicting to the fact that $W_{\Phi^{(n,k)}}$ is optimal. Hence
$\hat{W}_{\hat{\Phi}^{(n,k)}}$ is optimal, completing the proof of the
statement (1).
The proof statement (2) is the same as that of (2) in Theorem 3.2. $\Box$
Remark 5.2. By checking the proof of Theorem 5.1, one sees that the following
general result is true: Let $W$ be an entanglement witness on $H\otimes K$. If
there exist projections $P\in{\mathcal{B}}(H)$ and $Q\in{\mathcal{B}}(K)$ such
that $W=(P\otimes Q)W(P\otimes Q)$ and $W|_{(P\otimes Q)(H\otimes K)}$ is
optimal on $P(H)\otimes Q(K)$, then $W$ is optimal.
It was asked in [18] whether or not $\hat{\Phi}^{(n,\frac{n}{2})}$ is
indecomposable. The following proposition gives an answer to this question.
Proposition 5.3. Let $H$ and $K$ be infinite dimensional complex Hilbert
spaces. For any even positive integers $n\geq 4$, let
$\hat{\Phi}^{(n,\frac{n}{2})}$ be the positive map defined in Eq.(5.1). Then
$\hat{\Phi}^{(n,\frac{n}{2})}$ is decomposable.
Proof. Note that, for infinite dimensional systems, we have no one-to-one
correspondence between the set of positive linear maps and the set of
entanglement witnesses complemented by Choi-Jamiołkowski isomorphism. So, we
can not get the decomposability of $\hat{\Phi}^{(n,\frac{n}{2})}$ from the
decomposability of $\hat{W}_{\hat{\Phi}^{(n,\frac{n}{2})}}$ proved in Theorem
5.1.
Let $P$ and $Q$ be as that in the proof of Theorem 5.1. Then
$\hat{\Phi}^{(n,\frac{n}{2})}(A)=Q\hat{\Phi}^{(n,\frac{n}{2})}(PAP)Q$
for all $A\in{\mathcal{B}}(H)$. Thus, restricting to
${\mathcal{B}}(P(H)\otimes Q(K))$ we get
$\hat{\Phi}^{(n,\frac{n}{2})}|_{{\mathcal{B}}(P(H)\otimes
Q(K))}=\Phi^{(n,\frac{n}{2})}$. Since the positive map
$\Phi:{\mathcal{B}}(P(H))\rightarrow{\mathcal{B}}(Q(K))$ and its associated
entanglement witness $W_{\Phi}$ is one-to-one corresponded, and $\Phi$ is
decomposable if and only if $W_{\Phi}$ is decomposable. By applying Theorem
3.2, $W_{\Phi^{(n,\frac{n}{2})}}$ is decomposable. So $\Phi^{(n,\frac{n}{2})}$
is decomposable. Thus, there are completely positive maps $\Delta_{i}$
($i=1,2$) such that
$\Phi^{(n,\frac{n}{2})}(A)=\Delta_{1}(A)+\Delta_{2}(A^{T})$ for all
$A\in{\mathcal{B}}(P(H))$, where the transpose is taken with respect to the
basis $\\{|i\rangle\\}_{i=1}^{n}$. Extending $\Delta_{i}$ to
$\hat{\Delta}_{i}$ on ${\mathcal{B}}(H)$ by
$\hat{\Delta}_{i}(A)=Q\Delta_{i}(PAP|_{P(H)})Q$ regarding $Q$ as an operator
from $K$ onto $Q(K)$. Then
$\hat{\Delta}_{i}:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ is completely
positive for $i=1,2$ and
$\hat{\Phi}^{(n,\frac{n}{2})}(A)=\hat{\Delta}_{1}(A)+\hat{\Delta}_{2}(A^{T})$,
where the transpose is taken with respect to the given basis
$\\{|j^{\prime}\rangle\\}_{j=1}^{\infty}$. Hence,
$\hat{\Phi}^{(n,\frac{n}{2})}$ is decomposable. $\Box$
Furthermore, for each $i=1,2$, as $\Delta_{i}$ is an elementary operator, we
see that $\hat{\Delta}_{i}$ is an elementary operator, too. So, there exist
operators $C_{1},\ldots,C_{s},D_{1},\ldots,D_{t}\in{\mathcal{B}}(H,K)$ with
$C_{h}=QC_{h}P$ and $D_{l}=QD_{l}P$ such that
$\hat{\Phi}^{(n,\frac{n}{2})}(A)=\sum_{h=1}^{s}C_{h}AC_{h}^{\dagger}+(\sum_{l=1}^{t}D_{l}A^{T}D_{l}^{\dagger})$
for all $A$. By checking the proof of Proposition 5.3, we see that this fact
indeed hold for all finite rank positive elementary operators. Thus a question
is raised:
Question 5.4. Let $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a
decomposable positive elementary operator with infinite rank. Whether or not
there exist completely positive elementary operators $\Delta_{i}$, $i=1,2$,
such that $\Phi=\Delta_{1}+\Delta_{2}\circ{\bf T}$?
## 6\. Conclusions
We present a characterization of optimal entanglement witnesses in terms of
positive maps and provide a general approach of checking optimality of
entanglement witnesses. This allows us to show that a kind of indecomposable
entangled witnesses $W_{\Phi^{(n,k)}}$ corresponding to the positive maps
$\Phi^{(n,k)}:M_{n}\rightarrow M_{n}$ with $k\not=\frac{n}{2}$ are optimal,
where, for each $n\geq 3$ and $1\leq k\leq n-1$, $\Phi^{(n,k)}$ is defined by
$\Phi^{(n,k)}(a_{ij})={\rm
diag}((n-1)a_{11}+a_{1+k,1+k},(n-1)a_{22}+a_{2+k,2+k},\cdots,(n-1)a_{nn}+a_{kk})-(a_{ij}).$
The space spanned by all product vectors on which $W_{\Phi^{(n,k)}}$ has zero
mean values is of dimension $n^{2}-n+1$. So, in addition to the well-known
indecomposable entanglement witness acting on $M_{3}\otimes M_{3}$
corresponding to the Choi map, we get much more indecomposable optimal EWs
acting on $M_{n}\otimes M_{n}$ that have no spanning property. These also
allow us to get new examples of indecomposable optimal EWs of infinite
dimensional systems. Moreover, these optimal EWs give new examples supporting
a recent SPA conjecture posed in [15] saying that the so-called structural
physical approximations (SPA) to optimal positive maps (optimal EWs) give
entanglement breaking (EB) maps (separable states).
## References
* [1] R. Augusiak, J. Bae, L. Czekaj, M. Lewenstein, J. Phys. A: Math. Theor. 44 (2011) 185308.
* [2] I. Bengtsson, K. Zyczkowski, Cambridge University Press, Cambridge, 2006.
* [3] D. Bru${\ss}$, J. Math. Phys. 43 (2002) 4237.
* [4] M.-D. Choi, Lin. Alg. Appl. 10, 285 (1975); ibid 12, 95 (1975).
* [5] D. Chru$\acute{s}$ci$\acute{n}$ski and A. Kossakowski, Open Systems and Inf. Dynamics 14 (2007) 275; D. Chru$\acute{s}$ci$\acute{n}$ski and A. Kossakowski, J. Phys. A: Math. Theor. 41 (2008) 145301.
* [6] D. Chru$\acute{s}$ci$\acute{n}$ski, F. A. Wudarski, 2011, arXiv:1105.4821.
* [7] K. C. Ha, S. H. Kye, Phys. Rev. A 84 (2011) 024302; arXiv:1107.2720v1.
* [8] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223 (1996) 1.
* [9] J. Hou, J. Operator Theory, 39 (1998), 43-58.
* [10] J. Hou, J. Phys. A: Math. Theor. 43 (2010) 385201; arXiv[quant-ph]: 1007.0560v1.
* [11] J. Hou, X. Qi, Phys. Rev. A 81 (2010) 062351.
* [12] J. Hou, Y. Guo, Phys. Rev. A 82.052301 (2010).
* [13] M. A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197.
* [14] A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972).
* [15] J.K. Korbicz, M.L. Almeida, J. Bae, M. Lewenstein and A. Acin, Phys. Rev. A 78, 062105 (2008).
* [16] M. Lewensetein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62 (2001) 052310.
* [17] M. A. Nielsen, I. L. Chuang, Cambridge University Press, Cambridge, 2000.
* [18] X. Qi, J. Hou, J. Phys. A: Math. Theor. 43 (2011) 385201.
* [19] G. T$\acute{o}$th, O. G$\ddot{u}$hne, Phys. Rev. Lett. 94 (2005) 060501.
|
arxiv-papers
| 2011-09-11T09:37:52 |
2024-09-04T02:49:22.064095
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaofei Qi and Jinchuan Hou",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1109.2300"
}
|
1109.2499
|
# The Evolution of the Cuban HIV/AIDS Network
Charanpal Dhanjal Institut Telecom, LTCI UMR Telecom ParisTech/CNRS No. 5141,
46 rue Barrault, 75634 Paris Cedex 13, France Stéphan Clémençon Institut
Telecom, LTCI UMR Telecom ParisTech/CNRS No. 5141, 46 rue Barrault, 75634
Paris Cedex 13, France Hector De Arazoza Facultad de Matemática y
Computación, Universidad de la Habana, La Habana, Cuba Laboratoire Paul
Painlevé, UMR CNRS 8524, Université Lille 1, 59 655 Villeneuve d’Ascq Cedex,
France Fabrice Rossi Institut Telecom, LTCI UMR Telecom ParisTech/CNRS No.
5141, 46 rue Barrault, 75634 Paris Cedex 13, France Viet Chi Tran
Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, 59 655
Villeneuve d’Ascq Cedex, France Centre de Mathématiques Appliquées, UMR CNRS
7641, Ecole Polytechnique, Route de Saclay, 91 128 Palaiseau Cedex, France.
###### Abstract
An individual detected as HIV positive in Cuba is asked to provide a list of
his/her sexual contacts for the previous 2 years. This allows one to gather
detailed information on the spread of the HIV epidemic. Here we study the
evolution of the sexual contact graph of detected individuals and also the
directed graph of HIV infections. The study covers the Cuban HIV epidemic
between the years 1986 and 2004 inclusive and is motivated by an earlier study
on the static properties of the network at the end of 2004. We use a variety
of advanced graph algorithms to paint a picture of the growth of the epidemic,
including an examination of diameters, geodesic distances, community structure
and centrality amongst others characteristics. The analysis contrasts the HIV
network with other real networks, and graphs generated using the configuration
model. We find that the early epidemic starts in the heterosexual population
and then grows mainly through MSM (Men having Sex with Men) contact. The
epidemic exhibits a giant component which is shown to have degenerate chains
of vertices and after 1989, diameters are larger than that expected by the
equivalent configuration model graphs. In 1997 there is an significant
increase in the detection rate from 73 to 256 detections/year covering mainly
MSMs which results in a rapid increase of distances and diameters in the giant
component.
11footnotetext: Author for correspondence (dhanjal@telecom-paristech.fr)
## 1 Introduction
In Cuba, the HIV epidemic is recorded and controlled using detections made in
a number of different ways, including _contact tracing_ which is the process
of finding and testing the contacts of a person detected as HIV positive. This
detection system, set up in 1986, has been effective at keeping the HIV
epidemic under control and has also enabled the recording of a considerable
amount of epidemiological data. Any individual tested as HIV positive is
indexed and described through several attributes including gender, age, sexual
preference, way of detection and area of residence. Furthermore, a list of
indices corresponding to the sexual partners in the database she/he possibly
named is available, allowing one to construct a graph of sexual contacts along
which the epidemic spreads. Note that HIV most often spreads in Cuba through
sexual transmission, and infection cases by blood transfusion or injection
drug use have negligible impact on the evolution of the epidemic. In many
situations, a medical investigation allows one to obtain a plausible date of
infection together with the index of the supposed HIV transmitter.
A preliminary overview of the HIV/AIDS epidemics in Cuba is provided in [3]
and the characteristics of the network at the end of 2004 are studied in [18].
One of the conclusions of the study of [18] is that it is necessary to analyse
the time varying characteristics of the HIV epidemic. Hence we use recent
graph-mining techniques and statistical analysis to learn how the network has
evolved between the start of 1986 and the end of 2004. We aim to describe the
connectivity and communication properties of the network and understand the
impact of heterogeneity (with respect to the attributes observed) in the graph
structure and its evolution. We also compare the epidemic graph to patterns
observed in a variety of other real networks and a set of generated networks
with matching degrees to the real epidemic.
Our database contains 1109 women and 4280 men, a total of 5389 individuals and
4097 edges. Of this set 2122 people died by the year 2006. As a preprocessing
step on the graph of sexual contacts, we take the union of infection and
contact edges. This ensures that if an infection is recorded from one
individual from another then an edge exists between the corresponding vertices
for the sexual contact graph. Notice that due to the nature of the way data is
recorded, gender and sexual orientation can be merged into the groups women,
heterosexual men, and MSM (Men having Sex with Men: those who reported at
least one sexual contact with another man in the two years preceding HIV
detection). Note also that contact tracing starts not immediately after a
person gives her/his sexual partners (for the last two years), but takes some
time. When a person is found to be HIV positive, there is an epidemiological
interview, carried out by the Epidemiology Department of the municipality
where the person is residing or by the Regional Sanatorium for HIV/AIDS. Then
the contacts the person gives are forwarded to the corresponding
municipalities where they reside. This takes time, therefore some people that
should be in the contact tracing “road” to detection are detected some time
after the primary case (the one that gives him/her as contact) has been
detected.
To discuss the properties we will later examine, we first formalise the
representation of the epidemic graph (for an introduction to the study of
graphs, see [45, 24, 15, 53]). The _contact graph_ at time $t$ is
$G_{t}=(V_{t},E_{t})$. Here, $V_{t}=\\{v_{1},\ldots,v_{|V_{t}|}\\}$ is a set
of vertices, and $E_{t}\subseteq V_{t}\times V_{t}$ is a set of edges where
$|V_{t}|$ is the number of vertices at time $t$. Each vertex represents an
individual detected as having HIV and edges between vertices represent
recorded sexual contact. Since edges and vertices are not deleted with time,
$V_{t}\subseteq V_{t+1}$ and $E_{t}\subseteq E_{t+1}$ for all times $t$.
Furthermore, our graph is undirected which means that if $(v_{i},v_{j})$ is an
edge then $(v_{j},v_{i})$ is also an edge. As we also have information on the
possible infections we can construct an _infection graph_ which has directed
edges $D_{t}\subseteq V_{t}\times V_{t}$ to indicate infections. The set of
edges representing infections at time $t$ is a subset of the set of sexual
contact edges, i.e. $D_{t}\subseteq E_{t}$, since we assume an infection can
only occur between individuals who have had sexual contact.
We begin with some background to contextualise this work in terms of the
analysis of real and random graph growth, and the study of epidemics.
Following, in Section 3 we outline the main aims beyond a temporal exploration
of the data, and also the methods used for analysis. Section 4 presents the
resulting analysis on the epidemic data and in the final part of this paper we
summarise findings and provide concluding remarks.
## 2 Background
The _degree_ of a vertex in an undirected graph is the number of edges
incident to it, and in a directed graph one can talk about the number of
outgoing and incoming edges of a vertex as the out-degrees and in-degrees
respectively. An early observation on the characteristics of real graphs is
that the _degree distribution_ follows a particular pattern. The degree
distribution gives the probabilities $(p(k))_{k\geq 0}$ that a vertex selected
at random has a degree $k\geq 0$. In many real-world networks (e.g. [26, 50,
14, 1]) the degree distribution follows a _power law_ , which means that
$p(k)\propto k^{-\alpha}$ for some exponent $\alpha>1$. This means that many
vertices have few connections but a small fraction have large degrees. One of
the first network growth models to show a power law distribution was the
Albert-Barabasi model [8]. In this model vertices are added incrementally and
when a new vertex is added to the network, it is linked to an existing vertex
with a probability proportional to the degree. The degree exponent in the
Albert-Barabasi model is proven to be $3$. Modifications of the preferential
attachment model have been studied by many authors, for instance in the ageing
of nodes where the probability to connect to a node decreases with its age and
this is shown to change the exponent of the power distribution. See also for
example the works by Dorogovtsev and Mendes [23, 22, 21] and by Kumar and co-
authors [32, 35].
There have been a number of other random graph models which possess a power
law degree distribution. In [51] a random graph is constructed with a degree
exponent of 3 using random walks. Another random graph model which generates a
power law degree distribution is the copying model [33] which also results in
a community structure. For a detailed review of these models and their
variations, we refer the reader to [42]. One way of generating a graph with
given degree distribution, including the power law distribution, is provided
by the configuration model (CM, [11, 24, 43, 53]). In this model, for each
vertex $v_{i}$ a degree is sampled from a given degree distribution and one
can consider the vertices as having the corresponding number of “spokes”
sticking out of them. One then randomly pairs unconnected spokes in order to
form a graph under the required degree distribution.
### 2.1 Real Graph Growth
Of particular relevance to the present study is the study of the sexual
contact network in Sweden [40] in which 4781 individuals were surveyed about
their sexual behaviour. Using the graph constructed from the sexual contacts
up to 12 months before the survey, the authors noted a power law exponent of
2.4 for males and females. Two reasons accounting for the power law behaviour
are: increased skill in getting new partners as the number of previous
partners grows and different levels of attractiveness. In [27] a similar study
is conducted using heterosexual contact networks in Sweden, Uganda and the USA
and by fitting to the Yule distribution.
The growth characteristics of real networks have been studied in a wide
variety of domains. A review on the evolution of networks is presented in [20,
23]. In these works the authors detail a number of studies on citation and
collaboration networks, networks of internet routers, neural and protein
networks and electronic circuits amongst others. Further interesting studies
include [9] on the evolution of a scientific collaboration network, [34] which
studies a university social network, [25] which considers an Internet dating
community and [41] which examines 3 months of growth in the Flickr social
network.
In Leskovec et al. [38, 39] the evolution of a number of real graphs are
studied over time. The graphs considered include citation graphs for articles
and patents, a graph of routers within the Internet, and a set of bipartite
affiliation graphs of papers and authors. The authors find that the networks
become denser with time (vertices increase their out-degree with time),
encapsulated in the so-called Densification Power Law (DPL). Furthermore the
diameter, defined as the longest shortest path, decreases indicating a small-
world effect in which a large network has a low average path length between
pairs of vertices
The DPL was independently observed in a citation network in Katz [29]. It
states that the number of edges grows exponentially faster than the number of
vertices, $|E_{t}|\propto|V_{t}|^{\sigma}$ where $|E_{t}|$ and $|V_{t}|$ are
respectively the number of edges and vertices at time $t$, and the value of
$\sigma$ is often between $1$ and $2$. In order to better model these effects
[38] proposes two graph models: the community guided attachment and the forest
fire graph. This later model generates both the densification property and a
shrinking diameter. Another graph growth model which satisfies these
properties is the Kronecker graph [36, 37], which is based on successive
Kronecker multiplication of adjacency matrices.
### 2.2 Epidemic Networks
There is a considerable amount of research which studies stochastic models of
epidemic networks and their associated properties, and here we briefly recount
some interesting results. The interested reader is directed to [2] for a
detailed review. A well-known compartmentalized epidemic model is the
Susceptible Infected Removed (SIR) one which was introduced in [30]. A set of
differential equations describes how these groups of individuals change over
time. The spread of infections on spatial lattices have been for instance
considered by [31]. Similar studies involving mean-field approximations are
considered in [10, 24, 48].
The small world property of real networks is important in the spread of
disease as it implies that paths between pairs of vertices can be small and
the spread of an infectious disease could potentially spread quickly over a
population. Studies involving disease transmission over a small world network
are treated by Ball Neal [4] and Moore and Newman [44].
Propagation of epidemics on CM graphs has been considered for instance by Ball
and Neal [5]. Further work along these lines include those of Volz and Meyers
[54, 55], Decreusefond et al. [19] where approximations by ordinary
differential equations have been proposed in the case of large epidemics. In
Ball, Sirl and Trapman [6, 7] the authors propose a configuration graph which
also has the small world property of high clustering motivated in part due to
a “household” structure.
In line with this work is the notion of a random intersection graph,
introduced by [28]. These graphs have been used for epidemiology by Britton et
al. [13]. The authors investigate the effect of the clustering within the
graph on the epidemic threshold.
## 3 Aims and Method
As previously mentioned, a study has already been conducted on the static
graph characteristics of the epidemic and here we extend this analysis in a
temporal sense in order to understand how the epidemic evolves. The main aim
is to perform an exploration of the properties of the data, however, in
addition we would like to answer the following questions:
1. 1.
How do the properties of a set of growing CM graphs generated with degree
distributions of the real epidemic vary from those seen in the evolving HIV
graph?
2. 2.
How do the different detection methods change their roles over time?
3. 3.
Does the evolution of the epidemic network follow the same properties observed
in [38, 39]?
The first question identifies differences in the characteristics of the real
epidemic graph with a set of random graphs with the same degree distribution
over time. In order to conduct the analysis we generate 10 CM random graphs
with specified degree distributions identical to that of the HIV epidemic at 3
month intervals. Resulting characteristics are averaged over all random graphs
in order to make a comparison with the equivalent characteristics on the real
graph. This analysis is conducted on both the sexual contact graph and
infection graph. In the case of the infection graph, the CM is modified to
generate directed graphs and one changes the algorithm in an intuitive fashion
to allow both the in-degrees and out-degrees.
For the exploratory analysis and in order to address the remaining questions,
we compute statistics over both contact and infection graphs which are
described in the follow subsections. In most cases graph properties are
evaluated 90 day intervals starting from 03/01/86 and ending on 02/10/04. For
some properties the quantities of interest are recorded at 4 year intervals in
the same date range, namely the dates 31/12/89, 31/12/93, 31/12/97, 31/12/01
and 02/10/04. Most of the analysis code is written in Python using the APGL
graph library (http://packages.python.org/apgl) however clustering is
performed using the igraph package for R (http://igraph.sourceforge.net).
### 3.1 Global Properties and Connectivity
We begin with analyses which gives a global picture of the evolving structure
of the epidemic, and plot the number of vertices and edges in each graph. To
test if the DPL is exemplified in our data, we plot $\log(|E_{t}|)$ and
$\log(|D_{t}|)$ versus $\log(|V_{t}|)$ for which the slope is the exponent
$\sigma$. In the datasets used in [39], the values of the exponent fall
between $1$ and $2$. An exponent of 1 corresponds to a constant average degree
over time and $\sigma=2$ implies that on average each vertex has edges to a
constant fraction of all vertices.
Following, we perform a study of the components in the graph. For a graph $G$
two vertices are said to be _connected_ if there is a path between them. In
the same way a graph is said to be connected if all pairs of vertices are
connected. The HIV epidemic graph is not connected. However, one can talk
about _connected components_ of the graph i.e. sets of vertices which are
connected. The component distribution is recorded at 4 year intervals for both
the contact and infection graph (in the latter we consider the distribution of
trees), as is the number of components of different sizes. As connectivity is
related to the detection methods we explore their evolution over time and also
the total number of sexual contacts declared by each individual, as well as
the number tested and found positive.
We also study the mean degrees of the graphs to make the comparison to [38,
39] in which mean degrees are shown to always increase. Following, we examine
the degree change at an individual level by considering the difference in the
days between two individuals connected by an edge in the contact graph. We
call the absolute value of the difference in days between detection dates of
two connected persons their _detection distance_.
### 3.2 The Largest Component
In many networks there exists a _giant component_ which is a component of size
comparable to the entire graph. We observed a giant component at the end of
the period of the recorded epidemic, and we would like to know when a giant
component comes into formation and how it grows. To this effect, we record the
maximum component size and its number of edges at 90 day intervals. In the
infection graph we look at size evolution of the largest trees.
A common property observed in many real networks is the _small world_ effect.
This property states that even in large networks the shortest path between any
two vertices is generally small. One way of measuring if the epidemic network
has a small world property is to record the diameter of the graph. Diameter
however can be subject to degenerate structures such as long chains in the
graph. A more robust measure is the effective diameter, defined in the
following way:
###### Definition 3.1 (Effective Diameter).
For each natural number $d$ and a graph $G$, let $g(d)$ denote the fraction of
connected vertices which have a shortest path between them of length at most
$d$. Let $D$ be an integer for which $g(D-1)<q$ and $g(D)\geq q$, for some
user-defined value $q$. Then the graph $G$ has the effective diameter $D$.
In the experiments presented in [39], the diameter and effective diameter
exhibit similar behaviour. In particular it should be noted that effective
diameter is equivalent to diameter when $q=1$. We compute effective diameters
using $q=0.9$ in our experiments. Analogously, for the infection trees we
study the depths of the largest trees.
To extend the notion of measuring shortest paths within a graph, one can
consider the _average geodesic distance_ :
###### Definition 3.2 (Average Geodesic Distance).
For an undirected graph $G$, let $d(v_{i},v_{j})$ be the distance of the
shortest path between vertices $v_{i}$ and $v_{j}$, with $d(v_{i},v_{i})=0$,
and $d(v_{i},v_{j})=0$ if $v_{i}$ and $v_{j}$ are not connected. Then the
average geodesic distance is
$\ell(G)=\frac{2}{|V|(|V|+1)}\sum_{i\leq j}d(v_{i},v_{j}).$
In other words the average geodesic distance is a measure of the average
shortest path length between all pairs of connected vertices. The diameter and
geodesic distance give information on the set of all shortest paths between
vertices. We additionally consider a measure of the mean growth of the
neighbourhood of vertices with path length using a _hop plot_ [47].
###### Definition 3.3 (Hop plot).
Let $f(h)$ be the number of pairs of vertices that are reachable within path
lengths (hops) of $h$ or less, then the hop plot is a plot of $f(h)$ with $h$.
#### 3.2.1 Community Structure
A way of finding the community structure of vertices is to study those
subgraphs in which there are many edges within each subgraph and few between
different subgraphs. As we are dealing with a loosely connected graph, it is
natural to consider the giant component for this study, and this component has
already been considered in [18, 16, 17]. We use the maximization of the
modularity defined in [18], together with an algorithm by Noack and Rotta [46]
to cluster the giant component obtained in the whole data set. Notice that the
graph clustering does not take into account the temporal aspect of the
interactions. More precisely, we study the final graph that contains all
detected individuals, regardless of their detection date. In addition, each
edge of the graph (sexual contact) is considered to exist “forever”, in a
static final graph.
We measure the change in the sexual orientations of the clusters in the giant
component as the epidemic evolves, comparing to the distribution of
orientations in the whole population. To discover more about whether the
clustering reveals interesting temporal homogeneity we also plot the temporal
variation in the detection dates for several clusters. Finally, we look at the
distribution of detection distances for those edges within the clusters and
between clusters.
### 3.3 Gender, Sexual Orientation and Location
In the static analysis of the epidemic network it was clear that the gender
and orientation of individuals are key factors for understanding changes in
the network. Hence we study these properties as well location. We first look
at genders and orientations over the whole graph, and then study the breakdown
for those people detected using contact tracing and doctor recommendations.
The infections along edges are also examined with respect to gender and
orientation. Closely related to orientations is the study of the distribution
of triangle participations of vertices in the contact graph. For each vertex
we count how many triangles that vertex participates within, which implies a
set of 3 MSM vertices with edges between them. We would in general expect the
triangle participation distribution to be strongly peaked at zero and have a
heavy tail, however the distribution is interesting since it provides an
insight into the clustering of the MSM population, which is a key facet of the
epidemic spread.
Given a set of infection trees an interesting question is whether the
individuals within the tree are homogeneous in terms of their location and
sexual orientation. Hence we measure a quantity called the _information
entropy_ [52] of these characteristics for each tree which has a size greater
than 1. Intuitively, entropy is a measure of the unpredictability of a random
variable $X$. As an example a fair coin (each side lands with probability
$0.5$) has an entropy of $1$, whereas an unfair one which always lands on
heads has an entropy of $0$.
###### Definition 3.4 (Entropy).
Let $X$ have possible values $x_{1},\ldots,x_{k}$, then entropy is defined as:
$H_{b}(X)=\sum_{i=1}^{k}-p(x_{i})\log_{b}(p(x_{i})),$
where $b$ is the base of the logarithm and $b=2$ in our experiments.
### 3.4 Subgraph/subset Distances and Degrees
One of the problems with the average geodesic distance of Definition 3.2 is
that in graphs that are not connected, some of the distances between vertices
are infinite. Another way of measuring the mean distances in disconnected
graphs is to use the _harmonic mean geodesic distance_ in which infinite
distances contribute nothing to the sum:
###### Definition 3.5 (Harmonic Average Geodesic Distance).
Let $d(v_{i},v_{j})$ be the distance of the shortest path between vertices
$v_{i}$ and $v_{j}$. Then the harmonic average geodesic distance for an
undirected graph $G$ is
$\ell^{\prime}(G)^{-1}=\frac{2}{|V|(|V|+1)}\sum_{i\leq j}d(v_{i},v_{j})^{-1}.$
This is informative since it includes components other than the largest one.
In addition we look at the distance evolution in the subgraph of just men,
knowing that MSM contact is a main driver for the evolution. This is followed
by a study of the distances between the subset of MSM individuals and those
with the top 10% highest degrees.
We plot the degree distribution at various points in time for both the contact
and infection graph. For the contact graph we also compute the power law
exponent $\alpha$ as the graph grows. Estimation of $\alpha$ can be achieved
by minimizing Küllback-Leibler divergence or by computing Hill estimators, see
[18] for more details. In our case we use Küllback-Leibler minimization.
Notice that a large value of $\alpha$ implies a short tailed distribution in
which large degrees are less probable. In [39], for an email network the
degree exponent for the _power law degree distribution_ is almost constant
over time whereas with a citation graph the exponent decreases over time.
These effects are well predicted by Theorems 5.1 and 5.2 of the paper, and we
would like to know whether these theorems can predict the degree exponent in
the epidemic graph.
### 3.5 Centrality
The final set of analyses concerns the _centrality_ of vertices which is a
measure of their relative importance, and also the resilience of the graph. By
resilience we mean that if we remove a central vertex then a resilient graph
still has connectivity. There are several measures of centrality in existence
[11] and we consider the _eigenvector centrality_. Relative scores to all
vertices in the network are based on the principal that connections to high-
scoring vertices contribute more to the score of the vertex in question than
equal connections to low-scoring nodes. This method is used as a basis of
Google’s PageRank [12] for example. Eigenvector centrality is defined more
formally as:
###### Definition 3.6 (Eigenvector centrality).
For a graph $G$, let $\textbf{x}_{i}$ denote the score of vertex $v_{i}$, and
let A be the adjacency matrix of $G$ such that $\textbf{A}_{ij}=1$ if an edge
exists between $v_{i}$ and $v_{j}$ otherwise $\textbf{A}_{ij}=0$. The
centrality score $\textbf{x}_{i}$ is proportional to the sum of the scores of
the vertices connected to it, i.e.
$\displaystyle\textbf{x}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{v_{j}\in
n(v_{i})}\textbf{x}_{j}=\frac{1}{\lambda}\sum_{j=1}^{|V|}\textbf{A}_{ij}\textbf{x}_{j}.$
It follows that $\textbf{A}\textbf{x}=\lambda\textbf{x}$ where x and $\lambda$
are an eigenvector-eigenvalues pair of A, and since all scores must be
positive the eigenvector corresponding to the largest eigenvalue of A gives
the vector of scores.
One way of finding the largest eigenvector is by using Power Iteration [49].
## 4 Results and Analysis
We start by considering the growth in the number of vertices and edges in the
contact and infection graphs, Figure 1(a). The number of detections per unit
time increases over time and there are fewer edges in each graph than vertices
implying a low average connectivity. Since we are using data up until the end
of 2004, the end of the plots are flat as not all detected individuals towards
the end of this period have been entered into the database. The detection rate
increases from 125/year to 430/year after 1997 and there is an increasing
discrepancy between the number of vertices and edges in the contact graph.
Notice also the sudden increase in detections in 1989. The proportion of
contact edges in which an infection occurs decreases: it is 0.426 at 31/12/89
and 0.35 at 31/12/93 indicating more effective contact tracing at the start of
the epidemic.
(a) Vertices and edges
(b) $\log(|V|)$ versus $\log(|E|)$/$\log(|D|)$
Figure 1: The growth rate of edges and vertices in the graphs.
Figure 1(b) provides more insight into the relationship between the number of
vertices and edges in the graphs. A linear regression shows that for the
entire contact graph:
$\log(|V|)=1.294\log(|E|)-2.399,$
$R^{2}=97\%$, where both regression coefficients are significant. The slope
$\sigma$ of the curve of Figure 1(b) is significant and estimated at
$\widehat{\sigma}=1.294$ with the confidence interval $[1.241,1.346]$. This
value is closer to 1 than to 2 and later we shall see how this affects the
mean degree. On closer examination the growth exponent is 2.4 at the start of
the graph growth and 0.89 at the end, hence rate of growth of the number of
contact edges with the number of vertices slows over time and this corresponds
to the stabilization of the degree distribution. For the maximum component we
observed that the growth exponent was 1.59 at the start and 0.88 at the end of
the recorded period. Furthermore the corresponding plot for the edges in the
infection graph is close to that of the contact graph at the start of the
recorded period and then diverges as expected.
### 4.1 Component Distribution
One of the reasons for the apparent slow growth of contact edges with respect
to vertices is that there are a large number of isolated individuals detected
randomly, see Figure 2. There are sudden increases in gradient of the total
number of components after 1989, 1990, and to a larger extent after 1997. The
increases in 1989 and 1990 correspond to detections of isolated vertices.
Before 1997 a new component is formed approximately every 8 days, and
afterwards this period becomes 1.5 days. We can also see that the number of
components that are bigger than a pair is generally less than half the number
of pairs. On examination of the probability distribution of component sizes
the probability of an isolated vertex is surprisingly constant around $0.77$
at each 4 year interval. Together, isolated and paired vertices account for
approximately 90% of the total number of components in the graph as it changes
over time. In a way this is not surprising since many detections are not made
using contact tracing, and furthermore many pairs are husband and wife in
which the husband/wife does not disclose additional information about his/her
contacts.
Increasingly over time the number of components found using the configuration
graphs is less than the equivalent number on the real epidemic. At the end of
2004 there are $2111$ real components compared to $1994$ predicted using the
configuration model. Similar patterns are observed for components with sizes
$\geq 2$ and $\geq 3$. This implies than a new detection generally forms an
edge with a smaller component than that expected by random choice.
On analysis of the trees present within the infection graph, Figure 2(b),
there is a much higher number of trees than components: $3099$ compared to
$2111$ and this is to be expected as not all sexual contact from an infected
individual results in transmission of HIV. Ignoring scaling however, the
number of trees matches closely the corresponding number of components in the
contact graph. Whilst most trees are small we observed that the largest tree
sizes were respectively 27, 75, 91, 108 and 120 for the dates 31/12/89,
31/12/93, 31/12/97, 31/12/01 and 02/10/04. Over time approximately 87% of the
trees are pairs or isolated vertices and 96.7% have a depth of 2 or less. We
observed an increase in the number of deep trees between 31/12/97 and
31/12/01, in which 80 and 161 had depths of 2 or larger respectively. At the
end of 2004 there were 233 trees of this depth.
A perhaps surprising artefact of this figure is that the CM predicts closely
the number of trees of different sizes. However, some insight into this
scenario can be gained by noting that the number of predicted trees cannot be
fewer than the number of roots in the set of real trees. Furthermore, the
number of isolated vertices will be at least that of the real case, and trees
of size 2 or more are highly likely to remain greater than size 2 when
generating the CMs.
(a) Number of components
(b) Number of trees
Figure 2: The change in the number of components and trees. The difference
between the curves for components/trees greater than or equal to 1 and 2 is
the number of isolated individuals. The error bars indicate a difference of 1
standard deviation in the values of interest over the configuration model
graphs.
(a) Detections by type
(b) Average contacts
Figure 3: The total number of detections per year and the change in the mean
number of contacts for the two years preceding detection.
To explore further the trends in the number of components and trees, Figure
3(a) shows the trends of the most common detection methods. The sudden
increase in detections in 1989 can be attributed to increases contact tracing,
STD tests and “other” tests. There is steady growth in the number of non-
contact traced detectees, with doctor recommendations increasing most rapidly
after approximately 1997. Contact tracing plays a decreasing role in the
overall detection of infected individuals. In general the initial detected
epidemic was driven by contact tracing with doctor recommendations and
voluntary visits taking an increasing role several years after recording
began. The number of detections by voluntary visits is smaller than the
detections by doctor recommendations after 2001. The increasing number of non-
contact traced detections explains why the number of components in the graph
increases exponentially and also the slow rate of edge growth with vertices.
Figure 4: The mean proportion of HIV positive individuals of those tested.
Also collected in the HIV dataset is the total number of sexual contacts for
each individual during the 2 years preceding their detection, and the number
of tested and positive individuals in this group. Figure 3(b) shows the
evolution of the mean values of these quantities. The curves for the number of
contacts and tested contacts follow similar trends, and encouragingly at least
77% of contacts are tested on average over time. Both curves have a peak at
approximately 1992, at 6.5 and 5.5 respectively, and then decrease which is
also reflected in the decrease in the number of positive contacts. At the end
of 2004 the mean number of positive contacts is 0.96 per new detectee. The
increase in detections in 1989 manifests itself as a lower mean number of
contacts and tested contacts although a greater number of positive contacts.
This implies that newly detection individuals in this period declared fewer
contacts however the ones they did declare resulted in more positive
detections than average before this point. A breakdown of the number of
contacts by gender and orientation shows that early on in the epidemic it is
the MSMs who have the highest mean number of contacts with 22.4 in late 1986
however this number falls to 5.8 in late 2004. The mean number of contacts for
females starts at 1 and then rises to 6.2 in 1992 after which it stays
constant. Figure 4 shows the mean proportion of positive contacts found
against the number tested. It shows that at the start of the recorded epidemic
females had their contacts tested positive most often. There is a rise in the
mean proportion in 1989 for MSM and men, and another small increase just
before the end of 1991.
### 4.2 Mean Degrees
Figure 5: The temporal variation in the average degree.
To compare the epidemic evolution to the graphs studied in [39] we show the
mean degrees in Figure 5. The mean out-degrees do not continually rise (as in
the graphs in [39]) and are generally small for both the maximum component and
the entire graph. One reason for the peak in the plots is that the non-contact
tracing detections starts to become a significant fraction of the total number
of detections. The peak mean degrees are 1.8 in 1994 for all vertices and 3.3
for the largest component, after which they decrease slowly. A Chow test shows
that there is a breakpoint between 1996 and 1998 for each of these curves. In
the first period, there is a regular increase of the mean degree, which can be
due to the discovery of individuals with high degrees. After 1994, the mean
degree decreases which can be linked to exploration of leaves of the graph,
corresponding to individuals with lower degrees. Moreover, an increase in the
detection of isolated individuals or couples decreases the mean degree.
For the CM the mean degree for all vertices is slightly higher than that of
the observed graphs, $G_{1},\ldots,G_{T}$, at most time points. The reason for
this is subtle: in the generation of graphs it is guaranteed that the degree
sequence $\textbf{d}^{t}$ at time $t$ of $G_{t}$ is greater than or equal to
that of a generated CM graph $\hat{\textbf{d}}^{t}$, i.e.
$\textbf{d}^{t}_{i}\geq\hat{\textbf{d}}^{t}_{i}$ for all $i,t$. Indeed, we do
not explicitly store detection dates in the generated CM graphs, hence using
the configuration graph at the final time $T$ to compute the mean degree at
time $t$ we simply select the subgraph corresponding exactly to those edges in
$G_{t}$. It is possible that this subgraph has more edges than the equivalent
graph $G_{t}$ and we see it to be the case in Figure 5.
### 4.3 Detection Distance Analysis
One would hope that the degree of each vertex increases only for a period
whilst their contacts are being tested and then stops. Figure 6(a) gives the
distribution of the detection distance in the largest connected component of
the contact graph.
(a) Giant component
(b) All vertices
Figure 6: Detection distance distributions.
The distribution can be considered surprising as 1060 edges among 3168 have a
detection distance higher than two years, while the official temporal span of
the infection tracing procedure is two years. This can be explained partially
by the long time the tracing procedure can take in practice, and the
continuation of infection after detection. As shown in Figure 6(b), the
distribution of the detection distance in the full graph is not very different
from what happens in the largest connected component, yet the percentage of
high detection distances (more than two years) slightly drops from 0.33 to
0.3. An interesting question is triggered by this finding: can we relate high
detection distance to infection events that occurred after the detection of
the infecting person? It turns out that 345 edges in the contact graph connect
two persons $a$ and $b$ such that $a$ has been detected more than two years
before the probable infection date of $b$. Among those edges, 194 are
considered infectious among a total of 2287 infectious edges.
### 4.4 Evolution of the Largest Component
We have already touched upon some of the properties of the largest connected
component and here we study more of its characteristics. Figure 7(a) shows the
number of vertices and edges in the largest component over time. As an aside
we note that the size of the 2nd largest component is generally small compared
to the largest component being just 18 at the end of the recorded epidemic.
Before 1996, the number of nodes of the largest component increases as
$|V_{t}|=19.565+8.243(t-\verb"01/01/1986"),$
$R^{2}=99.36\%$, while after 1998,
$|V_{t}|=-1784.634+44.125(t-\verb"01/01/1986"),$
$R^{2}=98.69\%$, which implies a more than 5-fold increase in the rate of new
detections within the giant component before 1996 and 1998. It can be verified
with a Chow-Fisher test that there is a breakpoint between 1996 and 1998. The
curve for the number of edges in the largest component follows almost the same
shape as that of the number of vertices. Notice that there is an increase in
the number of edges relative to the number of vertices between 1991 and 1992.
Before 1991 there are 79 edges per year and afterwards there are 203, an
increase by a factor of 2.6. The equivalent CM results show a larger maximum
component with more edges although the curves have similar breakpoints.
Therefore, in practice a new detection is less likely to connect to the
largest connected component than if edges are chosen at random. This
corresponds with the trends seen in Figure 2.
(a) Size
(b) Mean geodesic distance
Figure 7: The size of the largest component and mean geodesic distances.
In the next series of analyses we consider several distance measures over the
largest component: Figure 7(b) shows the geodesic distance over time, and
Figure 8 displays the hop plot and diameter of the largest component. We
compared the diameter (not shown) with the effective diameter and found that
they both followed broadly similar trends although it is clear that there are
long chains in the giant component as the maximum diameter is 33 compared to a
maximum effective diameter of 15. The increasing diameter is in direct
contrast to the trends observed in the graphs in [39] and is explained in part
due since in general vertices do not continually increase their degrees after
detection (see Figure 6).
Four periods appear in Figure 8(b) (and also in Figure 7(b)): two periods of
stability in the size of the graph (diameter and geodesic distances) alternate
with two periods of expansion of the graph. The diameter rises from 5 to 10 in
1989 and this corresponds to a change in the geodesic distance from 3 to 6.
Upon further analysis we can see that in 1989 there is a sudden increase in
the largest component size from 25 to 102 and diameter from 5 to 10 in 1989
which implies that the largest component merges with one or more smaller
components at this point. This is coupled with a rise in the detection rate
during this period. From 1989 to 1997 the diameter and geodesic distances stay
constant at 9 (falling from 10 in 1991) and 6 respectively. There were indeed
fewer detections during this period due to the economic crisis following the
collapse of the Soviet Union. Furthermore, in this period the growth in the
largest component is nearly constant at approximately 73 individuals per year.
After 1997 this growth rate increases to approximately 256 new detections per
year, a significant increase.
The sudden increases in diameter and geodesic distances at 1989 and 1997 are
not reflected in the CM graphs. The diameter and geodesic distances rises
earlier from 1987 and remains nearly flat between 1989 and 1995. One intuition
is that geodesic distances decrease when there is a “long distance” edge
between communities and will increase when chains for vertices are added to
the largest component. Of note was that the number of vertices of degree 3 or
more is 19 in 18/12/88 and rises to 40 in 18/03/89, and this contributes to
the peak in the geodesic distance in this period. In the real epidemic the
edges formed by this set of vertices do not join communities to the same
extent. The number of vertices of degree 1 and 2 continue to increase with the
rate of additional degree 1 vertices increasing from 58 to 121 per year after
1997. We already know that the largest component for the CM graphs is
generally larger and has more edges than that on the real graphs and this
coincidences with the smaller distances after 1997.
(a) Hop plot. The initial gradients computed until $k=3$ are $0.313$, $0.460$,
$0.447$, $0.423$ and $0.395$ respectively in ascending order of dates.
(b) Effective diameter
Figure 8: The hop-plot and effective diameter of the largest component over
time.
The hop-plot in Figure 8(a) shows the extent of degenerate chains of vertices.
For example at 31/12/01 the number of pairs reachable with 15 hops is almost
identical to those reachable with 27 hops. At $k=0$ the number of reachable
pairs is exactly the number of vertices in the graph. The number of reachable
pairs of vertices with $k$ hops increases exponentially with $k$ for the
initial part of each curve, however the initial gradient at 31/12/89 is lower
than the other time points.
### 4.5 Evolution of the Largest Infection Trees
In analogy to the study of the largest component, we consider the largest
trees in the infection graph. Whereas the largest component was significantly
larger than the 2nd largest component, the largest tree is close in size to
the 2nd largest tree. Their sizes are respectively 120 and 105 in 2004 which
are 5% and 4.4% of the size of giant component. Both of these trees continue
to grow for nearly the whole recorded period. Observe that the 2nd largest
tree before 1999 becomes the largest tree after 1999 and vice versa. Note also
the rapid growths in 1989 and 1991 of the largest tree. After 1992 the growth
of the largest tree is approximately linear at 6.2 new detectees per year. The
CM graphs have much smaller trees of 59.8 and 47.8 at the end of 2004 for the
largest and 2nd largest trees however the standard deviations are large.
The tree depths indicate that the 2nd largest tree spawns long chains of
vertices, having a depth of 13 at its largest point. This is only slightly
shorter than the effective diameter of the largest component. The depth of
these trees is larger than that expected for the CM trees after 1989 for the
2nd largest tree and after 1997 for the largest tree. Clearly the large
diameters seen in the contact graph correspond to large chains of infections
in the infection trees.
As expected, early growth in the largest two trees result from MSM individuals
who account for 8 and 10 detections for the 1st and 2nd trees compared to just
2 for heterosexual infectees in 1989. In the period between 1990 and 1991
there is a sudden growth in the number of heterosexual detections for both
trees. The MSM population of the 1st tree is comparatively larger than the
second at the end of the displayed period. In the following section we show
how the trends in sexual orientation manifest themselves in the entire
detected population.
### 4.6 Gender, Sexual Orientation and Location
We plot the gender and orientation in Figure 9(a). After 1996 there are more
MSM detections than heterosexual ones and after 1997 the detection rate of MSM
individuals (along with males) more than doubles from 108 detections/year to
264 detections/year. In 2002 the MSM detection rate is 497/year in contrast
with the heterosexual detection rate of 156/year. These trends explain the
rising diameter and geodesic distance starting after 1997. Communities of
individuals are newly detected with few “long distance” edges which would
shorten the diameter and mean geodesic distances. The MSM population starts to
grow at a faster rate than the heterosexual one at approximately 1991 with a
detection rate of 86/year compared to that of 60/year for heterosexuals. To
quantify this increase, the number of newly detected heterosexual males from
13/03/90 to 02/10/04 is 410 whereas the change in the MSM population in the
same period is 3552. Women account for a small and decreasing proportion of
the contact graph, however a majority of the heterosexual population, 66.2%,
at the end of 2004. A slightly different picture is presented in Figure 9(b)
on the individuals detected using contact tracing. In this case we see a far
smaller difference in the number of people that are MSM versus those that are
heterosexual and it is only in 1999 that there are more MSM detections than
heterosexual ones. Detection rates before 1997 for all curves are similar. On
analysis of the breakdown of individuals detected using doctor
recommendations, there are just 5 people detected in late 1992 and it is in
1999 that there is an increase in detection rate for MSM individuals from 23
detections/year to 61 after 1999. By the end of 2004 there are 601 bisexuals
detected using this method versus 218 heterosexuals.
We study next in Figure 10(a) the frequency of infection based on the gender
and sexual orientation of the ego and alter. As one might suspect from
previous results the main means of infection occur between MSM individuals and
also from MSN to women. The epidemic before 1987 was driven almost exclusively
by heterosexual men infecting women. After 1987, there is an increase in MSM
to MSM infections and in 1988 we see that infections suddenly rise between MSM
individuals and women, and between MSM detectees. In 1990 the infection rates
MSM to MSM and MSM to women are 29 and 21 per year. In contrast at 1997 these
rates are respectively 102 and 26 infections per year.
(a) All detection methods
(b) Contact tracing
Figure 9: The number of detected individuals by sexual orientation and gender.
Again, breakpoints are detected between 1996 and 1998.
To investigate further the effects of the MSM population in the contact graph,
we compute the probability distribution of triangle participation at 4 year
intervals. Recall that only MSM contacts can form triangles. Most triangles
are seen at 31/12/93 in which 14.3% of individuals are present in at least 1
triangle, and the number of individuals without any triangles decreases from
31/12/89 to 31/12/93 and then increases. Due to the increase in isolated
individuals, the most rapid increase in triangles is between 31/12/89 to
31/12/93 in which the number of individuals with 1, 2 and 3 triangles
increases from 12 to 58, 5 to 22 and 3 to 10 respectively, and by the end of
1997 there is 1 individual in 11 triangles. The MSM population is small in
31/12/89 and increases in proportion which would explain this trend. Note that
the increase in MSM detection rate from 1997 does not impact the number of
individuals in triangles and conversely there are fewer individuals in
triangles at this point. One contribution to the trend in the number of
individuals with no triangles is the increase in the proportion of random
detections. At the end of 2004, 319 individuals participate in at least 1
triangle corresponding to 8.6% of the MSM population.
The entropies of the sexual orientations in the infection trees show a strong
homogeneity at 31/12/89 with a probability of entropy 0 tree being 0.76 versus
an average of 0.65 at the other time points (this compares with the
probability of entropy 1 as 0.07 versus 0.27 respectively). This is counter-
intuitive given that from Figure 9(a) the overall mix of sexual orientation is
most balanced at this point. It appears that although the epidemic population
becomes more MSM later on, it is at an earlier stage that infection trees are
more homogeneous.
(a) Sexual transmission characteristics
(b) Location
Figure 10: Gender/orientation of ego and alter along infection edges and the
change in the detections by location.
Next we recorded the number of detections per province for the 6 provinces
with the most detections in Figure 10(b). Unsurprisingly most detections are
in Havana City as it is the largest city in Cuba (with approximately 2 million
residents out of a total of 11 million). The epidemic starts mostly in Havana
with 31 detections at the end of 1986, and between the end of 1986 and 1987
there is an increase in detections from 6 to 30 in Sancti Spiritus. Between
the end of 1988 and 1989 the number of detections in Villa Clara increases
from 10 to 37, and 1990 and 1992 the detections rate in Pinar del Rio
increases from 7 detections/year to 32. The increases in detection rates in
these cities correspond to newly discovered communities. Most of the activity
however, including the increase in the detections starting from 1997, occur in
Havana and this change in detection rate is not seen in other cities. The
epidemic in Havana grows at a fast rate because it is difficult to manage
however in other cities contact tracing ensures the epidemic spread is under
control.
### 4.7 Subgraph/subset Distances and Degrees
Previously we looked at some distance measures over the contact graph and here
we conduct similar analyses for the whole contact graph and
subgraphs/subgroups of interest using the harmonic distance. The initial
contact graph starts with 1 node and the harmonic distance is undefined. As
later individuals are detected using contact tracing, this distance rapidly
decreases (see Figure 11(a)) and fluctuates when components become connected.
At 1989 the distance drops from 117 to 50 which corresponds to the merging of
two or more components and a sudden increase in detections as previously
observed. It is after 1992 that the detection rate in conjunction with contact
tracing falls which increases the harmonic distance but does not affect the
largest component for example.
(a) Complete graph and men subgraph.
(b) Subset of individuals
Figure 11: Harmonic mean geodesic distances.
Figure 11(a) also displays the mean harmonic geodesic distance on the
_subgraph_ of just men. In this case, connections can only be made between MSM
individuals. There is a dramatic fall in distance in 1987 as this corresponds
exactly to the point at which MSM to MSM contact starts. There is then an
approximately constant mean distance of 71 from 1989 to 1992. The peak at 1997
can be explained by the increase in the detection rate for contact traced MSM
individuals at this point.
In Figure 11(b) we consider distances between _subsets_ of individuals. For
those individuals who are within 10% of the highest degrees, one observes
broadly similar trends to the full detected population except that distances
are small in this case indicating a closer connectivity than the general
population. The rises and drops in distance before 1989 correspond to those in
Figure 11(a). The distance rises from under 4 in 1993 to just under 7 in 2004
which is less than the corresponding mean geodesic distances in the giant
component of 6 and just above 10. Given the high connectivity of these
individuals one would expect that there are many paths between them resulting
in low distances. For MSM individuals, distances are lower than the set of all
individuals. This said however, during the period 1993 to 2003, the distance
changes from 14 to 38 showing that MSM individuals grow further apart at a
faster rate than the population as a whole.
We study the properties of the degree distribution and found them to be
broadly constant at 4 year intervals. Note that a degree 1 individual
corresponds to someone in a pair or on the periphery of a connected component.
We also studied the trend in degree exponents $\alpha$ over time for the set
of detected individuals and their complete set of sexual contacts. We noticed
that $\alpha$ tends to increase over time from 1.5 to just over 3 in 2004
which implies that the degree distributions are becoming shorter tailed. We
also look at the power law exponents of various subsets of individuals
according to gender, orientation and detection method. We see that degrees do
not vary much across gender or sexual orientation. All the same, after doctor
recommendation begins in 1993, it allows one to find individuals who have a
slightly higher degree than the ones found by the other methods. Individuals
found by random methods have smaller degrees on average.
We also studied the out-degrees of the infection graph. The in-degree of each
non-root vertex in this graph is 1 as individuals are infected by a single
other person. As with the contact graph, the curves are similar at 4 year
intervals, however at 31/12/89 there is a slightly lower probability of degree
1 vertices. At this point approximately 70% of vertices have a degree of zero,
and 20% have a degree of one. The degrees therefore have a much more skewed
distribution than that of the contact graph. The largest number of infections
caused by a single person is 16 compared to a corresponding maximum of 25
detected contacts.
### 4.8 Temporal Analysis of the Clustering
The clustering obtained in [18] at the end of the recorded epidemic results in
39 subclusters of the giant-component. The first detected individual of this
component was tested seropositive on June 5th 1986. It can be seen that
between 1992 and 1995, few clusters are added: the detections in this period
mainly contribute to the densification of the existing communities. As a
consequence, the diameter does not change much. The expansion of the graph
diameter before 1990 and between 1997 and 1999 correspond to the initial
spread of the epidemic and to the discoveries of the new communities which
have become infectious between 1989 and 1997 when the research of new
infectious individuals was less active.
A study of their significance in terms of sexual orientation [18, 17] shows
that 21 out of the latter have atypical distributions: 9 (resp. 12) have less
(more) MSM than in the whole population (76%): they constitute the mixed
(resp. MSM) subgroups. The remaining clusters are called typical since their
composition looks like the one of the whole population in terms of sexual
orientation. The evolution of of the clustering of the giant component is
represented in Figure 12.
Figure 12: Evolution of the largest component and of the size of its
subgroups.
We can see that there have been two waves describing the growth of the giant
component. This is confirmed by the analysis of the evolution of the subgroups
(MSM, mixed and typical) exhibited for the giant component (see Figure 12). We
see that between 1986 and 1993, the growth of the giant component is largely
due to the subgroups mixed and typical, then, there is a large increase, after
1995 of all the subgroups sizes, including for the MSM subgroup. This latter
observation ties in with the trends of Figure 9(a). The late contamination of
the MSM community can be explained by the fact that the contamination has gone
from heterosexual men to women and MSM before propagating among MSM. Because
the latter community is relatively well connected, this can explain the higher
spread started in the 90’s (see [18]).
#### 4.8.1 Temporal Homogeneity
Next we study the temporal homogeneity of the obtained clusters. The detection
date of each person is used as the temporal information as this is the only
date variable available for each person (as opposed to the infection date,
death date, etc.). Detection dates are encoded as a number of days between the
earliest detection date in the database and the considered detection date. We
study therefore the distribution of this quantity conditioned by the cluster.
The spread of the detection date distributions in each cluster is measured by
standard deviations of the dates. Most of the clusters exhibit a large spread,
as shown in Figure 13. The median standard deviation is 1052.14. That said,
most of the clusters are more homogeneous in term of detection date than the
full connected component: 34 components out of 39 have a smaller standard
deviation than the full component.
Figure 13: Distribution of the standard deviations of the detections dates
within the clusters. The red line identifies the global standard deviation of
dates.
Some clusters gather persons detected at extreme dates, as shown in Figure 14,
which compares the distribution of detection days in the full largest
connected component to the conditional distribution in the cluster that
exhibits the worst time homogeneity. Typical clusters have a standard
deviation around 1000 days, roughly three years. Two examples are presented in
Figure 14.
Figure 14: Detection date distributions of the largest component, the least
time homogeneous cluster and 2 typical clusters.
Therefore, while the clustering is done without direct knowledge of the
temporal information, clusters tend to be homogeneous in term of detection
dates compared to the whole network. As indicated by the quite low value of
the standard deviation in most of the clusters compared to the global one,
this does not happen by chance. We verify this claim with a Monte Carlo
approach in which node assignments to clusters are randomized (in a way that
preserves the size of the clusters). Then we compute the standard deviations
of the detection dates in each cluster and the median of those deviations. The
mode of the distribution of this median over 1000 replications is 1615 which
is not close to the value obtained by the clustering of 1052.14.
#### 4.8.2 Detection Distances
One can also look at temporal homogeneity at the edge level. While it makes
sense intuitively to ask to clusters to contain persons detected at the same
period, there are numerous examples in the database of connections between
persons detected at quite far away dates, as seen in Section 4.3. In fact, the
detection distance distribution shown in this section partially explains why
the clustering exhibits naturally some temporal homogeneity. Indeed, the mode
of the distribution is around 40 days and its median is 396. Most of the
connections are between close persons (in term of detection distance), and
most of the connection between persons occur inside clusters (there is only
333 inter-cluster edges among 3168 edges). Additionally, as shown in Figure
15, inter-cluster edges correspond to higher detection distance. This should
reinforce the homogeneity effect.
Figure 15: Detection distance distributions in conjunction with the
clustering.
Differences are confirmed by elementary statistics: the median detection
distances between members of distinct clusters is 528 while it is only 379 for
members of similar clusters. A simple Monte Carlo study shows a similar trend:
only 1 random subset of the edges out of 1000 give a median detection distance
as high as the one observed in the intra-cluster edges (random edge subsets
have the same size as the intra-cluster edge set).
### 4.9 The Most Central Vertices
For the final set of analyses we consider those individuals who are most
central to the epidemic according to the maximum eigenvector coefficients of
the adjacency matrix. Recall from the definition of eigenvector centrality
that the coefficient for the $i$th vertex
$\textbf{x}_{i}=\sum_{j=1}^{|V|}\textbf{A}_{ij}\textbf{x}_{j}$ and hence the
sum of the scores of the neighbouring vertices. It follows that in the case a
vertex has no neighbours, its score will be zero. The distribution of
eigenvector coefficients is long tailed with few vertices having a coefficient
larger than 0.15, and the proportion of vertices with this level of centrality
decreases rapidly. At 31/12/93 for example 99.19% of the vertices have a
coefficient smaller than 0.15 compared to 99.98% in 02/10/04.
| 31/12/89 | 31/12/93 | 31/12/97 | 31/12/01 | 02/10/04
---|---|---|---|---|---
Female | 15.4 | 37.4 | 35.5 | 33.3 | 35.1
Male | 84.6 | 62.6 | 64.5 | 66.7 | 64.9
Heterosexual | 17.9 | 39.4 | 39.3 | 39.8 | 41.6
Bisexual | 82.1 | 60.6 | 60.7 | 60.2 | 58.4
Contact traced | 41.0 | 60.6 | 62.8 | 57.4 | 56.8
Blood donor | 5.1 | 3.0 | 3.3 | 3.4 | 2.8
RandomTest | 0.0 | 2.0 | 1.1 | 3.6 | 3.9
STD | 20.5 | 9.1 | 9.8 | 9.0 | 8.7
Prisoner | 2.6 | 5.1 | 6.6 | 11.6 | 12.6
Doctor recommendation | 0.0 | 1.0 | 1.1 | 2.6 | 2.8
Mean age (years) | 22.66 | 20.96 | 21.80 | 22.84 | 23.45
Holguin | 0.0 | 0.0 | 0.0 | 0.3 | 1.5
La Habana | 0.0 | 1.0 | 2.7 | 2.6 | 3.7
Havana City | 15.4 | 14.1 | 19.1 | 20.7 | 23.2
Pinar del Rio | 7.7 | 36.4 | 31.7 | 23.3 | 20.4
Sancti Spiritus | 15.4 | 3.0 | 3.8 | 5.4 | 6.9
Villa Clara | 48.7 | 37.4 | 35.0 | 37.0 | 34.7
Mean degree | 4.26 | 6.32 | 5.54 | 4.51 | 4.09
Std degree | 2.93 | 3.37 | 3.29 | 2.99 | 2.81
Table 1: The characteristics of the 10% most central vertices. Attributes are
represented as percentages in all cases except for age and degree.
Table 1 shows the characterisation of 10% of the vertices with the highest
eigenvector coefficients. Between 31/12/89 and 31/12/93 we see that the most
central vertices become more female, heterosexual, detected using contact
tracing, younger and are more likely to be located in Pinar del Rio and less
likely in Sancti Spiritus and Villa Clara. After 31/12/93 the changes are less
dramatic and the central population becomes more male, heterosexual, detected
less using contact tracing, and older at detection. Notice that before this
date, 20.5% of the central individuals are detected due to other STDs and this
figure declines after 31/12/93. These patterns mostly correspond with the
findings presented for the entire detected population however the highest
proportion of central vertices live in Villa Clara and not Havana City. On
analysis of the degree distributions of the central vertices we see as
expected that they have higher degrees than the general case.
## 5 Conclusions
Motivated by an earlier study into the static properties of the Cuban HIV
epidemic in [18], we examined the evolution of the epidemic from the period
beginning in 1986 to the end of 2004 and presented a number of analyses
characterising the complex behaviour of the network over time. In response to
the questions presented at the start of the study we can say that this
epidemic does not follow all of the trends of other networks observed in [38,
39]. In particular the growth exponent of the number of edges with vertices is
not constant but decreases over time, diameters increase and the mean degrees
do not continually increase in both the whole graph and largest connected
component. These findings can be explained in large part by the fact that upon
detection, additionally detected contacts are typically found within a few
years. Furthermore, although contact tracing plays an important role in the
detection of infected individuals other methods capture the majority of the
detections resulting a strongly disconnected graph with an increasing number
of isolated and paired vertices. In contrast to [38, 39] the giant component
was characterised by large geodesic distances and effective diameters and
contains degenerate chains of vertices. Analogously, a study of the infection
trees shows that the largest trees grow continuously.
A key comparison we made was between the growth of the epidemic and that of a
set of CM graphs generated using the evolving degree sequences of the real
epidemic. The CM graphs showed that many properties of the real networks are
captured purely in the evolving degree sequences. However, the real epidemic
has more components and a smaller giant component than one expected using the
CM. Furthermore, after 1997 the mean geodesic distances and effective
diameters were smaller in the CM graphs corresponding to the point in which
long chains of vertices appear in the largest component. These observations
are mirrored in the infection graph, as we saw that the real largest trees
were larger and deeper than those found using the CM graphs.
The main points in the temporal behaviour of the epidemic are summarised as
follows: in 1986 the spread of the epidemic occurs mainly between heterosexual
men and women and it is only later in 1987 that MSM individuals are detected
in significant numbers. In 1989 the network changes dramatically: one observes
a temporary increase in the detection rate caused mainly by people detected
through contact tracing, “other” and STD testing. There is a noticeable
increase in the size and diameter of the giant component in which smaller
components merge into it. This corresponds to a drop in the harmonic mean
geodesic distance of the entire graph from 120 to 50. The newly detected
people declare fewer contacts than those detected earlier on average, however
the MSMs result in a higher proportion of positive contact of those tested at
this point. In 1991 many new edges are added to the network between existing
vertices due to an increasing contact tracing rate and this causes the
diameter of the largest component to fall from 10 to 9. Between 1992 and 1995
existing communities are shown to gain more edges in the clustering of the
giant component resulting in a constant diameter. In 1992 we see a peak in the
mean number of contacts declared and also those tested for HIV. In the
following year the mean harmonic distance between MSM individuals reaches its
lowest point, followed in 1994 by the peak mean degree. The top 10% more
central vertices after 1993 are increasingly located in Havana City, however
Villa Clara still resides most of the central vertices. The year 1997
represents another large change in the epidemic: the rate of detection
increases from 73 to 256 detections/year and this is driven by an increase in
contact traced detections and also doctor recommendations of mostly MSM
individuals. This change is reflected in an increase in the size, mean
geodesic distance and diameter of the giant component, and also an increase in
the proportion of pairs of individuals in which an infection has occurred. One
explanation for the breakpoint at 1997 is that the lack of detections at the
beginning of the 90’s has increased the reservoir of undetected infectious
individuals and has shifted the epidemics to another trend after 1997.
## Acknowledgements
This work was supported by the French Agency for Research under grant ANR
Viroscopy (ANR-08-SYSC-016-03) and by AECID project D/030223/10.
## References
* [1] R. Albert, H. Jeong, and A.L. Barabási. Internet: Diameter of the world-wide web. Nature, 401(6749):130–131, 1999.
* [2] H. Andersson and T. Britton. Stochastic epidemic models and their statistical analysis. Springer Verlag, 2000.
* [3] B. Auvert, H. de Arazoza, S. Clémençon, J. Perez, and R. Lounes. The HIV/AIDS epidemic in Cuba: description and tentative explanation of its low HIV prevalence. BMC Infectious Diseases, 7(30), November 2007.
* [4] Frank Ball and Peter Neal. The great circle epidemic model. Stochastic Processes and their Applications, 107(2):233–268, 2003\.
* [5] Frank Ball and Peter Neal. Network epidemic models with two levels of mixing. Mathematical Biosciences, 212:69–87, 2008.
* [6] Frank Ball, David Sirl, and Pieter Trapman. Threshold behaviour and final outcome of an epidemic on a random network with household structure. Advances in Applied Probability, 41:765–796, 2009.
* [7] Frank Ball, David Sirl, and Pieter Trapman. Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Mathematical Biosciences, 224:53–73, 2010.
* [8] A-L Barabasi and Reka Albert. Emergence of scaling in random networks. 286:509–512, October 1999.
* [9] A.L. Barabâsi, H. Jeong, Z. Néda, E. Ravasz, A. Schubert, and T. Vicsek. Evolution of the social network of scientific collaborations. Physica A: Statistical Mechanics and its Applications, 311(3-4):590–614, 2002.
* [10] M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani. Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. Journal of Theoretical Biology, 235:275–288, 2005.
* [11] B. Bollobás. Random graphs. Cambridge University Press, 2 edition, 2001.
* [12] S. Brin and L. Page. The anatomy of a large-scale hypertextual Web search engine* 1. Computer networks and ISDN systems, 30(1-7):107–117, 1998.
* [13] Tom Britton, Maria Deijfen, Andreas N. Lageras, and Mathias Lindholm. Epidemics on random graphs with tunable clustering. Journal of Applied Probability, 45:743–756, 2008.
* [14] Q. Chen, H. Chang, R. Govindan, and S. Jamin. The origin of power laws in Internet topologies revisited. In INFOCOM 2002. Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, volume 2, pages 608–617. IEEE, 2002.
* [15] F. Chung and L. Lu. Complex graphs and networks. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 2006. Published for the Conference Board of the Mathematical Sciences, Washington D.C. by the American Mathematical Society.
* [16] S. Clémençon, H. De Arazoza, F. Rossi, and V.C. Tran. Hierarchical clustering for graph vizualization, April 2011.
* [17] S. Clémençon, H. De Arazoza, F. Rossi, and V.C. Tran. Visual mining of epidemic networks, June 2011.
* [18] Stéphan Clémençon, Hector De Arazoza, Fabrice Rossi, and Viet Chi Tran. A network analysis of the HIV/AIDS epidemics in cuba. 2011\. submitted.
* [19] Laurent Decreusefond, Jean-Stéphane Dhersin, Pascal Moyal, and Viet Chi Tran. Large graph limit for a SIR process in random network with heterogeneous connectivity. Annals of Applied Probability. to appear.
* [20] S.N. Dorogovtsev and J.F.F. Mendes. Evolution of networks. Arxiv preprint cond-mat/0106144, 2001.
* [21] S.N. Dorogovtsev and J.F.F. Mendes. Scaling properties of scale-free evolving networks: Continuous approach. Physical Review E, 63(056125), 2001.
* [22] S.N. Dorogovtsev and J.F.F. Mendes. Evolution of networks with aging of sites. Physical Review E, 62(1842), 2002.
* [23] S.N. Dorogovtsev and J.F.F. Mendes. Evolution of networks: from biological nets to the internet and www. Oxford University Press, Oxford, 2003.
* [24] R. Durrett. Random graph dynamics. Cambridge University Press, 2007.
* [25] Petter Holme, Christofer R. Edling, and Fredrik Liljeros. Structure and time evolution of an internet dating community. Social Networks, 26(2):155–174, 2004.
* [26] H. Jeong, S.P. Mason, A.L. Barabási, and Z.N. Oltvai. Lethality and centrality in protein networks. Nature, 411(6833):41–42, 2001.
* [27] J.H. Jones and M.S. Handcock. An assessment of preferential attachment as a mechanism for human sexual network formation. Proceedings of the Royal Society of London. Series B: Biological Sciences, 270(1520):1123, 2003.
* [28] Michal Karonski, Edward Scheinerman, and Karen Singer-Cohen. On random intersection graphs: the subgraph problem. Combinatorics, Probability and Computing, 8:131–159, 1999.
* [29] J.S. Katz. Scale-independent bibliometric indicators. Measurement: Interdisciplinary Research & Perspective, 3(1):24–28, 2005.
* [30] W. O. Kermack and A. G. McKendrick. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society of London. Series A, 115(772):700–721, August 1927.
* [31] A. Kleczkowski and B.T. Grenfell. Mean-field-type equations for spread of epidemics: The small world model. Physica A, 274:355–360, 1999.
* [32] J. Kleinberg, R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. The web as a graph: measurement, models and methods. Proceedings of the International Conference on Combinatorics and Computing, pages 1–18, 1999.
* [33] Jon M. Kleinberg, Ravi Kumar, Prabhakar Raghavan, Sridhar Rajagopalan, and Andrew S. Tomkins. The web as a graph: measurements, models, and methods. In COCOON’99: Proceedings of the 5th annual international conference on Computing and combinatorics, pages 1–17, Berlin, Heidelberg, 1999\. Springer-Verlag.
* [34] Gueorgi Kossinets and Duncan J. Watts. Empirical analysis of an evolving social network. Science, 311(5757):88–90, 2006.
* [35] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal. Stochastic models with the web graph. Proceedings of Foundations of Computer Science, pages 57–65, 2000\.
* [36] J. Leskovec, D. Chakrabarti, J. Kleinberg, and C. Faloutsos. Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication. Knowledge Discovery in Databases: PKDD 2005, pages 133–145, 2005\.
* [37] J. Leskovec, D. Chakrabarti, J. Kleinberg, C. Faloutsos, and Z. Ghahramani. Kronecker graphs: an approach to modeling networks. The Journal of Machine Learning Research, 11:985–1042, 2010.
* [38] J. Leskovec, J. Kleinberg, and C. Faloutsos. Graphs over time: densification laws, shrinking diameters and possible explanations. In Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, page 187. ACM, 2005.
* [39] J. Leskovec, J. Kleinberg, and C. Faloutsos. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD), 1(1):2, 2007.
* [40] F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, and Y. Åberg. The web of human sexual contacts. Nature, 411(6840):907–908, 2001.
* [41] A. Mislove, H.S. Koppula, K.P. Gummadi, P. Druschel, and B. Bhattacharjee. Growth of the flickr social network. In Proceedings of the first workshop on Online social networks, pages 25–30. ACM, 2008.
* [42] M. Mitzenmacher. A brief history of generative models for power law and lognormal distributions. Internet mathematics, 1(2):226–251, 2004.
* [43] Michael Molloy and Bruce Reed. A critical point for random graphs with a given degree sequence. Random structures and algorithms, 6:161–180, 1995.
* [44] C. Moore and M.E.J. Newman. Epidemics and percolation in small-world networks. Phys. Rev. E, 61:5678–5682, 2000.
* [45] M. E. J. Newman. The structure and function of complex networks. SIAM Review, 45(2):167–256, 2003.
* [46] Andreas Noack and Randolf Rotta. Multi-level algorithms for modularity clustering. In SEA ’09: Proceedings of the 8th International Symposium on Experimental Algorithms, pages 257–268, Berlin, Heidelberg, 2009. Springer-Verlag.
* [47] C.R. Palmer, P.B. Gibbons, and C. Faloutsos. ANF: A fast and scalable tool for data mining in massive graphs. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 81–90. ACM, 2002.
* [48] R. Pastor-Satorras and A. Vespignani. Epidemics and immunization in scale-free networks. In Handbook of Graphs and Networks: From the Genome to the Internet, pages 113–132, Berlin, 2002. Wiley-VCH.
* [49] David Poole. Linear Algebra: A Modern Introduction. Thomson Brooks/Cole, Pacific Grove, CA, USA, 2nd edition, 2003.
* [50] S. Redner. How popular is your paper? An empirical study of the citation distribution. The European Physical Journal B-Condensed Matter and Complex Systems, 4(2):131–134, 1998.
* [51] J. Saramäki and K. Kaski. Scale-free networks generated by random walkers. Physica A: Statistical Mechanics and its Applications, 341:80–86, 2004.
* [52] C.E. Shannon. A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review, 5(1):3–55, 2001.
* [53] R. van der Hofstad. Random graphs and complex networks. Lecture notes, 2010. ”http://www.win.tue.nl/ rhofstad”.
* [54] Erik M. Volz. SIR dynamics in random networks with heterogeneous connectivity. Mathematical Biology, 56:293–310, 2008.
* [55] Erik M. Volz and Lauren Ancel Meyers. Epidemic thresholds in dynamic contact networks. Journal of the Royal Society Interface, 6:233–241, 2009.
|
arxiv-papers
| 2011-09-12T15:20:50 |
2024-09-04T02:49:22.079696
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Charanpal Dhanjal, Stephan Clemencon, Hector De Arazoza, Fabrice\n Rossi, Viet Chi Tran",
"submitter": "Charanpal Dhanjal",
"url": "https://arxiv.org/abs/1109.2499"
}
|
1109.2503
|
# A new method of finding all Roots of simple quaternionic polynomials
Lianggui Feng and Kaiming Zhao
###### Abstract.
In this paper, we provide a new method to find all zeros of polynomials with
quaternionic coefficients located on only one side of the powers of the
variable (these polynomials are called simple polynomials). This method is
much more efficient and much simpler than the known one in [9]. We recover
several known results, and deduce several interesting consequences concerning
solving equations with all real coefficients or complex coefficients which do
not seem to be deduced easily from the results in [9]. We also give a
necessary and sufficient condition for a simple quaternionic polynomials to
have finitely many solutions (only isolated solutions).
K.M. Zhao is partially supported by NSERC and NSF of China (Grant 10871192),
and L.G. Feng is partially supported by HNSF of China (Grant 11JJ7002).
Keywords: quaternion, simple quaternionic polynomial, root
2000 Mathematics Subject Classification.11R52, 12E15, 12Y05, 65H05
## 1\. Introduction
The quaternion algebra plays an important role in many subjects, such as,
quaternionic quantum mechanics [1], and signal precessing [2, 8]. Because of
the noncommutativity of quaternionic multiplication, solving a quaternionic
equation of degree $n$ becomes a challenging problem [3-19].
Niven in [14, 15] made first steps in generalizing the fundamental theorem of
algebra onto quaternionic situation which led to the article by Eilenberg and
Niven [3] where the existence of roots for a quaternion equation of degree $n$
was proved using strongly topological methods. After that, Topuridze in [20],
also with help of topological method, showed that the zero set of polynomials
with quaternionic coefficients located on only one side of the power of the
single variable (these polynomials are called simple polynomials) consists of
a finite number of points and Euclidean spheres of corresponding dimension.
Concerning about the computation of roots of a quaternionic polynomial, the
first numerically working algorithm to find a root was presented by
Ser$\hat{o}$dio, Pereira, and Vit$\acute{o}$ria [18], and further
contributions were made by Ser$\hat{o}$dio and Siu [19], Pumpl$\ddot{u}$n and
Walcher [17], De Leo, Ducati, and Leonardi [13], Gentili and Struppa [6],
Gentili, Struppa, and Vlacci [7], Gentili and Stoppato [5]. A large
bibliography on quaternions in general was given by Gsponer and Hurni in 2006
[4]. Recently, Janovsk$\acute{a}$ and Opfer presented a method in [9] for
producing all zeros of a simple quaternionic polynomial, by using the real
companion polynomial introduced for the first time by Niven [14], the number
one introduced in [16], and the presentation of the powers of a quaternion as
a real, linear combination of the quaternion. Let us recall some notions
first.
Throughout this paper, let $\mathbb{N}$ be the set of positive integers,
$\mathbb{R}$ the real number field,
$\mathbb{C}=\mathbb{R}\oplus\mathbb{R}\mathbf{i}$ the complex number field,
and $\mathbb{H}$ the skew-field of real quaternions, that is, any element of
$\mathbb{H}$ is of the form
$q=a_{0}+a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}=(a_{0}+a_{1}\mathbf{i})+(a_{2}+a_{3}\mathbf{i})\mathbf{j}$,
where $\mathbf{i},\mathbf{j},\mathbf{k}$ are usual quaternionic imaginary
units, and $a_{0},a_{1},a_{2},a_{3}\in\mathbb{R}$, and the
$\mathbb{R}$-bilinear product is determined by
$\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=-1,$
$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\quad\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i},\quad\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}.$
For the above quaternion $q$, we denote by $\operatorname{Re}q$ the real part
of $q$, by $|q|$ the module of $q$ (i.e.
$|q|=\sqrt{a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$), and by $[q]$ the
conjugate class of $q$(i.e. $[q]=\\{aqa^{-1}|a\in\mathbb{H},a\neq 0\\}$). We
confirm that a quaternionic polynomial with the coefficients on the same side
of the power of the single variable are called a simple quaternionic
polynomial or a simple polynomial.
Definition 1.1. Let $z_{0}\in\mathbb{H}$ be a zero of a simple polynomial
$p(z)$ as given on the left-hand side in (2.1) (i.e., $p(z_{0})=0$). If
$z_{0}$ is not real and has the property that $p(z)=0$ for all $z\in[z_{0}]$,
then we will say that $z_{0}$ is a spherical zero of $p(z)$. If $z_{0}$ is
real or is not a spherical zero, it is called an isolated zero of $p(z)$.
In the present paper, we provide a new method for finding all zeros of simple
polynomials $p(z)$ of arbitrary degree $n$ (Theorems 1 and 4). Our proof is
based on two well-known techniques: the presentation of a quaternion as a
$2\times 2$ complex matrix, and the Jordan canonical form of a complex matrix.
We first write $p(z)$ so that its constant term is $1$ or $0$. Then introduce
derived polynomials $f_{1}(t)$ and $f_{2}(t)$ of $p(z)$ which have complex
coefficients, where $t$ is a real variable, such that
$p(t)=f_{1}(t)+f_{2}(t)\mathbf{j}$; and define the discriminant polynomial
$\tilde{p}(t)=f_{1}(t)\overline{f}_{1}(t)+f_{2}(t)\overline{f}_{2}(t)$ of
$p(z)$ which is a polynomial with real coefficients, where $t$ is considered
as a complex variable, the polynomials $\overline{f}_{1}(t)$ and
$\overline{f}_{2}(t)$ are obtained by only taking the conjugate coefficients
of $f_{1}(t)$ and $f_{2}(t)$ respectively. Then all zeros of $p(z)$ can be
obtained from complex zeros of the discriminant polynomial $\tilde{p}(t)$.
More precisely, let $z_{0}\in\mathbb{C}$ such that $\tilde{p}(z_{0})=0$. If
$z_{0}$ is real then it is an isolated zero of $p(z)$. If $z_{0}$ is not real
and $f_{1}(z_{0})=f_{2}(z_{0})=\bar{f}_{1}(z_{0})=\bar{f}_{2}(z_{0})=0$ then
it is a spherical zero of $p(z)$. If $z_{0}$ is not real and at least one of
$f_{1}(z_{0}),f_{2}(z_{0}),\bar{f}_{1}(z_{0}),\bar{f}_{2}(z_{0})$ is not zero,
let $\left(\begin{array}[]{cc}a\\\ b\end{array}\right)$ be a unit complex
solution of the linear system
$\left(\begin{array}[]{cc}f_{2}(\overline{z_{0}})&f_{1}(\overline{z_{0}})\\\
\bar{f}_{1}(\overline{z_{0}})&-\bar{f}_{2}(\overline{z_{0}})\\\
\end{array}\right)X=0.$
Then corresponding to this $z_{0}$ we have an isolated solution for $p(t)$:
${|a|^{2}z_{0}+|b|^{2}\overline{z_{0}}-2b}\overline{a}(\operatorname{Im}z_{0})\mathbf{k}.$
From the above three cases we obtain all zeros of $p(z)$.
The paper is organized as follows. In Section 2, we prove that our methods for
solving a simple polynomial equation in Theorems 1 and 4 are valid. Then we
recover several known results, and deduce several very interesting
consequences concerning solving equations with all real coefficients or
complex coefficients which do not seem to be deduced easily from the results
in [9] (see Corollary 2 and Corollary 3). We also give a necessary and
sufficient condition for a simple quaternionic polynomials to have finitely
many solutions (only isolated solutions). In Section 3 we give an algorithm to
find all zeros of a simple quaternionic equation, based upon our Theorems 1
and 4. In Section 4 we give three examples. In particular, we use our method
to redo Example 3.8 in [9]. In Section 5, we make some necessary numerical
considerations and compare our algorithm with that in [9].
## 2\. Finding all zeros of a simple polynomial
We consider the simple quaternionic polynomial equation:
(2.1) $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0\,\,(q_{n}\neq 0),$
where $x\in\mathbb{H}$ is the variable, $n\in\mathbb{N}$ and
$q_{i}\,(i=0,\ldots,n)\in\mathbb{H}$ are given. If $q_{0}\neq 0$, then
Eq.(2.1) can be written as
$q_{0}^{-1}q_{n}x^{n}+\cdots+q_{0}^{-1}q_{1}x+1=0.$
Hence, in order to solve Eq.(2.1), it suffices to solve the following equation
(2.2) $p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0,$
where $p_{i}\,(i=1,\ldots,n)\in\mathbb{H},p_{n}\neq 0,d_{0}=0\ {\text{or}}\
1$. We simply denote the left-hand side of (2.2) as $p(x)$.
Let $\sigma:\mathbb{H}\to\mathbb{C}^{2\times 2}$,
$q=z_{1}+z_{2}\mathbf{j}\mapsto\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right),$ where
$z_{1},z_{2}\in\mathbb{C}$. Then $\sigma$ is an $\mathbb{R}$-algebra
monomorphism from $\mathbb{H}$ to $\mathbb{C}^{2\times 2}$. Sometimes, this
monomorphism is also named as the derived mapping of $\mathbb{H}$, and
$\sigma(q)$ is denoted by $q^{\sigma}$. Obviously,
$a^{\sigma}=\left(\begin{array}[]{cc}a&0\\\ 0&a\\\ \end{array}\right)$ for any
$a\in\mathbb{R}$.
Let $p_{i}^{\sigma}=\left(\begin{array}[]{cc}t_{1}^{(i)}&t_{2}^{(i)}\\\
-\overline{t_{2}^{(i)}}&\overline{t_{1}^{(i)}}\\\ \end{array}\right)$ for
$i=1,\ldots,n,$ where $t_{1}^{(i)},t_{2}^{(i)}\in\mathbb{C}$. Then (2.2)
becomes the following matrix equation in the matrix variable
$Y=\left(\begin{array}[]{cc}z_{1}&z_{2}\\\ -\bar{z_{2}}&\bar{z_{1}}\\\
\end{array}\right)\,\,(z_{1},z_{2}\in\mathbb{C}$):
(2.3) $\begin{array}[]{c}\left(\begin{array}[]{cc}t_{1}^{(n)}&t_{2}^{(n)}\\\
-\overline{t_{2}^{(n)}}&\overline{t_{1}^{(n)}}\\\
\end{array}\right)Y^{n}+\cdots+\left(\begin{array}[]{cc}t_{1}^{(1)}&t_{2}^{(1)}\\\
-\overline{t_{2}^{(1)}}&\overline{t_{1}^{(1)}}\\\
\end{array}\right)Y+\left(\begin{array}[]{cc}d_{0}&0\\\ 0&d_{0}\\\
\end{array}\right)=0.\end{array}$
Now we introduce a matrix polynomial $P(t)$ in real variable $t$ (considered
as a real number) as follows:
(2.4) $P(t)\equiv p_{n}^{\sigma}t^{n}+\cdots+p_{1}^{\sigma}t+d_{0}I,$
where $I$ is the $2\times 2$ identity matrix.
Write $P(t)=Q(t)(tI-Y)+P_{l}(Y)$, where
$P_{l}(Y)=p_{n}^{\sigma}Y^{n}+\cdots+p_{1}^{\sigma}Y+d_{0}I,$
$Q(t)=p_{n}^{\sigma}t^{n-1}+(p_{n-1}^{\sigma}+p_{n}^{\sigma}Y)t^{n-2}+\cdots+(p_{1}^{\sigma}+p_{2}^{\sigma}Y+\cdots+p_{n}^{\sigma}Y^{n-1}).$
If $P_{l}(Y)=0$, then $P(t)=Q(t)(tI-Y)$, and hence
$\det P(t)=\det Q(t)\det(tI-Y)=\chi_{Y}(t)\det Q(t),$
where $\chi_{Y}(t)$ is the characteristic polynomial of $Y$.
Set $\tilde{p}(t)\equiv\det P(t)$. By Cayley-Hamilton Theorem, we see that
$\tilde{p}(Y)=0$ for every $Y$ satisfying $P_{l}(Y)=0$.
Notice that, $\tilde{p}(t)\equiv\det P(t)$ is a polynomial in real variable
$t$ of degree $2n$ with real coefficients, since
$\tilde{p}(t)=\det(p_{n}t^{n}+\cdots+p_{1}t+d_{0})^{\sigma}$, and
$\det(p_{n}t^{n}+\cdots+p_{1}t+d_{0})^{\sigma}\geqslant 0$ for any real value
of $t$. Then
(2.5)
$\tilde{p}(t)=b{(t-\xi_{1})^{2r_{1}}\cdots(t-\xi_{s})^{2r_{s}}}\cdot\hskip
56.9055pt$ $\hskip
56.9055pt\cdot{(t-\eta_{1})^{s_{1}}(t-\overline{\eta_{1}})^{s_{1}}}\cdots{(t-\eta_{k})^{s_{k}}(t-\overline{\eta_{k}})^{s_{k}}},$
where $b$ is a real number, $\xi_{1},\ldots,\xi_{s}$ are distinct real
numbers, $\eta_{1},\bar{\eta}_{1},\ldots,$ $\eta_{k},\bar{\eta}_{k}$ are
distinct nonreal complex numbers, and $r_{1},r_{2},$
$\cdots,r_{s},s_{1},\cdots,$ $s_{k}\in\mathbb{N}$. It is clear that $s+k\leq
n$.
Now suppose $Y$ is a solution of Equation (2.3). Then, $\tilde{p}(Y)=0$. Since
$Y$ is of form $\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right)$ where
$z_{1},z_{2}\in\mathbb{C}$, then
$\chi_{Y}(t)=(t-z_{1})(t-\bar{z}_{1})+z_{2}\bar{z}_{2}\geq 0$ for all real
values of $t$. Consequently, $Y$ has two equal real eigenvalues or two
conjugate complex eigenvalues. Hence, its Jordan canonical form, $J_{Y}$, has
to be of form $aI$ or $\left(\begin{array}[]{cc}c&0\\\ 0&\overline{c}\\\
\end{array}\right)$ where $a\in\mathbb{R}$ and $c$ is a complex with nonzero
imaginary part. Using (2.5) we know that, for a solution $Y$ of Equation
(2.3), $J_{Y}$ has to be one of the following $s+k$ matrices (up to the order
of the diagonal entries):
(2.6) $\xi_{1}I,\ldots,\xi_{s}I,\left(\begin{array}[]{cc}\eta_{1}&0\\\
0&\bar{\eta_{1}}\\\
\end{array}\right),\ldots,\left(\begin{array}[]{cc}\eta_{k}&0\\\
0&\bar{\eta_{k}}\\\ \end{array}\right).$
Next we will prove that each of the above cases can occur.
Case 1: $J_{Y}=\xi_{i}I$ for $i=1,\ldots,s$.
In this case $Y=J_{Y}$. Clearly, $Y$ is a solution of Equation (2.3) iff
$p_{n}^{\sigma}\xi_{i}^{n}+\cdots+p_{1}^{\sigma}\xi_{i}+d_{0}I=0$, iff
$\xi_{i}$ is a common real root of both
$t_{1}^{(n)}x^{n}+\cdots+t_{1}^{(1)}x+d_{0}=0$
and
$t_{2}^{(n)}x^{n}+\cdots+t_{2}^{(1)}x=0.$
But $\xi_{i}$ is a real root of $\tilde{p}(t)$, the equalities
$t_{1}^{(n)}\xi_{i}^{n}+\cdots+t_{1}^{(1)}\xi_{i}+d_{0}=0$ and
$t_{2}^{(n)}\xi_{i}^{n}+\cdots+t_{2}^{(1)}\xi_{i}=0$ hold naturally. Hence,
$\xi_{i}I$ $(i=1,\ldots,s)$ is a solution of the equation (2.3).
Case 2: $J_{Y}=\left(\begin{array}[]{cc}\eta_{i}&0\\\ 0&\overline{\eta_{i}}\\\
\end{array}\right)$ for $i=1,\ldots,k$.
We may assume that (see [22])
$Y=\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right)^{-1}$
for some $z_{1},z_{2}\in\mathbb{C}$ with $|z_{1}|^{2}+|z_{2}|^{2}\neq 0$. Then
$Y$ is a solution of (2.3) if and only if
$\begin{array}[]{c}p_{n}^{\sigma}\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}^{n}&0\\\
0&\overline{\eta_{i}}^{n}\\\
\end{array}\right)\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right)^{-1}+\cdots\\\ \ \ \
\ \ \ \ \ \ \ \ \ \ +p_{1}^{\sigma}\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)^{-1}+d_{0}I=0,\end{array}$
i.e.,
(2.7) $p_{n}^{\sigma}\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}^{n}&0\\\
0&\overline{\eta_{i}}^{n}\\\ \end{array}\right)+\cdots\hskip 56.9055pt$
$\hskip 28.45274pt\begin{array}[]{cc}&\ \ \ \ \ \ \
+p_{1}^{\sigma}\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)+d_{0}\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right)=0.\end{array}$
In other words, for any nonzero solution
$\left(\begin{array}[]{cc}z_{1}&z_{2}\\\ -\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)$ of (2.7), we obtain a solution for (2.3) of the following
form:
(2.8) $\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\overline{z_{2}}&\overline{z_{1}}\\\ \end{array}\right)^{-1}.$
We write (2.7) into two equations
$\left\\{\begin{array}[]{cc}t_{1}^{(n)}\eta_{i}^{n}z_{1}+t_{2}^{(n)}\eta_{i}^{n}(-\overline{z_{2}})+\cdots+t_{1}^{(1)}\eta_{i}z_{1}+t_{2}^{(1)}\eta_{i}(-\overline{z_{2}})+d_{0}z_{1}=0&\\\
t_{1}^{(n)}\overline{\eta_{i}}^{n}z_{2}+t_{2}^{(n)}\overline{\eta_{i}}^{n}\overline{z_{1}}+\cdots+t_{1}^{(1)}\overline{\eta_{i}}z_{2}+t_{2}^{(1)}\overline{\eta_{i}}\overline{z_{1}}+d_{0}z_{2}=0,&\end{array}\right.$
i.e.,
(2.9)
$\left\\{\begin{array}[]{cc}(t_{2}^{(n)}\overline{\eta_{i}}^{n}+\cdots+t_{2}^{(1)}\overline{\eta_{i}})\overline{z_{1}}+(t_{1}^{(n)}\overline{\eta_{i}}^{n}+\cdots+t_{1}^{(1)}\overline{\eta_{i}}+d_{0})z_{2}=0\\\
(\overline{t_{1}^{(n)}}\overline{\eta_{i}}^{n}+\cdots+\overline{t_{1}^{(1)}}\overline{\eta_{i}}+d_{0})\overline{z_{1}}-(\overline{t_{2}^{(n)}}\overline{\eta_{i}}^{n}+\cdots+\overline{t_{2}^{(1)}}\overline{\eta_{i}})z_{2}=0.\end{array}\right.$
Considering (2.9) as a linear system in the variables
$X=\left(\begin{array}[]{c}\overline{z_{1}}\\\ z_{2}\\\ \end{array}\right)$,
we see that its determinant of the coefficient matrix is
$D=\left|\begin{array}[]{cc}t_{2}^{(n)}\overline{\eta_{i}}^{n}+\cdots+t_{2}^{(1)}\overline{\eta_{i}}&t_{1}^{(n)}\overline{\eta_{i}}^{n}+\cdots+t_{1}^{(1)}\overline{\eta_{i}}+d_{0}\\\
\overline{t_{1}^{(n)}}\overline{\eta_{i}}^{n}+\cdots+\overline{t_{1}^{(1)}}\overline{\eta_{i}}+d_{0}&-(\overline{t_{2}^{(n)}}\overline{\eta_{i}}^{n}+\cdots+\overline{t_{2}^{(1)}}\overline{\eta_{i}})\end{array}\right|.$
Since $\eta_{i}$ and $\bar{\eta}_{i}$ are roots of
$\tilde{p}(t)=\det
P(t)=\left|\begin{array}[]{cc}t_{1}^{(n)}t^{n}+\cdots+t_{1}^{(1)}t+d_{0}&t_{2}^{(n)}t^{n}+\cdots+t_{2}^{(1)}t\\\
-(\overline{t_{2}^{(n)}}t^{n}+\cdots+\overline{t_{2}^{(1)}}t)&\overline{t_{1}^{(n)}}t^{n}+\cdots+\overline{t_{1}^{(1)}}t+d_{0}\end{array}\right|,$
it follows that $D=0$, which shows that (2.9) always has nonzero solutions.
Since all coefficients of (2.9) are known, the solution set of (2.9) in
$X=\left(\begin{array}[]{c}\overline{z_{1}}\\\ z_{2}\\\ \end{array}\right)$
can be given clearly, which will be denoted by $\Gamma_{i}$.
If $\eta_{i}$ and $\overline{\eta_{i}}$ simultaneously satisfy
$t_{1}^{(n)}x^{n}+\cdots+t_{1}^{(1)}x+d_{0}=0$ and
$t_{2}^{(n)}x^{n}+\cdots+t_{2}^{(1)}x=0$, then (2.9) becomes trivial, and
$\Gamma_{i}=\mathbb{C}^{2\times 1}$. Consequently, any element of the form
(2.8) is a solution of (2.3).
Now suppose (2.9) is nontrivial. Then $\Gamma_{i}$ is of dimension 1, and
$\Gamma_{i}=\left\\{z\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\
\end{array}\right)|z\in\mathbb{C}\right\\},$ in which
$\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\ \end{array}\right)$ is a fixed
nonzero solution of (2.9) with $|a^{(i)}|^{2}+|b^{(i)}|^{2}=1$. Up to now, we
have actually provided a method to find all the roots of (2.1) in
$\mathbb{H}$. To summarize our result as a theorem, we need to introduce some
notions.
Let $p_{i}=t_{1}^{(i)}+t_{2}^{(i)}\mathbf{j}\in\mathbb{H}$ for $i=1,\ldots,n$
where $t_{1}^{(i)},t_{2}^{(i)}\in\mathbb{C}$. We call the following four
polynomials the derived polynomials of Equation (2.2):
$\begin{array}[]{cc}f_{1}(t)=t_{1}^{(n)}t^{n}+\cdots+t_{1}^{(1)}t+d_{0},&f_{2}(t)=t_{2}^{(n)}t^{n}+\cdots+t_{2}^{(1)}t;\\\
\bar{f}_{1}(t)=\overline{t_{1}^{(n)}}t^{n}+\cdots+\overline{t_{1}^{(1)}}t+d_{0},&\bar{f}_{2}(t)=\overline{t_{2}^{(n)}}t^{n}+\cdots+\overline{t_{2}^{(1)}}t.\end{array}$
We define the discriminant polynomial of Equation (2.2) as
$\tilde{p}(t)=f_{1}(t)\bar{f}_{1}(t)+f_{2}(t)\bar{f}_{2}(t)$. We factor it as
in (2.5). Remark that these two $\tilde{p}(t)$ are essentially equal.
Introduce sets $T_{1}$ and $T_{2}$ as follows
$T_{1}=\\{\eta\in\\{\eta_{1},\ldots,\eta_{k}\\}\,|\,f_{1}(\eta)=f_{2}(\eta)=\bar{f}_{1}(\eta)=\bar{f}_{2}(\eta)=0\\},$
$T_{2}=\\{\eta_{1},\ldots,\eta_{k}\\}\backslash T_{1}.$
Now we can state our main result
###### Theorem 1.
With the above notations, the solution set of (2.2) over $\mathbb{H}$ is
(2.10) $\\{\xi_{1},\ldots,\xi_{s}\\}\dot{\cup}_{\eta_{i}\in
T_{2}}\left\\{\omega_{i}\right\\}\dot{\cup}_{\eta_{i}\in T_{1}}[\eta_{i}],$
where $\omega_{i}$ takes
(2.11)
$\frac{1}{|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}}\\{|f_{2}(\eta_{i})|^{2}\eta_{i}+|f_{1}(\eta_{i})|^{2}\overline{\eta_{i}}-2f_{2}(\eta_{i})\overline{f_{1}(\eta_{i})}({\rm{Im}}\eta_{i})\mathbf{k}\\}$
as its value if $|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}\neq 0$, otherwise
takes
(2.12)
$\frac{1}{|f_{1}(\overline{\eta_{i}})|^{2}+|f_{2}(\overline{\eta_{i}})|^{2}}\\{|f_{1}(\overline{\eta_{i}})|^{2}\eta_{i}+|f_{2}(\overline{\eta_{i}})|^{2}\overline{\eta_{i}}+2f_{2}(\overline{\eta_{i}})\overline{f_{1}(\overline{\eta_{i}})}({\rm{Im}}\eta_{i})\mathbf{k}\\}$
as its value, in which $\operatorname{Im}\eta_{i}$ means the imaginary part
(real number) of $\eta_{i}$. Moreover, the union of the first two parts in the
above set is the set of isolated solutions and the third part of the above set
is the set of spherical solutions.
###### Proof.
To finish the proof we need to continue the argument on the case when (2.9) is
nontrivial, i.e., $\eta_{i}\in T_{2}$. For any $\eta_{i}\in T_{2}$, let
$\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\ \end{array}\right)$ be a
nonzero complex solution of the system
$\left(\begin{array}[]{cc}f_{2}(\overline{\eta_{i}})&f_{1}(\overline{\eta_{i}})\\\
\bar{f}_{1}(\overline{\eta_{i}})&-\bar{f}_{2}(\overline{\eta_{i}})\\\
\end{array}\right)X=0$ with $|a^{(i)}|^{2}+|b^{(i)}|^{2}=1$. Since the set of
nonzero solutions of (2.9) in unknown
$X=\left(\begin{array}[]{c}\overline{z_{1}}\\\ z_{2}\\\ \end{array}\right)$ is
$\left\\{l\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\ \end{array}\right)\mid
l\neq 0,l\in\mathbb{C}\right\\}$, the solutions of (2.3) corresponding to the
Jordan canonical form $\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\ \end{array}\right)$ are
$\left(\begin{array}[]{cc}\overline{la^{(i)}}&lb^{(i)}\\\
-\overline{lb^{(i)}}&la^{(i)}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\overline{la^{(i)}}&lb^{(i)}\\\
-\overline{lb^{(i)}}&la^{(i)}\\\ \end{array}\right)^{-1}$
$=\left(\begin{array}[]{cc}\overline{a^{(i)}}&b^{(i)}\\\
-\overline{b^{(i)}}&a^{(i)}\\\
\end{array}\right)\left(\begin{array}[]{cc}\eta_{i}&0\\\
0&\overline{\eta_{i}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\overline{a^{(i)}}&b^{(i)}\\\
-\overline{b^{(i)}}&a^{(i)}\\\ \end{array}\right)^{-1},$
which is actually one value, and in $\mathbb{H}$ which can be written as
$(\overline{a^{(i)}}+b^{(i)}\mathbf{j})\eta_{i}(\overline{a^{(i)}}+b^{(i)}\mathbf{j})^{-1}=(\overline{a^{(i)}}+b^{(i)}\mathbf{j})\eta_{i}({a^{(i)}}-b^{(i)}\mathbf{j})$
$={|a^{(i)}|^{2}\eta_{i}+|b^{(i)}|^{2}\overline{\eta_{i}}+\overline{a^{(i)}}b^{(i)}(\overline{\eta_{i}}-\eta_{i})\mathbf{j}},$
which can not be a real number. Therefore, the solution set of
$p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0$ in the skew-field $\mathbb{H}$ is
(2.13) $\\{\xi_{1},\ldots,\xi_{s}\\}\dot{\cup}_{\eta_{i}\in
T_{2}}\left\\{{|a^{(i)}|^{2}\eta_{i}+|b^{(i)}|^{2}\overline{\eta_{i}}-2\overline{a^{(i)}}b^{(i)}(\operatorname{Im}\eta_{i})\mathbf{k}}\right\\}\dot{\cup}_{\eta_{i}\in
T_{1}}[\eta_{i}].$
Note that $|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}$ and
$|f_{1}(\overline{\eta_{i}})|^{2}+|f_{2}(\overline{\eta_{i}})|^{2}$ can not be
$0$ simultaneously for $\eta_{i}\in T_{2}$, then we can take
$\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\
\end{array}\right)=\left(\begin{array}[]{c}\frac{\overline{f}_{2}(\overline{\eta_{i}})}{\sqrt{|\overline{f}_{1}(\overline{\eta_{i}})|^{2}+|\overline{f}_{2}(\overline{\eta_{i}})|^{2}}}\\\
\frac{\overline{f}_{1}(\overline{\eta_{i}})}{\sqrt{|\overline{f}_{1}(\overline{\eta_{i}})|^{2}+|\overline{f}_{2}(\overline{\eta_{i}})|^{2}}}\\\
\end{array}\right)$
for the case $|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}\neq 0$, otherwise we
take
$\left(\begin{array}[]{c}a^{(i)}\\\ b^{(i)}\\\
\end{array}\right)=\left(\begin{array}[]{c}\frac{{f}_{1}(\overline{\eta_{i}})}{\sqrt{|{f}_{1}(\overline{\eta_{i}})|^{2}+|{f}_{2}(\overline{\eta_{i}})|^{2}}}\\\
\frac{-{f}_{2}(\overline{\eta_{i}})}{\sqrt{|{f}_{1}(\overline{\eta_{i}})|^{2}+|{f}_{2}(\overline{\eta_{i}})|^{2}}}\\\
\end{array}\right).$
After manipulations, the set in (2.13) becomes
$\\{\xi_{1},\ldots,\xi_{s}\\}\dot{\cup}_{\eta_{i}\in
T_{2}}\left\\{\omega_{i}\right\\}\dot{\cup}_{\eta_{i}\in T_{1}}[\eta_{i}],$
where $\omega_{i}$ takes
$\frac{1}{|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}}\\{|f_{2}(\eta_{i})|^{2}\eta_{i}+|f_{1}(\eta_{i})|^{2}\overline{\eta_{i}}-2f_{2}(\eta_{i})\overline{f_{1}(\eta_{i})}({\rm{Im}}\eta_{i})\mathbf{k}\\}$
as its value if $|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}\neq 0$; otherwise
it takes
$\frac{1}{|f_{1}(\overline{\eta_{i}})|^{2}+|f_{2}(\overline{\eta_{i}})|^{2}}\\{|f_{1}(\overline{\eta_{i}})|^{2}\eta_{i}+|f_{2}(\overline{\eta_{i}})|^{2}\overline{\eta_{i}}+2f_{2}(\overline{\eta_{i}})\overline{f_{1}(\overline{\eta_{i}})}({\rm{Im}}\eta_{i})\mathbf{k}\\}$
as its value.
Finally, it is clear that $[\eta_{i}]$ contains no real numbers for
$\eta_{i}\in T_{1}$. This completes the proof. ∎
Note that there is no repetition in the solution set given in (2.10) and one
can use (2.11) or (2.12) for $\omega_{i}$ if
$(|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2})(|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2})\neq
0.$
Theorem 1 shows, once we get a decomposition with the form (2.5) of the
discriminant polynomial $\tilde{p}(t)$, then we can produce all roots of the
quaternionic equation (2.2) by formula (2.10).
From Theorem 1 and the arguments before Theorem 1, we can easily see the
following well-known results.
###### Corollary 2.
1. (a).
Any simple quaternionic equation $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0(q_{n}\neq
0)$ has a root in $\mathbb{H}$.
2. (b).
The simple quaternionic equation $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0(q_{n}\neq
0)$ has a finite number of roots in $\mathbb{H}$ iff it has at most $n$
distinct roots in $\mathbb{H}$.
3. (c).
The roots of $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0(q_{n}\neq 0)$ are distributed
in at most $n$ conjugate classes, and there are at most $n$ real roots among
them.
###### Proof.
(a) is obvious.
(b) From Theorem 1 we need to only show that $[\eta_{i}]$ is an infinite set
if $\eta_{i}\in T_{1}$. It is well-known from [21] that
$u_{1},u_{2}\in\mathbb{H}$ are conjugate (i.e., there exists nonzero
$q\in\mathbb{H}$ such that $u_{1}=qu_{2}q^{-1}$) iff
$\operatorname{Re}u_{1}=\operatorname{Re}u_{2}$ and $|u_{1}|=|u_{2}|$. Since
$\eta_{i}\in T_{1}$ has a nonzero imaginary part, $[\eta_{i}]$ is an infinite
set. Thus $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$ has a finite number of roots in
$\mathbb{H}$ iff $T_{1}=\varnothing$, iff $p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0$
has at most $n$ roots in $\mathbb{H}$ since $s+k\leq n$ (See (2.5) for the
notations).
(c) follows from $s+k\leq n$. ∎
Now we give a quick method to solve simple quaternionic polynomials with all
real coefficients or with all complex coefficients. These results do not seem
to be deduced easily from the results in [9].
###### Corollary 3.
1. (a).
If all $q_{i}$ in $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0(q_{n}\neq 0)$ are real
numbers and the solution set of this equation in $\mathbb{C}$ is
$\\{\xi_{1},\ldots,\xi_{s},\zeta_{1},\overline{\zeta_{1}},$
$\ldots,\zeta_{t},\overline{\zeta_{t}}\\}$, where $\xi_{1},\ldots,\xi_{s}$ are
distinct real numbers, $\zeta_{1},\ldots,\zeta_{t}$ are distinct nonreal
complex numbers, then the solution set of this equation in $\mathbb{H}$ is
$\\{\xi_{1},\ldots,\xi_{s},\\}\cup[\zeta_{1}]\cup\cdots\cup[\zeta_{t}].$
2. (b).
More generally, if all $q_{i}$ in $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0(q_{n}\neq
0)$ are complex numbers, and the solution set of this equation in $\mathbb{C}$
is
$\\{\xi_{1},\ldots,\xi_{s},\eta_{1},\ldots,\eta_{k},\zeta_{1},\overline{\zeta_{1}},\ldots,\zeta_{t},\overline{\zeta_{t}}\\}$,
where $\xi_{1},\ldots,\xi_{s}$ are distinct real numbers,
$\eta_{1},\ldots,\eta_{k},\zeta_{1},\ldots,\zeta_{t}$ are distinct nonreal
complex numbers (each $\overline{\eta_{i}}$ is no longer the root of this
equation), then the solution set of this equation in $\mathbb{H}$ is
$\\{\xi_{1},\ldots,\xi_{s},\eta_{1},\ldots,\eta_{k},\\}\cup[\zeta_{1}]\cup\cdots\cup[\zeta_{t}].$
3. (c).
Let $f_{1}(t)$, $f_{2}(t)$, $\bar{f}_{1}(t)$ and $\bar{f}_{2}(t)$ be the
derived polynomials for (2.2). Then (2.2) has finitely many solutions iff the
complex polynomial $\gcd(f_{1}(t),f_{2}(t),\bar{f}_{1}(t),\bar{f}_{2}(t))$ has
no nonreal complex root iff the complex polynomial $\gcd(f_{1}(t),f_{2}(t))$
has no nonreal conjugate complex roots.
###### Proof.
(a) This is a special case of Part (b).
(b) When the equation considered has only complex coefficients, two of the
derived polynomials are $f_{1}(t)=q_{n}t^{n}+\cdots+q_{1}t+q_{0}$ (up to a
complex scalar), and $f_{2}(t)=0$. So, the roots of the discriminant
polynomial are
$\xi_{1},\ldots,\xi_{s},\eta_{1},\overline{\eta_{1}},\ldots,\eta_{k},\overline{\eta_{k}},\zeta_{1},\overline{\zeta_{1}},\ldots,\zeta_{t},\overline{\zeta_{t}}$.
For each $\eta_{i}\,(i=1,\ldots,k)$, since $\overline{f}_{1}(\eta_{i})\neq 0$,
$\eta_{i}$ is in $T_{2}$. We can take $\left(\begin{array}[]{c}a^{(i)}\\\
b^{(i)}\\\ \end{array}\right)=\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)$ as the unit complex solution of the system
$\left(\begin{array}[]{cc}f_{2}(\overline{\eta_{i}})&f_{1}(\overline{\eta_{i}})\\\
\bar{f}_{1}(\overline{\eta_{i}})&-\bar{f}_{2}(\overline{\eta_{i}})\\\
\end{array}\right)X=0$. Then,
$|a^{(i)}|^{2}\eta_{i}+|b^{(i)}|^{2}\overline{\eta_{i}}-2b^{(i)}\overline{a^{(i)}}(\operatorname{Im}\eta_{i})\mathbf{k}=\eta_{i}$.
It is easy to see that $\zeta_{i}\in T_{1}$. This completes the proof.
(c) Suppose $c$ is a nonreal complex root of
gcd$(f_{1}(t),f_{2}(t),\bar{f}_{1}(t),\bar{f}_{2}(t))$. Then both $c$ and
$\bar{c}$ are roots of the discriminant polynomial,
$f_{1}(c)=f_{2}(c)=\bar{f}_{1}(c)=\bar{f}_{2}(c)=0$ and
$f_{1}(\bar{c})=f_{2}(\bar{c})=\bar{f}_{1}(\bar{c})=\bar{f}_{2}(\bar{c})=0$,
which implies that at least one of $c$, $\bar{c}$ is in $T_{1}$. Thus,
$T_{1}=\varnothing$ iff gcd$(f_{1}(t),f_{2}(t),\bar{f}_{1}(t),\bar{f}_{2}(t))$
has no nonreal complex root. From Theorem 1 we see that (2.2) has finitely
many solutions iff $T_{1}=\varnothing$. The conclusions in the corollary
follow easily. ∎
Now we can give a simplified version of Theorem 1.
###### Theorem 4.
Consider the simple quaternionic equation
$p(x):=p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0,$ where $p_{i}\in\mathbb{H}$ with
$p_{n}\neq 0$ and $d_{0}=0\ {\text{or}}\ 1$. We write
$p(t)=g(t)(g_{1}(t)+g_{2}(t)\mathbf{j})$ where $t$ is considered as a real
variable, $g,g_{1},g_{2}\in\mathbb{C}[t]$ with $\gcd(g_{1},g_{2})=1$. Let the
complex solution sets for $g(t)$ and
$\tilde{g}(t)=g_{1}(t)\bar{g}_{1}(t)+g_{2}(t)\bar{g}_{2}(t)$ are
(2.14)
$\\{\xi_{1},\ldots,\xi_{s};\lambda_{1},\bar{\lambda}_{1},\lambda_{2},\bar{\lambda}_{2},\cdots,\lambda_{t},\bar{\lambda}_{t};\eta_{1},\ldots,\eta_{k_{1}}\\},$
(2.15)
$\\{\eta_{k_{1}+1},\bar{\eta}_{k_{1}+1}\ldots,\eta_{k},\bar{\eta}_{k}\\}$
respectively. We may assume that $\xi_{1},\ldots,\xi_{s}$ are distinct real
numbers; $\eta_{1},\ldots,$
$\eta_{k_{1}},\eta_{k_{1}+1},\bar{\eta}_{k_{1}+1}\ldots,\eta_{k},\bar{\eta}_{k},\lambda_{1},\bar{\lambda}_{1},\lambda_{2},\bar{\lambda}_{2},\cdots,\lambda_{t},\bar{\lambda}_{t}$
are distinct nonreal complex numbers after deleting those $\eta_{i}$’s and
$\bar{\eta}_{i}$’s in (2.15) if one of them appears in the set (2.14). Then
the quaternionic solution set of $p(x)$ is
(2.16)
$\\{\xi_{1},\ldots,\xi_{s},;\omega_{1},\cdots,\omega_{k}\\}\dot{\cup}_{i=1}^{t}[\lambda_{i}],$
where
$\omega_{i}=\frac{1}{|g_{1}(\bar{\eta}_{i})|^{2}+|g_{2}(\bar{\eta}_{i})|^{2}}\\{|g_{2}(\bar{\eta}_{i})|^{2}\bar{\eta}_{i}+|g_{1}(\bar{\eta}_{i})|^{2}{\eta_{i}}+2g_{2}(\bar{\eta}_{i})\overline{g_{1}(\bar{\eta}_{i})}({\rm{Im}}\eta_{i})\mathbf{k}\\}.$
###### Proof.
We continue to use the notations in Theorem 1. We know that
$f_{1}(t)=g(t)g_{1}(t)$, $f_{2}(t)=g(t)g_{2}(t)$,
$\tilde{p}(t)=g(t)\bar{g}(t)\tilde{g}(t)$, and $\tilde{g}(t)$ has no real
root.
From Theorem 1 we see that
$\\{\xi_{1},\ldots,\xi_{s}\\}\dot{\cup}_{i=1}^{t_{1}}[\lambda_{i}]$ are zeros
of $p(x)$. All other zeros come from
$\\{\eta_{1},\ldots,\eta_{k_{1}},\eta_{k_{1}+1},\bar{\eta}_{k_{1}+1}\ldots,\eta_{k},\bar{\eta}_{k}\\}$.
For each $\eta_{i}$, we see that
$|f_{1}(\bar{\eta}_{i})|^{2}+|f_{2}(\bar{\eta}_{i})|^{2}\neq 0$. Then using
(2.12) in Theorem 1 and simplifying we obtain $\omega_{i}$. This completes the
proof. ∎
Remark that the above theorem simplifies the computation for finding all zeros
of $p(x)$, and the following known result (see [16]) can follow easily from
the above theorem.
###### Corollary 5.
The spherical zeros of simple quaternionic polynomial
$p(x):=p_{n}x^{n}+\cdots+p_{1}x+p_{0}(p_{i}\in\mathbb{H},p_{n}\neq 0)$ are
distributed in at most INT$(\frac{n}{2})$ conjugate classes, where
INT$(\frac{n}{2})$ means the integral function value at $\frac{n}{2}$.
## 3\. Algorithm
Based on our Theorem 1, we now can give an algorithm to solve the quaternionic
equation $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0\,(q_{n}\neq 0)$, as follows.
Algorithm 1 (for solving the simple quaternionic polynomial equation
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$)
Step 1. Write the equation as $p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0$ with
$d_{0}=0$ or $1$ (in fact, if $q_{0}\neq 0$, simply multiply the equation by
$q_{0}^{-1}$ on the left). Write $p_{i}=t_{1}^{(i)}+t_{2}^{(i)}\mathbf{j}$ for
$i=1,\ldots,n$ with $t_{1}^{(i)},t_{2}^{(i)}\in\mathbb{C}$. Find the derived
polynomials and discriminant polynomial of $p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0$:
$\begin{array}[]{cc}f_{1}(t)=t_{1}^{(n)}t^{n}+\cdots+t_{1}^{(1)}t+d_{0},&f_{2}(t)=t_{2}^{(n)}t^{n}+\cdots+t_{2}^{(1)}t;\\\
\bar{f}_{1}(t)=\overline{t_{1}^{(n)}}t^{n}+\cdots+\overline{t_{1}^{(1)}}t+d_{0},&\bar{f}_{2}(t)=\overline{t_{2}^{(n)}}t^{n}+\cdots+\overline{t_{2}^{(1)}}t,\end{array}$
$\tilde{p}(t)=f_{1}(t)\bar{f}_{1}(t)+f_{2}(t)\bar{f}_{2}(t).\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Make sure the coefficients of $\tilde{p}(t)$ are real.
Step 2. Compute all distinct zeros (real or complex) of the discriminant
polynomial $\tilde{p}(t)$ (in MATLAB, use the command roots). Denote these
zeros by $\xi_{1},\ldots,\xi_{s}$, $\eta_{1},\ldots$, $\eta_{k}$,
$\overline{\eta_{1}},\ldots$, $\overline{\eta}_{k}$ such that
$\xi_{1},\ldots,\xi_{s}$ are distinct real numbers and $\eta_{1},\ldots$,
$\eta_{k}$ are distinct nonreal complex numbers. Then classify
$\eta_{1},\ldots,\eta_{k}$ into two sets $T_{1}$ and $T_{2}$:
$T_{1}=\\{\eta\in\\{\eta_{1},\ldots,\eta_{k}\\}\,|\,f_{1}(\eta)=f_{2}(\eta)=\bar{f}_{1}(\eta)=\bar{f}_{2}(\eta)=0\\}$,
$T_{2}=\\{\eta_{1},\ldots,\eta_{k}\\}\backslash T_{1}$.
When the simple quaternionic polynomial considered is a polynomial with real
coefficients, then directly set $T_{1}=\\{\eta_{1},\ldots,\eta_{k}\\}$ and
$T_{2}=\varnothing$.
Step 3. For each $\eta_{i}\in T_{2}$, compute $f_{1}(\eta_{i})$,
$f_{2}(\eta_{i})$. Then by Formula (2.10), output all roots of
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$.
If we use our Theorem 4, then we get a better version of Algorithm 1\.
Algorithm 1’ (for solving the simple quaternionic polynomial equation
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$)
Step 1. Write the equation as $p(x):=p_{n}x^{n}+\cdots+p_{1}x+d_{0}=0$ with
$d_{0}=0$ or $1$ (in fact, if $q_{0}\neq 0$, simply multiply the equation by
$q_{0}^{-1}$ on the left). Then write $p(t)=g(t)(g_{1}(t)+g_{2}(t)\mathbf{j})$
where $t$ is considered as a real variable, $g,g_{1},g_{2}\in\mathbb{C}[t]$
with $\gcd(g_{1},g_{2})=1$. Now we compute
$\tilde{g}(t)=g_{1}(t)\bar{g}_{1}(t)+g_{2}(t)\bar{g}_{2}(t)$.
Step 2. Compute all distinct zeros (real or complex) for $g(t)$ and
$\tilde{g}(t)$ respectively (in MATLAB, use the command roots):
(3.1)
$\\{\xi_{1},\ldots,\xi_{s};\lambda_{1},\bar{\lambda}_{1},\lambda_{2},\bar{\lambda}_{2},\cdots,\lambda_{t},\bar{\lambda}_{t};\eta_{1},\ldots,\eta_{k_{1}}\\},$
(3.2)
$\\{\eta_{k_{1}+1},\bar{\eta}_{k_{1}+1}\ldots,\eta_{k},\bar{\eta}_{k}\\},$
where $\xi_{1},\ldots,\xi_{s}$ are distinct real numbers;
$\\{\lambda_{1},\bar{\lambda}_{1},\lambda_{2},\bar{\lambda}_{2},\cdots,\lambda_{t},\bar{\lambda}_{t}\\}$
and
$\\{\eta_{1},\ldots,\eta_{k_{1}},\eta_{k_{1}+1},\bar{\eta}_{k_{1}+1}\ldots,\eta_{k},\bar{\eta}_{k}\\}$
are two sets of distinct nonreal complex numbers. Delete those $\eta_{i}$’s
and $\bar{\eta}_{i}$’s in (3.2) if one of them appears in the set (3.1).
Step 3. For each $\eta_{i}$, we compute
(3.3)
$\omega_{i}=\frac{1}{|g_{1}(\bar{\eta}_{i})|^{2}+|g_{2}(\bar{\eta}_{i})|^{2}}\\{|g_{2}(\bar{\eta}_{i})|^{2}\bar{\eta}_{i}+|g_{1}(\bar{\eta}_{i})|^{2}{\eta_{i}}+2g_{2}(\bar{\eta}_{i})\overline{g_{1}(\bar{\eta}_{i})}({\rm{Im}}\eta_{i})\mathbf{k}\\}.$
Then the quaternionic solution set of $p(x)$ is
(3.4)
$\\{\xi_{1},\ldots,\xi_{s},;\omega_{1},\cdots,\omega_{k}\\}\dot{\cup}_{i=1}^{t}[\lambda_{i}].$
Prior to our method, Janovsk$\acute{a}$ and Opfer gave an algorithm in [9] for
solving the same quaternionis equation
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0\,(q_{n}\neq 0)$. In order to introduce their
algorithm precisely, let us first recall the known concept, companion
polynomial. For
$p(x)=\sum_{j=0}^{n}q_{j}x^{j},q_{j}\in\mathbb{H},j=0,\ldots,n,\
q_{0},q_{n}\neq 0,$
following Niven [14], or more recently Janovsk$\acute{a}$ and Opfer [9], its
companion polynomial is defined by
(3.5)
$q_{2n}(x)=\sum_{j,k=0}^{n}\bar{q}_{j}q_{k}x^{j+k}=\sum_{k=0}^{2n}b_{k}x^{k},\textrm{where}\
b_{k}=\sum_{j=max(0,k-n)}^{min(k,n)}\bar{q}_{j}q_{k-j}\in\mathbb{R}.$
We remark that $q_{2n}$ is equal to the discriminant $\tilde{p}(t)$ in our
case. From Pogorui and Shapiro[16], we know that all powers $x^{j}$,
$j\in\mathbb{Z}$ of a quaternion $x$ have the form $x^{j}=\alpha x+\beta$ with
real $\alpha$, $\beta$. In particular, $x^{2}=2({\rm{Re}}x)\ x-|x|^{2}$. In
order to determine the numbers $\alpha$, $\beta$, Janovsk$\acute{a}$ and Opfer
in [9] set up the following iteration:
(3.6) $\left\\{\begin{array}[]{c}x^{j}=\alpha_{j}x+\beta_{j},\
\alpha_{j},\beta_{j}\in\mathbb{R},j=0,1,\ldots,\\\
\alpha_{0}=0,\beta_{0}=1,\\\ \alpha_{j+1}=2Rex\ \alpha_{j}+\beta_{j},\\\
\beta_{j+1}=-|x|^{2}\alpha_{j},j=0,1,\dots.\end{array}\right.$
Now by means of the first line of Iteration (3.6), the polynomial $p(x)$ can
be rewritten as
$p(x)=\sum_{j=0}^{n}q_{j}(\alpha_{j}x+\beta_{j})=\left(\sum_{j=0}^{n}q_{j}\alpha_{j}\right)x+\left(\sum_{j=0}^{n}q_{j}\beta_{j}\right)\equiv
A(x)x+B(x),$
where
(3.7) $A(x)=\sum_{j=0}^{n}q_{j}\alpha_{j},\
B(x)=\sum_{j=0}^{n}q_{j}\beta_{j}.$
With these in hand, we can state the algorithm given in [9], as follows.
Algorithm 2 (for solving the simple quaternionic equation
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$)
Step 1. Write $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$ as
$p(x)=a_{n}x^{n}+\cdots+a_{1}x+a_{0}=0$ with $a_{n}=1$. For this $p(x)$,
compute the real coefficients $b_{0}$, $b_{1}$, $\ldots$, $b_{2n}$ of the
companion polynomial $q_{2n}(x)$ by formula (3.5).
Step 2. Compute all $2n$ (real and complex) zeros of $q_{2n}(x)$, denote these
zeros by $z_{1}$, $z_{2}$, $\ldots$, $z_{2n}$ and order them (if necessary)
such that $z_{2j-1}=\overline{z_{2j}}$, $j=1$, $2$, $\ldots$, $n$.
Step 3. Define an integer vector $\mathbf{ind}$ (like indicator) of length
$n$, and set all components to zero. Define a quaternionic vector $Z$ of
length $n$, and set all components to zero.
$\mathbf{For}$ $\mathbf{j:=1:n}$ $\mathbf{do}$
(a) $\mathbf{Put}$ $z:=z_{2j-1}$.
(b) $\mathbf{if}$ $z$ is real, $Z(j):=z$; go to the next step; $\mathbf{end}$
$\mathbf{if}$
(c) $\mathbf{Compute}$ $v:=\overline{A(z)}B(z)$ by formula (3.7), with the
help of Iteration (3.6).
(d) $\mathbf{if}$ $v=0$, put $\mathbf{ind}(j):=1;Z(j):=z$; go to the next
step; $\mathbf{end}$ $\mathbf{if}$
(e) $\mathbf{if}$ $v\neq 0$, let $(v_{1},v_{2},v_{3},v_{4}):=v$. Compute
$|w|:=\sqrt{v_{2}^{2}+v_{3}^{2}+v_{4}^{2}}$, and put
(3.8)
$Z(j):=\left(\textrm{Re}z,-\frac{|\textrm{Im}z|}{|w|}v_{2},-\frac{|\textrm{Im}z|}{|w|}v_{3},-\frac{|\textrm{Im}z|}{|w|}v_{4}\right).$
$\mathbf{end}$ $\mathbf{if}$
$\mathbf{end}$ $\mathbf{for}$
In this algorithm, corresponding to a real $z$ the expression $Z(j)$ produces
a real isolated zero $z$, corresponding to “$v=0$” the expression $Z(j)$
produces a spherical zero $[z]$, and corresponding to “$v\neq 0$” the
expression $Z(j)$ produces an isolated zero
$\textrm{Re}z-\frac{|\textrm{Im}z|}{|w|}v_{2}\mathbf{i}-\frac{|\textrm{Im}z|}{|w|}v_{3}\mathbf{j}-\frac{|\textrm{Im}z|}{|w|}v_{4}\mathbf{k}$.
The output results of $Z(j)$ produce all zeros of polynomial
$q_{n}x^{n}+\cdots+q_{1}x+q_{0}$. Algorithm 2’s original edition is Section 7
of [9], where $|w|:=\sqrt{v_{2}^{2}+v_{3}^{3}+v_{4}^{2}}$ is false, which is a
misprint.
## 4\. Examples
Example 1 In $\mathbb{H}$, solve the equation
$p(x):=\mathbf{i}x^{3}+\mathbf{j}x^{2}+\mathbf{k}x+1=0$.
Solution 1. Use our Algorithm 1 to do this. Write the coefficients
$\mathbf{i},\mathbf{j},\mathbf{k}$ into the form:
$\mathbf{i}=\mathbf{i}+0\mathbf{j},\ \mathbf{j}=0+1\mathbf{j},\
\mathbf{k}=0+\mathbf{i}\mathbf{j}$. Then the derived polynomials of this
equation are
$\begin{array}[]{cc}f_{1}(t)=\mathbf{i}t^{3}+1,&f_{2}(t)=t^{2}+\mathbf{i}t,\\\
\bar{f}_{1}(t)=-\mathbf{i}t^{3}+1,&\bar{f}_{2}(t)=t^{2}-\mathbf{i}t,\end{array}$
the discriminant polynomial is
$\tilde{p}(t)=f_{1}\bar{f}_{1}+f_{2}\bar{f}_{2}=(t-\mathbf{i})(t+\mathbf{i})(t-e^{\mathbf{i}\frac{\pi}{4}})(t-e^{\mathbf{i}\frac{3\pi}{4}})(t-e^{\mathbf{i}\frac{5\pi}{4}})(t-e^{i\frac{7\pi}{4}}),$
which has no real root. It is easy to see that
$T_{1}=\varnothing\ \textrm{and}\
T_{2}=\\{\eta_{1}=\mathbf{i},\eta_{2}=e^{\mathbf{i}\frac{\pi}{4}},\eta_{3}=e^{\mathbf{i}\frac{3\pi}{4}}\\}.$
For $\eta_{1}=\mathbf{i}$, then $f_{1}(\mathbf{i})=2\neq 0$ and
$f_{2}(\mathbf{i})=-2$, and we get an isolated zero by Formula (2.10):
$\frac{1}{8}\cdot(4\mathbf{i}-4\mathbf{i}-2\cdot(-2)\cdot 2\cdot
1\cdot\mathbf{k})=\mathbf{k}$;
For
$\eta_{2}=e^{\mathbf{i}\frac{\pi}{4}}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\mathbf{i}$,
then $f_{1}(\eta_{2})=\frac{2-\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathbf{i}$ and
$f_{2}(\eta_{2})=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}+2}{2}\mathbf{i}$, and we
get another isolated zero by Formula (2.10):
$\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k}$;
Similarly for $\eta_{3}=e^{\mathbf{i}\frac{3\pi}{4}}$ we get the isolated
zero: $\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k}$.
Thus, the solution set is
$\left\\{\mathbf{k},\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k},-\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k}\right\\}$.
Solution 2. Use our Algorithm 2 to do this. Write
$p(t)=(\mathbf{i}t^{3}+1)+(t^{2}+\mathbf{i}t)\mathbf{j}:=f_{1}+f_{2}\mathbf{j}$,
then
$g=\gcd(f_{1},f_{2})=t+\mathbf{i}$, $g_{1}=\mathbf{i}t^{2}+t-\mathbf{i}$,
$g_{2}=t$, $\tilde{g}=g_{1}\bar{g}_{1}+g_{2}\bar{g}_{2}=t^{4}+1$,
Compute the zeros of $g$: $\eta_{1}=-\mathbf{i}$.
Compute the zeros of $\tilde{g}$: $\eta_{2}=e^{\mathbf{i}\frac{\pi}{4}}$,
$\overline{\eta_{2}}$, $\eta_{3}=e^{\mathbf{i}\frac{3\pi}{4}}$,
$\overline{\eta_{3}}$.
Now for $\eta_{i}$($i=1,2,3$), compute $g_{1}(\overline{\eta_{i}})$,
$|g_{1}(\overline{\eta_{i}})|^{2}$, $g_{2}(\overline{\eta_{i}})$, and
$|g_{2}(\overline{\eta_{i}})|^{2}$, by formula (3.3) we get the solution set
of $\mathbf{i}x^{3}+\mathbf{j}x^{2}+\mathbf{k}x+1=0$:
$\left\\{\mathbf{k},\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k},-\frac{\sqrt{2}}{2}+\frac{1}{2}\mathbf{i}+\frac{1}{2}\mathbf{k}\right\\}.$
Example 2 Solve the equation $x^{3}+x^{2}+x+1=0$ in $\mathbb{H}$.
Solution 1. We will use our method to do this first. Since
$x^{3}+x^{2}+x+1=(x+1)(x+\mathbf{i})(x-\mathbf{i})$, from Corollary 3(a), we
directly know the solution set is $\\{-1\\}\cup[\mathbf{i}]$.
Solution 2. Now we use the method in [9] to do this.
First Step. By formula (3.5), we get
$q_{6}(x)=(x^{3}+x^{2}+x+1)^{2}=x^{6}+2x^{5}+3x^{4}+4x^{3}+3x^{2}+2x+1.$
Second Step. Compute all zeros of $q_{6}(x)$:
$-1,-1,\mathbf{i},\mathbf{i},-\mathbf{i},-\mathbf{i}.$
Third Step. For $-1$ we get a real isolated zero $-1$. For $\mathbf{i}$ we
need give the following expansion by Iteration (3.6):
$\mathbf{i}^{0}=0\mathbf{i}+1,\mathbf{i}^{1}=1\mathbf{i}+0,\mathbf{i}^{2}=0\mathbf{i}+(-1),\mathbf{i}^{3}=(-1)\mathbf{i}+0.$
Then by formula (3.7) get
$A(\mathbf{i})=0+1+0+(-1)=0,B(\mathbf{i})=1+0+(-1)+0=0,$
and $v:=\overline{A(\mathbf{i})}B(\mathbf{i})=0.$ It leads to a spherical zero
$[\mathbf{i}]$. Finally, for $-\mathbf{i}$ we have to repeat the same process
as that made for $\mathbf{i}$:
$(-\mathbf{i})^{0}=0(-\mathbf{i})+1,(-\mathbf{i})^{1}=1(-\mathbf{i})+0,$
$(-\mathbf{i})^{2}=0(-\mathbf{i})+(-1),(-\mathbf{i})^{3}=-1(-\mathbf{i})+0,$
$A(-\mathbf{i})=0+1+0+(-1)=0,B(-\mathbf{i})=1+0+(-1)+0=0,$
and $v:=\overline{A(-\mathbf{i})}B(-\mathbf{i})=0$. It also produces a
spherical zero $[-\mathbf{i}]$. Note that $[-\mathbf{i}]=[\mathbf{i}]$, so the
solution set is $\\{-1\\}\cup[\mathbf{i}]$.
At last let us solve the same polynomial in Example 3.8 of [9].
Example 3. Find all zeros of
$p(z)=z^{6}+\mathbf{j}z^{5}+\mathbf{i}z^{4}-z^{2}-\mathbf{j}z-\mathbf{i}.$
Solution 1. We first use the method in this paper to do this.
We have
$\mathbf{i}p(z)=\mathbf{i}z^{6}+\mathbf{i}\mathbf{j}z^{5}-z^{4}-\mathbf{i}z^{2}-\mathbf{i}\mathbf{j}z+1.$
Then
$\mathbf{i}p(t)=(\mathbf{i}t^{6}-t^{4}-\mathbf{i}t^{2}+1)+(\mathbf{i}t^{5}-\mathbf{i}t)\mathbf{j},$
$g=t^{4}-1,\,\,\,g_{1}=\mathbf{i}t^{2}-1,\,\,\,g_{2}=\mathbf{i}t,$
$\tilde{g}=g_{1}\bar{g}_{1}+g_{2}\bar{g}_{2}=(\mathbf{i}t^{2}-1)(-\mathbf{i}t^{2}-1)+\mathbf{i}t\cdot(-\mathbf{i}t)=t^{4}+t^{2}+1.$
The all distinct zeros (real or complex) for $g$ and $\tilde{g}$ are
respectively $\\{1,-1,\mathbf{i},-\mathbf{i}\\}$ and
$\\{e^{-\mathbf{i}\frac{\pi}{3}},e^{\mathbf{i}\frac{\pi}{3}},e^{-\mathbf{i}\frac{2\pi}{3}},e^{\mathbf{i}\frac{2\pi}{3}}\\}$.
Now we take
$\eta_{1}=e^{-\mathbf{i}\frac{\pi}{3}}=1/2-\sqrt{-3}/2$
$\eta_{2}=e^{-\mathbf{i}\frac{2\pi}{3}}=-1/2-\sqrt{-3}/2$,
$g_{1}(\bar{\eta}_{1})=\mathbf{i}e^{\mathbf{i}\frac{2\pi}{3}}-1=-1-\frac{\sqrt{3}}{2}-\frac{1}{2}\mathbf{i}$,
$|g_{1}(\bar{\eta}_{1})|^{2}=2+\sqrt{3}$;
$g_{2}(\bar{\eta}_{1})=\mathbf{i}e^{\mathbf{i}\frac{\pi}{3}}=-\frac{\sqrt{3}}{2}+\frac{1}{2}\mathbf{i}$,
$|g_{2}(\bar{\eta}_{1})|^{2}=1$;
$g_{1}(\bar{\eta}_{2})=\mathbf{i}e^{\mathbf{i}\frac{4\pi}{3}}-1=-1+\frac{\sqrt{3}}{2}-\frac{1}{2}\mathbf{i}$,
$|g_{1}(\bar{\eta}_{2})|^{2}=2-\sqrt{3}$;
$g_{2}(\bar{\eta}_{2})=\mathbf{i}e^{\mathbf{i}\frac{2\pi}{3}}=-\frac{\sqrt{3}}{2}-\frac{1}{2}\mathbf{i}$,
$|g_{2}(\bar{\eta}_{1})|^{2}=1$.
So by formula (3.3), we get
$\displaystyle\omega_{1}=$
$\displaystyle\frac{1}{3+\sqrt{3}}\\{(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(2+\sqrt{3})(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})$
$\displaystyle\hskip
28.45274pt+2(-\frac{\sqrt{3}}{2}+\frac{1}{2}\mathbf{i})(-1-\frac{\sqrt{3}}{2}+\frac{1}{2}\mathbf{i})(-\frac{\sqrt{3}}{2})\mathbf{k}\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{2}\mathbf{i}-\frac{1}{2}\mathbf{j}-\frac{1}{2}\mathbf{k}$
$\displaystyle\omega_{2}=$
$\displaystyle\frac{1}{3-\sqrt{3}}\\{(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(2-\sqrt{3})(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})$
$\displaystyle\hskip
28.45274pt+2(-\frac{\sqrt{3}}{2}-\frac{1}{2}\mathbf{i})(-1+\frac{\sqrt{3}}{2}+\frac{1}{2}\mathbf{i})(-\frac{\sqrt{3}}{2})\mathbf{k}\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{1}{2}\mathbf{i}-\frac{1}{2}\mathbf{j}-\frac{1}{2}\mathbf{k}.$
Hence, the set of roots of $p(z)$ is
$\\{1,-1\\}\cup\\{\frac{1}{2}-\frac{1}{2}\mathbf{i}-\frac{1}{2}\mathbf{j}-\frac{1}{2}\mathbf{k},-\frac{1}{2}+\frac{1}{2}\mathbf{i}-\frac{1}{2}\mathbf{j}-\frac{1}{2}\mathbf{k}\\}\cup[\mathbf{i}].$
Solution 2. We use the method in [9] to redo this.
Step 1 Compute the companion polynomial $q_{12}(x)$ by formula (3.5):
$\displaystyle q_{12}(x)=$ $\displaystyle
x^{12}+\mathbf{j}x^{11}+\mathbf{i}x^{10}-x^{8}-\mathbf{j}x^{7}-\mathbf{i}x^{6}$
$\displaystyle-\mathbf{j}x^{11}+x^{10}-\mathbf{ji}x^{9}+\mathbf{j}x^{7}-x^{6}+\mathbf{ji}x^{5}$
$\displaystyle-\mathbf{i}x^{10}-\mathbf{k}x^{9}+x^{8}+\mathbf{i}x^{6}+\mathbf{k}x^{5}-x^{4}$
$\displaystyle-x^{8}-\mathbf{j}x^{7}-\mathbf{i}x^{6}+x^{4}+\mathbf{j}x^{3}+\mathbf{i}x^{2}$
$\displaystyle+\mathbf{j}x^{7}-x^{6}+\mathbf{ji}x^{5}-\mathbf{j}x^{3}+x^{2}-\mathbf{ji}x$
$\displaystyle+\mathbf{i}x^{6}+\mathbf{k}x^{5}-x^{4}-\mathbf{i}x^{2}-\mathbf{k}x+1$
$\displaystyle=$ $\displaystyle
x^{12}+0+x^{10}+0-x^{8}+0-2x^{6}+0-x^{4}+0+x^{2}+1$ $\displaystyle=$
$\displaystyle x^{12}+x^{10}-x^{8}-2x^{6}-x^{4}+x^{2}+1.$
Step 2 Compute the zeros of $q_{12}(x)$.
$\displaystyle q_{12}(x)=$ $\displaystyle
x^{8}(x^{4}+x^{2}-1)-x^{6}-x^{4}+x^{2}-x^{6}+1$ $\displaystyle=$
$\displaystyle x^{8}(x^{4}+x^{2}-1)-x^{2}(x^{4}+x^{2}-1)-x^{6}+1$
$\displaystyle=$ $\displaystyle(x^{4}+x^{2}-1)(x^{8}-x^{2})-(x^{6}-1)$
$\displaystyle=$ $\displaystyle(x^{4}-1)(x^{2}+1)(x^{6}-1).$
The $12$ zeros of $q_{12}$ are
$1,1,-1,-1,\mathbf{i},\mathbf{i},-\mathbf{i},-\mathbf{i},\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i},\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i},-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i},-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}.$
Step 3 For $1$ and $-1$, we get isolated zeros: $1$, $-1$.
For $x=\mathbf{i},-\mathbf{i}$, then
$\begin{array}[]{ll}\mathbf{i}^{0}=0\mathbf{i}+1&(-\mathbf{i})^{0}=0(-i)+1\\\
\mathbf{i}^{1}=1\mathbf{i}+0&(-\mathbf{i})^{1}=1(-\mathbf{i})+0\\\
\mathbf{i}^{2}=0\mathbf{i}+(-1)&(-\mathbf{i})^{2}=0(-\mathbf{i})+(-1)\\\
\mathbf{i}^{3}=(-1)\mathbf{i}+0&(-\mathbf{i})^{3}=(-1)(-\mathbf{i})+0\\\
\mathbf{i}^{4}=0\mathbf{i}+1&(-\mathbf{i})^{4}=0(-\mathbf{i})+1\\\
\mathbf{i}^{5}=1\mathbf{i}+0&(-\mathbf{i})^{5}=1(-\mathbf{i})+0\\\
\mathbf{i}^{6}=0\mathbf{i}+(-1)&(-\mathbf{i})^{6}=0(-\mathbf{i})+(-1)\\\ \\\
A(\mathbf{i})=1\cdot 0+\mathbf{j}\cdot 1+\mathbf{i}\cdot
0&A(-\mathbf{i})=1\cdot 0+\mathbf{j}\cdot 1+\mathbf{i}\cdot 0\\\ \hskip
8.5359pt+0\cdot(-1)+(-1)\cdot 0&\hskip 8.5359pt+0\cdot(-1)+(-1)\cdot 0\\\
\hskip 8.5359pt+(-\mathbf{j})\cdot 1+(-\mathbf{i})\cdot 0=0&\hskip
8.5359pt+(-\mathbf{j})\cdot 1+(-\mathbf{i})\cdot 0=0\\\ \\\
v=\overline{A(\mathbf{i})}B(\mathbf{i})=0&v=\overline{A(-\mathbf{i})}B(-\mathbf{i})=0.\end{array}$
They produce the same spherical zero $[\mathbf{i}]$ since
$[-\mathbf{i}]=[\mathbf{i}]$.
For $x=\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i}$,
$\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$, we have
$\small\begin{array}[]{ll}(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{0}=0(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{0}=0(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{1}=1(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+0&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{1}=1(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+0\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{2}=1(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(-1)&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{2}=1(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+(-1)\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{3}=0(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(-1)&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{3}=0(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+(-1)\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{4}=(-1)(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+0&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{4}=(-1)(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+0\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{5}=(-1)(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{5}=(-1)(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{6}=0(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{6}=0(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
\\\ A(x)=1\cdot 0+\mathbf{j}\cdot-1+\mathbf{i}\cdot-1&A(x)=1\cdot
0+\mathbf{j}\cdot-1+\mathbf{i}\cdot-1\\\ \hskip 8.5359pt+0\cdot 0-1\cdot
1+-\mathbf{j}\cdot 1+-\mathbf{i}\cdot 0&\hskip 8.5359pt+0\cdot 0+\cdots\\\
\hskip 8.5359pt=-1-\mathbf{i}-2\mathbf{j}&\hskip
8.5359pt=-1-\mathbf{i}-2\mathbf{j}\end{array}$
$\small\begin{array}[]{ll}B(x)=1\cdot 1+\mathbf{j}\cdot 1+\mathbf{i}\cdot
0&B(x)=1\cdot 1+\mathbf{j}\cdot 1\\\ \hskip 8.5359pt-1\cdot-1+-\mathbf{j}\cdot
0+-\mathbf{i}\cdot 1+0\cdot-1&\hskip 8.5359pt+\mathbf{i}\cdot 0+\cdots\\\
\hskip 8.5359pt+0\cdot-1=2-\mathbf{i}+\mathbf{j}&\hskip
8.5359pt=2-\mathbf{i}+\mathbf{j}\\\ \\\
v=\overline{A(x)}B(x)=-3+3\mathbf{i}+3\mathbf{j}+3\mathbf{k}&v=\overline{A(x)}B(x)=-3+3\mathbf{i}+3\mathbf{j}+3\mathbf{k}\\\
|w|=3\sqrt{3}&|w|=3\sqrt{3}.\end{array}$
They produce the same isolated zero:
$\frac{1}{2}-\frac{\frac{\sqrt{3}}{2}}{3\sqrt{3}}\cdot
3\mathbf{i}-\frac{\frac{\sqrt{3}}{2}}{3\sqrt{3}}\cdot
3\mathbf{j}-\frac{\frac{\sqrt{3}}{2}}{3\sqrt{3}}\cdot 3\mathbf{k}$, which is
$0.5-0.5\mathbf{i}-0.5\mathbf{j}-0.5\mathbf{k}$.
Finally for $x=-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i}$,
$-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i}$, we still have to finish the
following work:
$\small\begin{array}[]{cc}(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{0}=0(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{0}=0(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{1}=1(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+0&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{1}=1(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+0\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{2}=-1(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(-1)&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{2}=-1(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+(-1)\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{3}=0(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{3}=0(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{4}=1(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+0&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{4}=1(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+0\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{5}=(-1)(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+(-1)&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{5}=(-1)(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+-1\\\
(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})^{6}=0(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathbf{i})+1&(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})^{6}=0(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathbf{i})+1\\\
\\\ {\small{\small\begin{array}[]{c}A(x)=1\cdot
0+\mathbf{j}\cdot-1+\mathbf{i}\cdot 1+0\cdot 0\\\ \hskip
8.5359pt-1\cdot-1+-\mathbf{j}\cdot 1+-\mathbf{i}\cdot 0\\\ \hskip
8.5359pt=1+\mathbf{i}-2\mathbf{j}\end{array}}}&{\small{\small\begin{array}[]{c}A(x)=1\cdot
0+\mathbf{j}\cdot-1+\mathbf{i}\cdot 1\\\ \hskip 8.5359pt+0\cdot 0+\cdots\\\
\hskip 8.5359pt=1+\mathbf{i}-2\mathbf{j}\end{array}}}\\\ \\\
\begin{array}[]{c}B(x)=1\cdot 1+\mathbf{j}\cdot-1+\mathbf{i}\cdot 0+0\cdot
1\\\ \hskip 8.5359pt-1\cdot-1+-\mathbf{j}\cdot 0+-\mathbf{i}\cdot 1\\\ \hskip
8.5359pt=2-\mathbf{i}-\mathbf{j}\end{array}&\begin{array}[]{c}B(x)=1\cdot
1+\mathbf{j}\cdot-1+\mathbf{i}\cdot 0\\\ \hskip 8.5359pt+0\cdot 1+\cdots\\\
\hskip 8.5359pt=2-\mathbf{i}-\mathbf{j}\end{array}\\\ \\\
v=\overline{A(x)}B(x)=3-3\mathbf{i}+3\mathbf{j}+3\mathbf{k}&v=\overline{A(x)}B(x)=3-3\mathbf{i}+3\mathbf{j}+3\mathbf{k}\\\
\\\ |w|=3\sqrt{3}&|w|=3\sqrt{3}.\end{array}$
By formula (3.8), They produce the same isolated zero:
$-0.5+0.5\mathbf{i}-0.5\mathbf{j}-0.5\mathbf{k}$. The set of roots of $p(z)$
is
$\\{1,\,\,-1,\,\,0.5-0.5\mathbf{i}-0.5\mathbf{j}-0.5\mathbf{k},\,\,-0.5+0.5\mathbf{i}-0.5\mathbf{j}-0.5\mathbf{k}\\}\cup[\mathbf{i}].$
In Example $3.8$ of [9], the authors said that the $12$ zeros of $q_{12}$ were
1 (twice), -1 (twice), $\pm\mathbf{i}$ (twice each), $0.5(\pm
1\pm\mathbf{i})$. We point out that $0.5(\pm 1\pm\mathbf{i})$ should be
corrected as $0.5(\pm 1\pm\sqrt{3}\mathbf{i})$.
## 5\. Numerical consideration and algorithm comparison
The polynomial in Example 1 has the property that its discriminant polynomial
$\tilde{p}$ can be factored by hands as
$(t-\mathbf{i})(t+\mathbf{i})(t-e^{\mathbf{i}\frac{\pi}{4}})(t-e^{\mathbf{i}\frac{3\pi}{4}})(t-e^{\mathbf{i}\frac{5\pi}{4}})(t-e^{\mathbf{i}\frac{7\pi}{4}})$.
In general, one can not always expect to get the zeros of the discriminant
polynomial by hands, one has to rely on machine computations. In Algorithm 1,
if we compute the zeros of the discriminant polynomial, $\tilde{p}(x)$, of
Example 3 in Section 4 by MATLAB 7.12.0(R2011a), we find the 12 zeros are as
following:
(5.1) $\begin{array}[]{ccc}1&-1.000000000000001&+0.000000002066542i\\\
2&-1.000000000000001&-0.000000002066542i\\\
3&-0.500000000000000&+0.866025403784440i\\\
4&-0.500000000000000&-0.866025403784440i\\\ 5&\ \
0.999999990102304&+0.000000000000000i\\\ 6&\ \
1.000000009897694&-0.000000000000000i\\\ 7&\ \
0.500000000000000&+0.866025403784439i\\\ 8&\ \
0.500000000000000&-0.866025403784439i\\\ 9&\ \
0.000000000016075&+1.000000008531051i\\\ 10&\ \
0.000000000016075&-1.000000008531051i\\\
11&-0.000000000016074&+0.999999991468949i\\\
12&-0.000000000016074&-0.999999991468949i\end{array}$
which are nearly same as Table 1 in [9]. This is simply because our
$\tilde{p}(t)$ is exactly the companion polynomial $q_{12}(t)$. Hence, the
similar measures to [9] can be made to obtain machine precision for the zeros
with multiplicity 2(e.g., by Newton’s method). It is interesting to note that
the discriminant polynomial given in this paper is always equal to the
companion polynomial after considering the variable as a real variable.
In Algorithm 1, if we have found by MATLAB the zeros of discriminant
polynomial, we usually by comparing
$|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}$ with
$|f_{1}(\overline{\eta_{i}})|^{2}+|f_{2}(\overline{\eta_{i}})|^{2}$ decide
which one $\omega_{i}$ should take between (2.11) and (2.12). If
$|f_{1}(\eta_{i})|^{2}+|f_{2}(\eta_{i})|^{2}$ is greater, then $\omega_{i}$
takes (2.11), otherwise takes (2.12). However, our Algorithm 1’ does not need
to make such a decision.
In Step 2 of Algorithm 1, we need decide whether a zero $\xi$ is real. In our
experience (the same as [9]), a test of the form $|{\rm{Im}}\xi|<10^{-5}$ is
appropriate. There is another delicate decision to make in our Algorithm 1.
That is, in Step 2 one has to decide whether a zero $\eta$ satisfies
$f_{1}(\eta)=f_{2}(\eta)=\bar{f}_{1}(\eta)=\bar{f}_{2}(\eta)=0$. For this, a
test for $f(\eta)=0$ can be carried out in the form $|f(\eta)|<10^{-10}$(This
was also used to test $v=0$ in [9]). Our Algorithm 1’ avoids making this
delicate decision.
Both Algorithm 1 and Algorithm 2 need to find all zeros of the companion
polynomial(note that the discriminant polynomial is always equal to the
companion polynomial). Unlike Algorithm 2, our algorithms no longer need the
use of any iterations. Using our algorithms, one can easily produce all zeros
of the simple quaternionic polynomial from the zeros of the companion
polynomial. Our algorithm has less workload, as shown in Section 4. This has a
great advantage for a simple quaternionic polynomial with high degree. In
fact, when one use Iteration (3.6) to compute $A(z)$ and $B(z)$ for a nonreal
complex root $z$ with $|z|\neq 1$, if the degree of the simple quaternionic
polynomial considered is high, the workload is huge, even interferes one’s
deciding whether $v$ is zero.
In [Page 252, 9] D. Janovsk$\acute{a}$ and G. Opfer said “We made some hundred
tests with polynomials $p_{n}$ of degree $n\leqslant 50$ with random integer
coefficients in the range $[-5,5]$ and with real coefficients in the range
$[0,1]$. In all cases we found only (nonreal) isolated zeros $z$. The test
cases showed $|p_{n}(z)|\thickapprox 10^{-13}$. Real zeros and spherical zeros
did not show up. If n is too large, say $n\thickapprox 100$, then usually it
is not any more possible to find all zeros of the companion polynomial by
standard means (say roots in MATLAB) because the coefficients of the companion
polynomial will be too large.” But this can be easily avoid by using Algorithm
1’ in many cases. For example, to solve a real coefficient polynomial of
degree of $50$, we can use Corollary 3 to do this easily.
Example 4. Find all zeros of $p(z)=z^{1000}-2$ in $\mathbb{H}$.
Solution. We know that the complex solution set of $p(t)$ is
$\\{\pm\sqrt[1000]{2}\\}\cup_{k=1}^{499}\\{\eta_{k}=\sqrt[1000]{2}e^{\frac{k\pi\mathbf{i}}{500}},\bar{\eta}_{k}\\}.$
Applying Corollary 3, we obtain the solution set for $p(z)$:
$\\{\pm 1\\}\cup_{k=1}^{499}[\sqrt[1000]{2}e^{\frac{k\pi\mathbf{i}}{500}}].$
According to the above comments D. Janovsk$\acute{a}$ and G. Opfer made on
their method, it is impossible to find all solutions of this example since the
degree of $p(z)$ is much larger than $100$ and all roots except for
$\pm\sqrt[1000]{2}$ are spherical.
## 6\. Conclusion
The method introduced in Section 2 can also be used to find all zeros of
simple polynomials with quaternionic coefficients located on only right side
of the power of the variable, since $x^{n}q_{n}+\cdots+xq_{1}+q_{0}=0$ is
equivalent to $\overline{q_{n}}\ \overline{x}^{n}+\cdots+\overline{q_{1}}\
\overline{x}+\overline{q_{0}}=0$. In retrospect, Eilenberg and Niven in [3]
for the first time proved that quaternionic $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$
has always a root, but the method they used is topological. For this result,
here we have provided a new alternate proof, without using any topological
tools. Based on the derived polynomial and discriminant polynomial introduced
in this paper, we have given the zeros in an explicit form for a simple
quaternionic polynomial. Comparing with the methods provided in [9] to seek
roots of $q_{n}x^{n}+\cdots+q_{1}x+q_{0}=0$, our method is different, and has
great advantages. Using our method, we have recovered several known results,
and deduced several very interesting consequences concerning seeking the zeros
of a simple quaternionic polynomial which do not seem to be deduced easily
from the results in [9].
We would like to conclude our paper by making the following remark. We hope
the method in the proof of our main theorem can be useful to seeking a method
to find all solutions of a quaternionic polynomial equation with coefficients
on both sides of the powers of the variable.
## References
* [1] S.L. Adler, Quaternionic quantum field theory, Commun. Math. Phys., 104(1986), pp. 611-656.
* [2] N. Le Bihan and J. Mars, Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing, Signal Process, 84(2004), pp. 1177–1199.
* [3] S. Eilenberg and I. Niven, The ’Fundamental Theorem of Algebra’ for quaternions, Bull. Amer. Math. Soc., 50(1944), pp. 246-248.
* [4] A. Gsponer and J.-P. Hurni, Quaternions in Mathematical Physics (2): Analytical Bibliography, Independent Scientific Research Institute report ISRI-05-05.26, 2008; also available online at http://arxiv.org/abs/math-ph/0511092v3.
* [5] G. Gentili and C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable, Michigan Math. J., 56(2008), pp. 655-667.
* [6] G. Gentili and D. C. Struppa, On the multiplicity of zeros of polynomials with quaternionic coefficients, Milan J. Math., 76(2007), pp. 1-10.
* [7] G. Gentili, D.C. Struppa and F. Vlacci, The fundamental theorem of algebra for Hamilton and Cayley numbers, Math. Z., 259(2008), pp. 895-902.
* [8] D. Janovsk$\acute{a}$ and G. Opfer, Linear equations in quaternions, Numerical mathematics and advanced applications, Springer, Berlin, 2006, pp. 945-953.
* [9] D. Janovsk$\acute{a}$ and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48(1)(2010), pp. 244-256.
* [10] D. Janovsk$\acute{a}$ and G. Opfer, The classification and the computation of the zeros of quaternionic two-sided polynomials, Numer. Math., 115(1)(2010), pp. 81-100.
* [11] D. Janovsk$\acute{a}$ and G. Opfer, Computing quaternionic roots in Newton’s method, Electron. Trans. Numer. Anal., 26(2007), pp. 82-102.
* [12] D. Janovsk$\acute{a}$ and G. Opfer, Linear equations in quaternionic variables, Mitt. Math. Ges. Hamburg, 27(2008), pp. 223-234.
* [13] S. De Leo, G. Ducati and V. Leonardi, Zeros of unilateral quaternionic polynomials, Electron. J. Linear Algebra, 15(2006), pp. 297-313.
* [14] I. Niven, Equations in quaternions, Amer. Math. Monthly, 48(1941), pp. 654-661.
* [15] I. Niven, The roots of a quaternion, Amer. Math. Monthly, 49(1942), pp. 386-388.
* [16] A. Pogorui and M. Shapiro, On the structure of the set of zeros of quaternionic polynomials, Complex Var. Elliptic Funct., 49(2004), pp. 379-389.
* [17] S. Pumpl$\ddot{u}$n and S. Walcher, On the zeros of polynomials over quaternions, Comm. Algebra, 30(2002), pp. 4007-4018.
* [18] R. Ser$\hat{o}$dio, E. Pereira and J. Vit$\acute{o}$ria, Computing the zeros of quaternionic polynomials, Comput. Math. Appl., 42(2001), pp. 1229-1237.
* [19] R. Ser$\hat{o}$dio and L.S. Siu, Zeroes of quaternion polynomials, Appl. Math. Lett., 14(2)(2001), pp. 237-239.
* [20] N. Topuridze,On the roots of polynomials over division algebras, Georgian Math. J., 10(4)(2003), pp. 745-762.
* [21] F.Z. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251(1997), pp. 21-57.
* [22] F.Z. Zhang and Y. Wei, Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices, Comm. Algebra, 29(6)(2001), pp. 2363-2375.
L.G. Feng: Department of Mathematics and System Science, National University
of Defense Technology, Changsha 410073, Hunan, P.R.China. Email:
fenglg2002@sina.com
K.M. Zhao: Department of Mathematics, Wilfrid Laurier University, Waterloo,
ON, Canada N2L 3C5, and College of Mathematics and Information Science, Hebei
Normal (Teachers) University, Shijiazhuang 050016, Hebei, P. R. China. Email:
kzhao@wlu.ca
|
arxiv-papers
| 2011-09-12T15:25:09 |
2024-09-04T02:49:22.093754
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lianggui Feng and Kaiming Zhao",
"submitter": "Kaiming Zhao",
"url": "https://arxiv.org/abs/1109.2503"
}
|
1109.2624
|
# High-power broadband laser source tunable from 3.0 $\mu$m to 4.4 $\mu$m
based on a femtosecond Yb:fiber oscillator
Tyler W. Neely1,∗ Todd A. Johnson,1,2 and Scott A. Diddams1,3
###### Abstract
We describe a tunable broadband mid-infrared laser source based on difference-
frequency mixing of a 100 MHz femtosecond Yb:fiber laser oscillator and a
Raman-shifted soliton generated with the same laser. The resulting light is
tunable over 3.0 $\mu$m to 4.4 $\mu$m, with a FWHM bandwidth of 170 nm and
maximum average output power up to 125 mW. The noise and coherence properties
of this source are also investigated and described.
1National Institute of Standards and Technology, Time and Frequency Division,
Mail Stop 847, 325 Broadway, Boulder, Colorado 80305, USA
2Present address: Saint John’s University, Collegeville, MN 56321, USA
3email: scott.diddams@nist.gov
∗Corresponding author: tneely@nist.gov
140.3070, 190.4410, 190.7110
High repetition-rate mid-infrared (MIR) femtosecond lasers and frequency combs
present useful light sources for absorption spectroscopy, combining broad
bandwidth with high spectral resolution and brightness [1, 2]. Such sub-
picosecond sources have typically targeted the region from 2 $\mu$m to 20
$\mu$m, where many molecules exhibit strong fundamental transitions. In
particular, several schemes using single-pass difference frequency generation
(DFG) have been developed and are attractive due to their relative simplicity
and the benefit of passive carrier-envelope offset (CEO) frequency
stabilization [3].
DFG schemes based on Er:fiber oscillators have been successful; up to 1.1 mW
of average power has been achieved in the range of 3.2 $\mu$m to 4.8 $\mu$m
[4]; 3 mW average power has been achieved at 3 $\mu$m [5]; 1.5 $\mu$W was
achieved in the region of 9.7 $\mu$m to 14.9 $\mu$m [6]; and 100 $\mu$W of
average power has been produced in the range of 5 $\mu$m to 12 $\mu$m [7]. The
power levels achieved thus far are however not comparable with watt-level
powers available from optical parametric oscillators (OPOs) [8]. We present
here a 100 MHz DFG-based system that for the first time achieves output power
levels that are competitive with OPO techniques. At the same time, we examine
the noise properties of our source and demonstrate that the nonlinear Raman
shifting we employ can suffer from excess amplitude and phase noise, with the
result of reduced pulse-to-pulse coherence in the MIR light. These results
should be relevant for related DFG approaches to broadband MIR generation [6,
7].
We generate a MIR frequency comb with straightforward difference frequency
generation (DFG) between the amplified output of a 100 MHz repetition rate
mode-locked Yb:fiber laser and a Raman-shifted soliton [9]. The oscillator
consists of both fiber and free space elements [10, 11] and generates a
spectrum centered at 1.04 $\mu$m with 140 mW average power. The oscillator
operates in the similariton regime [12] and produces strongly chirped pulses
with a duration of about 1.5 ps. Additional chirp is introduced with an 8 m
length of positive second-order, negative third-order dispersion fiber.
Amplification occurs in a double-clad Yb:fiber reverse-pumped with up to 8 W
of multimode light centered around 976 nm. Average output power and pulse
duration after isolation and compression are 2.4 W and 130 fs.
[width=8.3cm]optical_setup.eps
Figure 1: (color online) Experimental setup. PBS polarizing beam splitting
cube, PCF photonic crystal fiber, DBS dichroic beam splitter, PPLN fan-out
periodically-poled lithium niobate, Ge DBS anti-reflection coated germanium
filter.
A simplified diagram of the experimental setup is shown in Fig. 1. Part of the
amplifier output is coupled with an efficiency of $\sim$ 43 % into a photonic
crystal fiber (PCF) (1 m, 3.2 $\mu$m core, 945 nm zero-dispersion wavelength),
primarily generating a red-shifted Raman soliton through asymmetrical
broadening. Adjustment of both the input polarization and launch power into
the PCF results in a signal pulse tunable from 1.10 $\mu$m to 1.63 $\mu$m, as
shown in Fig. 2(a). DFG occurs in a fan-out periodically poled MgO-doped
lithium niobate (MgO:PPLN) crystal with poling period varying from 21 $\mu$m
to 35 $\mu$m and corresponding poled length varying from 1.2 mm to 2 mm,
allowing for continuous tuning of the phase-matching condition. The crystal
temperature is held at $100~{}^{\circ}{\rm C}$ to ensure that photorefractive
damage is avoided. The signal is combined with the original amplified pump
centered at 1.04 $\mu$m, resulting in an idler wave in the MIR
($\lambda_{i}^{-1}=\lambda_{p}^{-1}-\lambda_{s}^{-1}$). Temporal overlap of
the pulses is accomplished by introducing extra optical delay into the pump
path, compensating for the dispersion-induced delay in the PCF. The pulse
overlap is monitored using coincidental non-phase-matched sum-frequency
generation (SFG) between the soliton and pump pulses
($\lambda_{sum}^{-1}=\lambda_{p}^{-1}+\lambda_{s}^{-1}$); the resulting sum
light is centered around 630 nm and is measured with a silicon photodiode. A
5.8 kHz dither of the Yb:fiber amplifier pump diode current causes a
propagation delay dither at the output of the PCF and a subsequent dither in
SFG power. Lock-in detection of the SFG power generates an error signal that
drives both a moving mirror pair at sub-hertz bandwidth (Fig. 1) and the
Yb:fiber amplifier pump diodes at bandwidths of a few hundred hertz.
The resulting MIR idler is thus tunable from 3.0 $\mu$m to 4.4 $\mu$m as shown
in Fig. 2(b), with a FWHM spectral width of $\sim$ 170 nm and maximum average
power of 128 mW at $\sim$ 3.2 $\mu$m. A Gaussian fit to this spectrum gives a
peak spectral power density of $0.67\pm 0.01$ mW/nm. As shown in the inset of
Fig. 2(b) for the 3.2 $\mu$m tuning point, the output power is approximately
linear with pump power above a given threshold, with a slope of $93\pm 3$
mW/W. This suggests that high-output MIR powers may be possible with pump-
power scaling; 1 W of average idler power might be expected with 11.2 W of
pump power, comparable to the output of PPLN-based OPO systems [8].
[width=8.3cm,height=9cm]spectrum.eps
Figure 2: (color online) (a) Soliton shift as a function of launch power into
the PCF, with each spectrum individually normalized. (b) MIR idler spectra,
and conversion efficiency relative to pump photons. The total average power is
also noted above each spectrum. The absorption feature at 4.4 $\mu$m is due to
air-path absorption of CO2 before the monochromator. (inset): Average idler
power as function of pump power at the $\sim$ 3.2 $\mu$m tuning point.
[width=8.3cm,height=8.8cm]RIN.eps
Figure 3: (color online) (a) Relative intensity noise (RIN) for the MIR tuning
point of 3.196 $\mu$m. Measurement of the MIR RIN is limited to 10 MHz due to
photodiode bandwidth. (b) Integrated RIN (10 Hz - 10 MHz).
The decreased photon conversion efficiency at longer wavelengths is not
understood, although it is consistent with results from similar systems based
on Er:fiber oscillators [4, 7]. Throughout the tuning range, the soliton pulse
lengths varied between 97 fs and 117 fs, pulse powers varied between 33 mW and
51 mW, and spot size at the PPLN varied between $\omega_{0}=78$ $\mu$m and
$\omega_{0}=91$ $\mu$m. Taken together, the variation in these parameters is
however insufficient to explain the measured variation in output MIR power,
and we have been unable to tie this behavior to an experimental parameter.
The utility of this source for producing a MIR comb with passive CEO
stabilization depends on both on the phase (PM) and amplitude (AM) noise of
the generated light. It is known that asymmetrical broadening, as employed
here, can lead to excessive AM and PM noise [13]. Additionally, AM noise on
the amplifier output couples via dispersion to PM noise on the soliton, which
couples back to AM noise on the MIR light. We have attempted to quantify the
origin and magnitude of noise sources by heterodyne and relative intensity
noise (RIN) measurements at each stage of the system.
RIN measurements of the system are shown in Fig. 3. The RIN of the MIR light
(0.3 % integrated, 10 Hz to 10 MHz) is found to be about a factor of 5 to 10
greater than the RIN on the soliton. An estimated RIN-induced timing jitter of
the soliton of 4.5 fs (10 Hz to 10 MHz) is obtained from the product of the
integrated amplifier RIN (0.00021), the average PCF launch power (350 mW), the
measured PCF dispersion (70 fs/nm), and the average power-dependent soliton
wavelength shift (1 nm/mW). However, the impact of this amount of jitter on
the cross-correlation signal between a Gaussian-approximated pump (130 fs) and
soliton (100 fs) amounts to MIR power variation of only 0.04 %, which is
insufficient to account for the measured results in the range of 0.3 %. We
thus attribute the observed increase in MIR RIN to excess intrinsic timing
noise on the soliton that arises in the nonlinear spectral broadening process.
Measurements of the SFG RIN are additionally included in Fig. 3. The SFG also
depends on pulse timing overlap, but its RIN measurement is not limited by
photodiode bandwidth. The reason for the measured increase in RIN for the SFG
(compared to the MIR RIN) is not clear at this point.
As an additional characterization we measured the heterodyne beat between the
soliton and a tunable CW diode laser with a balanced InGaAs detector. Fig.
4(a) shows the results of the measurement of the free-running beat notes at
the soliton center wavelength of 1.36 $\mu$m, with a 300 kHz integration
bandwidth. As the soliton shift is increased, the contrast of the beat note
decreases to zero, as shown in Fig. 4(b), suggesting poor pulse-to-pulse
coherence. The MIR spectrum was also investigated by doubling the light
generated at 3.1 $\mu$m in a AgGaS2 crystal, resulting in $\sim$ 100 $\mu$W of
light at 1.55 $\mu$m. The heterodyne between this doubled light and the CW
laser is shown in Fig. 4(c), and is compared with similarly doubled light from
a MgO:PPLN-based OPO in the lab. Relatively high beat-note contrast is seen in
the OPO, and no beat note is observed in the DFG system. Since the amplifier
output generates a beat note with a Nd:YAG laser at 1064 nm (20 dB contrast in
a 300 kHz bandwidth with similar light levels), we conclude that the excess
noise is due to the soliton.
Although limitations are evident in the Raman soliton generation process, the
simplicity, power and brightness of this source are still useful for ongoing
broadband spectroscopic studies, including high-resolution trace gas
spectroscopy [14]. The system may be improved with careful choice of non-
linear fiber and/or pulse parameters. In particular, recently developed
suspended-core fibers have shown promise for highly coherent supercontinuum
generation in the tunable range utilized in this experiment; such fibers may
allow the development of high-power comb sources based on these techniques
[15].
We thank M. Hirano, Y. Kobayashi, I. Hartl, L. Nugent-Glandorf, and F. Adler
for their contributions and helpful comments. This work was funded by NIST and
the United States Department of Homeland Security’s Science and Technology
Directorate. As a contribution of NIST, an agency of the US government, this
work is not subject to copyright in the US.
[width=8.3cm,height=8cm]coherence.eps
Figure 4: (color online) Coherence properties of the system. (a) Free-running
beat note between the 1.36 $\mu$m soliton and CW laser with the max-hold
function enabled over several seconds of measurement; the spread in beat-note
frequency is due to variation in the unlocked repetition rate of the Yb:fiber
oscillator. (b) Beat note signal-to-noise ratio across the soliton tuning
range. (c) MIR beat notes.
## References
* [1] F. Adler, P. Masłowski, A. Foltynowicz, K. Cossel, T. Briles, I. Hartl, and J. Ye, Opt. Express 18, 21861 (2010).
* [2] K. Vodopyanov, E. Sorokin, I. Sorokina, and P. Schunemann, Opt. Lett. 36, 2275 (2011).
* [3] A. Baltuška, T. Fuji, and T. Kobayashi, Phys. Rev. Lett. 88, 133901 (2002).
* [4] C. Erny, K. Moutzouris, J. Biegert, D. Kühlke, F. Adler, A. Leitenstorfer, and U. Keller, Opt. Lett. 32, 1138 (2007).
* [5] P. Maddaloni, P. Malara, G. Gagliardi, and P. D. Natale, New Journal of Physics 8, 262 (2006).
* [6] D. Winters, P. Schlup, and R. Bartels, Opt. Lett. 35, 2179 (2010).
* [7] A. Gambetta, R. Ramponi, and M. Marangoni, Opt. Lett. 33, 2671 (2008).
* [8] F. Adler, K. Cossel, M. Thorpe, I. Hartl, M. Fermann, and J. Ye, Opt. Lett. 34, 1330 (2009).
* [9] F. Mitschke and L. Mollenauer, Opt. Lett. 11, 659 (1986).
* [10] L. Nugent-Glandorf, T. Johnson, Y. Kobayashi, and S. Diddams, Opt. Lett. 36, 1578 (2011).
* [11] X. Zhou, D. Yoshitomi, Y. Kobayashi, and K. Torizuka, Opt. Express 16, 7055 (2008).
* [12] F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
* [13] J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
* [14] T. Johnson and S. Diddams, to appear in App. Phys. B, arXiv:1108.0943v1. (2011).
* [15] A. Ruehl, K. Cossel, M. Martin, H. McKay, B. Thomas, D. Ling, M. Fermann, J. Dudley, I. Hartl, and J. Ye, arXiv:1105.2093v1 (2011).
|
arxiv-papers
| 2011-09-12T20:55:21 |
2024-09-04T02:49:22.110651
|
{
"license": "Public Domain",
"authors": "Tyler W. Neely, Todd A. Johnson, Scott A. Diddams",
"submitter": "Tyler Neely",
"url": "https://arxiv.org/abs/1109.2624"
}
|
1109.2857
|
arxiv-papers
| 2011-09-13T17:30:53 |
2024-09-04T02:49:22.125052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Edson Sampaio",
"submitter": "Jose Edson Sampaio",
"url": "https://arxiv.org/abs/1109.2857"
}
|
|
1109.2866
|
# Extensão de aplicações na esfera de um espaço vetorial com produto interno
Jose Edson Sampaio
Instituto de Federal de Educação, Ciência e Tecnologia do Ceará
###### Abstract
Being $\mathbb{E}$ a vector space with inner product and
$\mathbb{S}_{\mathbb{\mathbb{E}}}$ the sphere of $\mathbb{E}$, will be given a
demonstration that every application of the sphere
$\mathbb{S}_{\mathbb{\mathbb{E}}}$ itself it such that preserve inner product
is the restriction of a linear isometry in $\mathbb{E}$.
## 1 Introdução
Neste texto apresentaremos uma demonstração diferente da feita em [5] e [4] de
que uma isometria na esfera de um espaço vetorial com produto interno se
extende a uma isometria linear em todo espaço, desde que a isometria na esfera
preserve produto interno, para espaços de dimenção infinita e finita
respectivamente. Em um certo sentido, isso generaliza o teorema 2 de [5] que
prova esse fato para espaços $l^{p}$ com $p>1$. Para isso, apresentaremos
algumas definições antes dos resultados.
###### Definição 1.1.
Sejam $(\mathbb{E},\|.\|_{1})$ e $(\mathbb{F},\|.\|_{2})$ dois espaços
vetoriais normados e uma aplicação linear
$F:\mathbb{E}\longrightarrow\mathbb{F}$. Dizemos que $F$ é uma isometria se
$\|F(x)-F(y)\|_{2}=\|x-y\|_{1}$ para quaisquer $x,y\in\mathbb{E}$.
###### Definição 1.2.
Sejam $(\mathbb{E},\langle,\rangle_{1})$ e $(\mathbb{F},\langle,\rangle_{2})$
dois espaços vetoriais com produto interno e uma aplicação $F:A\longrightarrow
B$, onde $A\subset\mathbb{E}$ e $B\subset\mathbb{F}$. Dizemos que $F$ preserva
produto interno se $\langle F(x),F(y)\rangle_{2}=\langle x,y\rangle_{1}$ para
quaisquer $x,y\in A$.
###### Definição 1.3.
Dado $(\mathbb{E},\|.\|)$ um espaço vetorial normado, a esfera de $\mathbb{E}$
será denotada por $\mathbb{S}_{\mathbb{E}}=\\{x\in\mathbb{E};\|x\|=1\\}$.
###### Lema 1.1.
Se $\varphi:\mathbb{S}^{n}\longrightarrow\mathbb{S}^{n}$ diferenciável é tal
que a diferencial $d\varphi$ preserva produto interno então a própria
$\varphi$ preserva produto interno.
###### Proof.
Pelo teorema 4.3 do capitulo 6, do livro Elementary differential geometry [2]
temos que $\varphi$ preserva distância intrinseca. E ainda no mesmo livro, no
exemplo 1.9(b) do capitulo 8, temos que a distância intrinseca de dois pontos
$p,q\in\mathbb{S}^{n}$ é justamente o ângulo entre $p$ e $q$ já que estamos
com a esfera unitária. E como $\varphi$ preserva distância intrinseca, temos
portanto que $\varphi$ preserva ângulos e assim preserva produto interno. ∎
###### Lema 1.2.
Seja $(\mathbb{E},\langle,\rangle)$ um espaço vetorial com produto interno. Se
$F:\mathbb{E}\longrightarrow\mathbb{E}$ preserva distância e $F(0)=0$ então
$F$ é uma isometria.
###### Proof.
Façamos primeiramente para o caso em que $dim\mathbb{E}=\infty$. Como $F$
preserva distância, temos que $\|F(x)-F(y)\|=\|x-y\|$ para quaisquer
$x,y\in\mathbb{E}$ e como $F(0)=0$ temos que $\|F(x)\|=\|x\|$ para todo
$x\in\mathbb{E}$. Mas para todo $x,y\in\mathbb{E}$ temos
$\|F(x)-F(y)\|^{2}=\|F(x)\|^{2}-2\langle F(x),F(y)\rangle+\|F(y)\|^{2}$
e
$\|x-y\|^{2}=\|x\|^{2}-2\langle x,y\rangle+\|y\|^{2}.$
E então, $\langle F(x),F(y)\rangle=\langle x,y\rangle$ quaisquer que sejam
$x,y\in\mathbb{E}$. Sejam agora, $\\{e_{j}\\}_{j=1}^{\infty}\subset\mathbb{E}$
uma base ortonormal, $v=(v_{j})_{j=1}^{\infty}$ e $w=(w_{j})_{j=1}^{\infty}$
vetores quaisquer em $\mathbb{E}$. Daí, sendo $\lambda\in\mathbb{R}$ temos,
$F(v+\lambda w)=F(\sum\limits_{j=1}^{\infty}v_{j}+\lambda w_{j})e_{j}).$
E como $F$ preserva produto interno temos que
$\\{F(e_{j})\\}_{j=1}^{\infty}\subset\mathbb{E}$ também é base ortonormal e
assim,
$F(v+\lambda w)=\sum\limits_{j=1}^{\infty}\langle F(v+\lambda
w),F(e_{j})\rangle F(e_{j}).$
E então para mostrar que $F$ é linear e portanto isometria (pois $F$ preserva
distância), é suficiente mostrar que $\langle F(v+\lambda
w),F(e_{j})\rangle=v_{j}+\lambda w_{j}$ para todo$j\in\mathbb{N}$. Mas isso é
justamente o que ocorre, já que $\langle F(v+\lambda
w),F(e_{j})\rangle=\langle v+\lambda w,e_{j}\rangle=v_{j}+\lambda w_{j}$ para
todo $j\in\mathbb{N}$. E para o caso em que $dim\mathbb{E}=n+1<\infty$ e neste
caso $\mathbb{E}\cong\mathbb{R}^{n+1}$ basta tomar a base ortonormal como
sendo a base canônica do $\mathbb{R}^{n+1}$ e o resto é análogo ao feito no
caso acima. ∎
## 2 Resultado principal
###### Teorema 2.1.
Seja $(\mathbb{E},\langle,\rangle)$ um espaço vetorial com produto interno. Se
$\varphi:\mathbb{S}_{\mathbb{E}}\longrightarrow\mathbb{S}_{\mathbb{E}}$
preserva produto interno então existe uma isometria
$F:\mathbb{E}\longrightarrow\mathbb{E}$ tal que
$\varphi=F|_{\mathbb{S}_{\mathbb{E}}}$.
###### Proof.
Defina $F:\mathbb{E}\longrightarrow\mathbb{E}$ por
$F(x)=\left\\{\begin{array}[]{ll}\|x\|\varphi(\frac{x}{\|x\|}),&x\not=0\\\
0,&x=0\end{array}\right.$
É claro que $\|F(x)\|=\|x\|$ para todo $x\in\mathbb{E}$. E para $x$ e $y$
diferentes de zero em $\mathbb{E}$ temos que,
$\displaystyle\|F(x)-F(y)\|^{2}$ $\displaystyle=$
$\displaystyle\|F(x)\|^{2}-2\langle F(x),F(y)\rangle+\|F(y)\|^{2}$ (1)
$\displaystyle=$
$\displaystyle\|x\|^{2}-2\|x\|\|y\|\langle\varphi(\frac{x}{\|x\|}),\varphi(\frac{y}{\|y\|})\rangle+\|y\|^{2}$
E como $\varphi$ preserva produto interno, temos que,
$\langle\varphi(\frac{x}{\|x\|}),\varphi(\frac{y}{\|y\|})\rangle=\langle\frac{x}{\|x\|},\frac{y}{\|y\|}\rangle$
(2)
E assim pelas igualdades (1) e (2) temos,
$\|F(x)-F(y)\|^{2}=\|x-y\|^{2}$ (3)
Logo, F preserva distância e então pelo lema (1.2) $F$ é isometria. ∎
###### Corolário 2.2.
Se $\varphi:\mathbb{S}^{n}\longrightarrow\mathbb{S}^{n}$ diferenciável é tal
que a diferencial $d\varphi$ preserva produto interno então existe uma $F\in
O(n+1)$ tal que $\varphi=F|_{\mathbb{S}^{n}}$.
###### Proof.
Decorre direto do lema (1.1) e do teorema (2.1). ∎
## References
* [1] Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
* [2] Barrett O’Neill. Elementary differential geometry. Elsevier, rev. 2nd edition, 2006.
* [3] Bryan P. Rynne. Linear Funtional Analysis. Springer, 2nd edition, 2008.
* [4] Daryl Tingley. Isometries of the unit sphere. Geometriae Dedicata, 22(3):371–378, 1987.
* [5] Ji Jin YI; Rui Dong WANG. On extension of isometries between the unit spheres of normed space E and lp. Acta Mathematica Sinica, English Series, 25(7):1139–1144, 2009\.
|
arxiv-papers
| 2011-09-13T17:55:54 |
2024-09-04T02:49:22.129157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Edson Sampaio",
"submitter": "Jose Edson Sampaio",
"url": "https://arxiv.org/abs/1109.2866"
}
|
1109.2982
|
# Statistical and dynamical fluctuations of Binder ratios in heavy ion
collisions
Zhiming Li, Fengbo Xiong, and Yuanfang Wu Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China Key Laboratory of Quark
and Lepton Physics (Central China Normal University),Ministry of Education
###### Abstract
Higher moments of net-proton Binder ratio, which is suggested to be a good
observation to locate the QCD critical point, is measured in relativistic
heavy ion collisions. We firstly estimate the effect of statistical
fluctuations of the third and forth order Binder ratios. Then the dynamical
Binder ratio is proposed and investigated in both transport and statistical
models. The energy dependence of dynamical Binder ratios with different system
sizes at RHIC beam scan energies are presented and discussed.
###### pacs:
25.75.Gz,25.75.Nq
## I Introduction
One of the main goals of current relativistic heavy ion collisions is to map
the QCD phase diagram QCDdiagram . At vanishing baryon chemical potential
$\mu_{B}=0$, finite temperature Lattice QCD calculations predict that a cross-
over transition from hadronic phase to the Quark Gluon Plasma (QGP) phase will
occur around a temperature of 170 - 190 MeV Lattice ; Crossover . QCD based
model calculations indicate that the transition could be a first order at
large $\mu_{B}$ firstorder . The point where the first order phase transition
ends is the so-called QCD Critical Point (QCP) QCP1 ; QCP2 . Attempts are
being made to locate the QCP both experimentally and theoretically searchQCP .
Lattice QCD calculations at finite $\mu_{B}$ face numerical challenges in
computing. Thus the location of the QCP are highly uncertain in theoretically.
In experimental aspect, the RHIC beam energy scan program RHICbes has been
motivated to search for the QCP in experiment. By decreasing the collision
energy down to a center of mass of 5 GeV, RHIC will be able to vary the baryon
potential from $\mu_{B}\sim 0$ to 500 MeV.
Fluctuations of conserved charges, which behave differently between the
hadronic and QGP phase, are generally considered to be sensitive indicators
for the transition conserv ; fluctuation1 . The singularity at the QCP, at
which the transition is believed to be second order, may cause enhancement of
fluctuations if fireballs created by heavy ion collisions pass near the
critical point during the time evolution fluctuation2 . It has been shown that
near the critical point, the density-density correlator of baryon number
follow the same power law behavior as the correlator of the sigma field which
is associated with the chiral order parameter fluctuation2 ; baryon .
Therefore, the baryon number is considered as an equivalent order parameter of
formed system in nuclear collisions. In experiment, net-proton multiplicity
distribution is much easier to measure than the net-baryon numbers.
Theoretical calculations have shown that in QCD with exact isospin invariance,
the relevant corrections due to isospin breaking are small and the net-proton
fluctuations can reflect the singularity of the baryon number susceptibility
as expected at the QCP netproton . Hence, the net-proton number is used in
current heavy ion experiment STARorder .
It is suggested recently that the Binder-like ratios are good identification
of critical behavior in relativistic heavy ion collisionsbinder ; binderwu ;
binder2 . The third and forth order Binder ratios are defined as
$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\frac{<M^{3}>}{<M^{2}>^{3/2}}$ $\displaystyle B_{4}$
$\displaystyle=$ $\displaystyle\frac{<M^{4}>}{<M^{2}>^{2}},$ (1)
where $M$ can be a conserved net-charge, e.g. the net-baryon number. The
Binder ratio is a new observable in heavy ion collisions. The difference
between Binder ratios and the well-known higher momentsSTARorder ; highmoment
is that Binder ratios are the normalized higher raw moments while the higher
moments are central moments.
The universality argument indicates that the static critical exponents of the
second order phase transition are determined by the dimensionality and
symmetry of the system. The QCD critical point of deconfinement phase
transition belongs to the same universality class as liquid-gas phase
transition and the 3D-Ising model fluctuation2 ; baryon ; university . Its
universal critical properties are discussed to be valid in various of models
and relevant to heavy ion collisions fluctuation2 ; univer1 ; univer2 , in
particular the event-by-event fluctuations of baryon numbers fluctuation2 .
In the calculations of the 3D-Ising modelbinderwu with external field $h=0$,
Binder ratios of $B_{3}$ and $B_{4}$ as a function of temperature ($T$) show a
step jump from a lower platform to a higher one near the vicinity of critical
point. If we could map the parameters ($T,h$) of the Ising model onto the
parameters ($T,\mu$) along the freeze-out curve in QCD and find a path that
correspond to $h=0$, the critical behaviors of 3D-Ising model is expected for
the QCD critical point. Therefore, if the formed system in heavy ion
collisions reaches the critical point and the freeze-out curve is close to the
transition line, the step function liked behavior of Binder ratios in the
Ising model may serve as a probe of QCP in current heavy ion collisions, where
critical incident energy is difficult to assign precisely in priori.
In the mean time, the effect of trivial statistical fluctuations statistical
due to insufficient number of particles should be studied and properly
eliminated in higher moment calculations. Therefore, we should discuss the
statistical contributions from the measured fluctuations firstly, then we
could identify the dynamical part which is more relevant to the critical point
of the QCD phase transition.
In this paper, we firstly investigate the statistical and dynamical
fluctuations of net-proton Binder ratios by using the AMPT and THERMINATOR
models. Then, the energy dependence of dynamical Binder ratios in Au + Au
collisions at various RHIC beam scan energies are studied.
In our analysis, two versions of a multi-phase transport (AMPT) model ampt
are used. One is the AMPT default and the other one is the AMPT with string
melting. In both versions, the initial conditions are obtained from the heavy
ion jet interaction generator (HIJING) model, and then the scattering among
partons is given by the Zhangs parton cascade (ZPC) model. In the AMPT default
model, the partons recombine with their parent strings when they stop
interacting, and the resulting strings are converted to hadrons using the Lund
string fragmentation model, whereas in the AMPT model with string melting,
quark coalescence is used in combining partons into hadrons. The dynamics of
the hadronic matter is described by the ART model. The THERMINATOR statistical
model therminator is a Monte Carlo event generator designed for studying of
particle production in relativistic heavy ion collisions from SPS to LHC
energies. It implements thermal models of particle production with single
freeze out.
In order to make our calculations convenient for comparison with the RHIC beam
energy scan data, we choose the mid-rapidity ($|y|<0.5$) region with
transverse momentum $0.4<p_{T}<0.8$ GeV/c. This phase space is where the STAR
experiment can do the particle identification for proton numbers with its main
tracking detector - the Time Projection Chamber STARorder . The number of
events used in this analysis is around 6 million. This statistics is needed
for the calculation of the dynamical Binder ratios of net-proton to ensure the
statistical errors under control.
## II Statistical fluctuations of Binder ratios
Figure 1: Binder ratios of $B_{3}$ (a) and $B_{4}$ (b) as a function of
number of participants from AMPT default (solid circle), AMPT with string
melting (open circle), and THERMINATOR (open square) models in Au + Au
collisions at $\sqrt{s_{NN}}=200$ GeV. The dashed lines represent the
corresponding statistical fluctuations.
In the measurement of the net-proton Binder ratios, finite number of protons
and antiprotons will cause non-negligible statistical fluctuations. If the
produced protons and antiprotons are two independent Poisson-like
distributions statistical ; possion , the net-protons then obey a Skellam (SK)
distribution skellam .
According to the definition of Eq. (1), the statistical fluctuations of the
net-proton Binder ratios can be directly deduced from the Skellam distribution
$\displaystyle B_{3,stat}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{3}+6\mu\Delta+\Delta}{(\Delta^{2}+2\mu)^{3/2}}$
$\displaystyle B_{4,stat}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{4}+12\mu\Delta^{2}+4\Delta^{2}+12\mu^{2}+2\mu}{(\Delta^{2}+2\mu)^{2}},$
(2)
where $\Delta=\langle N_{p}\rangle-\langle N_{\bar{p}}\rangle$ is the average
number of net-protons, and $\mu=\left(\langle N_{p}\rangle+\langle
N_{\bar{p}}\rangle\right)/2$ is the mean value of protons and antiprotons in
the event sample. More details of the calculation of this formula could be
found in the appendix.
In Fig. 1 (a) and (b), we show the results of $B_{3}$ and $B_{4}$ as a
function of number of participants ($N_{\rm part}$) from AMPT default (solid
circle), AMPT with string melting (open circle), and THERMINATOR (open square)
models in Au + Au collisions at $\sqrt{s_{NN}}=200$ GeV, respectively. For
comparison, the statistical fluctuations of the Binder ratios, calculated from
Eq. (2), are presented as dashed lines. We can see that in both transport and
statistical models, the statistical fluctuations give main contributions to
the Binder ratios. It shows that the influence of statistical fluctuations are
not negligible in the measurement of net-proton Binder ratios at RHIC energy.
## III Dynamical net-proton Binder ratios
Figure 2: The third (a) and forth (b) order dynamical Binder ratios as a
function of $N_{\rm part}$ from AMPT default (solid circle), AMPT with string
melting (open circle), and THERMINATOR (open square) models in Au + Au
collisions at $\sqrt{s_{NN}}=200$ GeV.
In the THERMINATOR model, it is well-known that the fluctuations are thermal.
From Fig. 1, we observe it gives a good agreement with the Skellam statistical
fluctuations. It is difficult to disentangle purely statistical effects from
thermal fluctuations which follow the physics of a hadron resonance gas. Since
neither of them is associate with the QCP behavior, we suggest to eliminate
these statistical or thermal fluctuations in order to get the dynamical part.
As shown in section II, the statistical fluctuations of Binder ratio can be
expressed by Eq. (2) given proton and antiproton obey independent Poisson
distributions. We define the so-called dynamical Binder ratios as,
$\displaystyle B_{3,dyn}$ $\displaystyle=$ $\displaystyle B_{3}-B_{3,stat}$
$\displaystyle B_{4,dyn}$ $\displaystyle=$ $\displaystyle B_{4}-B_{4,stat}.$
(3)
We suggest to measure these dynamical Binder ratios instead of the original
definition given by Eq. (1) in relativistic heavy ion experiment.
The dynamical net-proton $B_{3}$ and $B_{4}$ as a function of $N_{\rm part}$
from AMPT default (solid circle), AMPT with string melting (open circle), and
THERMINATOR (open square) in Au + Au collisions at $\sqrt{s_{NN}}=200$ GeV are
shown in Fig .2 (a) and (b), respectively. We find that both the third and
forth order Binder ratios from THERMINATOR are zero at all centralities. This
is because that THERMINATOR is based on the hadron resonance gas model and the
produced net-protons in the final state obey the Skellam distribution hrg .
While, in transport models, both dynamical $B_{3}$ and $B_{4}$ are larger than
zero in peripheral collisions, then tend to be zero in central collisions. The
results from AMPT string melting are larger than that from the default model.
This is due to different mechanisms of hadronization scheme used for finite
state particles in different versions of AMPT models.
## IV Energy dependence of dynamical Binder ratios in transport models
Figure 3: Energy dependence of the dynamical $B_{3}$ (left panel) and $B_{4}$
(right panel) at RHIC energies from AMPT default model (upper panel) and
string melting (lower panel), respectively. The nine different symbols
represent nine collision sizes, which goes from most peripheral (70-80%
central) to most central (0-5% central) collisions. The lines are used only to
guide eyes.
In the upper panels of Fig. 3 (a) and (b), we show the energy dependence of
the dynamical $B_{3}$ and $B_{4}$ at six RHIC energies, 7.7, 9.2, 11.5, 39,
62.4, and 200 GeV from AMPT default model. The nine different symbols
represent nine collision sizes (denoted by centralities in experiments). In
all beam energies, when centrality goes from most peripheral (70-80% central)
to most central (0-5% central) collisions, the dynamical Binder ratios
decrease and are close to zero in the most central collisions. It means that
both dynamical $B_{3}$ and $B_{4}$ are system size dependent. From 7.7 GeV to
200 GeV, we observe no platform from the AMPT default model. The lower panels
of (c) and (d) for the string melting version give similar results.
Therefore, there is no step function behavior observed in both two versions of
the AMPT models. This is understandable since there is no QCD critical
mechanism implemented in these transport models.
## V Summary and outlook
In this paper, the statistical and dynamical Binder ratios of net-proton are
studies in Au + Au collisions at RHIC energies. Using transport and
statistical models, it is shown that statistical fluctuations are not
negligible in the measurement of higher Binder ratios in relativistic heavy
ion collisions.
In order to obtain a clean signature which may be related to the critical
point, we suggest to use the dynamical Binder ratio in experimental
measurement. The dynamical net-proton Binder ratio is found to be zero in the
THERMINATOR model but larger than zero in peripheral collisions in AMPT model.
The energy dependence of dynamical Binder ratios with different system sizes
shows no step function behavior either in AMPT default or string melting
models.
Whether the critical behavior of 3D-Ising model without external filed
suggested in ref. [15] correspond to the QCD critical point could be
discussed. Future work on the study of projecting the QCD parameters onto the
ones in the Ising model and find a path in the phase diagram that corresponds
to the vanishing external field are needed. The alternative way is that one
can include the external field into the Ising model and then relate it to the
Binder ratios to explore the QCD critical point. F. Karsch et al have explored
this way in the 3-state Potts model potts . The analysis in the O(N) models
including the 3D-Ising model is thus called for.
It is interesting to investigate the behavior of the dynamical Binder ratios
in the coming high energy collisions at RHIC, SPS, and FAIR experiments, where
the critical incident energy of QCD phase transition may be covered. Our model
study can serve as a background study of the behavior expected from known
physics effects for the experimental search for the QCD critical point.
## VI Acknowledgments
We thank Dr. Lizhu Chen and Shusu Shi for valuable discussions and remarks.
This work is supported in part by the NSFC of China with project No. 11005046
and No. 10835005.
## VII Appendix: Binder ratios from Skellam distribution
Given the distributions of proton and antiproton are independent Poisson
distributions with mean values are $\langle N_{p}\rangle$ and $\langle
N_{\bar{p}}\rangle$, the net-proton will follow a Skellam distribution. If we
define the net-proton as $M=N_{p}-N_{\bar{p}}$, then the probability
distribution function of $M$ is
$\displaystyle f(M;\langle N_{p}\rangle,\langle N_{\bar{p}}\rangle)$
$\displaystyle=$ $\displaystyle e^{-(\langle N_{p}\rangle+\langle
N_{\bar{p}}\rangle)}\left(\frac{\langle N_{p}\rangle}{\langle
N_{\bar{p}}\rangle}\right)^{M/2}I_{|M|}\left(2\sqrt{\langle
N_{p}\rangle\langle N_{\bar{p}}\rangle}\right),$
where $I_{|M|}\left(2\sqrt{\langle N_{p}\rangle\langle
N_{\bar{p}}\rangle}\right)$ is the modified Bessel function of the first kind.
The $n^{th}$ moment of $M$, which is defined as $\langle
M^{n}\rangle=\int_{-\infty}^{\infty}M^{n}f(M;\langle N_{p}\rangle,\langle
N_{\bar{p}}\rangle)dM$, can be calculated from the above distribution
function. We obtain
$\displaystyle\langle M\rangle$ $\displaystyle=$ $\displaystyle\Delta$
$\displaystyle\langle M^{2}\rangle$ $\displaystyle=$
$\displaystyle\Delta^{2}+2\mu$ $\displaystyle\langle M^{3}\rangle$
$\displaystyle=$ $\displaystyle\Delta^{3}+6\mu\Delta+\Delta$
$\displaystyle\langle M^{4}\rangle$ $\displaystyle=$
$\displaystyle\Delta^{4}+12\mu\Delta^{2}+4\Delta^{2}+12\mu^{2}+2\mu,$
where $\Delta=\langle N_{p}\rangle-\langle N_{\bar{p}}\rangle$ is the average
number of net-protons, and $\mu=\left(\langle N_{p}\rangle+\langle
N_{\bar{p}}\rangle\right)/2$ is the mean value of proton and antiproton.
By the definitions of the third and forth Binder ratios of Eq. (1), we get the
Binder ratios of the Skellam statistical distribution as
$\displaystyle B_{3,stat}$ $\displaystyle=$
$\displaystyle\frac{<M^{3}>}{<M^{2}>^{3/2}}=\frac{\Delta^{3}+6\mu\Delta+\Delta}{(\Delta^{2}+2\mu)^{3/2}},$
$\displaystyle B_{4,stat}$ $\displaystyle=$
$\displaystyle\frac{<M^{4}>}{<M^{2}>^{2}}=\frac{\Delta^{4}+12\mu\Delta^{2}+4\Delta^{2}+12\mu^{2}+2\mu}{(\Delta^{2}+2\mu)^{2}}.$
## References
* (1) J. Adams et al., Nucl. Phys. A 757, 102 (2005).
* (2) Y. Aoki et al., Nature 443, 675 (2006).
* (3) Y. Aoki et al., Phys. Lett. B 643, 46 (2006); M. Cheng et al., Phys. Rev D 74, 054507 (2006).
* (4) E.S. Bowman and J.I. Kapusta, Phys. Rev. C 79, 015202 (2009); S. Ejiri, Phys. Rev. D 78, 074507 (2008).
* (5) M.A. Stephanov, Prog. Theor. Phys. Suppl. 153, 139 (2004); Z. Fodor and S.D. Katz, JHEP 0404, 50 (2004).
* (6) R.V. Gavai and S. Gupta, Phys. Rev. D 78, 114503 (2008); Phys. Rev. D 71, 114014 (2005).
* (7) B. Mohanty, Nucl. Phys. A 830, 899c (2009).
* (8) B.I. Abelev et al., Phys. Rev. C 81, 024911 (2010); STAR internal Note - SN0493 (2009).
* (9) V. Koch, et al. Phys. Rev. Lett. 95, 182301 (2005); M. Asakawa, et al. Phys. Rev. Lett. 85, 2072 (2000).
* (10) M. Asakawa, U.W. Heinz, and B. Muller, Phys. Rev. Lett. 85, 2072 (2000); S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000).
* (11) M.A. Stephanov, K. Rajagopal, and E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998); Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003).
* (12) J. Kapusta, arXiv:1005.0860; D. Bower and S. Gavin, Phys. Rev. C 64, 051902 (2001).
* (13) Y. Hatta and A. Stephanove, Phys. Rev. Lett. 91, 102003 (2003); M. Asakawa, Baryon Number Cumulants and Proton Number Cumulants in Relativistic Heavy Ion Collisions, talks given in the 7th International workshop on critical point and onset of deconfinement (CPOD2011), Nov. 2011, Wuhan, China.
* (14) M.M. Aggarwal et al. (STAR Coll.), Phys. Rev. Lett. 105, 022302 (2010).
* (15) Lizhu Chen et al., arXiv:1010.1166.
* (16) K. Binder, Rep. Prog. Phys. 60, 487 (1997); K. Binder, Z. Phys. B 43, 119 (1981).
* (17) K. Kanaya and S. Kaya, Phys. Rev. D 51, 2404 (1995).
* (18) M.A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009); M. Asakawa, S. Ejiri, and M. Kitazawa, Phys. Rev. Lett. 103, 262301 (2009).
* (19) J. Garca, and J.A. Gonzalo, Physica A 326, 464 (2003); P. Forcrand, and O. Phylipsen, Phy. Rev. Lett. 105, 152001 (2010); M. Asakawa, J Phys. G 36 064042 (2009).
* (20) J. Berges, K. Rajagopal, Nucl. Phys. B 538 215 (1999).
* (21) M.A. Halasz et al., Phys. Rev. D 58 096007 (1998).
* (22) C. Athanasiou, K. Rajagopal, and M. Stephanov, Phys. Rev. D 82, 074008 (2010); A. Bialas and R. Peschanski, Nucl. Phys. B 273, 703 (1986); A. Bialas and R. Peschanski, Phys. Lett. B 207, 59 (1988).
* (23) Z.W. Lin, C.M. Ko, B.A. Li et al., Phys. Rev. C 72, 064901 (2005).
* (24) A. Kisiel et al., Comput. Phys. Commun. 174, 559 (2006).
* (25) C. Pruneau, S. Gavin, S. Voloshin, Phys. Rev. C 66, 044904 (2002); STAR Coll., Phys. Rev. C 79, 024906 (2009).
* (26) J.G. Skellam, Journal of the Royal Statistical Socienty 109, 296 (1946); X.F. Luo, B. Mohanty, H.G. Ritter, N. Xu, J. Phys. G 37, 094061 (2010).
* (27) P. Braun-Munizinger et al., arXiv:1107.4267.
* (28) F. Karsch, and S. Stickan, Phys. Letts. B 488, 319 (2000); F. Karsch, Ch. Schmidt, and S. Stickan, Comput. Phys. Commun. 147, 451 (2002).
|
arxiv-papers
| 2011-09-14T03:16:21 |
2024-09-04T02:49:22.145623
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhiming Li, Fengbo Xiong, and Yuanfang Wu",
"submitter": "Zhiming Li",
"url": "https://arxiv.org/abs/1109.2982"
}
|
1109.3215
|
PARTIAL DIFFERENTIAL EQUATIONS/EQUATIONS AUX DERIVEES PARTIELLES
MATHEMATICAL PHYS/PHYSIQUE MATHEMATIQUE
Problèmes de Cauchy avec des conditions modifiées pour les équations d’Euler-
Poisson- Darboux
Cheikh Ould Mohamed El-hafedh et Mohamed Vall Ould Moustapha
Cauchy problems with the modified conditions for the Euler-Poisson-Darboux
equations
Abstract. Nowadays the Euler-Poisson-Darboux equation is extensively studied
in several settings. The main questions on every spaces are explicit solutions
for the classical Cauchy problems with the second data null. In this note we
will generalize and unify several results on Euler-Poisson-Darboux equation.
We consider the Cauchy problems with modified conditions for the classical and
radial Euler-Poisson-Darboux equations. We give the explicit solutions in
terms of the Gauss ${}_{2}F_{1}$ and Appell $F_{4}$ hypergeometric functions.
The main results have many applications such as the classical and radial wave
equation as well as the Tricomi operator [1].
1\. INTRODUCTION. On considère la famille classique d’équations d’Euler-
Poisson-Darboux dans $R^{n}$
$\Delta_{n}U(t,x)=(\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\mu}{t}\frac{\partial}{\partial
t})U(t,x),~{}~{}~{}~{}~{}t>0~{}~{}~{}~{}~{}~{}~{}(E_{1})$
et la famille radiale d’équations d’Euler-Poisson-Darboux
$(\frac{\partial^{2}}{\partial x^{2}}+\frac{1-2\nu}{x}\frac{\partial}{\partial
x})U(t,x)=(\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\mu}{t}\frac{\partial}{\partial
t})U(t,x),~{}~{}~{}~{}~{}t>0,~{}x>0~{}~{}~{}~{}(E_{2})$
avec les conditions initiales modifiées
$U\left(0,x\right)=f\left(x\right),~{}~{}~{}~{}\lim_{t\rightarrow
0}t^{1-2\mu}\frac{\partial}{\partial
t}U\left(t,x\right)=g\left(x\right)~{}(C_{1})$
$U\left(0,x\right)=A^{q}_{x}f\left(x\right),~{}\lim_{t\rightarrow
0}t^{1-2\mu}\frac{\partial}{\partial
t}U\left(t,x\right)=A^{q}_{x}g\left(x\right)~{}(C_{2})$
où $A^{q}_{x}$ est la $q^{\mbox{i\\`{e}me}}$ puissance de l’opérateur
$A_{x}=\left\\{\begin{array}[]{rr}\Delta_{n}~{}\mbox{si}~{}x\in R^{n}\\\
\Lambda_{x}~{}\mbox{si}~{}x\in R^{+}\\\ \end{array}\right.$,
$\Delta_{n}=\frac{\partial^{2}}{\partial
x^{2}_{1}}+\frac{\partial^{2}}{\partial
x^{2}_{2}}+...+\frac{\partial^{2}}{\partial x^{2}_{n}}$ et
$\Lambda_{x}=\frac{\partial^{2}}{\partial
x^{2}}+\frac{1-2\nu}{x}\frac{\partial}{\partial x},~{}\nu,~{}\mu$ et $q$ sont
des paramètres réels.
L’équation $(E_{1})$ a été étudiée pour les valeurs entières de
$k,~{}(1-2\mu=k)$ par A.Weinstein [9] et son école de Maryland, D.W.Bresters
$[3]$ a exprimé la solution de l’équation $(E_{1})$ avec les conditions
$U(0,x)=f(x),~{}U_{t}(0,x)=g(x)~{}~{}~{}~{}~{}~{}(C)$
Mais il a été contraint -semble t-il- de prendre la deuxième donnée nulle
$(g=0)$ à cause de la singularité en $t=0$, les conditions modifiées $(C_{1})$
et $(C_{2})$ permettent de remédier à ce problème et de pouvoir prendre la
deuxième donnée une fonction non nulle $g$, tout en recouvrant les conditions
classiques $(C)$: ainsi pour $\mu=\frac{1}{2}$ on retrouve les problèmes de
Cauchy pour les équations classiques et radiales des ondes (voir $[4]$ et
$[2]$).
$(E_{1})$ et $(E_{2})$ sont des équations des ondes avec potentiels dépendants
du temps respectivement: $-\frac{1-2\mu}{t}\frac{\partial}{\partial t}$ et
$\frac{1-2\nu}{x}\frac{\partial}{\partial
x}-\frac{1-2\mu}{t}\frac{\partial}{\partial t}$.
L’interêt des équations $(E_{1})$ et $(E_{2})$ vient du fait que les
potentiels correspondants sont homogènes de degré $-2$ et donc les opérateurs
de gauche et droite se comportent de la même manière. L’une des difficultés
majeures dans le cas du potentiel dépendant du temps est l’absence de relation
entre les semi-groupes engendrés par l’ équation de Schrodinger et les
propriétés spectrales de l’opérateur $H=-\Delta+V$. Rappelons que pour un
potentiel indépendant du temps $V$ on a $g(H)=\int g(\lambda)dE(\lambda)f$ où
$dE(\lambda)$ est la mesure spectrale associée à l’opérateur H; ce qui n’est
pas valable dans le cas d’un potentiel dépendant du temps. Les résultats
principeaux de cet article sont les suivants:
THÉORÈME 1. Pour $0<\mu<\frac{1}{2}$, le problème de Cauchy
$(E_{1}),~{}(C_{1})$ admet la solution unique donnée par:
$U\left(t,x\right)=\alpha_{n,-\mu}t^{2\mu}\left(\frac{\partial}{t\partial
t}\right)^{\frac{n-1}{2}}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{-\mu-\frac{1}{2}}dx^{\prime}\\\
+\frac{1}{2\mu}\alpha_{n,\mu}\left(\frac{\partial}{t\partial
t}\right)^{\frac{n-1}{2}}\int_{\left|x^{\prime}-x\right|<t}g\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\mu-\frac{1}{2}}dx^{\prime}$
si $n$ est impair,
$U\left(t,x\right)=\beta_{n}t^{2\mu}\left(\frac{\partial}{t\partial
t}\right)^{\frac{n}{2}}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{-\mu}dx^{\prime}\\\
+\frac{1}{2\mu}\beta_{n}\left(\frac{\partial}{t\partial
t}\right)^{\frac{n}{2}}\int_{\left|x^{\prime}-x\right|<t}g\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\mu}dx^{\prime}$
si $n$ est pair
avec
$\alpha_{n,\mu}=\frac{\Gamma\left(1+\mu\right)}{2^{\frac{n-1}{2}}\pi^{\frac{n}{2}}\Gamma\left(\frac{1}{2}+\mu\right)}$
et $\beta_{n}=\frac{1}{(2\pi)^{\frac{n}{2}}}.$
THÉORÈME 2. Pour $0<\mu<\frac{1}{2}$ et
$-\frac{n}{2}<q<-\frac{\mu}{2}-\frac{n}{4}$, le problème de Cauchy
$(E_{1}),~{}(C_{2})$ admet la solution unique donnée par:
$U(t,x)=\int_{R^{n}}f(x^{\prime})N_{-\mu}(t,x,x^{\prime})dx^{\prime}+\frac{t^{2\mu}}{2\mu}\int_{R^{n}}g(x^{\prime})N_{\mu}(t,x,x^{\prime})dx^{\prime}$
où $N_{\mu}(t,x,x^{\prime})=\\\
\left\\{\begin{array}[]{rr}\frac{2^{2q+\frac{n}{2}}\i^{2q}\Gamma(q+\frac{n}{2})}{(2\pi)^{n}\Gamma(-q)}\left|x-x^{\prime}\right|^{-2q-n}~{}_{2}F_{1}(q+\frac{n}{2},q+1,\mu+1,\frac{t^{2}}{\left|x-x^{\prime}\right|^{2}})~{}\mbox{si}~{}0<t<\left|x-x^{\prime}\right|\\\
\frac{2^{2q+\frac{n}{2}}\i^{2q}\Gamma(1+\mu)\Gamma(q+\frac{n}{2})}{(2\pi)^{n}\Gamma(\frac{n}{2})\Gamma(1+\mu-q-\frac{n}{2})}t^{-2q-n}~{}_{2}F_{1}(q+\frac{n}{2},q+\frac{n}{2}-\mu,\frac{n}{2},\frac{\left|x-x^{\prime}\right|^{2}}{t^{2}})~{}\mbox{si}~{}\left|x-x^{\prime}\right|<t\\\
\end{array}\right.$
avec
${}_{2}F_{1}(a,b,c,z)=\sum^{\infty}_{n=0}\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}z^{n}$
la fonction hypergéométrique de Gauss
et $(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(a)}$ le symbole de Pochhammer.
THÉORÈME 3. Pour $\nu>-\frac{1}{2}$ et $0<\mu<\frac{1}{2}$, le problème de
Cauchy $(E_{2}),~{}(C_{1})$ admet la solution unique donnée par:
$U(t,x)=\int^{+\infty}_{0}f(x^{\prime})t^{2\mu}K_{-\mu}(t,x,x^{\prime})x^{\prime
1-2\nu}dx^{\prime}+\frac{1}{2\mu}\int^{+\infty}_{0}g(x^{\prime})K_{\mu}(t,x,x^{\prime})x^{\prime
1-2\nu}dx^{\prime}$
où $K_{\mu}(t,x,x^{\prime})=\\\
\left\\{\begin{array}[]{rrrrr}0~{}~{}~{}\mbox{pour}~{}0<x^{\prime}<x-t~{}\mbox{ou}~{}x^{\prime}>x+t,\\\
\frac{2^{\mu-\frac{1}{2}}\Gamma(1+\mu)}{\sqrt{\pi}\Gamma(\frac{1}{2}+\mu)}(xx^{\prime})^{\nu+\mu-1}(1-z)^{\mu-\frac{1}{2}}~{}_{2}F_{1}(\frac{1}{2}-\nu,\frac{1}{2}+\nu,\frac{1}{2}+\mu,\frac{1-z}{2})\\\
\mbox{pour}~{}\left|x-t\right|<x^{\prime}<x+t,\\\
\frac{2^{\mu-\nu}\Gamma(1+\mu)\Gamma(1-\mu+\nu)\sin[(\mu-\nu)\pi]}{\pi\Gamma(\nu+1)}(xx^{\prime})^{\nu+\mu-1}z^{\mu-\nu-1}\\\
\times_{2}F_{1}(\frac{\nu-\mu+1}{2},\frac{\nu-\mu}{2}+1,\nu+1,\frac{1}{z^{2}})~{}\mbox{pour}~{}0<x^{\prime}<t-x\\\
\end{array}\right.$ avec $z=\frac{x^{2}+x^{\prime 2}-t^{2}}{2xx^{\prime}}.$
THÉORÈME 3 bis. Pour $0<\mu<1$ et les données initiales analytiques
$f(x)=\sum^{\infty}_{l=0}a_{l}x^{l}$ et $g(x)=\sum^{\infty}_{l=0}b_{l}x^{l}$,
le problème $(E_{2}),~{}(C_{1})$ admet la solution unique donnée par:
$U(t,x)=\sum^{\infty}_{l=0}a_{l}U_{l}+\sum^{\infty}_{l=0}b_{l}V_{l}$
où
$U_{l}=x^{l}~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1-\mu,\frac{t^{2}}{x^{2}})$
et
$V_{l}=\frac{t^{2\mu}}{2\mu}x^{l}~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1+\mu,\frac{t^{2}}{x^{2}})$.
THÉORÈME 4. Pour $\nu>-\frac{1}{2},~{}0<\mu<\frac{1}{2}$ et
$-\frac{1}{2}<q<-\frac{\mu}{2}-\frac{1}{4}$, le problème de Cauchy
$(E_{2}),~{}(C_{2})$ admet la solution unique donnée par:
$U(t,x)=K(\nu,q)\int^{+\infty}_{0}f(x^{\prime})H_{-\mu}(t,x,x^{\prime})x^{\prime
1-2\nu}dx^{\prime}\\\
+\frac{K(\nu,q)t^{2\mu}}{2\mu}\int^{+\infty}_{0}g(x^{\prime})H_{\mu}(t,x,x^{\prime})x^{\prime
1-2\nu}dx^{\prime}$ avec
$H_{\mu}(t,x,x^{\prime})=x^{2\nu}x^{\prime-2(q+1)}F_{4}(q+1,q+1+\nu,1+\mu,1+\nu,\frac{t^{2}}{x^{\prime
2}},\frac{x^{2}}{x^{\prime 2}}),$
$K(\nu,q)=\frac{2^{2q+1}\i^{2q}\Gamma(q+1+\nu)}{\Gamma(1+\nu)\Gamma(-q)}$
et $F_{4}$ la fonction hypergéométrique de deux variables d’Appell définie
par[8]:
$F_{4}(a,b,c,d,x,y)=\sum^{\infty}_{m,n=0}\frac{(a)_{m+n}(b)_{m+n}}{(c)_{m}(d)_{n}m!n!}x^{m}y^{n}.$
2\. PRÉLIMINAIRES.
Rappelons l’équation de Bessel $[6]~{}P106$
$[\frac{\partial^{2}}{\partial
x^{2}}+\frac{1-2\alpha}{x}\frac{\partial}{\partial x}+(\beta\gamma
x^{\gamma-1})^{2}+\frac{\alpha^{2}-\nu^{2}\gamma^{2}}{x^{2}}]V=0$
dont deux solutions indépendantes sont $t^{\alpha}J_{\nu}(\beta t^{\gamma})$
et $t^{\alpha}Y_{\nu}(\beta t^{\gamma})$ avec $J_{\nu}$ et $Y_{\nu}$ des
fonctions de Bessel du premier espèce. On définit la transformation de
Fourier-Bessel-Hankel d’une fonction $f$ par:
$\widehat{f}(\lambda)=\int^{+\infty}_{0}f(x)(\lambda x)^{\nu}J_{\nu}(\lambda
x)x^{1-2\nu}dx.$
Dans la suite, on aura besoin des lemmes suivants:
Lemme 1. $([6]~{}P~{}134-135)$ Pour $\mu>0$ on a les comportements
asymptotiques
i $J_{\mu}(Z)\approx\frac{Z^{\mu}}{2^{\mu}\Gamma(\mu+1)}$ et
$Y_{\mu}(Z)\approx\frac{-2^{\mu}\Gamma(\mu)}{\pi.Z^{\mu}}$ en zéro,
ii $J_{\mu}(Z)\approx\sqrt{\frac{2}{\pi
Z}}\cos(Z-\frac{1}{2}\mu\pi-\frac{1}{4}\pi)$ et
$Y_{\mu}(Z)\approx\sqrt{\frac{2}{\pi
Z}}\sin(Z-\frac{1}{2}\mu\pi-\frac{1}{4}\pi)$ à l’infini.
Lemme 2.
i $\widehat{\Lambda_{x}f}(\lambda)=-\lambda^{2}\widehat{f}(\lambda)$.
ii La transformation inverse de Fourier-Bessel-Hankel est donnée par:
$f(x)=\int^{+\infty}_{0}\widehat{f}(\lambda)(\lambda x)^{\nu}J_{\nu}(\lambda
x)\lambda^{1-2\nu}d\lambda.$
Lemme 3 $([5]~{}P~{}675)$.
i La transformation de Fourier d’une fonction radiale est donnée par:
$\int_{I\\!R^{n}}f(\left|\xi\right|)\exp(i\xi.x)d\xi=\left|x\right|^{1-\frac{n}{2}}\int^{+\infty}_{0}f(r)J_{\frac{n}{2}-1}(r\left|x\right|)r^{\frac{n}{2}}dr.$
ii.
$\int^{+\infty}_{0}r^{-\rho}J_{\mu}(ar)J_{\nu}(br)dr=\frac{2^{-\rho}a^{\rho-\nu-1}b^{\nu}\Gamma(\frac{1+\nu+\mu-\rho}{2})}{\Gamma(1+\nu)\Gamma(\frac{1-\nu+\mu+\rho}{2})}\\\
\times~{}_{2}F_{1}(\frac{1+\nu+\mu-\rho}{2},\frac{1+\nu-\mu-\rho}{2},\nu+1,\frac{b^{2}}{a^{2}})$
où $\nu+\mu-\rho+1>0,~{}\rho>-1,~{}a>b>0.$
Lemme 4 $([5]~{}P~{}677)$.
$\int^{+\infty}_{0}\lambda^{2a-1-\mu}J_{\mu}(\lambda t)J_{\nu}(\lambda
x)J_{\nu}(\lambda x^{\prime})d\lambda=\\\
\frac{2^{2a-1-\mu}\Gamma(a+\nu)}{\Gamma(1+\mu)\Gamma(1+\nu)\Gamma(1-a)}t^{\mu}x^{\nu}x^{\prime-\nu-2a}.F_{4}(a,a+\nu,1+\mu,1+\nu,\frac{t^{2}}{x^{\prime
2}},\frac{x^{2}}{x^{\prime 2}})$
pour $-\nu<a<\frac{5}{4}+\frac{\mu}{2},~{}x>0,~{}t>0$ et $x^{\prime}>x+t$;
et pour $-\nu<a<\frac{3}{4}+\frac{\mu}{2}$ l’integrale converge absolument
et par suite, elle prolonge $F_{4}$ pour $0<x^{\prime}<x+t$.
Lemme 5. Pour
$W_{n,\mu}(t,x,x^{\prime})=C_{n,\mu}\left[t^{2}-\left|x^{\prime}-x\right|^{2}\right]^{\mu-\frac{n}{2}}$
et
$C_{n,\mu}=\frac{\Gamma(1+\mu)}{\pi^{\frac{n}{2}}\Gamma(1+\mu-\frac{n}{2})}$,
on a
i
$W_{n,\mu}(t,x,x^{\prime})=\left\\{\begin{array}[]{rr}\alpha_{n,\mu}(\frac{\partial}{t\partial
t})^{\frac{n-1}{2}}\left[t^{2}-\left|x^{\prime}-x\right|^{2}\right]^{\mu-\frac{1}{2}}~{}\mbox{si
n est impair}\\\ \beta_{n}(\frac{\partial}{t\partial
t})^{\frac{n}{2}}\left[t^{2}-\left|x^{\prime}-x\right|^{2}\right]^{\mu}~{}\mbox{si
n est pair}\\\ \end{array}\right.$
ii
$\Delta_{n}W_{\mu}(t,x,x^{\prime})=2(\mu-\frac{n}{2})C_{n,\mu}(t^{2}-\left|x-x^{\prime}\right|^{2})^{\mu-\frac{n}{2}-2}[2(\mu-1)\left|x-x^{\prime}\right|^{2}-nt^{2}]$,
iii $[\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\mu}{t}\frac{\partial}{\partial t}]W_{\mu}(t,x,x^{\prime})=\\\
4(\mu-\frac{n}{2})C_{n,\mu}(t^{2}-\left|x-x^{\prime}\right|^{2})^{\mu-\frac{n}{2}-2}[(1-\mu)(t^{2}-\left|x-x^{\prime}\right|^{2})+(\mu-\frac{n}{2}-1)t^{2}]$,
Preuve. Une simple vérification suffit.
3\. ÉQUATION CLASSIQUE D’EULER-POISSON-DARBOUX.
Preuve du théorème 1. D’après le lemme 5 on a
$U(t,x)=t^{2\mu}\int_{\left|x^{\prime}-x\right|<t}f(x^{\prime})W_{-\mu}(t,x,x^{\prime})dx^{\prime}+\frac{1}{2\mu}\int_{\left|x^{\prime}-x\right|<t}g(x^{\prime})W_{\mu}(t,x,x^{\prime})dx^{\prime}$,
et $(\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\mu}{t}\frac{\partial}{\partial t}-\Delta_{n})W_{\mu}=0$,
soit $V$ une solution de l’équation $(E_{1})$, remarquons que si
$V(t,x)=t^{2\mu}w_{-\mu}(t,x)$ alors
$\Delta w_{-\mu}(t,x)=[\frac{\partial^{2}}{\partial
t^{2}}+\frac{1+2\mu}{t}\frac{\partial}{\partial t}]w_{-\mu}(t,x),$
donc il suffit de montrer que $W_{\mu}$ vérifie l’équation $(E_{1})$, ce qui
est réalisé.
Pour voir les conditions initiales, on utilise les coordonnées polaires
centrées en $x$
$x^{\prime}=x+r\omega,~{}\omega\in S^{n-1},~{}S^{n-1}=\left\\{\omega\in
R^{n},~{}\left|\omega\right|=1\right\\}$ et le changement des variables
$r=ts,~{}0<s<1$, on obtient
$U\left(t,x\right)=C_{n,-\mu}\int^{1}_{0}f^{\\#}_{x}(ts)(1-s^{2})^{-\mu-\frac{n}{2}}s^{n-1}ds+\frac{C_{n,\mu}}{2\mu}t^{2\mu}\int^{1}_{0}g^{\\#}_{x}(ts)(1-s^{2})^{\mu-\frac{n}{2}}s^{n-1}ds$
avec $f^{\\#}_{x}(r)=\int_{S^{n-1}}f(x+r\omega)d\sigma(\omega)$,
et
$\int_{S^{n-1}}d\sigma(\omega)=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$.
à la limite on obtient la première donnée initiale à savoir que
$\int^{1}_{0}(1-s^{2})^{-\mu-\frac{n}{2}}s^{n-1}ds=\frac{1}{2}B(1-\mu-\frac{n}{2},\frac{n}{2})=\frac{\Gamma(1-\mu-\frac{n}{2})\Gamma(\frac{n}{2})}{2\Gamma(1-\mu)}.$
De même on obtient la deuxième donnée initiale.
Preuve du théorème 2. On pose $F(x)=\Delta^{q}_{x}f(x)$ et
$G(x)=\Delta^{q}_{x}g(x)$, en utilisant la transformation de Fourier et les
lemmes 1, 2 et quelques propriétés des fonctions $J_{\nu}$ et $Y_{\nu}$ (voir
$[6]$ et $[7]$) on obtient
$\widehat{U}(t,\xi)\approx\frac{\left|\xi\right|^{\mu}C_{1}(\xi)}{2^{\mu}\Gamma(\mu+1)}t^{2\mu}-\frac{2^{\mu}\Gamma(\mu)}{\pi\left|\xi\right|^{\mu}}C_{2}(\xi)\Rightarrow\widehat{U}(0,\xi)=-\frac{2^{\mu}\Gamma(\mu)C_{2}(\xi)}{\pi\left|\xi\right|^{\mu}},$
on obtient
$C_{2}(\xi)=-\frac{\pi\left|\xi\right|^{\mu}\widehat{F}(\xi)}{2^{\mu}\Gamma(\mu)}$,
soit $Z=\left|\xi\right|t$ on a
$\frac{\partial}{\partial
t}\widehat{U}(t,\xi)=\left|\xi\right|^{1-\mu}C_{1}(\xi)Z^{\mu}J_{\mu-1}(Z)-\frac{\pi\left|\xi\right|\widehat{F}(\xi)}{2^{\mu}\Gamma(\mu)}Z^{\mu}Y_{\mu-1}(Z)$
$=\left|\xi\right|^{1-\mu}C_{1}(\xi)Z^{\mu}\left\\{\cos[(1-\mu)\pi]J_{1-\mu}(Z)-\sin[(1-\mu)\pi]Y_{1-\mu}(Z)\right\\}\\\
-\frac{\pi\left|\xi\right|\widehat{F}(\xi)}{2^{\mu}\Gamma(\mu)}Z^{\mu}\left\\{\sin[(1-\mu)\pi]J_{1-\mu}(Z)+\cos[(1-\mu)\pi]Y_{1-\mu}(Z)\right\\}$,
$\frac{\partial}{\partial
t}\widehat{U}(t,\xi)\approx\frac{t}{2^{1-\mu}\Gamma(2-\mu)}\left\\{\cos[(1-\mu)\pi]\left|\xi\right|^{2-\mu}C_{1}(\xi)-\frac{\pi\sin[(1-\mu)\pi]}{2^{\mu}\Gamma(\mu)}\left|\xi\right|^{2}\widehat{F}(\xi)\right\\}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{2^{1-\mu}\Gamma(1-\mu)}{\pi}t^{2\mu-1}\left\\{\sin[(1-\mu)\pi]\left|\xi\right|^{\mu}C_{1}(\xi)+\frac{\pi\cos[(1-\mu)\pi]}{2^{\mu}\Gamma(\mu)}\left|\xi\right|^{2\mu}\widehat{F}(\xi)\right\\}$,
$\lim_{t\rightarrow 0}t^{1-2\mu}\frac{\partial}{\partial
t}\widehat{U}(t,\xi)=\\\
\frac{2^{1-\mu}\Gamma(1-\mu)}{\pi}\left\\{\sin[(1-\mu)\pi]\left|\xi\right|^{\mu}C_{1}(\xi)+\frac{\pi\cos[(1-\mu)\pi]}{2^{\mu}\Gamma(\mu)}\left|\xi\right|^{2\mu}\widehat{F}(\xi)\right\\}=\widehat{G}(\xi)$
$\Rightarrow
C_{1}(\xi)=-\frac{\pi\left|\xi\right|^{\mu}\widehat{F}(\xi)}{2^{\mu}\tan[(1-\mu)\pi]\Gamma(\mu)}+\frac{\pi\left|\xi\right|^{-\mu}\widehat{G}(\xi)}{2^{1-\mu}\sin[(1-\mu)\pi]\Gamma(1-\mu)}~{}~{}~{}~{}$,
par suite on a
$\widehat{U}(t,\xi)=\\\ \\\
\left\\{-\frac{\pi\left|\xi\right|^{\mu}\widehat{F}(\xi)}{2^{\mu}\tan[(1-\mu)\pi]\Gamma(\mu)}+\frac{\pi\left|\xi\right|^{-\mu}\widehat{G}(\xi)}{2^{1-\mu}\sin[(1-\mu)\pi]\Gamma(1-\mu)}\right\\}t^{\mu}J_{\mu}(\left|\xi\right|t)-\frac{\pi\left|\xi\right|^{\mu}\widehat{F}(\xi)}{2^{\mu}\Gamma(\mu)}t^{\mu}Y_{\mu}(\left|\xi\right|t)\\\
\\\
=\frac{\pi\left|\xi\right|^{-\mu}\widehat{G}(\xi)}{2^{1-\mu}\sin[(1-\mu)\pi]\Gamma(1-\mu)}t^{\mu}J_{\mu}(\left|\xi\right|t)-\frac{\pi\left|\xi\right|^{\mu}\widehat{F}(\xi)}{2^{\mu}\Gamma(\mu)}t^{\mu}\left\\{\frac{1}{\tan[(1-\mu)\pi]}J_{\mu}(\left|\xi\right|t)+Y_{\mu}(\left|\xi\right|t)\right\\}$,
or
$\frac{1}{\tan[(1-\mu)\pi]}J_{\mu}(\left|\xi\right|t)+Y_{\mu}(\left|\xi\right|t)=-\frac{J_{-\mu}(\left|\xi\right|t)}{\sin[(1-\mu)\pi]}~{}\mbox{et}~{}\Gamma(\mu)\Gamma(1-\mu)=\frac{\pi}{\sin[(1-\mu)\pi]},$
alors
$\widehat{U}(t,\xi)=2^{-\mu}\Gamma(1-\mu)t^{\mu}\left|\xi\right|^{\mu}J_{-\mu}(\left|\xi\right|t)\widehat{F}(\xi)+2^{\mu-1}\Gamma(\mu)t^{\mu}\left|\xi\right|^{-\mu}J_{\mu}(\left|\xi\right|t)\widehat{G}(\xi)$,
donc
$\widehat{U}(t,\xi)=2^{-\mu}\i^{2q}\Gamma(1-\mu)t^{\mu}\left|\xi\right|^{2q+\mu}J_{-\mu}(\left|\xi\right|t)\widehat{f}(\xi)+2^{\mu-1}\i^{2q}\Gamma(\mu)t^{\mu}\left|\xi\right|^{2q-\mu}J_{\mu}(\left|\xi\right|t)\widehat{g}(\xi).$
La transformation inverse de Fourier, l’interversion des intégrales et le
lemme 3 nous donnent le résultat du théorème 2.
Remarque. On justifie l’interversion des intégrales à l’aide de Fubini, car
les intégrales qui représentent les noyaux convergent absolument
(voir le lemme 3 et les comportements asymptotiques des fonctions de Bessel).
4\. ÉQUATION RADIALE D’EULER-POISSON-DARBOUX.
\- Preuve du théorème 3.
Soit $\varphi$ une solution de l’équation $(E_{1})$, remarquons que si
$\varphi(t,x)=t^{2\mu}k_{-\mu}(t,x)$ alors
$[\frac{\partial^{2}}{\partial x^{2}}+\frac{1-2\nu}{x}\frac{\partial}{\partial
x}]k_{-\mu}(t,x)=[\frac{\partial^{2}}{\partial
t^{2}}+\frac{1+2\mu}{t}\frac{\partial}{\partial t}]k_{-\mu}(t,x)~{}(4.1),$
donc il suffit de montrer que le noyaux $K_{\mu}(t,x,x^{\prime})$ vérifie
l’équation $(E_{1})$, pour cela on fait le changement des fonctions
$\varphi(t,x)=x^{\nu+\mu-1}\psi(t,x)$, on obtient
$[\frac{\partial^{2}}{\partial x^{2}}+\frac{2\mu-1}{x}\frac{\partial}{\partial
x}+\frac{(\mu-1)^{2}-\nu^{2}}{x^{2}}]\psi(t,x)=[\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\mu}{t}\frac{\partial}{\partial t}]\psi(t,x)~{}~{}(4.2),$
on pose $\psi(t,x)=F(z)$ avec $z=\frac{x^{2}+x^{\prime
2}-t^{2}}{2xx^{\prime}}$ alors
$[(1-z^{2})\frac{\partial^{2}}{\partial
z^{2}}+(2\mu-3)z\frac{\partial}{\partial
z}+\nu^{2}-(\mu-1)^{2}]F(z)=0~{}~{}(4.3),$
finalement, pour $F(z)=(1-z^{2})^{\frac{\mu}{2}-\frac{1}{4}}G(z)$ on obtient
l’équation de Legendre $[7]~{}P~{}198$
$[(1-z^{2})\frac{\partial^{2}}{\partial z^{2}}-2z\frac{\partial}{\partial
z}+(\nu^{2}-\frac{1}{4})-\frac{(\frac{1}{2}-\mu)^{2}}{1-z^{2}}]G(z)=0~{}(4.4),$
dont deux solutions sont $P^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}(z)$ et
$Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}(z)$ où
$P^{\mu}_{\nu}(z)=\frac{1}{\Gamma(1-\mu)}(\frac{1+z}{1-z})^{\frac{\mu}{2}}~{}_{2}F_{1}(-\nu,\nu+1,1-\mu,\frac{1-z}{2})~{}\mbox{pour}~{}\left|z-1\right|<2$,
et
$Q^{\mu}_{\nu}(z)=e^{i\pi\mu}\frac{\sqrt{\pi}\Gamma(\nu+\mu+1)}{2^{\nu+1}\Gamma(\nu+\frac{3}{2})}(z^{2}-1)^{\frac{\mu}{2}}z^{-\nu-\mu-1}~{}_{2}F_{1}(\frac{\nu+\mu}{2}+1,\frac{\nu+\mu+1}{2},\nu+\frac{3}{2},\frac{1}{z^{2}})$
lorsque $\left|z\right|>1$.
Pour les conditions initiales on prend $t<x$, on obtient
$U(t,x)=\\\
\frac{2^{-2\mu-1}\Gamma(1-\mu)}{\sqrt{\pi}\Gamma(\frac{1}{2}-\mu)}t^{2\mu}\int^{x+t}_{x-t}f(x^{\prime})(xx^{\prime})^{\nu-\mu-1}X^{-\mu-\frac{1}{2}}~{}_{2}F_{1}(\frac{1}{2}-\nu,\frac{1}{2}+\nu,\frac{1}{2}-\mu,X)x^{\prime
1-2\nu}dx^{\prime}\\\
+\frac{4^{\mu-1}\Gamma(\mu)}{\sqrt{\pi}\Gamma(\frac{1}{2}+\mu)}\int^{x+t}_{x-t}g(x^{\prime})(xx^{\prime})^{\nu+\mu-1}X^{\mu-\frac{1}{2}}~{}_{2}F_{1}(\frac{1}{2}-\nu,\frac{1}{2}+\nu,\frac{1}{2}+\mu,X)x^{\prime
1-2\nu}dx^{\prime}$,
le changement des variables $x^{\prime}=x+ts$ donne
$U(t,x)=\frac{\Gamma(1-\mu)}{\sqrt{\pi}\Gamma(\frac{1}{2}-\mu)}\\\
\times\int^{1}_{-1}f(x+ts)x^{\nu-\frac{1}{2}}(x+ts)^{-\nu+\frac{1}{2}}(1-s^{2})^{-\mu-\frac{1}{2}}~{}_{2}F_{1}(\frac{1}{2}-\nu,\frac{1}{2}+\nu,\frac{1}{2}-\mu,\frac{t^{2}(1-s^{2})}{4x(x+ts)})ds\\\
+\frac{\Gamma(\mu)t^{2\mu}}{2\sqrt{\pi}\Gamma(\frac{1}{2}+\mu)}\\\
\times\int^{1}_{-1}g(x+ts)x^{\nu-\frac{1}{2}}(x+ts)^{-\nu+\frac{1}{2}}(1-s^{2})^{\mu-\frac{1}{2}}~{}_{2}F_{1}(\frac{1}{2}-\nu,\frac{1}{2}+\nu,\frac{1}{2}+\mu,\frac{t^{2}(1-s^{2})}{4x(x+ts)})ds$,
à la limite on obtient la première donnée initiale à savoir que
$\int^{1}_{-1}(1-s^{2})^{-\mu-\frac{1}{2}}ds=2^{-2\mu}B(\frac{1}{2}-\mu,\frac{1}{2}+\mu)=\frac{2^{-2\mu}[\Gamma(\frac{1}{2}-\mu)]^{2}}{\Gamma(1-2\mu)}=\frac{\sqrt{\pi}\Gamma(\frac{1}{2}-\mu)}{\Gamma(1-\mu)}.$
De même on obtient la deuxième donnée initiale.
\- Preuve du théorème 3 bis. D’après le principe de superposition, il suffit
d’étudier les problèmes de Cauchy [2]
$\Lambda^{\nu}_{x}U_{l}=\Lambda^{\mu}_{t}U_{l},~{}\Lambda^{\nu}_{x}=\Lambda_{x},~{}U_{l}(0,x)=x^{l},~{}\lim_{t\rightarrow
0}t^{1-2\mu}\frac{\partial}{\partial
t}U\left(t,x\right)=0~{}~{}~{}~{}(P_{1}).$
$\Lambda^{\nu}_{x}V_{l}=\Lambda^{\mu}_{t}V_{l},~{}V_{l}(0,x)=0,~{}\lim_{t\rightarrow
0}t^{1-2\mu}\frac{\partial}{\partial
t}V\left(t,x\right)=x^{l}~{}~{}~{}~{}(P_{2}).$
Pour résoudre $(P_{1})$, on pose $U_{l}=x^{l}\phi(Z)$ avec
$Z=\frac{t^{2}}{x^{2}}$ et $\left|Z\right|<1$, on obtient l’équation
$Z(1-Z)\frac{\partial^{2}\phi}{\partial
Z^{2}}+[1-\mu-(\nu-l+1)Z]\frac{\partial\phi}{\partial
Z}+\frac{l}{2}(\nu-\frac{l}{2})\phi=0~{}~{}(4.5).$
Or $1-\mu\notin Z$, la solution générale de $(4.5)$ s’écrit sous la forme
$[6]P248$
$\phi(Z)=A~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1-\mu,Z)+BZ^{\mu}~{}_{2}F_{1}(\mu-\frac{l}{2},\nu+\mu-\frac{l}{2},1+\mu,Z)$.
Les conditions initiales pour $U_{l}$ donnent $A=1$ et $B=0$, et par suite on
obtient
$~{}~{}~{}~{}~{}~{}U_{l}=x^{l}~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1-\mu,Z).$
De la même manière pour résoudre $(P_{2})$, on pose
$V_{l}=\frac{t^{2\mu}}{2\mu}x^{l}\psi(Z)$
avec $Z=\frac{t^{2}}{x^{2}}$ et $\left|Z\right|<1$, on obtient l’équation
$Z(1-Z)\frac{\partial^{2}\psi}{\partial
Z^{2}}+[1+\mu-(\nu-l+1)Z]\frac{\partial\phi}{\partial
Z}+\frac{l}{2}(\nu-\frac{l}{2})\phi=0~{}~{}(4.6),$
$1+\mu\notin Z$, la solution générale de $(4.6)$ s’écrit sous la forme
$\psi(Z)=A^{\prime}~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1+\mu,Z)+B^{\prime}Z^{-\mu}~{}_{2}F_{1}(-\mu-\frac{l}{2},\nu-\mu-\frac{l}{2},1-\mu,Z).$
Les conditions initiales pour $V_{l}$ donnent $A^{\prime}=1$ et
$B^{\prime}=0$, et par suite on obtient
$V_{l}=\frac{t^{2\mu}}{2\mu}x^{l}~{}_{2}F_{1}(-\frac{l}{2},\nu-\frac{l}{2},1+\mu,Z).$
\- Preuve du théorème 4. par une méthode analogue à celle procedée dans la
preuve du théorème 2 on obtient
$\widehat{U}(t,\lambda)=2^{-\mu}\Gamma(1-\mu)t^{\mu}\lambda^{\mu}J_{-\mu}(\lambda
t)\widehat{F}(\lambda)+2^{\mu-1}\Gamma(\mu)t^{\mu}\lambda^{-\mu}J_{\mu}(\lambda
t)\widehat{G}(\lambda).$
La transformation inverse de Fourier-Bessel-Hankel, l’interversion des
intégrales et le lemme 4 nous donnent le résultat du théorème 4.
PROPOSITION.
$U(t,x)=x^{\alpha}(x^{2}-t^{2})^{\beta}F_{4}(\frac{-\alpha}{2},\frac{-\alpha}{2}+\nu,1-\mu,\gamma,\frac{t^{2}}{x^{2}},\frac{(x^{2}-t^{2})^{2}}{x^{2}})$
vérifie l’équation $(E_{2})$ avec $\beta=\mu+\nu-\alpha-1$
Preuve. On rappelle d’abord que la fonction $F_{4}(a,b,c,d,x,y)$ vérifie le
système de deux équations [8]
$\left\\{\begin{array}[]{rr}y(1-y)\frac{\partial^{2}}{\partial
y^{2}}-z^{2}\frac{\partial^{2}}{\partial
z^{2}}-2yz\frac{\partial^{2}}{\partial y\partial
z}+[c-(a+b+1)y]\frac{\partial}{\partial y}-(a+b+1)z\frac{\partial}{\partial
z}-ab=0~{}(1)\\\ z(1-z)\frac{\partial^{2}}{\partial
z^{2}}-y^{2}\frac{\partial^{2}}{\partial
y^{2}}-2yz\frac{\partial^{2}}{\partial y\partial
z}+[d-(a+b+1)z]\frac{\partial}{\partial z}-(a+b+1)y\frac{\partial}{\partial
y}-ab=0~{}(2).\\\ \end{array}\right.$
On cherche maintenant une solution de $(E_{2})$ sous la forme
$V(t,x)=x^{\alpha}(x^{2}-t^{2})^{\beta}W(t,x)$, on obtient que $W$ vérifie
l’équation
$x^{2}\frac{\partial^{2}W}{\partial
x^{2}}+(1-2\nu+2\alpha+4\beta\frac{x^{2}}{x^{2}-t^{2}})x\frac{\partial
W}{\partial x}=\\\ x^{2}\frac{\partial^{2}W}{\partial
t^{2}}+(\frac{1-2\mu}{t}-4\beta\frac{t}{x^{2}-t^{2}})x^{2}\frac{\partial
W}{\partial t}+\alpha(2\nu-\alpha)W~{}~{}(4.7)$;
en posant $W(t,x)=F(y,z)$ avec $y=\frac{t^{2}}{x^{2}}$ et
$z=\frac{(x^{2}-t^{2})^{2}}{x^{2}}$ on obtient que $F$ vérifie l’équation
$(1)$ du système.
5\. APPLICATIONS ET PERSPECTIVES.
Corollaire 1 ( Équation des ondes en dimension $n$ [4] ).
Pour $\mu\rightarrow\frac{1}{2}$ dans le théorème 1, on retrouve la solution
du problème de Cauchy pour l’équation des ondes classique en dimension $n$
$U(t,x)=b(N)\frac{\partial}{\partial t}(\frac{1}{t}\frac{\partial}{\partial
t})^{N-1}[t^{2N-1}\int_{\left\\{\left|y\right|=1\right\\}}\Phi(x-ty)d\sigma(y)]+\\\
\\\ b(N)(\frac{1}{t}\frac{\partial}{\partial
t})^{N-1}[t^{2N-1}\int_{\left\\{\left|y\right|=1\right\\}}\Psi(x-ty)d\sigma(y)]$
si $n$ est impair $(n=2N+1)$
où
$b(N)=2^{-1}[1.3.5...(2N-1)]^{-1}\pi^{-N-\frac{1}{2}}\Gamma(N+\frac{1}{2})=\frac{1}{2(2\pi)^{N}}$
et $d\sigma(y)$ est la mesure de surface $\left\\{\left|y\right|=1\right\\}$ ,
$U(t,x)=2b(N)\frac{\partial}{\partial t}(\frac{1}{t}\frac{\partial}{\partial
t})^{N-1}[t^{2N-1}\int_{\left\\{\left|y\right|<1\right\\}}\frac{\Phi(x-ty)}{\sqrt{1-\left|y\right|^{2}}}dy]+\\\
\\\ 2b(N)(\frac{1}{t}\frac{\partial}{\partial
t})^{N-1}[t^{2N-1}\int_{\left\\{\left|y\right|<1\right\\}}\frac{\Psi(x-ty)}{\sqrt{1-\left|y\right|^{2}}}dy]$
si $n$ est pair $(n=2N)$.
Preuve. On distingue deux cas:
\- Cas $n$ impair $(n=2N+1)$. $U(t,x)=I_{1}(t,x)+J_{1}(t,x),\\\
I_{1}(t,x)=\frac{\Gamma(\frac{1}{2})}{2^{N}\pi^{N+\frac{1}{2}}}\lim_{\mu\rightarrow\frac{1}{2}}\frac{1}{\Gamma(\frac{1}{2}-\mu)}t(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{-\mu-\frac{1}{2}}dx^{\prime},\\\
et~{}J_{1}(t,x)=\frac{1}{2(2\pi)^{N}}\left(\frac{\partial}{t\partial
t}\right)^{N}\int_{\left|x^{\prime}-x\right|<t}g\left(x^{\prime}\right)dx^{\prime}$,
$I_{1}(t,x)=\frac{1}{2(2\pi)^{N}}\lim_{\mu\rightarrow\frac{1}{2}}t(\frac{\partial}{t\partial
t})^{N}\frac{1}{t}\frac{\partial}{\partial
t}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\frac{1}{2}-\mu}dx^{\prime}\\\
=\frac{1}{2(2\pi)^{N}}\frac{\partial}{\partial t}(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)dx^{\prime}$
et$(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)dx^{\prime}=(\frac{\partial}{t\partial
t})^{N-1}\frac{1}{t}\frac{\partial}{\partial
t}\left\\{t^{2N+1}\int^{1}_{0}[\int_{\left|y\right|=1}f\left(x-tsy\right)d\sigma(y)]s^{2N}ds\right\\}$
et$\frac{1}{t}\frac{\partial}{\partial
t}\left\\{t^{2N+1}\int^{1}_{0}[\int_{\left|y\right|=1}f\left(x-tsy\right)d\sigma(y)]s^{2N}ds\right\\}=\\\
(2N+1)t^{2N-1}\int^{1}_{0}[\int_{\left|y\right|=1}f\left(x-tsy\right)d\sigma(y)]s^{2N}ds\\\
-t^{2N}\int^{1}_{0}[\int_{\left|y\right|=1}f^{\prime}\left(x-tsy\right)yd\sigma(y)]s^{2N+1}ds$
et$(2N+1)t^{2N-1}\int^{1}_{0}[\int_{\left|y\right|=1}f\left(x-tsy\right)d\sigma(y)]s^{2N}ds=\\\
t^{2N-1}\left\\{[s^{2N+1}\int_{\left|y\right|=1}f\left(x-tsy\right)d\sigma(y)]^{1}_{0}+t\int^{1}_{0}[\int_{\left|y\right|=1}f^{\prime}\left(x-tsy\right)yd\sigma(y)]s^{2N+1}ds\right\\}$,
donc $I_{1}(t,x)=\frac{1}{2(2\pi)^{N}}\frac{\partial}{\partial
t}(\frac{\partial}{t\partial
t})^{N-1}\left\\{t^{2N-1}\int_{\left|y\right|=1}f\left(x-ty\right)d\sigma(y)\right\\},\\\
et~{}J_{1}(t,x)=\frac{1}{2(2\pi)^{N}}(\frac{\partial}{t\partial
t})^{N-1}\left\\{t^{2N-1}\int_{\left|y\right|=1}g\left(x-ty\right)d\sigma(y)\right\\}$.
\- Cas $n$ pair $(n=2N)$. $U(t,x)=I_{2}(t,x)+J_{2}(t,x),\\\
I_{2}(t,x)=\frac{1}{(2\pi)^{N}}t(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{-\frac{1}{2}}dx^{\prime},\\\
et~{}J_{2}(t,x)=\frac{1}{(2\pi)^{N}}(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}g\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\frac{1}{2}}dx^{\prime}$,
$I_{2}(t,x)=\frac{1}{(2\pi)^{N}}t(\frac{\partial}{t\partial
t})^{N}\frac{1}{t}\frac{\partial}{\partial
t}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\frac{1}{2}}dx^{\prime}\\\
=\frac{1}{(2\pi)^{N}}\frac{\partial}{\partial t}(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\frac{1}{2}}dx^{\prime}$
et $(\frac{\partial}{t\partial
t})^{N}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{\frac{1}{2}}dx^{\prime}=\\\
(\frac{\partial}{t\partial
t})^{N-1}\int_{\left|x^{\prime}-x\right|<t}f\left(x^{\prime}\right)\left(t^{2}-\left|x^{\prime}-x\right|^{2}\right)^{-\frac{1}{2}}dx^{\prime}$,
donc $I_{2}(t,x)=\frac{1}{(2\pi)^{N}}\frac{\partial}{\partial
t}(\frac{\partial}{t\partial
t})^{N-1}\left\\{t^{2N-1}\int_{\left|y\right|<1}f\left(x-ty\right)\left(1-\left|y\right|^{2}\right)^{-\frac{1}{2}}dy\right\\}$
et $J_{2}(t,x)=\frac{1}{(2\pi)^{N}}(\frac{\partial}{t\partial
t})^{N-1}\left\\{t^{2N-1}\int_{\left|y\right|<1}g\left(x-ty\right)\left(1-\left|y\right|^{2}\right)^{-\frac{1}{2}}dy\right\\}$.
Corollaire 2 ( Théorème 1.1 [4] ).
Pour $\nu=-\alpha$ et $\mu\rightarrow\frac{1}{2}$ dans le théorème 3, on
retrouve la solution du problème de Cauchy pour l’équation radiale des ondes
$U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{+\infty}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
+\left\\{\begin{array}[]{rr}\frac{1}{2}x^{-\alpha-\frac{1}{2}}[f(x-t)(x-t)^{\frac{1}{2}+\alpha}+f(x+t)(x+t)^{\frac{1}{2}+\alpha}]~{}\mbox{pour}~{}t<x\\\
\frac{1}{2}x^{-\alpha-\frac{1}{2}}[-\sin\pi\alpha.f(t-x)(t-x)^{\frac{1}{2}+\alpha}+f(t+x)(t+x)^{\frac{1}{2}+\alpha}]~{}\mbox{pour}~{}x<t\\\
\end{array}\right.$
où $K(t,x,x^{\prime})=K_{\frac{1}{2}}(t,x,x^{\prime})x^{\prime 1+2\alpha}=\\\
\left\\{\begin{array}[]{rrrr}0~{}~{}~{}\mbox{pour}~{}0<x^{\prime}<x-t~{}\mbox{ou}~{}x^{\prime}>x+t,\\\
\frac{1}{2}x^{-\alpha-\frac{1}{2}}x^{\prime\frac{1}{2}+\alpha}~{}_{2}F_{1}(\frac{1}{2}-\alpha,\frac{1}{2}+\alpha,1,\frac{t^{2}-(x^{\prime}-x)^{2}}{4xx^{\prime}})~{}\mbox{pour}~{}\left|x-t\right|<x^{\prime}<x+t,\\\
\frac{2^{-2\alpha-1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\alpha)\Gamma(\alpha+1)}x^{-\alpha-\frac{1}{2}}x^{\prime\alpha+\frac{1}{2}}(\frac{4xx^{\prime}}{t^{2}-(x^{\prime}-x)^{2}})^{\alpha+\frac{1}{2}}\\\
\times{2}F_{1}(\alpha+\frac{1}{2},\alpha+\frac{1}{2},2\alpha+1,\frac{4xx^{\prime}}{t^{2}-(x^{\prime}-x)^{2}})~{}\mbox{pour}~{}0<x^{\prime}<t-x.\\\
\end{array}\right.$
Preuve. D’après les relations $[7]P41$
$\frac{d}{dX}[X^{c-1}~{}_{2}F_{1}(a,b,c,X)]=(c-1)X^{c-2}~{}_{2}F_{1}(a+1,b,c-1,X)$
et
$\frac{d}{dY}[Y^{a}~{}_{2}F_{1}(a,b,c,Y)]=aY^{a-1}~{}_{2}F_{1}(a+1,b,c,Y)~{}~{}~{}~{}~{}~{}$,
on obtient
$U(t,x)=\lim_{\mu\rightarrow\frac{1}{2}}\frac{4^{-\mu-\frac{1}{2}}\Gamma(1-\mu)}{\sqrt{\pi}\Gamma(\frac{3}{2}-\mu)}t^{2\mu}\\\
\times\int^{x+t}_{\left|x-t\right|}f(x^{\prime})(xx^{\prime})^{-\alpha-\mu-1}\frac{d}{dX}[X^{\frac{1}{2}-\mu}~{}_{2}F_{1}(\frac{1}{2}-\alpha,\frac{1}{2}+\alpha,\frac{3}{2}-\mu,X)]x^{\prime
1+2\alpha}dx^{\prime}\\\ +[\int^{t-x}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}~{}si~{}t>x]\\\
+\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime},~{}X=\frac{1-z}{2}$
on distingue deux cas:
\- Pour t $<$ x, on obtient
$U(t,x)=\lim_{\mu\rightarrow\frac{1}{2}}\frac{4^{-\mu-\frac{1}{2}}\Gamma(1-\mu)}{\sqrt{\pi}\Gamma(\frac{3}{2}-\mu)}t^{2\mu}x^{-\alpha-\mu-1}\\\
\times\int^{x+t}_{x-t}f(x^{\prime})x^{\prime\alpha-\mu}X^{\frac{1}{2}-\mu}\frac{d}{dX}~{}_{2}F_{1}(\frac{1}{2}-\alpha,\frac{1}{2}+\alpha,\frac{3}{2}-\mu,X)dx^{\prime}\\\
+\lim_{\mu\rightarrow\frac{1}{2}}\frac{4^{-\mu-\frac{1}{2}}\Gamma(1-\mu)(\frac{1}{2}-\mu)}{\sqrt{\pi}\Gamma(\frac{3}{2}-\mu)}t^{2\mu}x^{-\alpha-\mu-1}\\\
\times\int^{x+t}_{x-t}f(x^{\prime})x^{\prime\alpha-\mu}X^{-\frac{1}{2}-\mu}~{}_{2}F_{1}(\frac{1}{2}-\alpha,\frac{1}{2}+\alpha,\frac{3}{2}-\mu,X)dx^{\prime}+\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}$,
alors
$~{}~{}~{}~{}U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{+\infty}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
+\lim_{\mu\rightarrow\frac{1}{2}}\frac{\Gamma(1-\mu)}{2\sqrt{\pi}\Gamma(\frac{3}{2}-\mu)}t^{2\mu}x^{-\alpha-\frac{1}{2}}\\\
\times\int^{x+t}_{x-t}\frac{f(x^{\prime})x^{\prime\alpha+\frac{1}{2}}}{x-x^{\prime}}~{}_{2}F_{1}(\frac{1}{2}-\alpha,\frac{1}{2}+\alpha,\frac{3}{2}-\mu,X)\frac{d}{dx^{\prime}}[t^{2}-(x^{\prime}-x)^{2}]^{\frac{1}{2}-\mu}dx^{\prime}$,
une intégration par parties montre que la valeur de la dernière limite est
$\frac{1}{2}x^{-\alpha-\frac{1}{2}}[f(x-t)(x-t)^{\frac{1}{2}+\alpha}+f(x+t)(x+t)^{\frac{1}{2}+\alpha}]~{}~{}~{}~{}$,
d’où
$U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{+\infty}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
+\frac{1}{2}x^{-\alpha-\frac{1}{2}}[f(x-t)(x-t)^{\frac{1}{2}+\alpha}+f(x+t)(x+t)^{\frac{1}{2}+\alpha}]$.
\- Pour x $<$ t, le changement des variables
$x^{\prime}=\sqrt{t^{2}-(1-z^{2})x^{2}}+zx,$ donne
$U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{t-x}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
-\frac{1}{2}tx^{-\alpha-\frac{1}{2}}\lim_{\mu\rightarrow\frac{1}{2}}\int^{1}_{-1}\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}\frac{d}{dz}[(1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}P^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)]dz$,
en utilisant la relation $[7]~{}P167$
$P^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)=\frac{1}{\cos(\mu-\frac{1}{2})\pi}[\frac{\Gamma(\alpha+\mu)}{\Gamma(\alpha+1-\mu)}P^{\frac{1}{2}-\mu}_{\alpha-\frac{1}{2}}(z)+\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)],$
on voit que la valeur de la dernière limite est
$\int^{t+x}_{t-x}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
-\frac{1}{2}tx^{-\alpha-\frac{1}{2}}lim_{\mu\rightarrow\frac{1}{2}}\int^{1}_{-1}\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}\frac{d}{dz}[(1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}(\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z))]dz$,
soit alors
$U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{+\infty}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
-\frac{1}{2}tx^{-\alpha-\frac{1}{2}}lim_{\mu\rightarrow\frac{1}{2}}[\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}(1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)]^{1}_{-1}\\\
+\frac{1}{2}tx^{-\alpha-\frac{1}{2}}lim_{\mu\rightarrow\frac{1}{2}}\int^{1}_{-1}\frac{d}{dz}[\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}](1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)dz$,
d’après le comportement asymptotique de la fonction
$Q^{\mu}_{\nu}~{}[7]~{}P196-197$
$Q^{\mu}_{\nu}(z)\approx
2^{-1-\frac{1}{2}\mu}\Gamma(-\mu)\frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu-\mu+1)}(1-z)^{\frac{\mu}{2}}~{}pour~{}z\approx
1,~{}\mu<0,\\\ Q^{\mu}_{\nu}(z)\approx
2^{-1-\frac{1}{2}\mu}\Gamma(-\mu)\cos[\pi(\nu+\mu)]\frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu-\mu+1)}(1+z)^{\frac{\mu}{2}}~{}pour~{}z\approx-1,~{}\mu<0$,
on a
$~{}~{}~{}~{}~{}~{}[\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}(1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)]^{1}_{-1}=\\\
\frac{1}{2}x^{-\alpha-\frac{1}{2}}[-\sin\pi\alpha.f(t-x)(t-x)^{\frac{1}{2}+\alpha}+f(t+x)(t+x)^{\frac{1}{2}+\alpha}]$;
et d’après le théorème de convergence dominée de Lebesgue on a
$\lim_{\mu\rightarrow\frac{1}{2}}\int^{1}_{-1}\frac{d}{dz}[\frac{f(x^{\prime})x^{\prime\frac{1}{2}+\alpha}}{\sqrt{t^{2}-(1-z^{2})x^{2}}}](1-z^{2})^{\frac{1}{4}-\frac{\mu}{2}}\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)dz=0$
à savoir que
$\frac{2}{\Gamma(\mu-\frac{1}{2})\Gamma(\frac{3}{2}-\mu)}Q^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)=P^{\mu-\frac{1}{2}}_{\alpha-\frac{1}{2}}(z)-\frac{1}{\cos(\mu-\frac{1}{2})\pi}\frac{\Gamma(\alpha+\mu)}{\Gamma(\alpha+1-\mu)}P^{\frac{1}{2}-\mu}_{\alpha-\frac{1}{2}}(z)$
d’où
$~{}~{}~{}~{}U(t,x)=\int^{+\infty}_{0}g(x^{\prime})K(t,x,x^{\prime})dx^{\prime}+\int^{+\infty}_{0}f(x^{\prime})\frac{\partial}{\partial
t}K(t,x,x^{\prime})dx^{\prime}\\\
+\frac{1}{2}x^{-\alpha-\frac{1}{2}}[-\sin\pi\alpha.f(t-x)(t-x)^{\frac{1}{2}+\alpha}+f(t+x)(t+x)^{\frac{1}{2}+\alpha}]$.
Corollaire 3 ( Théorème 2.1.1 [2] ).
Pour $\mu=\frac{1}{2},~{}\nu=\frac{k+1}{2}$ dans le théorème 3 bis, on
retrouve la solution exacte de l’équation homogène d’Euler-Poison-Darboux
$U(t,x)=\sum^{\infty}_{l=0}a_{l}U_{l}+\sum^{\infty}_{l=0}b_{l}V_{l}$
où
$U_{l}=x^{l}~{}_{2}F_{1}(\frac{-l}{2},\frac{k+1-l}{2},\frac{1}{2},\frac{t^{2}}{x^{2}})$
et
$V_{l}=tx^{l}~{}_{2}F_{1}(\frac{-l}{2},\frac{k+1-l}{2},\frac{3}{2},\frac{t^{2}}{x^{2}})$.
Exemples.
1. Le problème $\left\\{\begin{array}[]{rr}(\frac{\partial^{2}}{\partial x^{2}}-\frac{3}{x}\frac{\partial}{\partial x})U(t,x)=\frac{\partial^{2}}{\partial t^{2}}U(t,x)\\\ U(0,x)=0,~{}~{}U_{t}(0,x)=x\\\ \end{array}\right.$
admet la solution unique $U(t,x)=t\sqrt{x^{2}-t^{2}}$.
2. Le problème $\left\\{\begin{array}[]{rr}(\frac{\partial^{2}}{\partial x^{2}}+\frac{1}{x}\frac{\partial}{\partial x})U(t,x)=\frac{\partial^{2}}{\partial t^{2}}U(t,x)\\\ U(0,x)=x,~{}~{}U_{t}(0,x)=0\\\ \end{array}\right.$
admet la solution unique $U(t,x)=\sqrt{x^{2}-t^{2}}+t\arcsin\frac{t}{x}$.
En perspective, on étudiera les équations d’Euler-Poisson-Darboux à conditions
modifiées dans les espaces hyperboliques et elliptiques.
6\. RÉFÉRENCES BIBLIOGRAPHIQUES.
${\bf[1]}$\- J. Barros-Neto: Hypergeometric functions and the Tricomi
operator, arXiv:math/0310480v1 [math.AP] 30 Oct 2003.
${\bf[2]}$\- A.Bentrad: Exact solutions for a different version of the
monhomogeneouse E-P-D equation.Complex variables and elliptic
equations,vol.51.No.3 March 2006,243-253.
${\bf[3]}$\- D.W.Bresters: On the equation of Euler-Poisson-Darboux.Siam
J.Math.Anal.1973 no.1, 31-41.
${\bf[4]}$\- L.Colzani: Radial solutions to the wave equation. Annali di
matematica 181, 25-54 (2002).
${\bf[5]}$\- I.S. Gradshteyn and I.M. Ryzhik: Table of Integrals, Series, and
Products; sixth edition. Academic press 2000.
${\bf[6]}$\- N.N.Lebedev: Special functions And their applications. Dover
Publications,Inc New york 1972.
${\bf[7]}$\- W. Magnus, F. Oberhettinger, and R. P. Soni: Formulas and
Theorems for the special Functions of Mathematical Physics, Springer-Verlag,
New York, 1966.
${\bf[8]}$\- Raimundas Vidunas: Specialization of Appell’s functions to
univariate hypergeometric functions. J. Math. Anal. Appl. 355 (2009) 145-163.
${\bf[9]}$\- A. Weinstein: On the wave equation and the equation of Euler-
Poisson, proc. Symposia Appl. Math, vol. 5, McGraw-Hill, New York, 1954, pp.
137-147.
Université Gaston Berger de Saint-Louis B.P: 234. Sénégal.
E-mail adresse: cheikh976@yahoo.fr.
Université de Nouakchott Faculté des sciences et techniques B.P: 5026.
Mauritanie.
E-mail adresse: mohamedvall.ouldmoustapha230@gmail.com.
|
arxiv-papers
| 2011-09-14T21:52:32 |
2024-09-04T02:49:22.158944
|
{
"license": "Public Domain",
"authors": "Cheikh Ould Mohamed El-hafed and Mohamed Vall Ould Moustapha",
"submitter": "Ould Moustapha Mohamed Vall",
"url": "https://arxiv.org/abs/1109.3215"
}
|
1109.3233
|
# Studies of multi-parton interactions in photon+jets events at D0
Dmitry Bandurin (for the DO Collaboration) Department of Physics, Florida
State University, Tallahassee, FL, USA
###### Abstract
We consider sample of inclusive $\gamma+{\rm 3~{}jet}$ events collected by the
D0 experiment. The double parton fraction ($f_{\rm DP}$) and effective cross
section $\sigma_{\rm eff}$, a process-independent scale parameter related to
the parton density inside the nucleon, are measured in three intervals of the
second (ordered in $p_{T}$) jet transverse momentum $p_{T}^{\rm jet2}$ within
the $15\leq p_{T}^{\rm jet2}\leq 30$ GeV range. Also we measured cross
sections as a function of the angle in the plane transverse to the beam
direction between the transverse momentum ($p_{T}$) of the $\gamma+$leading
jet system and $p_{T}$ of the other jet for $\gamma+{\rm 2~{}jet}$, or $p_{T}$
sum of the two other jets for $\gamma+{\rm 3~{}jet}$ events. The results are
compared to different models of multiple parton interactions (MPI) in the
pythia and sherpa Monte Carlo (MC) generators.
## I Introduction
Many features of high energy inelastic hadron collisions depend directly on
the parton structure of hadrons. The inelastic scattering of nucleons occurs
mainly through a single parton-parton interaction but the contribution from
double (or multiple) parton collisions can be significant. Information about
DP rates is needed for understanding of nature of MPI events and correct
estimating background to many rare processes, especially with multi-jet final
state.
## II Double parton interactions in $\gamma$+3 jet events
The cross section of DP events production directly proportional to cross
sections of two processes A and B and should be normalized by some scaling
parameter $\sigma_{\rm eff}$ in cross section’s units.
$\displaystyle\sigma_{DP}\equiv\frac{\sigma^{A}\sigma^{B}}{\sigma_{\rm eff}}.$
(1)
In general sense, $\sigma_{\rm eff}$ is a factor which characterizes a size of
the effective interaction region.
We have used a sample of $\gamma+3$ jets events collected by the DO experiment
with an integrated luminosity of about 1 fb-1. The D0 detector is a general
purpose detector described in D0_det . The events should pass triggers based
on the identification of high $p_{T}$ cluster in the EM calorimeter with loose
shower shape requirements for photons. Jets are reconstructed using the D0 Run
II iterative midpoint cone algorithm Run2Cone with a cone size $0.7$. Each
event must contain at least one $\gamma$ in the rapidity region $|y|<1.0$ or
$1.5<|y|<2.5$ and at least three jets with $|y|<3.0$. Events are selected with
$\gamma$ transverse momentum $60<p^{\gamma}_{T}<80$ GeV, leading (in $p_{T}$)
jet $p_{T}>25$ GeV, while the next-to-leading (second) and third jets must
have $p_{T}>15$ GeV. The DP fractions and $\sigma_{\rm eff}$ are determined in
three $p_{T}^{\rm jet2}$ bins: 15–20, 20–25, and 25–30 GeV.
We use rates of double interactions (DI) from two separate $p\bar{p}$
collisions and DP from a single $p\bar{p}$ collision to extract $\sigma_{\rm
eff}$ from their ratio prd1 . The DI events differ from the DP events by the
fact that the second parton scattering happens at a separate $p\bar{p}$
collision vertex. Data events with a single $p\bar{p}$ collision vertex, which
compose the sample of DP candidates, are selected separately from events with
two vertices which compose a sample of DI candidates. A distinctive feature of
the DP events is a presence of two independent parton-parton scatterings
within the same $p\bar{p}$ collision. We define the variable sensitive to the
kinematics of the DP events:
$\displaystyle\Delta
S\equiv\Delta\phi\left(\vec{p}_{T}^{\gamma,jet1},~{}\vec{p}_{T}^{jet2,jet3}\right),$
(2)
where $\Delta\phi$ is an azimuthal angle between the $p_{T}$ vectors of the
total transverse momenta of the two two-body systems,
$\vec{p}_{T}^{\gamma,jet1}$ and $\vec{p}_{T}^{jet2,jet3}$, in $\gamma+{\rm
3~{}jet}$ events. This angle is schematically shown in Fig. 1.
Figure 1: A possible orientation of photon and jets transverse momenta vectors
in $\gamma+{\rm 3~{}jet}$ events. Vectors $\vec{P}_{T}^{1}$ and
$\vec{P}_{T}^{2}$ are the $p_{T}$ imbalance vectors of $\gamma$+jet and jet-
jet pairs. The figure illustrates a general case for the production of
$\gamma$+3 jets +X events.
We consider the data-driven method to extract the DP fractions $f_{\rm DP}$.
Specifically, we consider data in two adjacent $p_{T}$ intervals of the second
jet. The distribution for $\Delta S$ variable in data can be expressed as a
sum of signal and background distributions. If we known properties of data and
DP model, the only unknown parameter is the fraction of DP events in one
$p_{T}^{\rm jet2}$ bin. It is obtained from a minimization. The found $f_{\rm
DP}$ values with total uncertainties are $0.466\pm 0.041$ for $15<p_{T}^{\rm
jet2}<20$ GeV, $0.334\pm 0.023$ for $20<p_{T}^{\rm jet2}<25$ GeV, and
$0.235\pm 0.027$ for $25<p_{T}^{\rm jet2}<30$ GeV. They are shown in Fig. 2
(three sets of the points correspond to three possible definitions for the
$\Delta S$ variable prd1 ). The values of $\sigma_{\rm eff}$ are shown in Fig.
3. The main systematic uncertainty are caused by determinations of the DI and
DP fractions giving a total systematic uncertainty of $(20.5-32.2)\%$. The
obtained $\sigma_{\rm eff}$ values in different $p_{T}^{\rm jet2}$ bins agree
with each other within their uncertainties and highly uncorrelated, and are
used to calculate the average value:
$\sigma_{\rm eff}^{\rm ave}=16.4\pm 0.3(\rm stat)\pm 2.3(\rm syst){~{}~{}\rm
mb}.$ (3)
This average value is in the range of those found in previous measurements AFS
; UA2 ; CDF93 ; CDF97 performed at different energy scales of parton
interactions.
Figure 2: Fractions of $\gamma+{\rm 3~{}jet}$events with double parton
interactions in the three $p_{T}^{\rm jet2}$ intervals.
Figure 3: Effective cross section $\sigma_{\rm eff}$ (mb) measured in the
three $p_{T}^{\rm jet2}$ intervals.
## III Azimuthal decorrelations and multiple parton interactions in
$\gamma+{\rm 2~{}jet}$ and $\gamma+{\rm 3~{}jet}$ events in $p\bar{p}$
collisions
Samples of $\gamma+2(3)$ jets events with the same cuts as prd1 are
considered. The next modifications are applied: each event must contain at
least one $\gamma$ in the pseudorapidity region $|y|<1.0$ or $1.5<|y|<2.5$ and
at least two (or three) jets with $|y|<3.5$. Events are selected with $\gamma$
transverse momentum $50<p^{\gamma}_{T}<90$ GeV, leading jet $p_{T}>30$ GeV,
and the second jet $p_{T}>15$ GeV. If there is a third jet with $p_{T}>15$ GeV
that passes the selection criteria, the event is also considered for the
$\gamma+3$ jet analysis. By measuring differential cross sections vs. the
azimuthal angles in $\gamma+2(3)$ jet events we can better tune MPI models in
events with high $p_{T}$ jets. We present the four measurements of normalized
differential cross sections, $\Delta S$ in a single bin $15<p_{T}^{\rm
jet2}<30$ for $\gamma+{\rm 3~{}jet}$ events (see Fig. 1), and $\Delta\phi$ in
three $p_{T}^{\rm jet2}$ bins, 15-20, 20-25, and 25-30 GeV, for $\gamma+{\rm
2~{}jet}$ events. The $\Delta\phi$ is an angle between the $p_{T}$ vector
obtained by pairing the $\gamma$ and the leading jet $p_{T}$ vectors and the
second jet $p_{T}$ vector prd2 ). It is shown in Fig. 4. We consider a few MPI
models and two models without MPI simulated by pythia and sherpa MC
generators. Figure 5 shows the measured cross section for the two angular
variables $\Delta S$ (left plot) and $\Delta\phi$ (right plot). The data have
a good sensitivity to the various MPI models, which predictions vary
significantly and differ from each other by up to a factor 2 at small $\Delta
S$ and $\Delta\phi$, i.e. in the region where the relative DP contribution is
expected to be highest.
Figure 4: A possible orientation of photon and jets transverse momenta vectors
in $\gamma+{\rm 2~{}jet}$ events.
Figure 5: Left: Normalized differential cross section in the $\gamma+3$-jet
events, $(1/\sigma_{\gamma 3j})\sigma_{\gamma 3j}/d\Delta S$, in data compared
to MC models and the ratio of data over theory, only for models including MPI,
in the range $15<p_{T}^{jet2}<30$ GeV. Right: Normalized differential cross
section in $\gamma+2$-jet events, $(1/\sigma_{\gamma 2j})\sigma_{\gamma
2j}/d\Delta\phi$, in data compared to MC models and the ratio of data over
theory, only for models including MPI, in the range $15<p_{T}^{jet2}<20$ GeV.
From these two plots we may conclude that: (a) a large difference between
single parton-parton interaction (SP) models and data confirms a presence of
DP events in the data sample; (b) the data favor the predictions of the MPI
models with P0, S0 and Sherpa MPI tunes with $p_{T}$-ordered showers; (c) the
predictions from tune A and DW MPI models are disfavored. It is important that
our preferable choice of MPI models is stable for all our measurements.
In $\gamma+{\rm 2~{}jet}$ events in which the second jet is produced in the
additional independent parton interaction, the $\Delta\phi$ distribution
should be flat. Using this fact and also SP prediction for $\Delta\phi$ we can
get the DP fractions from a fit to data. The distributions in data, SP, and DP
models, as well as a sum of the SP and DP distributions, weighted with their
respective fractions for $15<p_{T}^{\rm jet2}<20$ GeV, are shown in the left
plot of Figure 6. The DP fractions in the $\gamma+{\rm 2~{}jet}$ samples
decrease in the bins of $p_{T}^{\rm jet2}$ as $(11.6\pm 1.0)\%$ for $15-20$
GeV, $(5.0\pm 1.2)\%$ for $20-25$ GeV, and $(2.2\pm 0.8)\%$ for $25-30$ GeV.
To determine the fractions as a function of $\Delta\phi$, we perform a fit in
the different $\Delta\phi$ regions by excluding the bins at high $\Delta\phi$.
We find that they grow significantly towards the smaller angles and are higher
for smaller $p_{T}^{\rm jet2}$ (right plot of Figure 6).
Figure 6: Left: the $\Delta\phi$ distribution in data, SP, and DP models, and
the sum of the SP and DP contributions weighted with their fractions for
$15<p_{T}^{\rm jet2}<20$ GeV. Right: the fractions of DP events with total
uncertainties in $\gamma+{\rm 2~{}jet}$final state as a function of the upper
limit on $\Delta\phi$ for the three $p_{T}^{\rm jet2}$ intervals.
We also estimate the fraction of $\gamma+{\rm 3~{}jet}$ events from triple
parton interactions (TP) in data as a function of $p_{T}^{\rm jet2}$. In
$\gamma+{\rm 3~{}jet}$ TP events, the three jets come from three different
parton interactions, one $\gamma+$ jet and two dijet final states. In each of
the two dijet events, one of the jets is either not reconstructed or below the
15 GeV $p_{T}$ selection threshold. The fractions of TP events in the
$\gamma+{\rm 3~{}jet}$ data have been estimated and are shown in Fig. 7. As we
see, they vary in the $p_{T}^{\rm jet2}$ bins as $(5.5\pm 1.1)\%$ for $15-20$
GeV, $(2.1\pm 0.6)\%$ for $20-25$ GeV, and $(0.9\pm 0.3)\%$ for $25-30$ GeV.
Figure 7: Fractions of $\gamma+{\rm 3~{}jet}$events with triple parton
interactions in the three $p_{T}^{\rm jet2}$ intervals.
## References
* (1) V.M. Abazov et al. (DØ Collaboration), Nucl. Instrum. Methods Phys. Res. A 565, 463 (2006).
* (2) G.C. Blazey et al., arXiv:hep-ex/0005012 (2000).
* (3) V. M. Abazov et al. (DØ Collaboration), Phys.Rev. D 81, 052012 (2010).
* (4) F. Abe et al. (CDF Collaboration), Phys. Rev. D 56, 3811 (1997).
* (5) T. Akesson et al. (AFS Collaboration), Z. Phys. C 34, 163 (1987).
* (6) J. Alitti et al. (UA2 Collaboration), Phys. Lett. B 268, 145 (1991).
* (7) F. Abe et al. (CDF Collaboration), Phys. Rev. D 47, 4857 (1993).
* (8) V. M. Abazov et al. (DØ Collaboration), Phys.Rev. D 83, 052008 (2011).
|
arxiv-papers
| 2011-09-15T00:27:26 |
2024-09-04T02:49:22.166806
|
{
"license": "Public Domain",
"authors": "Dmitry Bandurin (for the D0 Collaboration)",
"submitter": "Dmitry Bandurin V",
"url": "https://arxiv.org/abs/1109.3233"
}
|
1109.3253
|
††thanks: On behalf of the Daya Bay Collaboration
# Daya Bay Neutrino Experiment: Goal, Progress and Schedule
Zhe Wang Physics Department, Brookhaven National Laboratory, USA
###### Abstract
Daya Bay Neutrino Experiment is dedicated to measuring the last unobserved
neutrino mixing angle $\theta_{13}$. The predicted precision on
$\sin^{2}2\theta_{13}$ is 0.01 at 90% confidence level. This document briefly
reviews the measurement method and detector construction status. The first two
anti-neutrino detectors’ dry run result is also discussed. The Daya Bay near
hall data taking is expected to commence in the summer of 2011 and the data
taking of all of the three halls in the summer of 2012.
## I Introduction
Based on a three-generation assumption, a $3\times 3$ mixing matrix was
proposed to explain the oscillation in neutrino propagation, which is usually
called Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix). Eq. 1 shows one
possible parametrization of the PMNS matrix which, for dirac neutrino
assumption, includes three mixing angles, $\theta_{12}$, $\theta_{23}$,
$\theta_{13}$ and one CP-violation phase angle, $\delta$.
$PMNS=\left[\begin{array}[]{ccc}1&0&0\\\ 0&\cos\theta_{23}&\sin\theta_{23}\\\
0&-\sin\theta_{23}&\cos\theta_{23}\end{array}\right]\left[\begin{array}[]{ccc}\cos\theta_{13}&0&\sin\theta_{13}e^{-i\delta}\\\
0&1&0\\\
-\sin\theta_{13}e^{i\delta}&0&\cos\theta_{13}\end{array}\right]\left[\begin{array}[]{ccc}\cos\theta_{12}&\sin\theta_{12}&0\\\
-\sin\theta_{12}&\cos\theta_{12}&0\\\ 0&0&1\end{array}\right]$ (1)
Like the mixing matrix to describe quark mixing, these parameters cannot be
predicted by theory, and their values must be determined by experimental
measurement. A global fit global_fit and citations therein show $\theta_{12}$
is close to 34o and $\theta_{23}$ is close to 45o. In the past Chooz Chooz
obtained the best limit of 0.17 in $\sin^{2}2\theta_{13}$ for $\Delta
m^{2}_{31}=2.5\times 10^{-3}eV^{2}$ at the 90% confidence level. The
particular pattern of the PMNS matrix, two big mixing angles and a small one,
is an intriguing picture, which might indicate some discrete family
symmetries, like the Tri-bimaximal mixing model Tri-bimaximal , etc. Besides
the interest of its value in theory, $\theta_{13}$ is also a critical input
for other neutrino experiments. As can be seen from Eq. 1, the CP-violation
phase term, $e^{i\delta}$ is always multiplied by $\sin\theta_{13}$. A zero
$\theta_{13}$ will make any CP-violation undetectable. $\theta_{13}$ also
affects the sensitivity to the neutrino mass hierarchy in future long baseline
neutrino experiment LongBaseline . $\theta_{13}$ is also present in the
effective electron neutrino mass term, which is usually the direct measurement
of neutrinoless double beta decay experiments. To really understand the
majorana mass of neutrino, a reasonable estimation of $\theta_{13}$ is always
necessary. Ultimately the unitarity of the PMNS matrix can be tested, i.e. the
3-flavor mixing framework, possibly leading to further exciting discoveries.
The Daya Bay Neutrino Experiment Dyb_arxiv exploits reactor generated anti-
electron-neutrinos, $\bar{\nu}_{e}$, with gadolinium (Gd) loaded liquid
scintillator (LS), measures their disappearance probability, then extracts the
mixing angle, $\theta_{13}$. The Daya Bay Experiment plans to measure
$\sin^{2}2\theta_{13}$ to a precision of 0.01 at 90% confidence level with
three years’ data taking.
In the second section the measurement method will be briefly reviewed. The
status of the detector construction is discussed in the third section. In the
fourth section the detector dry run results will be shown, then a short
summary and outlook are given in the last section.
## II Measurement Method
### II.1 Detection of anti-electron-neutrinos
Low energy $\bar{\nu}_{e}$, mostly less than 10MeV, are emitted by the fission
of nuclei in reactors. They can be detected by the inverse-beta-decay (IBD)
process, i.e. $\bar{\nu}_{e}$ are captured by the free protons (hydrogen) in
the target region of the Daya Bay detector.
$\bar{\nu}_{e}+p\rightarrow e^{+}+n$ (2)
where the energy loss and annihilation of $e^{+}$ gives a prompt signal. This
process then is followed by the capture of the neutron which gives a delayed
signal. At Daya Bay two dominant types of captures can happen. One type is
capture on Gd:
$n+Gd\rightarrow Gd^{*}\rightarrow Gd+\gamma^{\prime}s$ (3)
The capture process on Gd will release on average three or four photons, and
their total energy is about 8MeV. For liquid scintillator with 0.1% Gd in
mass, the average capture lifetime is 28$\mu s$. The other type of capture is
on hydrogen:
$n+p\rightarrow D+\gamma$ (4)
which will release a 2.2 MeV single gamma. Its average capture lifetime is
180$\mu s$.
### II.2 Precise Measurement of $\sin^{2}2\theta_{13}$
The concept of the measurement of $\sin^{2}2\theta_{13}$ at Daya Bay can be
described by a very short equation of the form:
$\frac{N_{f}}{N_{n}}=\left(\frac{N_{p,f}}{N_{p,n}}\right)\left(\frac{L_{n}}{L_{f}}\right)^{2}\left(\frac{\epsilon_{f}}{\epsilon_{n}}\right)\left[\frac{P_{sur}(E,L_{f})}{P_{sur}(E,L_{n})}\right]$
(5)
where $N_{f}$ and $N_{n}$ are measured numbers of IBD event at far and near
sites respectively. Then $N_{p,f}$ and $N_{p,n}$ are the numbers of free
protons, $L_{f}$ and $L_{n}$ are the baselines, $\epsilon_{f}$ and
$\epsilon_{n}$ the IBD event detection efficiencies for far and near site
respectively. The $P_{sur}(E,L_{f})$ indicates the survival probability of
$\bar{\nu}_{e}$ with energy, E, at baseline $L_{f}$. Similarly
$P_{sur}(E,L_{n})$ is for near site.
Earlier attempts to measure $\sin^{2}2\theta_{13}$ usually had only one
detector, measuring the $\bar{\nu}_{e}$ flux at some distance, then comparing
that flux to a prediction estimated based on the thermal power output of the
reactor. The intrinsic uncertainty in the neutrino flux prediction made on the
basis of thermal power is around 2-3% which is not sufficient for a precise
measurement of $\theta_{13}$. The Daya Bay Experiment has near detectors that
are close to the reactor and can absolutely calibrate the flux before
oscillation.
The discrepancy between near and far detectors needs to be minimized. The Daya
Bay Experiment is trying to make detector modules for every site identical in
all aspects from mechanical design, manufacture and liquid scintillator
synthesis. Many systematic errors can be canceled in this way. Detector
swapping is also planned to crosscheck their performance.
The large statistics given by the powerful reactor and high target mass is
another precondition for making a precise measurement.
All the modules are located underground. Cosmic ray muon flux is greatly
suppressed and in addition an active muon veto detector is deployed to
identify muon-induced background.
The Daya Bay Experiment plans to measure $\sin^{2}2\theta_{13}$ to a precision
of 0.01 at 90% confidence level with three years’ data taking. This can be
interpreted as follows: if the true value of $\sin^{2}2\theta_{13}$ is zero,
the final 90% coverage uncertainty of it contributed by all statistical and
systematic error sources is 0.01. The real setup of the Daya Bay Experiment is
a little more complicated than this simple two-detector situation. Details
about the sensitivity prediction of the Daya Bay Neutrino Experiment can be
found in Dyb_arxiv .
## III Detector Construction
### III.1 Site
The Daya Bay Nuclear Power Plant is a complex composed of three sites, all
located along the coast line. They are called “Daya Bay”, “Ling Ao I” and
“Ling Ao II” respectively. Each site includes two reactor cores. By the summer
of 2011 all reactors were put into commercial running. The total nominal
thermal power is 6$\times$2.9 GW, which is one of the most powerful nuclear
power plants in the world.
The Daya Bay Neutrino Experiment has three experimental halls. Two near halls,
”Daya Bay Near Hall” and ”Ling Ao Near Hall”, are used to calibrate the
$\bar{\nu}_{e}$ flux from reactors. A far hall, ”Far Hall”, is located at a
distance which has maximum sensitivity for $\bar{\nu}_{e}$ disappearance
probability detection. The baselines and overburdens of the three halls are
tabulated in Tab. 1. The simulation-estimated muon fluxes and IBD event rates
at three halls are also listed in Tab. 1.
Each near hall has two anti-neutrino detector (AD) modules. They are immerged
in an octagonal water pool serving as active muon veto. On the top of the
water pool is a layer of resistive plate chambers, RPCs, also serving as a
muon veto and giving better position information. The Far Hall has a similar
design, but to increase statistics, four ADs are needed and correspondingly
larger water pool and RPC module arrays.
Table 1: Target mass, baseline, overburden, estimated muon flux and IBD rate at three halls. | Daya Bay near hall | Ling Ao near hall | Far hall
---|---|---|---
Target mass | 40 tons | 40 tons | 80 tons
Baseline | about 360 m | about 500 m | about 1600 m
| (to Daya Bay reactors) | (to Ling Ao I, II reactors) | (to Daya Bay reactors)
| | | about 1900 m
| | | (to Ling Ao I, II reactors)
Overburden | 98 m | 112 m | 350 m
Estimated Muon flux | 1.2 Hz/m2 | 0.73 Hz/m2 | 0.04 Hz/m2
Estimated IBD rate | 840/day/module | 740/day/module | 90/day/module
### III.2 Anti-neutrino Detector
The anti-neutrino detector is a sealed stainless steel cylinder. It has a
three-layer internal structure. In the center is the Gd-loaded liquid
scintillator serving as the target region. The second layer is pure liquid
scintillator to catch the gamma energy leaking from the target region.
Scintillation light is detected by photo multiplier tubes, PMTs, that are
installed in the outer layer which is filled with mineral oil to shield
against the radioactivity from the stainless steel vessel, the PMT supporting
structure and the PMTs themselves. 192 PMTs are deployed in 8 rings in the
mineral oil region, along the cylinder wall, facing inward, to measure the
energy deposited in the AD. A sectional drawing is in Fig. 1. There are no
PMTs on the top and bottom, instead highly reflective panels are mounted to
increase the light collection efficiency. The inside of a fully populated AD
is shown in Fig. 2 before the AD lid and top reflector were put on. The two
inner acrylic vessels and some PMTs are visible.
Figure 1: AD sectional drawing. From inside to outside are the volumes for Gd-
loaded liquid scintillator, liquid scintillator and mineral oil. Eight rings
of PMTs are deployed along the outer stainless steel wall. On the top are the
three ACU units. Figure 2: The inside of a fully populated AD before top
reflector and lid were put on.
Three Auto-Calibration-Units (ACU) are installed on the AD lid. They can be
used to scan the detector response along three axes, the central axis, A, an
off-center axis in Gd-loaded target region, B, and another off-center axis in
the pure liquid scintillator region, C. The optional sources are LED,
germanium-68 and americium-carbon with cobalt-60.
The synthesis of 185 tons Gd-loaded liquid scintillator and 180 tons pure
liquid scintillator are finished, which is all needed for three halls. The Gd-
loaded liquid Scintillator, currently stored in 5 40-ton tanks, is continually
circulated and nitrogen purged. The precision of the target mass is ensured by
redundant mass measurements during AD filling, including a 20-ton calibrated
filling tank and a coriolis mass flowmeter. The uncertainty of the mass
measurement is only 4kg out of 20 tons. Long term monitoring of the optical
and scintillation properties shows all liquid sample are stable.
The first two ADs have finished construction and liquid filling and are
already locked down to their positions in the Daya Bay Near Hall (see Fig. 3).
### III.3 Muon Veto
The signal IBD event consists of a prompt and a delayed signal. The energy
range of the paired signals and the coincidence window are so special that
muons by themselves can hardly make a background. But the muon-induced
spallation background, neutrons and other long-lifetime heavy nuclei can be
very serious. An active muon veto detector is still needed to tag these muons.
During offline study the coincidence of muon and muon-induced background will
be reconstructed and their contribution will be calculated.
As previously mentioned the muon veto includes two parts, the water pool and
RPCs. PMTs are installed in the water pool to detect the cerenkov light
generated by cosmic ray muons in water. The water pool itself is separated
into two regions, inner and outer, by reflective tyvek sheets. This increases
the light collection efficiency and provides a redundancy in muon detection.
Fig. 3 shows the status of the Daya Bay Near Hall: Two ADs are installed.
Water is filled to its nominal level. The inner water pool is visible while
outer pool is completely blocked by white tyvek sheets. RPC modules are
assembled and installed in parallel with water pool construction. The RPC
modules will cover the whole top surface of the water pool to provide better
position information for part of the down-going muons.
Figure 3: Filled Daya Bay near hall water pool with two ADs installed.
### III.4 Electronics
All PMT electronics for the Daya Bay Near Hall running are installed and ready
to use. The PMT front end electronics can provide a timing accuracy of
1.5625ns. The shaping time for charge measurement is 300ns. Several types of
triggers can be issued determined by multiplicity, number of fired PMTs within
a time window, analog energy sum, external signals etc. All on-site clocks are
synchronized to a GPS signal, so the Daya Bay Experiment can serve as a
supernova observatory. The RPC electronics are still under commissioning and
will come online soon.
## IV AD Dry Run
Before the filling with liquid, all PMTs and other instruments in the ADs were
tested with the full data processing chain. Raw PMT signals were processed by
electronics, and the charge and time of each PMT hit were recorded if a
trigger command was issued. Data files were automatically transferred to two
major computing sites simultaneously, Lawrence Berkeley National Laboratory in
the US and the Institute of High Energy Physics in China, with a latency of
only a few minutes. Data quality and detector performance were monitored
automatically. The ADs and electronics show good results with these intensive
tests.
Some initial offline analyses were done to further understand the properties
of the ADs. In the following the dry run results for AD No. 1 and 2 will be
shown. A LED enclosed by a plastic diffuser ball was lowered into AD to study
the response of PMTs. Fig. 4 shows the result when one off-center LED (ACUB)
was flashing at the half height of the detector. What shown is the charge
distribution in ring and column (a projection of an unrolled AD), and each
spot is color-coded according to the total charge of a PMT in that run. Ring 0
is for 2-inch liquid monitor PMTs.
Figure 4: 192-PMT charge distribution in ring and column for off-center LED
source at half height of AD.
Light is more concentrated around ring 4, 5 and column 17 where PMTs are
closer to the LED diffuser ball. With no liquid filled the charge distribution
basically shows the geometric effect between LED and PMT positions. It is also
affected by the optical properties of all kinds of surfaces in the AD, i.e.
the surfaces of two acrylic vessel, top and bottom reflectors, etc. In Fig. 5
the charge ratio in each ring (total charge in one ring divided by total
charge in whole detector), is plotted for the same LED configuration. Results
of AD No. 1 and 2 and simulation are overlaid together, where good consistency
can be observed.
Figure 5: AD No. 1, 2 and Monte-Carlo charge ratio comparison.
Another indication of the identicalness of AD No. 1 and 2 is shown in Fig. 6.
When a high energy muon flies through the thin acrylic wall of the two inner
acrylic vessels, cerekov light is emitted which is bright enough to trigger
the detector. Since AD No. 1 and 2 were tested side by side, it is believed
that the cosmic ray muon within two detectors should have identical fluxes and
momentum spectra. In Fig. 6 the multiplicity of these muon events of AD No. 1
and 2 are plotted. They are again identical within uncertainty. Note that the
cut-off around 50 is caused by trigger threshold.
Figure 6: AD No. 1 and 2 multiplicity distribution comparison.
Besides all kinds of functionality tests for the ADs, a long stability run was
also arranged. All PMTs, high voltage supply, front end electronics, trigger
system and data acquisition system were run for over 72 hours. For example, in
Fig. 7 shows the average gain of 192 PMTs in one AD within this testing period
where PMT gain is presented in units of ADC which describes the final digital
output of the PMT signal after electronic shaping. The variance of the average
PMTs’ gain is less than 1%.
Figure 7: Average gain of 192 PMTs in a stability run.
## V Summary and Outlook
To achieve the challenging goal of measuring $\sin^{2}2\theta_{13}$ to a
precision 0.01 at 90% C.L., great efforts were put into the design and
construction of the Daya Bay Neutrino Experiment. The analysis of AD No. 1 and
2 dry run data shows the good status of the detectors. The beginning of data
taking with two ADs in the Daya Bay Near Hall is anticipated in the summer of
2011 and data taking with all eight ADs in three halls in the summer of 2012.
###### Acknowledgements.
It is a pleasure to thank the organizers of ”DPF 2011” for the interesting
conference at Brown University. I am also very grateful for all the people who
contributed to the Daya Bay Neutrino Experiment. At the same time I’d like to
thank Laurence Littenberg, Steven Kettell, David Jaffe and David Webber for
discussing the contents of this talk and a careful reading of the manuscript.
## References
* (1) Thomas Schwetz, Mariam Tortola, and J. W. F. Valle, arXiv:1103.0734v2 [hep-ph]
* (2) M. Apollonio et al., Eur. Phys. J. C 27, 331 (2003)
* (3) P.F. Harrison, D.H. Perkins and W.G. Scott, Phys. Lett. B 530 (2002) 167
* (4) Adam Bernstein, et. al., arXiv:0907.4183v2 [hep-ex], 16 Sep, 2005
* (5) Xinheng Guo, et. al., arXiv:0701029v1 [hep-ex], 15 Jan, 2007
|
arxiv-papers
| 2011-09-15T03:45:02 |
2024-09-04T02:49:22.172731
|
{
"license": "Public Domain",
"authors": "Zhe Wang (on behalf of the Daya Bay Collaboration)",
"submitter": "Zhe Wang",
"url": "https://arxiv.org/abs/1109.3253"
}
|
1109.3340
|
# 325 MHz VLA Observations of Ultracool Dwarfs TVLM 513-46546 and 2MASS
J0036+1821104
T. R. Jaeger1,2, R. A. Osten3, T. J. Lazio4, N. Kassim2, and R. L. Mutel5 1
National Research Council Postdoctoral Research Associate 2 US Naval Research
Laboratory, Washington, DC 20375 3 Space Telescope Science Institute,
Baltimore, MD 21218 4 Jet Propulsion Laboratory, Pasadena, CA 91109 5
University of Iowa, Iowa City, IA 52242 ted.jaeger.ctr@nrl.navy.mil
###### Abstract
We present 325 MHz (90 cm wavelength) radio observations of ultracool dwarfs
TVLM 513-46546 and 2MASS J0036+1821104 using the Very Large Array (VLA) in
June 2007. Ultracool dwarfs are expected to be undetectable at radio
frequencies, yet observations at 8.5 GHz (3.5 cm) and 4.9 GHz (6 cm) of have
revealed sources with $>$ 100 $\mu\mathrm{Jy}$ quiescent radio flux and $>$ 1
mJy pulses coincident with stellar rotation. The anomalous emission is likely
a combination of gyrosynchrotron and cyclotron maser processes in a long-
duration, large-scale magnetic field. Since the characteristic frequency for
each process scales directly with the magnetic field magnitude, emission at
lower frequencies may be detectable from regions with weaker field strength.
We detect no significant radio emission at 325 MHz from TVLM 513-46546 or
2MASS J0036+1821104 over multiple stellar rotations, establishing $2.5\sigma$
total flux limits of 795 $\mu\mathrm{Jy}$ and 942 $\mu\mathrm{Jy}$
respectively. Analysis of an archival VLA 1.4 GHz observation of 2MASS
J0036+1821104 from January 2005 also yields a non-detection at the level of
$<$ 130 $\mu\mathrm{Jy}$ . The combined radio observation history (0.3 GHz to
8.5 GHz) for these sources suggests a continuum emission spectrum for
ultracool dwarfs which is either flat or inverted below 2-3 GHz. Further, if
the cyclotron maser instability is responsible for the pulsed radio emission
observed on some ultracool dwarfs, our low-frequency non-detections suggest
that the active region responsible for the high-frequency bursts is confined
within 2 stellar radii and driven by electron beams with energies less than 5
keV.
††: AJ
## 1 Introduction
Ultracool dwarfs (UCDs) describe a subsection of stellar objects located on
the boundary between more massive stars and sub-stellar bodies such as gas
giant planets. It includes fully convective, very low mass stars (M7 and
lower) and all brown dwarfs. X-ray and H$\alpha$ intensity (typical proxies
for magnetic activity) for UCDs is weak, dropping substantially after spectral
class M7 (Liebert et al., 1999; Audard et al., 2007). Radio emission is
expected to be undetectable at the $\mu\mathrm{Jy}$ level, based on empirical
X-ray scaling laws (Guedel & Benz, 1993; Benz & Guedel, 1994), and the
assumption that highly neutral UCD atmospheres are incapable of sustaining
magnetic stresses which pervade the atmospheres of solar-type stars. However,
a growing number of UCDs have been discovered that display significant
emission at cm-wavelengths and suggest the presence of persistent kG-scale
magnetic fields (Berger et al., 2001; Burgasser & Putman, 2005; Hallinan et
al., 2008; Osten et al., 2009). Furthermore, the quiescent radio emission is
nearly constant from spectral types M0 to L5 (Berger et al., 2005; Berger,
2006). It appears that, for at least some UCDs, the typical indicators of
magnetic activity are not well correlated with decreased radio flux.
Radio emission from these peculiar UCDs is typically broadband and unpolarized
with high brightness temperature ($10^{8}-10^{9}$K) during quiescence, and can
exceed $10^{11}$K with nearly 100% circular polarization during bursts
(Berger, 2002; Antonova et al., 2008; Hallinan et al., 2008). There is also
evidence of long-term radio variability (Antonova et al., 2007). The assumed
radiation mechanism was initially incoherent gyrosynchrotron (Berger, 2002;
Berger et al., 2005) from populations of mildly relativistic electrons with a
power-law energy distribution. However, the high brightness temperature and
high circular polarization seen during burst events suggested a coherent
radiation mechanism such as the cyclotron maser instability (CMI, Melrose et
al., 1984; Winglee, 1985). First suggested by Hallinan et al. (2006), CMI has
also been used to describe burst emission from the polar, low density - high
magnetic field regions of magnetized planets (Zarka, 1998; Ergun et al.,
2000), Algol (Mutel et al., 1998), and late-type flare stars (Bingham et al.,
2001; Kellett et al., 2002). The CMI model may also explain the quiescent
emission, possibly created via depolarization of persistent maser sources
(Hallinan et al., 2006; Littlefair et al., 2008; Yu et al., 2011).
The exact mechanism(s) responsible for UCD radio emission (flaring on top of a
quiescent background) is unclear, but it is likely due to a combination of
both gyrosynchrotron emission and CMI. The emission frequency for each
mechanism scales directly with the local magnetic field strength, emitting at
the electron cyclotron frequency ($\Omega_{ce,MHz}=2.8\cdot B_{G}$). Gigahertz
radio observations then require kG-scale fields, while detectable radio
emission may exist at megahertz frequencies from regions of weaker field
strength ($\sim$ 116 Gauss at 325 MHz). A vast majority of UCD observations
have been limited to gigahertz frequencies where radio instruments are
historically the most sensitive. Constraining the full spectral profile with
addition of low frequency observations may provide valuable information to
distinguish between the suggested emission processes, as well as reveal key
differences in the atmospheres of UCDs compared to high mass M-dwarfs.
Further, as UCDs are uniquely placed on the boundary between stars and sub-
stellar objects, low frequency measurements also provide a guide for future
observations of extrasolar planets.
## 2 Target History
TVLM 513-46546 (hereafter TVLM513 ) is a spectral type M9 dwarf with mass
equaling $0.09~{}M_{sun}$ and age $>$ 1 Gyr (Reid et al., 2000). Radio
emission from this near-by source ($\sim$ 10.6 pc, Dahn et al. (2002)) was
first detected by Berger (2002). Berger’s observations at 8.5 GHz revealed
both persistent stellar emission and occasional ($2\%-10\%$ duty cycle) pulses
with flux densities exceeding 1 mJy. Detected bursts were highly circularly
polarized and lasted multiple minutes. Both the persistent and burst emission
features were later confirmed by Osten et al. (2006), Hallinan et al. (2006),
Hallinan et al. (2007), Berger et al. (2008), Forbrich & Berger (2009), and
Doyle et al. (2010) through observations at 8.5 GHz and 4.9 GHz. Further,
Hallinan et al. (2007) and Lane et al. (2007, optical) detected a pulse
periodicity of $\sim$ 2 hr, consistent with the stellar rotation rate. Osten
et al. (2006) made a 5$\sigma$ detection of TVLM513 at 1.4 GHz and it is
undetected at lower frequencies at sensitivity levels similar to the GHz
detections (Antonova, 2007).
2MASS J0036+1821104 (2M0036 onward) is a L3.5 brown dwarf with mass $\sim
0.06-0.074~{}M_{sun}$ (Schweitzer et al., 2001), age $>$ 0.8 Gyr (Burrows et
al., 2001), and is located at 8.8 pc (Dahn et al., 2002). Radio observatons at
8.5 GHz by Berger (2002) revealed quiescent emission with occasional bursts,
similar to those witnessed on TVLM513. Subsequent measurements at 4.9 GHz by
Hallinan et al. (2008) revealed strong (5x the quiescent level) circularly
polarized pulses, lasting 5-20 min and with a 3 hr period corresponding to the
stellar rotation rate. However, observations at 8.5 GHz by Berger et al.
(2005); Forbrich & Berger (2009) detected no significant activity over
multiple stellar rotation periods, indicating a possible absence of
periodicity above 4.9 GHz. To date, there are no published observations of
2M0036 below 4.9 GHz.
## 3 Observations and Data Reduction
Radio observations of UCDs TVLM513 and 2M0036 were conducted June $24-26$,
2007 using the NRAO Very Large Array (VLA). Each source was observed for
$\sim$ 10.5 hr (10 s integration) using 2 x 6.25 MHz bands centered at 327.5
MHz and 321.6 MHz. Each frequency band was split into 15 spectral channels for
the purpose of radio frequency interference removal and to mitigate bandwidth
smearing. At observation time, the array utilized 23, 25 m antennas positioned
in A configuration (maximum baseline $\sim$ 35 km), resulting in a $\sim$ 2.5
deg field of view and $\sim$ 6” x 5” angular resolution. The radio flux
density scale was set using amplitude calibrator 3C 286 (assumed 24.49 Jy at
327.5 MHz). Phase calibration and the receiver bandpass correction was
performed using standard sources 1513+236 and 0042+233 for TVLM513 and 2M0036
respectively. Measurements of each UCD and its corresponding phase calibrator
were intertwined, performing 2 x 30 min primary target scans followed by a 5
min calibrator scan, then repeating.
Data reduction and imaging were performed using both AIPS 111Astronomical
Image Processing System, release 31DEC10 and Obit 222Obit is developed and
maintained by Bill Cotton at The National Radio Astronomy Observatory in
Charlottesville, Virginia, USA and is made available under the GNU General
Public License. version 1.1.269-6-64b. software packages. The visibility data
was calibrated using standard AIPS tasks. Automated data flagging, visibility
self-calibration, and imaging was performed in Obit. Lightcurves of TVLM513
and 2M0036 were made in both total (Stokes I) and circularly polarized (Stokes
V) flux using AIPS task DFTPL.
For comparison with published 1.4 GHz quiescent emission observations of
TVLM513, we analyzed VLA archival data from January 10, 2005 which contained
an $\sim$ 8 hr observation of 2M0036. The data consisted of 2 x 50 MHz bands
centered on 1465 MHz and 1385 MHz obtained in the BnA hybrid configuration
(maximum baseline $\sim$ 21.2 km NS and 12.2 km EW). Amplitude and phase
offsets were determined using calibrators 3C 147 (assumed 21.39 Jy at 1465
MHz) and 0042+233. Data editing, calibration and imaging were performed using
standard AIPS routines.
## 4 Results
Figure 1: Peak luminosity of TVLM513 from all published radio observations,
scaling the radio flux density to a distance of 10.6 pc. Diamonds indicate
measured quiescent emission and stars indicate burst/pulse emission. Reported
error bars are roughly the size of each marker. Bursts reported as lower
limits are displayed with upward arrows. Non-detection limits (quiescent
solid, 10% duty bursts dashed) are indicated with downward arrows ending at
the 1$\sigma$ intensity. See Table 1 for specific radio measurements and
author sources.
Figure 2: (top) Measured circularly polarized radio flux at the position of
TVLM513 folded with a period of 1.96 hr. Dots indicate the 10 s resolution
measurements. The solid line indicates the 5 min median value of the folded
data. (bottom) Lomb-Scargle periodogram of the 10 s resolution flux values.
Dashed lines indicate false alarm probabilities of 0.01 (99%), 0.1 (90%), and
0.5 (61%).
Figure 3: Peak luminosity of 2M0036 from all published observations, scaling
the measured radio flux density to a distance of 8.8 pc. Indicators are the
same as in Figure 2. See Table 1 for specific radio measurements and author
sources.
Figure 4: (top) Measured radio flux (Stokes V) at the position of 2M0036
folded with a period of 3.08 hr. Dots represent the 10 s resolution radio
intensity. A solid line indicates the 5 min median value of the 10 s
measurements. (bottom) A Lomb-Scargle periodogram of the 10 s resolution
circularly polarized flux values. Dashed lines indicate false alarm
probabilities of 0.01 (99%), 0.1 (90%), and 0.5 (61%).
### 4.1 TVLM513
We observe no significant unpolarized or circularly polarized radio emission
associated with TVLM513 over the 10.5 hr observation. The position of TVLM513
is well determined ($<$ 1”), based on closely spaced radio (Berger et al.,
2008) and infrared (Cutri et al., 2003) detections and estimates of TVLM513’s
proper motion (Schmidt et al., 2007). Given the positional accuracy, we report
a $2.5\sigma$ non-detection limit to the total (Stokes I) and circularly
polarized (Stokes V) quiescent radio flux. These limits are 795
$\mu\mathrm{Jy}$ for I and 743 $\mu\mathrm{Jy}$ for V. The relationship
between our $2.5\sigma$ upper limit for TVLM513 and previously measured peak
flux density values is shown in Figure 2 (scaled to 10.6 pc) and listed in
Table 1.
Previous observations of TVLM513 indicate a pulse period of $\sim$ 1.96 hr
(Hallinan et al., 2006; Lane et al., 2007). To search for burst emission and
any potential periodicity, we constructed lightcurves at the known position of
TVLM513 in both unpolarized and circularly polarized intensity with time
resolutions between 10 s and 10.5 hr. We detect no significant variation to
the measured radio intensity on any timescale. Also, no periodicity was found
within the noise over $\sim$ 5 stellar rotations, performing both a blind
period search and by folding the flux values at the expected period.
Lightcurves of the circularly polarized flux with 10 s and 5 min temporal
resolution, along with the corresponding Lomb-Scargle periodogram of the 10 s
measurements are shown in Figure 2. Our 325 MHz non-detection implies a pulse
flux density upper limit of 2.4 mJy (Stokes I) assuming a 10% duty cycle. High
frequency observations by Berger (2002); Hallinan et al. (2007) observe pulse
duty cycles in the range of 2-10%. We choose a 10% duty cycle to account for
potential pulse dispersion at lower frequencies.
### 4.2 2M0036
We searched the anticipated location of 2M0036 for polarized and unpolarized
radio emission at observation frequencies of 325 MHz and 1.4 GHz. The position
of 2M0036 is known to an accuracy smaller than each of the synthesized beams
(see Sec. 3) and is therefore well constrained. No significant radio emission
was observed at either frequency. Our 325 MHz non-detection establishes a
$2.5\sigma$ upper limit on the the quiescent flux $<$ 942 $\mu\mathrm{Jy}$ in
total intensity (Stokes I) and $<$ 870 $\mu\mathrm{Jy}$ in circularly
polarized (Stokes V) intensity for the 10.5 hr observation. The non-detection
at 1400 MHz ($<$ 130 $\mu\mathrm{Jy}$ in Stokes I, $<$ 95 $\mu\mathrm{Jy}$
Stokes V, $2.5\sigma$) sets an upper limit on the quiescent flux which is
slightly lower than the predicted extrapolation from higher frequencies when
assuming a flat spectrum (See Fig. 4).
No notable burst activity in the 325 MHz total flux or polarized flux
measurements was observed on timescales from 10 s to 10.5 hr. The
corresponding burst flux density upper limit is 2.8 mJy (Stokes I) assuming a
10% duty cycle We also detect no periodicity in the measured flux over $>$ 3
stellar rotations, assuming a stable rotation rate of 3.08 hr (Hallinan et
al., 2008). A lightcurve of the 325 MHz circular polarization measurements
folded at the expected pulse period and a Lomb-Scargle periodogram of 10 s
resolution data are shown in Figure 4. A similar flux variability search using
the 1.4 GHz observation was not possible due to significant radio frequency
interference which persisted for approximately 3 hrs during the session.
Table 1: Measurement summary listing observing frequency $\nu$ in GHz, observation length $\tau$ in hr, and the recorded flux S in $\mu\mathrm{Jy}$ . Measurements are made using the VLA unless noted otherwise. Source | Date | $\nu$ | $\tau$ | S | Reference | Notes
---|---|---|---|---|---|---
TVLM513 | 30.03.2008 | 8.5 | 7 | $539\pm 19$ | Forbrich & Berger (2009) |
| 30.03.2008 | 8.5 | 7 | $230\pm 47$ | | VLBA
| 26.07.2007 | 0.3 | 10 | $<$ 795 | This paper |
| 01.07.2007 | 8.4 | 8 | $318\pm 9$ | Doyle et al. (2010) |
| 01.07.2007 | 8.4 | $-$ | $>2900$ | | Burst
| 20.04.2007 | 8.5 | 9 | $208\pm 18$ | Berger et al. (2008) |
| 20.04.2007 | 8.5 | 8 | $353\pm 14$ | |
| 20.04.2007 | 8.5 | $-$ | $>5500$ | | Burst
| 12.02.2007 | 0.6 | 4 | $<225$ | Antonova (2007) | GMRT
| 20.05.2006 | 8.4 | 10 | $464\pm 9$ | Hallinan et al. (2007) |
| 20.05.2006 | 8.4 | $-$ | $>5000$ | | Burst
| 20.05.2006 | 4.9 | 10 | $368\pm 16$ | |
| 20.05.2006 | 4.9 | $-$ | $>2000$ | | Burst
| 13.01.2005 | 8.4 | 5 | $396\pm 16$ | Hallinan et al. (2006) |
| 13.01.2005 | 4.9 | 5 | $405\pm 18$ | |
| 24.01.2004 | 8.4 | 4 | $228\pm 11$ | Osten et al. (2006) |
| 24.01.2004 | 4.8 | 4 | $284\pm 13$ | |
| 24.01.2004 | 1.4 | 4 | $260\pm 46$ | |
| 23.09.2001 | 8.5 | $-$ | $980\pm 40$ | Berger (2002) | Burst
2M0036 | 30.03.2008 | 8.5 | 7 | $144\pm 22$ | Forbrich & Berger (2009) |
| 24.06.2007 | 0.3 | 10 | $<$ 942 | This paper |
| 24.09.2006 | 4.9 | 12 | $241\pm 14$ | Hallinan et al. (2008) |
| 24.09.2006 | 4.9 | $-$ | $>500$ | | Burst
| 10.01.2005 | 4.9 | 8 | $152\pm 9$ | Berger et al. (2005) |
| 10.01.2005 | 1.4 | 8 | $<$ 130 | This paper |
| 28.09.2002 | 8.5 | 8 | $134\pm 16$ | Berger et al. (2005) |
| 28.09.2002 | 4.9 | 8 | $259\pm 19$ | |
| 09.10.2001 | 8.5 | 3 | $327\pm 14$ | Berger (2002) | Burst
| 09.10.2001 | 8.5 | $-$ | $720\pm 40$ | | Burst
| 23.09.2001 | 8.5 | 2 | $135\pm 14$ | |
## 5 Discussion
The absence of both flaring and quiescent emission from TVLM513 and 2M0036
implies constraints on the plasma environments (i.e. electron energy, spacial
distribution, etc.) of UCDs. Of the two sources, TVLM513 may offer the most
stringent constraints, given the numerous GHz observations of stable pulse
activity in the year preceding our June 2007 measurements (the closest burst
is within 25 days, Doyle et al., 2010). Here, we explore what limits can be
inferred by our non-detections within the context of behavior seen on active
stars and magnetized planets.
### 5.1 Flare Emission Constraints
Hallinan et al. (2008) have argued that the primary mechanism responsible for
time variable emission from UCDs is the cyclotron maser instability (CMI). The
CMI mechanism requires (1) a continuous source of energetic beamed electrons
accelerated by a parallel electric field and (2) a converging magnetic field
topology. As the electron beam propagates toward increasing field strength,
conservation of the first adiabatic invariant causes transfer of parallel to
perpendicular energy. The resulting velocity distribution $f(v)$ becomes
increasingly unstable ($\partial f/\partial v_{\bot}>0$), leading to
exponential wave growth at the electron cyclotron frequency.
The CMI mechanism is quenched if the relativistic RX-mode cutoff frequency
exceeds the cyclotron frequency. This constraint can be written in terms of an
upper limit on the plasma $\beta$, the ratio of electron plasma to cyclotron
frequencies (Mutel et al., 2006),
$\beta=\frac{\omega_{pe}}{\Omega_{ce}}<\sqrt{\frac{\gamma-1}{\gamma}},$ (1)
with $\gamma$ representing the electron beam Lorentz factor. For $\gamma-1\ll
1$, this can be recast as
$\beta\sim 5\times 10^{-3}\cdot\frac{n_{e}}{B_{G}^{2}\ E_{keV}}<1$ (2)
where $n_{e}$ is the electron density per cubic centimeter, $B_{G}$ is the
magnetic field in Gauss, and $E_{keV}$ is the beam energy in keV.
By assuming a magnetic field configuration, an electron density profile, and a
mean beam energy, this constraint can be mapped to a coronal volume for which
the CMI mechanism is viable. We assume a dipole field with surface equatorial
field strength $B_{o}=5$ kG, consistent with estimates of 3 kG at the 8 GHz
source altitude (Reiners & Basri, 2007; Hallinan et al., 2008) but below the
maximum of 10 kG based on dynamo models (Browning, 2008). The electron density
profile is more problematic. Stellar coronal loop models suggest either a low-
order power-law or exponential dependence with radial distance, with a scale
height of order the loop size (Rosner et al., 1978; Collier Cameron, 1988). We
assume an example surface electron density $n_{o}=10^{9}\ \rm{cm}^{-3}$
(Fludra et al., 1999; Yu et al., 2011) and an exponential dependence with a
density scale height equal to the stellar radius.
Figure 5: (a) Masing criteria above the surface of a ultracool dwarf, assuming
a dipole field ($B_{o}=5$ kG), an exponentially decreasing electron density
($n_{o}=10^{9}$ cm-3), and an electron beam energy of 100 keV. Solid arcs are
used to illustrate example magnetic field lines. Dashed curves indicate where
the magnetic field magnitude is sufficient to produce cyclotron emission at
325 MHz, 610 MHz, 1.4 GHz, 4.9 GHz, and 8.5 GHz. White to black shading
indicates the value of beta ($\beta=\omega_{pe}/\Omega_{ce}$). (b) Plasma beta
versus electron cyclotron frequency at magnetic latitude 65o for electron beam
energies of 1 keV, 10 keV, and 100 keV. Beta has been normalized to its
maximum allowed value for positive CMI growth as given by equation (1). Note
that at 325 MHz, only beam energies $E>10$ keV result in CMI growth
($\beta<1$), while above a few GHz, the CMI mechanism has positive growth at
much lower energies.
Figure 5a illustrates the masing criteria ($\beta$) in a model UDC
magnetosphere for 100 keV electron beams. Presumably, the periodic bursts
observed on UCDs are generated in an active region extending along an co-
rotating field line and emitting at a frequency consistent with the local
magnetic field strength. Detections at both 8.5 GHz and 4.9 GHz imply an
active region extending to at least 1.5 stellar radii. This is consistent with
estimates from 2M0036 by Hallinan et al. (2008). Lower frequency observations
probe larger radii, hence the highest observation frequency where $\beta<1$
and pulsed emission is absent defines the region’s uppermost vertical extent.
Assuming observations of 2M0036 and TVLM513 are typical of all UCDs and that
the year-long history of pulse activity on TVLM513 continued throughout our
325 MHz observations (i.e. nearly 100% pulse duty cycle), the absent pulse
emission at 1.4 GHz (Osten et al., 2006), 610 MHz (Antonova, 2007), and 325
MHz (this work, for strong flat spectrum bursts) suggests that the pulse
source is confined within 2 stellar radii.
Figure 5b shows ratio of $\beta$ to the critical $\beta$ given in Equations
1-2 versus electron cyclotron frequency for a magnetic latitude of 65o. For
325 MHz, the CMI mechanism is only viable for electron beam energies $E>20$
keV ($E>5$keV at 1.4 GHz), whereas only above a few GHz beam energies below 1
keV can drive the CMI mechanism. This may also account for our non-detection:
If CMI is driven by relatively low-energy beams, it could account for CMI-
induced bursts at high frequencies, but suppression at lower frequencies.
### 5.2 Quiescent Emission Constraints
The multi-frequency observations (325 MHz - 8.5 GHz) are consistent with a
flat gyrosynchrotron spectrum with indication in the 1.4 GHz measurements of
2M0036 of a possible low-frequency spectral break. While our 325 MHz
observational limits are the most stringent to date, they are not sensitive
enough to imply any real constraints on the quiescent emission mechanism for
UCDs, requiring at least a factor of 5-10 better sensitivity to rival the high
frequency detections. The ability to make these needed measurements will rely
on the completion of LOFAR (36 stations, $\sim$ 300 $\mu\mathrm{Jy}$
2.5$\sigma$ at 250 MHz in 10 hr) or the extension of the EVLA Low-Band333The
EVLA 74 MHz Low-Band system is scheduled to be completed Fall 2012 retrofit to
350 MHz ($\sim$ 100 $\mu\mathrm{Jy}$ 2.5$\sigma$ in 10 hr with 15% fractional
bandwidth).
## 6 Summary and Conclusion
We searched ultracool dwarfs TVLM 513-46546 and 2MASS J0036+1821104 for radio
emission at 325 MHz. Measurements of the total (Stokes I) and circularly
polarized (Stokes V) flux were made on timescales between 10 s and 10.5 hr.
While strong continuous emission and multi-minute, circularly polarized pulses
were previously recorded at 4.9 GHz and 8.5 GHz, no significant emission was
measured at 325 MHz. We set $2.5\sigma$ total flux limits of 795
$\mu\mathrm{Jy}$ and 942 $\mu\mathrm{Jy}$ for TVLM 513-46546 and 2MASS
J0036+1821104 , respectively, for the quiescent emission over 10.5 hr,
consistent with a flat gyrosynchrotron spectrum with a potential spectral
break near 2-3 GHz as indicated by (Osten et al., 2006). Furthermore, we
observe no significant variation in the 325 MHz radio flux at the expected
location of each source. The absence of variable radio flux from TVLM
513-46546 below 1.4 GHz suggests that pulse emission from these UCDs (assuming
a CMI electron acceleration model) originates from a source region confined
below 2 stellar radii and/or is driven by electron beam energies less than a
few keV.
## 7 Acknowledgments
This paper utilizes data from VLA programs AO218 and AB1169. We would like to
thank Bill Cotton for assistance with Obit. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. This research was
performed while the primary author held a National Research Council Research
Associateship Award a the US Naval Research Laboratory. Basic research in
radio astronomy at the Naval Research Laboratory is supported by 6.1 base
funding. Part of this research was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract with the
National Aeronautics and Space Administration. The LUNAR consortium is funded
by the NASA Lunar Science Institute to investigate concepts for astrophysical
observatories on the Moon.
## References
* Antonova (2007) Antonova, A. 2007, Ph.D. thesis, The Queen’s University of Belfast
* Antonova et al. (2008) Antonova, A., Doyle, J. G., Hallinan, G., Bourke, S., & Golden, A. 2008, A&A , 487, 317. 0805.4574
* Antonova et al. (2007) Antonova, A., Doyle, J. G., Hallinan, G., Golden, A., & Koen, C. 2007, A&A , 472, 257. 0707.0634
* Audard et al. (2007) Audard, M., Osten, R. A., Brown, A., Briggs, K. R., Güdel, M., Hodges-Kluck, E., & Gizis, J. E. 2007, A&A , 471, L63. 0707.1882
* Benz & Guedel (1994) Benz, A. O., & Guedel, M. 1994, A&A , 285, 621
* Berger (2002) Berger, E. 2002, ApJ , 572, 503. arXiv:astro-ph/0111317
* Berger (2006) — 2006, ApJ , 648, 629. arXiv:astro-ph/0603176
* Berger et al. (2001) Berger, E., Ball, S., Becker, K. M., Clarke, M., Frail, D. A., Fukuda, T. A., Hoffman, I. M., Mellon, R., Momjian, E., Murphy, N. W., Teng, S. H., Woodruff, T., Zauderer, B. A., & Zavala, R. T. 2001, Nature , 410, 338. arXiv:astro-ph/0102301
* Berger et al. (2008) Berger, E., Gizis, J. E., Giampapa, M. S., Rutledge, R. E., Liebert, J., Martín, E., Basri, G., Fleming, T. A., Johns-Krull, C. M., Phan-Bao, N., & Sherry, W. H. 2008, ApJ , 673, 1080. 0708.1511
* Berger et al. (2005) Berger, E., Rutledge, R. E., Reid, I. N., Bildsten, L., Gizis, J. E., Liebert, J., Martín, E., Basri, G., Jayawardhana, R., Brandeker, A., Fleming, T. A., Johns-Krull, C. M., Giampapa, M. S., Hawley, S. L., & Schmitt, J. H. M. M. 2005, ApJ , 627, 960. arXiv:astro-ph/0502384
* Bingham et al. (2001) Bingham, R., Cairns, R. A., & Kellett, B. J. 2001, A&A , 370, 1000
* Browning (2008) Browning, M. K. 2008, ApJ , 676, 1262. 0712.1603
* Burgasser & Putman (2005) Burgasser, A. J., & Putman, M. E. 2005, ApJ , 626, 486. arXiv:astro-ph/0502365
* Burrows et al. (2001) Burrows, A., Hubbard, W. B., Lunine, J. I., & Liebert, J. 2001, Rev. Mod. Phys., 73, 719
* Collier Cameron (1988) Collier Cameron, A. 1988, M.N.R.A.S., 233, 235
* Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., Beichman, C. A., Carpenter, J. M., Chester, T., Cambresy, L., Evans, T., Fowler, J., Gizis, J., Howard, E., Huchra, J., Jarrett, T., Kopan, E. L., Kirkpatrick, J. D., Light, R. M., Marsh, K. A., McCallon, H., Schneider, S., Stiening, R., Sykes, M., Weinberg, M., Wheaton, W. A., Wheelock, S., & Zacarias, N. 2003, 2MASS All Sky Catalog of point sources. (NASA/IPAC Infrared Science Archive)
* Dahn et al. (2002) Dahn, C. C., Harris, H. C., Vrba, F. J., Guetter, H. H., Canzian, B., Henden, A. A., Levine, S. E., Luginbuhl, C. B., Monet, A. K. B., Monet, D. G., Pier, J. R., Stone, R. C., Walker, R. L., Burgasser, A. J., Gizis, J. E., Kirkpatrick, J. D., Liebert, J., & Reid, I. N. 2002, AJ , 124, 1170. arXiv:astro-ph/0205050
* Doyle et al. (2010) Doyle, J. G., Antonova, A., Marsh, M. S., Hallinan, G., Yu, S., & Golden, A. 2010, A&A , 524, A15+
* Ergun et al. (2000) Ergun, R. E., Carlson, C. W., McFadden, J. P., Delory, G. T., Strangeway, R. J., & Pritchett, P. L. 2000, ApJ , 538, 456
* Fludra et al. (1999) Fludra, A., Del Zanna, G., Alexander, D., & Bromage, B. J. I. 1999, JGR , 104, 9709
* Forbrich & Berger (2009) Forbrich, J., & Berger, E. 2009, ApJ Let. , 706, L205. 0910.1349
* Guedel & Benz (1993) Guedel, M., & Benz, A. O. 1993, ApJ Let. , 405, L63
* Hallinan et al. (2006) Hallinan, G., Antonova, A., Doyle, J. G., Bourke, S., Brisken, W. F., & Golden, A. 2006, ApJ , 653, 690. arXiv:astro-ph/0608556
* Hallinan et al. (2008) Hallinan, G., Antonova, A., Doyle, J. G., Bourke, S., Lane, C., & Golden, A. 2008, ApJ , 684, 644. 0805.4010
* Hallinan et al. (2007) Hallinan, G., Bourke, S., Lane, C., Antonova, A., Zavala, R. T., Brisken, W. F., Boyle, R. P., Vrba, F. J., Doyle, J. G., & Golden, A. 2007, ApJ Let. , 663, L25. 0705.2054
* Kellett et al. (2002) Kellett, B. J., Bingham, R., Cairns, R. A., & Tsikoudi, V. 2002, MNRAS , 329, 102
* Lane et al. (2007) Lane, C., Hallinan, G., Zavala, R. T., Butler, R. F., Boyle, R. P., Bourke, S., Antonova, A., Doyle, J. G., Vrba, F. J., & Golden, A. 2007, ApJ Let. , 668, L163. 0709.1045
* Liebert et al. (1999) Liebert, J., Kirkpatrick, J. D., Reid, I. N., & Fisher, M. D. 1999, ApJ , 519, 345
* Littlefair et al. (2008) Littlefair, S. P., Dhillon, V. S., Marsh, T. R., Shahbaz, T., Martín, E. L., & Copperwheat, C. 2008, MNRAS , 391, L88. 0809.2193
* Melrose et al. (1984) Melrose, D. B., Dulk, G. A., & Hewitt, R. G. 1984, JGR , 89, 897
* Mutel et al. (2006) Mutel, R. L., Menietti, J. D., Christopher, I. W., Gurnett, D. A., & Cook, J. M. 2006, Journal of Geophysical Research (Space Physics), 111, A10203. arXiv:astro-ph/0609802
* Mutel et al. (1998) Mutel, R. L., Molnar, L. A., Waltman, E. B., & Ghigo, F. D. 1998, ApJ , 507, 371
* Osten et al. (2006) Osten, R. A., Hawley, S. L., Bastian, T. S., & Reid, I. N. 2006, ApJ , 637, 518. arXiv:astro-ph/0509762
* Osten et al. (2009) Osten, R. A., Phan-Bao, N., Hawley, S. L., Reid, I. N., & Ojha, R. 2009, ApJ , 700, 1750. 0905.4197
* Reid et al. (2000) Reid, I. N., Kirkpatrick, J. D., Gizis, J. E., Dahn, C. C., Monet, D. G., Williams, R. J., Liebert, J., & Burgasser, A. J. 2000, AJ , 119, 369. arXiv:astro-ph/9909336
* Reiners & Basri (2007) Reiners, A., & Basri, G. 2007, ApJ , 656, 1121. arXiv:astro-ph/0610365
* Rosner et al. (1978) Rosner, R., Tucker, W. H., & Vaiana, G. S. 1978, ApJ , 220, 643
* Schmidt et al. (2007) Schmidt, S. J., Cruz, K. L., Bongiorno, B. J., Liebert, J., & Reid, I. N. 2007, AJ , 133, 2258. arXiv:astro-ph/0701055
* Schweitzer et al. (2001) Schweitzer, A., Gizis, J. E., Hauschildt, P. H., Allard, F., & Reid, I. N. 2001, ApJ , 555, 368. arXiv:astro-ph/0103402
* Winglee (1985) Winglee, R. M. 1985, JGR , 90, 9663
* Yu et al. (2011) Yu, S., Hallinan, G., Doyle, J. G., MacKinnon, A. L., Antonova, A., Kuznetsov, A., Golden, A., & Zhang, Z. H. 2011, A&A , 525, A39+. 1009.1548
* Zarka (1998) Zarka, P. 1998, JGR , 103, 20159
|
arxiv-papers
| 2011-09-15T13:08:18 |
2024-09-04T02:49:22.182696
|
{
"license": "Public Domain",
"authors": "T. R. Jaeger, R. A. Osten, T. J. Lazio, N. Kassim, and R. L. Mutel",
"submitter": "Theodore Jaeger",
"url": "https://arxiv.org/abs/1109.3340"
}
|
1109.3354
|
# Stochastic oscillations in models of epidemics on a network of cities
G. Rozhnova1, A. Nunes1 and A. J. McKane1,2 1Centro de Física da Matéria
Condensada and Departamento de Física, Faculdade de Ciências da Universidade
de Lisboa, P-1649-003 Lisboa Codex, Portugal
2Theoretical Physics Division, School of Physics and Astronomy, University of
Manchester, Manchester M13 9PL, United Kingdom
###### Abstract
We carry out an analytic investigation of stochastic oscillations in a
susceptible-infected-recovered model of disease spread on a network of $n$
cities. In the model a fraction $f_{jk}$ of individuals from city $k$ commute
to city $j$, where they may infect, or be infected by, others. Starting from a
continuous time Markov description of the model the deterministic equations,
which are valid in the limit when the population of each city is infinite, are
recovered. The stochastic fluctuations about the fixed point of these
equations are derived by use of the van Kampen system-size expansion. The
fixed point structure of the deterministic equations is remarkably simple: a
unique non-trivial fixed point always exists and has the feature that the
fraction of susceptible, infected and recovered individuals is the same for
each city irrespective of its size. We find that the stochastic fluctuations
have an analogously simple dynamics: all oscillations have a single frequency,
equal to that found in the one city case. We interpret this phenomenon in
terms of the properties of the spectrum of the matrix of the linear
approximation of the deterministic equations at the fixed point.
###### pacs:
05.40.-a, 87.10.Mn, 02.50.Ey
## I Introduction
Two of the ideas that are currently dominating the discussion of modeling
epidemic spread are those of stochasticity and network structure Pastor-
Satorras and Vespignani (2001); Keeling and Eames (2005); Gross et al. (2006);
Volz and Meyers (2007); Colizza et al. (2007); Simoes et al. (2008).
Deterministic models of the Susceptible-Infected-Recovered (SIR) type have a
long history Anderson and May (1991); Keeling and Rohani (2007) and have been
thoroughly investigated Schenzle (1985); Keeling and Grenfell (2002) along
with many extensions of the models such as age classes or higher-order
nonlinear interaction terms. Although stochasticity, due to random processes
at the level of individuals, and networks, either between individuals or towns
and cities, were recognised early on as significant and important factors, the
tendency was to model them through computer simulations. This is not
surprising: it is rather straightforward to deal with stochastic behavior in
simulations, and similarly the analytic methods available to investigate
complex networks, especially adaptive networks, are limited. There has also
been a tendency towards developing extremely detailed agent-based models to
study disease spread Eubank et al. (2004); Meyers et al. (2005); Christensen
et al. (2010); Aleman et al. (2011), which are the antithesis of the simple
analytic approach based on the original SIR deterministic model.
In parallel with these developments, however, there have been several efforts
to extend analytic studies into the realm of stochastic and network dynamics.
The SIR model can be formulated as an individual-based model (IBM) which can
form a starting point for both an analytical treatment, based on the master
equation (continuous-time Markov chain) van Kampen (2007); Risken (1989), and
numerical simulations, based on the Gillespie algorithm Gillespie (1976). The
analytical studies use the system-size expansion of van Kampen to reduce the
master equation to the set of deterministic equations previously studied,
together with a set of stochastic differential equations for the deviations
from the deterministic result. As long as one is not too close to the fade-out
boundary, there is no need to go beyond next-to-leading order in the expansion
parameter, $1/\sqrt{N}$, where $N$ is the number of individuals in the system.
This already gives results which are, in general, in almost perfect agreement
with the results of simulations McKane and Newman (2005); Alonso et al.
(2007).
This approach has been used to study the stochastic version of the standard
SIR model Alonso et al. (2007), the Susceptible-Exposed-Infected-Recovered
(SEIR) model Rozhnova and Nunes (2010), both these models with annual forcing
Rozhnova and Nunes (2010); Black and McKane (2010a), staged-models Black et
al. (2009); Black and McKane (2010b), the pair-approximation in networked
models Rozhnova and Nunes (2009a, b), amongst others. In this paper we extend
the treatment to a metapopulation model for disease spread, which consists of
$n$ cities (labeled $j=1,\ldots,n$), each of which contains $N_{j}$
individuals. A fraction of the population of city $k$, $f_{jk}$, commutes to
city $j$ and this defines the strength of the link from node $k$ to $j$ in the
network of cities. We will show that the methods used in the case of one city
carry over to the case where the system comprises of a network of cities, and
that a surprisingly simple set of results can be derived which allow us to
make quite general predictions for a class of stochastic networked models of
epidemics.
The starting point for our analysis is a specification of how commuters move
between cities in the network. As will become clear, the model we arrive at
will not depend on the details of how and when these exchanges take place. We
then write down transition rates for the usual SIR process, now taking account
the city of origin of the infector and infected individuals. From the
resulting equation we can immediately find the deterministic equations
corresponding to the stochastic model when $N_{j}\to\infty$ for all $j$.
Deterministic models of this type began to be considered long ago Lajmanovich
and Yorke (1976) and the existence and stability properties of the endemic
equilibrium were studied for different formulations of the coupling between
the cities and of the disease dynamics Lloyd and May (1996); Keeling and
Rohani (2002); Arino and van den Driessche (2003). Stochastic effects in these
systems have also been analyzed from the point of view of the relation between
spatial heterogeneity, disease extinction and the threshold for disease onset
Lloyd and May (1996); Hagenaars et al. (2004); Colizza and Vespignani (2008);
Barthelemy et al. (2010).
Some rather strong and general results on the uniqueness and global stability
of the fixed points of the deterministic model are known Guo et al. (2008). We
will use these results and then go beyond this leading-order analysis to
determine the linear stochastic corrections that characterize the quasi-
stationary state of the finite system. As expected, the qualitative
predictions of the deterministic model are shown to be incorrect; instead
large stochastic cycles are found, although their form is much simpler than
might naively have been expected. We show that this is, in part, a reflection
of the special nature of the fixed points of the deterministic model.
The outline of the paper is as follows. In Section II we describe the basic
model and apply it to the case of two cities. The generalization to the
$n$-city case in given in Section III. The results for the form of the
sustained oscillations are given in Section IV and we conclude in Section V.
Two appendices contain technical details which are too cumbersome to include
in the main text.
## II Two-city model
In this section we will formulate the model when there are only two cities;
the general $n$ city case described in Section III does not introduce any new
points of principle and is easily explained once the two city case has been
understood.
The SIR model consists of three classes of individuals: susceptibles, infected
and recovered. The number of individuals in the three classes belonging to
city $j$ will be denoted by $S_{j},I_{j}$ and $R_{j}$ respectively. We assume
that births and deaths are coupled at the individual level, so that when an
individual dies another (susceptible) individual is born. This means that the
number of individuals belonging to any one city, $N_{j}$, does not change with
time, and so the number of recovered individuals is not an independent
variable: $R_{j}=N_{j}-S_{j}-I_{j}$, where $j=1,2$ Alonso et al. (2007).
There are four processes in the SIR model which cause transitions to a new
state: infection, recovery, death of an infected individual and death of a
recovered individual. The death of a susceptible individual does not cause a
transition, since it is immediately replaced by another, newborn, individual
which is by definition susceptible. Of the four listed processes, the final
three only involve one individual and so only involve one city. The transition
rates are Alonso et al. (2007):
* (a)
Recovery of an infective individual (and creation of a recovered individual)
$\displaystyle T_{1}$ $\displaystyle\equiv
T(S_{1},I_{1}-1,S_{2},I_{2}|S_{1},I_{1},S_{2},I_{2})=\gamma I_{1},$
$\displaystyle T_{2}$ $\displaystyle\equiv
T(S_{1},I_{1},S_{2},I_{2}-1|S_{1},I_{1},S_{2},I_{2})=\gamma I_{2}.$ (1)
* (b)
Death of an infected individual (and birth of a susceptible individual):
$\displaystyle T_{3}$ $\displaystyle\equiv
T(S_{1}+1,I_{1}-1,S_{2},I_{2}|S_{1},I_{1},S_{2},I_{2})=\mu I_{1},$
$\displaystyle T_{4}$ $\displaystyle\equiv
T(S_{1},I_{1},S_{2}+1,I_{2}-1|S_{1},I_{1},S_{2},I_{2})=\mu I_{2}.$ (2)
* (c)
Death of a recovered individual (and birth of a susceptible individual):
$\displaystyle T_{5}$ $\displaystyle\equiv
T(S_{1}+1,I_{1},S_{2},I_{2}|S_{1},I_{1},S_{2},I_{2})=\mu(N_{1}-S_{1}-I_{1}),$
$\displaystyle T_{6}$ $\displaystyle\equiv
T(S_{1},I_{1},S_{2}+1,I_{2}|S_{1},I_{1},S_{2},I_{2})=\mu(N_{2}-S_{2}-I_{2}).$
Here $\gamma$ and $\mu$ are parameters which respectively characterize the
rate of recovery and of birth/death.
The infection processes introduce the role of the commuters. We let $f_{21}$
be the fraction of the population from city $1$ which commutes to city $2$,
leaving a fraction $(1-f_{21})$ of the population as residents of city $1$.
Similarly, for commuters from city $2$, as illustrated in Fig. $1$. We note
that the number of individuals in city $j$ is
$M_{j}=(1-f_{kj})N_{j}+f_{jk}N_{k}$, where $j\neq k$. We will not specify the
nature of the commute in more detail and assume that the $f_{jk}$ are a
property of the corresponding pair of cities that defines the overall average
fraction of time that an individual from one city spends in the other city.
These coefficients will be taken as a coarse-grained measure of the
demographic coupling between the cities that will be applied to all
individuals independently of disease status and do not discriminate between
different types of stays with their typical frequencies and durations.
To see the nature of the infective interactions that occur, we first fix our
attention on those involving susceptible individuals from city $1$. There are
four types of term:
* (i)
Infective residents in city $1$ infect susceptible residents in city $1$.
* (ii)
Infective commuters from city $2$ infect susceptible residents in city $1$.
* (iii)
Infective residents in city $2$ infect susceptible commuters from city $1$.
* (iv)
Infective commuters from city $1$ infect susceptible commuters from city $1$
in city $2$.
The rates for these to occur according to the usual prescription for the SIR
model Alonso et al. (2007) are:
* (i)
$\beta\,\left(1-f_{21}\right)S_{1}\,\left(1-f_{21}\right)I_{1}/M_{1}$,
* (ii)
$\beta\,\left(1-f_{21}\right)S_{1}\,f_{12}I_{2}/M_{1}$,
* (iii)
$\beta\,f_{21}S_{1}\,\left(1-f_{12}\right)I_{2}/M_{2}$,
* (iv)
$\beta\,f_{21}S_{1}\,f_{21}I_{1}/M_{2}$,
where $\beta$ is the parameter which sets the overall rate of infection.
Adding these rates together we obtain the total transition rate for infection
of $S_{1}$ individuals as
$\beta\left[c_{11}\frac{S_{1}I_{1}}{N_{1}}+c_{12}\frac{S_{1}I_{2}}{N_{2}}\right],$
where
$\displaystyle c_{11}$ $\displaystyle=$
$\displaystyle\frac{\left(1-f_{21}\right)^{2}N_{1}}{M_{1}}+\frac{f^{2}_{21}N_{1}}{M_{2}},$
$\displaystyle c_{12}$ $\displaystyle=$
$\displaystyle\frac{\left(1-f_{21}\right)f_{12}N_{2}}{M_{1}}+\frac{f_{21}\left(1-f_{12}\right)N_{2}}{M_{2}}.$
A similar analysis can be made for the transitions involving susceptible
individuals from city 2. Putting these results together we obtain the
transition rates for infection as
* (d)
Infection of a susceptible individual:
$\displaystyle T_{7}$ $\displaystyle\equiv
T(S_{1}-1,I_{1}+1,S_{2},I_{2}|S_{1},I_{1},S_{2},I_{2})$
$\displaystyle=\beta\left[c_{11}\frac{S_{1}I_{1}}{N_{1}}+c_{12}\frac{S_{1}I_{2}}{N_{2}}\right],$
$\displaystyle T_{8}$ $\displaystyle\equiv
T(S_{1},I_{1},S_{2}-1,I_{2}+1|S_{1},I_{1},S_{2},I_{2})$
$\displaystyle=\beta\left[c_{21}\frac{S_{2}I_{1}}{N_{1}}+c_{22}\frac{S_{2}I_{2}}{N_{2}}\right],$
(4)
where
$\displaystyle c_{11}$ $\displaystyle=$
$\displaystyle\dfrac{(1-f_{21})^{2}}{1-f_{21}+f_{12}q}+\dfrac{f_{21}^{2}}{f_{21}+(1-f_{12})q},$
$\displaystyle c_{12}$ $\displaystyle=$
$\displaystyle\dfrac{(1-f_{21})f_{12}q}{1-f_{21}+f_{12}q}+\dfrac{f_{21}(1-f_{12})q}{f_{21}+(1-f_{12})q},$
$\displaystyle c_{21}$ $\displaystyle=$
$\displaystyle\dfrac{(1-f_{12})f_{21}}{f_{21}+(1-f_{12})q}+\dfrac{f_{12}(1-f_{21})}{1-f_{21}+f_{12}q},$
$\displaystyle c_{22}$ $\displaystyle=$
$\displaystyle\dfrac{(1-f_{12})^{2}q}{f_{21}+(1-f_{12})q}+\dfrac{f_{12}^{2}q}{1-f_{21}+f_{12}q},$
(5)
and $q=N_{2}/N_{1}$. We assume that $N_{1}$ and $N_{2}$ are not too different,
so that $q$ is neither very small nor very large.
Figure 1: A fraction $f_{jk}$ of residents of city $k$ commute to city $j$,
where $j,k=1,2$.
The model is defined by the transitions rates in Eqs. (1)-(4). It is
interesting that the transitions due to infection depend on the fractions
$f_{jk}$ only through the constants $c_{jk}$ defined in Eq. (5). Other ways of
accounting for commuting individuals would typically still give rise to the
form given in Eq. (4), but with the constants $c_{jk}$ defined in a different
way.
Since our counting of the ways that infection takes place was exhaustive, we
expect that the constants $c_{jk}$ are not independent. It is straightforward
to check that they obey the following relations:
$c_{11}+c_{12}=1,\ \ c_{21}+c_{22}=1,\ \ c_{12}=c_{21}q.$ (6)
So there are only two independent parameters in additional to the usual SIR
parameters $\beta,\gamma$ and $\mu$ found in the single city case, and we
choose these to be $c_{12}$ and $q=N_{2}/N_{1}$. Our results will be given in
terms of these two parameters. It is easy to see that, for each $q$, the range
of $c_{12}$ is the interval $[0,q/(q+1)]$ where the maximum is attained for
$f_{21}+f_{12}=1$. While exploring the general behavior of the system we will
consider the $c_{jk}$ independently of the underlying microscopic model as
positive parameters that take values in the wider admissible range defined by
the constraints (6).
Having specified the model it may be investigated in two ways as indicated in
Section I. First, it can simulated with Gillespie’s algorithm Gillespie
(1976), or some equivalent method. Second, it can be studied analytically by
constructing the master equation and performing van Kampen’s system size
expansion on this equation. This will be the main focus of this paper. For
notational convenience we will denote the states of the system by
$\sigma\equiv\\{S_{1},I_{1},S_{2},I_{2}\\}$, recalling that the number of
recovered individuals from each city may be written in terms of these
variables. The master equation gives the time evolution of $P(\sigma,t)$, the
probability distribution for finding the system in state $\sigma$ at time $t$.
It takes the form van Kampen (2007); Risken (1989)
$\frac{dP(\sigma,t)}{dt}=\sum_{\sigma^{\prime}\neq\sigma}\sum^{8}_{a=1}\left[T_{a}(\sigma|\sigma^{\prime})P(\sigma^{\prime},t)-T_{a}(\sigma^{\prime}|\sigma)P(\sigma,t)\right],$
(7)
where $T_{a}(\sigma|\sigma^{\prime})$, $a=1,\ldots,8$ are the transition rates
from the state $\sigma^{\prime}$ to the state $\sigma$ given explicitly in
Eqs. (1)-(4).
The full master equation (7) cannot be solved, but the van Kampen system-size
expansion when taken to next-to-leading order usually gives results which are
in excellent agreement with simulations. We will see that this will also be
the case in the extensions of the method which we are exploring in this paper.
The system-size expansion starts by making the following ansatz van Kampen
(2007):
$S_{j}=N_{j}s_{j}+N_{j}^{1/2}x_{j},\ \ I_{j}=N_{j}i_{j}+N_{j}^{1/2}y_{j},$ (8)
where $j=1,2$. Here $s_{j}=\lim_{N_{j}\to\infty}S_{j}/N_{j}$ and
$i_{j}=\lim_{N_{j}\to\infty}I_{j}/N_{j}$ are the fraction of individuals from
city $j$ which are respectively susceptible and infected in the deterministic
limit. The quantities $x_{j}$ and $y_{j}$ are the stochastic deviations from
these deterministic results, suitably scaled so that they also become
continuous in the limit of large population sizes. The ansatz (8) is
substituted into Eq. (7) and powers of $\sqrt{N_{j}}$ on the left- and right-
hand sides matched up. The leading order contribution gives the deterministic
equations of the model and the next-to-leading order linear stochastic
differential equations for $x_{j}$ and $y_{j}$. We shall not describe the
method in great detail, since it is described clearly in van Kampen’s book van
Kampen (2007) and in many papers, including several on the SIR model Alonso et
al. (2007); Simoes et al. (2008); Black et al. (2009). Instead we will outline
the main results of the approximation in the remainder of this section, and
give some explicit intermediate formulas in Appendix A.
The deterministic equations which are found to first order in the system-size
expansion can also be obtained by multiplying Eq. (7) by $S_{1},I_{1},S_{2}$
and $I_{2}$ in turn and then summing over all states $\sigma$. This yields
$\displaystyle\dfrac{ds_{1}}{dt}$ $\displaystyle=$ $\displaystyle-\beta
s_{1}\left(c_{11}i_{1}+c_{12}i_{2}\right)+\mu(1-s_{1}),$
$\displaystyle\dfrac{ds_{2}}{dt}$ $\displaystyle=$ $\displaystyle-\beta
s_{2}\left(c_{21}i_{1}+c_{22}i_{2}\right)+\mu(1-s_{2}),$
$\displaystyle\dfrac{di_{1}}{dt}$ $\displaystyle=$ $\displaystyle\beta
s_{1}\left(c_{11}i_{1}+c_{12}i_{2}\right)-(\gamma+\mu)i_{1},$
$\displaystyle\dfrac{di_{2}}{dt}$ $\displaystyle=$ $\displaystyle\beta
s_{2}\left(c_{21}i_{1}+c_{22}i_{2}\right)-(\gamma+\mu)i_{2}.$ (9)
For the case of cities with equal population sizes, these have been previously
found and analyzed in Keeling and Rohani (2002). In the context of the present
work we are mainly interested in the fixed points of these equations. We will
not discuss these here, instead we will wait until Section III, where the case
of $n$ cities will be discussed when we can give a more general treatment.
Of more interest to us in this paper are the variables $x_{j}$ and $y_{j}$
which describe the linear fluctuations around trajectories of the
deterministic set of equations (9). For convenience we will introduce the
vector of these fluctuations $\mathbf{z}=(x_{1},x_{2},y_{1},y_{2})$. Our focus
will be on fluctuations in the stationary state, that is, about the non-
trivial fixed point of the deterministic equations (which we will show in the
next section is unique). The fluctuations obtained through the system-size
expansion obey a linear Fokker-Planck equation, which is equivalent to a set
of stochastic differential equations of the form Risken (1989)
$\frac{dz_{J}}{dt}=\sum_{K=1}^{4}A_{JK}z_{K}+\eta_{J}(t),\ \ J=1,\ldots,4,$
(10)
where $\eta_{J}(t)$ are Gaussian noise terms with zero mean which satisfy
$\langle\eta_{J}(t)\eta_{K}(t^{\prime})\rangle=B_{JK}\delta(t-t^{\prime})$.
Since the fluctuations are about the fixed point, the $4\times 4$ matrices $A$
and $B$ are independent of time, and completely characterize the fluctuations.
They are given explicitly in Appendix A.
The fluctuations will be analyzed in detail in Section IV, when they will also
be compared to the results of numerical simulations. Before discussing this,
we will generalize the discussion of this section to an arbitrary network of
$n$ cities.
## III Arbitrary network structure
In this section we will generalize the content of Section II to $n$ cities and
also find the fixed points of the deterministic dynamics in this case.
Figure 2: Individuals commute between $n$ cities, illustrated for a particular
network when $n=4$.
### III.1 $n$-city model
We use the same notation as in Section II, labeling the cities by $j$ and $k$
which now run from $1$ to $n$. It will be convenient to introduce the quantity
$f_{j}=\sum_{k\neq j}f_{kj},$ (11)
so that the number of individuals in city $j$ may be written as
$\displaystyle M_{j}$ $\displaystyle=$ $\displaystyle\Bigl{[}1-\sum_{k\neq
j}f_{kj}\Bigr{]}N_{j}+\sum_{k\neq j}f_{jk}N_{k}$ (12) $\displaystyle=$
$\displaystyle\left(1-f_{j}\right)N_{j}+\sum_{k\neq j}f_{jk}N_{k}.$
There are, once again, four types of term in the process of infection (see
Figure 2) and we again fix our attention on those involving susceptible
individuals from city 1:
* (i)
Infective residents in city $1$ infect susceptible residents in city $1$. This
gives a rate of $\beta(1-f_{1})S_{1}(1-f_{1})I_{1}/M_{1}$.
* (ii)
Infective commuters from city $j$, $j=2,\ldots,n$, infect susceptible
residents in city $1$. This gives a rate, summing over all $j$, of
$\beta(1-f_{1})S_{1}\sum_{j\neq 1}f_{1j}I_{j}/M_{1}$.
* (iii)
Infective residents in city $j$, $j=2,\ldots,n$, infect susceptible commuters
from city $1$. This gives a rate, summing over all $j$, of $\beta\sum_{j\neq
1}(1-f_{j})I_{j}f_{j1}S_{1}/M_{j}$.
* (iv)
Infective commuters from city $k$ (including city $1$) infect susceptible
commuters from city $1$ in city $j$. This gives a total rate of
$\beta\sum_{j\neq 1}f_{j1}S_{1}\sum_{k\neq j}f_{jk}I_{k}/M_{j}$.
Since the transition rates for recovery and birth/death are simple extensions
of those for two cities we can now write down the transition rates for the
$n$-city model as:
* (a)
Recovery of an infective individual (and creation of a recovered individual)
$T_{j}\equiv
T(S_{1},I_{1},\ldots,S_{j},I_{j}-1,\ldots,S_{n},I_{n}|\sigma)=\gamma I_{j},$
(13)
* (b)
Death of an infected individual (and birth of a susceptible individual):
$T_{n+j}\equiv
T(S_{1},I_{1},\ldots,S_{j}+1,I_{j}-1,\ldots,S_{n},I_{n}|\sigma)=\mu I_{j},$
(14)
* (c)
Death of a recovered individual (and birth of a susceptible individual):
$\displaystyle T_{2n+j}$ $\displaystyle\equiv
T(S_{1},I_{1},\ldots,S_{j}+1,I_{j}\ldots,S_{n},I_{n}|\sigma)$
$\displaystyle=\mu(N_{j}-S_{j}-I_{j}),$ (15)
* (d)
Infection of a susceptible individual:
$\displaystyle T_{3n+j}$ $\displaystyle\equiv
T(S_{1},I_{1},\ldots,S_{j}-1,I_{j}+1\ldots,S_{n},I_{n}|\sigma)$
$\displaystyle=\beta\sum^{n}_{k=1}c_{jk}\frac{S_{j}I_{k}}{N_{k}},$ (16)
where $\sigma\equiv\\{S_{1},I_{1},\ldots,S_{j},I_{j}\ldots,S_{n},I_{n}\\}$ and
where $j=1,\ldots,n$. The coefficients $c_{jk}$ in Eq. (16) may be read off
from the terms (i)-(iv), but they are sufficiently complicated to write down
in full that we only list them in Appendix B. In that Appendix we also show
that relations between the $c_{jk}$, analogous to those given in Eq. (6) for
the two-city case hold, and are given by
$c_{jj}+\sum_{k\neq j}c_{jk}=1;\ \
c_{kj}=\left(\frac{N_{j}}{N_{k}}\right)c_{jk};\ j,k=1,\ldots,n.$ (17)
So in the $n$-city model, there are $n(n-1)/2$ independent coupling parameters
$c_{jk}$ and $(n-1)$ parameters for city sizes in additional to the usual
epidemiological parameters. Note that if all city sizes are equal the second
relation in Eq. (17) reduces to $c_{kj}=c_{jk}$. This symmetry will be used in
the subsequent analysis.
Following the same path as in Section II, having specified the model by giving
the transition rates, we move on to the dynamics. The process is Markovian and
so satisfies the master equation (7) except now the sum on $a$ goes from $1$
to $4n$. As detailed in Appendix A, invoking the van Kampen ansatz (8) gives
the following deterministic equations to leading order:
$\displaystyle\dfrac{ds_{j}}{dt}$ $\displaystyle=$ $\displaystyle-\beta
s_{j}\sum^{n}_{k=1}c_{jk}i_{k}+\mu\left(1-s_{j}\right),$
$\displaystyle\dfrac{di_{j}}{dt}$ $\displaystyle=$ $\displaystyle\beta
s_{j}\sum^{n}_{k=1}c_{jk}i_{k}-\left(\gamma+\mu\right)i_{j},$ (18)
where $j=1,\ldots,n$. At next-to-leading order the fluctuations are found to
satisfy the linear stochastic differential equation (10), but with
$J,K=1,\ldots,2n$. The two matrices $A$ and $B$ are given explicitly in
Appendix A. They are independent of time, since they are evaluated at the
fixed points of the dynamics (18). For the rest of this section we will
investigate the fixed point structure of these equations.
### III.2 The fixed points of the deterministic equations
The fixed points of the deterministic equations (18) will be denoted by
asterisks. Adding the two sets of equations we immediately see that
$\left(\gamma+\mu\right)i^{*}_{j}=\mu\left(1-s^{*}_{j}\right),\ \
j=1,\ldots,n.$ (19)
Using this equation to eliminate the $i^{*}_{j}$, and also using Eq. (17), one
finds that
$s^{*}_{j}\left[\left(\beta+\gamma+\mu\right)-\beta\sum^{n}_{k=1}c_{jk}s^{*}_{k}\right]=\left(\gamma+\mu\right),\
\ j=1,\ldots,n.$ (20)
Two fixed points can be found by inspection. First, suppose one of the
$i^{*}_{j}$ is zero, for instance $i^{*}_{\ell}=0$. Then from Eq. (19)
$s^{*}_{\ell}=1$. From Eq. (18) we see immediately that $\sum^{n}_{k=1}c_{\ell
k}i^{*}_{k}=0$. Since the coefficients $c_{jk}$ are non-negative (see Appendix
B), then $i^{*}_{k}=0$ for all $k$ as long as $c_{\ell k}\neq 0$. Using the
$i^{*}_{k}$ which are zero as input into Eq. (18), in the same way as we did
originally for $i^{*}_{\ell}$, we see that as long as the cities are connected
by non-zero $c_{jk}$, then they will have no infected individuals present.
From Eq. (19) it follows that $s^{*}_{k}=1$ for these cities. This is the
trivial solution where no infection is present anywhere in this cluster of
connected cities. We will assume that all the cities are connected either
directly or indirectly, so that $i^{*}_{k}=0,s^{*}_{k}=1$ for all $k$.
Of more interest is what we will call “the symmetric fixed point”. This has
$s^{*}_{k}=s^{*}$, a constant, for all $k$. From Eq. (19) one sees that the
$i^{*}_{k}$ are also independent of $k$, and we denote them by $i^{*}$. Using
Eq. (17), $s^{*}$ and $i^{*}$ are found to satisfy the equations
$\displaystyle s^{*}\left[\left(\beta+\gamma+\mu\right)-\beta
s^{*}\right]=\left(\gamma+\mu\right),$
$\displaystyle\left(\gamma+\mu\right)i^{*}=\mu\left(1-s^{*}\right),$ (21)
which are the fixed point equations for the ordinary ‘one-city’ SIR model
Anderson and May (1991); Keeling and Rohani (2007). As is well known these may
be solved to give for the non-trivial fixed point
$s^{*}=\dfrac{\gamma+\mu}{\beta},\ \
i^{*}=\dfrac{\mu\left[\beta-(\gamma+\mu)\right]}{\beta(\gamma+\mu)}.$ (22)
Due to a remarkable theorem, we can assert that the symmetric solution given
by Eq. (22) is the only non-trivial fixed point of the deterministic equations
(18) Guo et al. (2008). This is proved by finding a Liapunov function for the
$n$-city SIR model. In fact the result is more general than we require and was
proved for the SEIR model; in Appendix B we give the explicit form of the
Liapunov function for the SIR model and a brief outline of the proof following
the argument in Ref. Guo et al. (2008) for this simpler case. The theorem also
tells us that the non-trivial fixed point (22) is globally stable. Therefore
we can now go on to study stochastic fluctuations about this well
characterized attractor.
## IV Spectrum of the Stochastic Fluctuations
Based on previous studies of stochastic fluctuations in the SIR model in
different contexts, we would expect that the fixed point behavior predicted in
the deterministic limit is replaced by large stochastic oscillations McKane
and Newman (2005); Alonso et al. (2007). In effect, the noise due to the
randomness of the processes in the IBM, sustains the natural tendency for
cycles to occur, and amplifies them through a resonance effect. Since the
oscillations are stochastic, straightforward averaging will simply wipe out
the cyclic structure; to understand the nature of the fluctuations we need to
Fourier transform them and then pick out the dominant frequencies.
So we begin by taking the Fourier transform of the linear stochastic
differential equation Eq. (10) (generalized to the case of $n$ cities) to find
$\sum^{2n}_{K=1}\left(-i\omega\delta_{JK}-A_{JK}\right)\tilde{z}_{K}(\omega)=\tilde{\eta}_{J}(\omega),\
\ J=1,\ldots,2n,$ (23)
where the $\tilde{f}$ denotes the Fourier transform of the function $f$.
Defining the matrix $-i\omega\delta_{JK}-A_{JK}$ to be $\Phi_{JK}(\omega)$,
the solution to Eq. (23) is
$\tilde{z}_{J}(\omega)=\sum^{2n}_{K=1}\Phi^{-1}_{JK}(\omega)\tilde{\eta}_{K}(\omega).$
(24)
The power spectrum for fluctuations carrying the index $J$ is defined by
$P_{J}(\omega)\equiv\left\langle|\tilde{z}_{J}(\omega)|^{2}\right\rangle=\sum^{2n}_{K=1}\sum^{2n}_{L=1}\Phi^{-1}_{JK}(\omega)B_{KL}\left(\Phi^{{\dagger}}\right)^{-1}_{LJ}(\omega).$
(25)
Since $\Phi=-i\omega I-A$, where $I$ is the $2n\times 2n$ unit matrix, and
since $A$ and $B$ are independent of $\omega$, the structure of
$P_{J}(\omega)$ is that of a polynomial of degree $4n-2$ divided by another
polynomial of degree $4n$. The explicit form of the denominator is
$|\det\Phi(\omega)|^{2}$.
Oscillations with well-defined frequencies should show up as peaks in the
power-spectrum. The structure of the power spectrum described above — with the
ratio of polynomials of high order potentially giving rise to many maxima —
might lead us to suppose that the spectrum of fluctuations would have a rather
complex structure. In fact numerical simulations indicate that only a single
peak is present for a large range of parameter values. An example is shown in
Fig. 3, where typical values for measles Bauch and Earn (2003); Keeling and
Grenfell (2002); Anderson and May (1991) were chosen for the epidemiological
parameters (we shall keep these values fixed throughout this section).
Figure 3: (Color online) Power spectrum for the fluctuations of infectives
from simulation of a three-city model with equal population sizes plotted as a
function of the frequency $\nu=\omega/(2\pi)$ 1/y. The spectrum shown
corresponds to city 1, the spectra for the other cities are very similar.
Metapopulation model parameters: $N_{1}=N_{2}=N_{3}=10^{6}$, $c_{12}=0.06$,
and $c_{13}=c_{23}=0.02$. Epidemiological parameters: $\gamma=365/13$ 1/y,
$\mu=1/50$ 1/y, and $\beta=17(\gamma+\mu)$.
To understand how this comes about, we first note that the number of peaks in
the power spectrum is given by the form of the denominator,
$|\det\Phi(\omega)|^{2}$; the numerator essentially just shifts the position
of these peaks somewhat. Therefore we can understand the number and nature of
the peaks by studying the eigenvalues of $\Phi_{JK}$, which are those of the
matrix $A_{JK}$ shifted by $-i\omega$.
Each pair of complex conjugate eigenvalues of $A_{JK}$,
$\lambda_{c},\lambda_{c}^{*}$, will give rise to a factor in
$|\det\Phi(\omega)|^{2}$ of the form
$(|\lambda_{c}|^{2}-\omega^{2})^{2}+\left[2\operatorname{Re}(\lambda_{c})\omega\right]^{2}$,
and each real eigenvalue of $A_{JK}$, $\lambda_{r}$, yields a factor of the
form $(\lambda_{r}^{2}+\omega^{2})^{2}$. Peaks in the power spectrum are
associated with complex eigenvalues $\lambda_{c}$ of $A_{JK}$ with small real
parts, and their position is approximately given by
$\omega\approx\operatorname{Im}(\lambda_{c})$. In the trivial case of one
city, $n=1$, $A_{JK}$ has a pair of complex conjugate eigenvalues
$\lambda_{1}^{\pm}$ with
$\operatorname{Re}(\lambda_{1}^{\pm})=-\beta\mu/(2(\gamma+\mu))$ and
$|\lambda_{1}^{\pm}|=\sqrt{\mu(\beta-\gamma-\mu)}$ (see Appendix B). The
conditions for a pronounced peak for $\omega$ close to
$\operatorname{Im}(\lambda_{1}^{\pm})\approx|\lambda_{1}^{\pm}|$ are fulfilled
because $\mu$, the death-birth rate, is small. This carries over to the
general $n$ city case since, as shown in Appendix B, $\lambda_{1}^{\pm}$
always belong to the set of eigenvalues of $A_{JK}$. For the parameter values
of Fig. 3 the numerical values of the common eigenvalue pair are
$\lambda_{1}^{\pm}=-0.17\pm i\,2.99$, so we expect a peak to be located close
to $\nu=\operatorname{Im}(\lambda_{1}^{\pm})/(2\pi)\approx 0.48$ 1/y.
Figure 4: (Color online) An Argand diagram of the eigenvalues for the two-city
model with $q=N_{2}/N_{1}=3/2$ and $c_{12}\in[0,1]$. The large black dots are
the common eigenvalue pair $\lambda_{1}^{\pm}$. The sets of smaller dark gray
(blue) and light gray (green) dots are the remaining eigenvalues
$\lambda_{2}^{\pm}$ computed on a uniform grid of values of $c_{12}$ in the
interval. The eigenvalues with $\operatorname{Re}\lambda_{2}^{-}<-6$ are not
shown in the plot, they are found for $c_{12}>0.15$. The asterisks show the
eigenvalues for the parameter values used in Fig. 5.
For large demographic coupling, the $n$ city system will behave as well mixed
system comprising all the cities and we expect to find in that limit a power
spectrum similar to the case $n=1$, where each city contributes proportionally
to its size to the overall spectral density. In the opposite limit, the $n$
cities uncouple and we will find for each city the power spectrum of the one
city case. In order to understand why additional peaks do not show up in
simulations for intermediate coupling strengths, it is useful to consider the
case $n=2$, for which the eigenvalues of $A_{JK}$ can be determined
analytically and depend on a single coupling parameter $c_{12}$ and the ratio
of the population sizes $q=N_{2}/N_{1}$ (see Eq. (6) and Appendix B). An
Argand diagram of the two pairs of eigenvalues, $\lambda_{1}^{\pm}$ and
$\lambda_{2}^{\pm}$, for the two-city model is shown in Fig. 4. It can be seen
that as the coupling increases, $\lambda_{2}^{\pm}$ follow the circle ${\cal
C}$ centered at zero that goes through $\lambda_{1}^{\pm}$, moving away from
the imaginary axis. Real and imaginary parts become of the same order for very
small values of the coupling, and so we expect the power spectrum to be always
dominated by the peak associated with $\lambda_{1}^{\pm}$ that characterizes
the spectrum in the uncoupled case. This behavior carries over to the $n$-city
case with symmetric coupling, for which a complete analysis of the eigenvalues
of $A_{JK}$ can also be given, see Appendix B. In particular, it can be shown
that apart from the common eigenvalue pair $\lambda_{1}^{\pm}$ $A_{JK}$, has a
single $(n-1)$-fold degenerate additional eigenvalue pair that behaves as a
function of the coupling parameter as described above for $n=2$.
For the coupling parameter that corresponds to the values of
$\lambda_{2}^{\pm}$ marked with asterisks in Fig. 4 and for a certain choice
of population sizes, the infective fluctuations power spectra for the two-city
model obtained from simulations and from Eq. (25) are shown in Fig. 5. We find
a nearly perfect match between the results of numerical simulations and the
analytical calculations. In agreement with the above argument the power
spectra of city $1$ and city $2$ are very similar to the power spectrum of the
one city case, which in turn is very similar to the spectrum shown in Fig. 3
for 3 cities with small coupling. In all cases the functional form of the
spectral density is dominated by the peak associated with the common
eigenvalue pair $\lambda_{1}^{\pm}$. As for the amplitudes of the power
spectra $P_{J}(\nu)$, their ratio with respect to the one city case,
$r_{J}(\nu)$, decreases as the coupling increases. For two cities and $q=1$,
the power spectra $P_{3}$ and $P_{4}$ of city $1$ and city $2$ are equal and
the relative peak amplitudes $r_{3,4}(\nu_{\rm max})$ decrease with the
coupling strength $c_{12}$ down to $0.5$. For other values of $q$, as in Fig.
5, the different peak amplitudes in two cities reflect the symmetry
$P_{3}(\nu;c_{12},q)=P_{4}(\nu;c_{12}/q,1/q)$. Depending on $q$, the ratio
$r_{3,4}(\nu)$ may become even smaller than $0.5$, but due to the symmetry
that relates $P_{3}$ and $P_{4}$, the amplitude of at least one of these peaks
is always comparable to that of the uncoupled case. More precisely, it is easy
to check that $1\leq r_{3}(\nu;c_{12},q)+r_{4}(\nu;c_{12},q)\leq 2$, where the
second inequality is satisfied strictly for $c_{12}=0$ and the lower bound
corresponds to the large coupling limit $c_{12}=1$ and to $\nu=\nu_{\rm max}$
.
Figure 5: (Color online) Power spectra for the fluctuations of infectives from
simulation of the two-city model [(red) dots] and analytic calculation (black
solid curve) plotted as a function of the frequency $\nu=\omega/(2\pi)$ 1/y.
The population sizes were chosen to be $N_{1}=10^{6}$ and $N_{2}=1.5\times
10^{6}$ so that their ratio is $3/2$. The coupling coefficient $c_{12}=0.1$.
The location of the eigenvalues for this choice of parameters is indicated in
Fig. 4 by asterisks and large dots. Figure 6: (Color online) An Argand diagram
of the eigenvalues for a three-city model with equal population sizes,
$c_{12}\in[0,0.98]$ and the other parameters as in Fig. 3. The large black
dots are the common eigenvalue pair $\lambda_{1}^{\pm}$. The sets of smaller
dark gray (blue) and light gray (green) dots are the remaining eigenvalues
$\lambda_{2}^{\pm}$ (left panel) and $\lambda_{3}^{\pm}$ (right panel)
computed on a uniform grid of values of $c_{12}$ in the interval. The
eigenvalues with $\operatorname{Re}\lambda_{2}^{-}<-6$ are not shown in the
plot, they are found for $c_{12}>0.12$.
The general case of three cities with no symmetry can also in principle be
treated analytically because finding the eigenvalues of $A_{JK}$ reduces to
finding the roots of a fourth order polynomial. However, the problem now
depends on 3 independent coupling parameters and 2 parameters for city sizes
and closed form expressions are too lengthy to be useful. An approximate,
concise description of the behavior of the eigenvalues of $A_{JK}$ can be
given in terms of only two parameters that measure coupling strength and
coupling asymmetry, see Appendix B. In this approximation, we assume that all
the $c_{jk}$, $j\neq k$, are of order $\sqrt{\mu}$ and treat $\mu$ as the
small parameter of the system. Simple expressions for the real parts and the
absolute values of the additional eigenvalue pairs $\lambda_{2}^{\pm}$,
$\lambda_{3}^{\pm}$ of $A_{JK}$ up to terms of order $\mu$ can be derived [see
Eqs. (57) and (59)]. These show that, in this approximation, both eigenvalue
pairs behave as described for the symmetric case. As the coupling increases,
both eigenvalue pairs follow the circle ${\cal C}$ centered at zero that goes
through $\lambda_{1}^{\pm}$, moving away from the imaginary axis. The real and
imaginary parts become of the same order within the scope of the
approximation. Equation (57) also shows how the asymmetry lifts the degeneracy
of the two pairs $\lambda_{2}^{\pm}$, $\lambda_{3}^{\pm}$. As the coupling
increases, the two eigenvalue pairs move along the circle ${\cal C}$ at
different speeds. We have checked that Eqs. (57) and (59) give a good
approximation to the exact results in the regime when the eigenvalues are
complex.
The same behavior is illustrated in Fig. 6, where a plot of the exact
solutions for $\lambda_{2,3}^{\pm}$ is shown for parameter values that
correspond to taking those of Fig. 3 and allowing one of the coupling
coefficients to span the whole admissible range. One of the eigenvalues is
shown only up to $c_{12}=0.12$, where its real part becomes smaller than $-6$.
Figure 7: An Argand diagram of the eigenvalues for a four-city model with the
coupling strength $x\in[0,0.52]$. The large black dots are the common
eigenvalue pair $\lambda_{1}^{\pm}$. The remaining eigenvalues
$\lambda_{2,3,4}^{\pm}$ computed on a uniform grid of values of $x$ in the
interval are shown as sets of smaller gray dots. As in the previous figures we
only show eigenvalues whose real part is larger than $-6$. Metapopulation
model parameters: $N_{2}/N_{1}:N_{3}/N_{1}:N_{4}/N_{1}=2:3:4$,
${\hat{c}}_{12}=1/\sqrt{\mu}=2{\hat{c}}_{13}=5{\hat{c}}_{14}/2=5{\hat{c}}_{23}/2=3{\hat{c}}_{24}=4{\hat{c}}_{34}$.
In Fig. 7 we show numerical results for the behavior of the eigenvalues of
$A_{JK}$ in the case of 4 cities with different population sizes and a certain
choice of the coupling coefficients $c_{jk}$, $j,k=1,2,3,4$. We make use of
the following notation for the diagonal and off diagonal coupling coefficients
(see Appendix B): $c_{jj}=1-{\hat{c}}_{jj}\ x\ \sqrt{\mu}$ and
$c_{jk}={\hat{c}}_{jk}\ x\ \sqrt{\mu}$, respectively. We then calculate the
set of three non-trivial eigenvalue pairs as the coupling strength $x$ varies
in a suitable interval, keeping the ${\hat{c}}_{jk}$ fixed. These results
suggest that the behavior of the eigenvalues of $A_{JK}$ is essentially given
by the description of the symmetric case, and that more general couplings
break the degeneracy as in the case $n=3$, with no effects in the
contributions to the peaks in the power spectrum.
## V Discussion and conclusions
In this paper we have extended the analysis of a metapopulation model of
epidemics into the stochastic domain. Frequently epidemic models involving a
spatial component, such as the interaction between several cities, are studied
purely deterministically Keeling and Rohani (2002); Arino and van den
Driessche (2003) or through computer simulations Lloyd and May (1996); Simoes
et al. (2008); Christensen et al. (2010). We have demonstrated how a
stochastic metapopulation model can be studied analytically by using a
relatively straightforward extension of the methodology which was used to
study a well-mixed population in a single city.
We adopted a simple specification of residents and commuters in order to set
up the model. However, the coefficients which appear in the dynamical
equations are generic and would appear in the same form if residents and
commuters were included in a different way. It is evident that there are many
ways of characterizing the interchange of individuals between cities which
will result in the same model; only the identification of the coefficients
with the underlying structure will be different.
The deterministic form of the model predicts that the system will reach a
stable fixed point where the proportion of infected, susceptible and recovered
individuals is the same in every city. The stochastic version of the model
also predicts a clean simple result: that the large sustained oscillations
which replace the deterministic predictions of constant behavior, have a
single frequency which is the same for every city. Moreover, for small, large
and intermediate coupling between the cities, the form of the power spectrum
of these fluctuations is closely approximated by the power spectrum of the
single city system.
It is remarkable that such a simple result occurs in what is a quite
complicated stochastic nonlinear metapopulation model. We hope to explore the
range of validity of this result and its robustness to the addition of new
features to the model in the future. In any case, we believe that the work
presented here will give a firm foundation to possible future work, including
comparisons with the data available on childhood diseases.
###### Acknowledgements.
Financial support from the Portuguese Foundation for Science and Technology
(FCT) under Contract No. POCTI/ISFL/2/261 is gratefully acknowledged. G.R. was
also supported by FCT under Grant No. SFRH/BPD/69137/2010.
## Appendix A System-size expansion
Here we give some of the key steps in the application of the system-size
expansion to the model explored in this paper. The method has been extensively
discussed in the literature van Kampen (2007); Alonso et al. (2007); Simoes et
al. (2008); Rozhnova and Nunes (2009a, b); Rozhnova and Nunes (2010); Black et
al. (2009); Black and McKane (2010a, b), and so we confine ourselves to a
brief outline and to displaying the most important intermediate results in the
derivation. We will assume that we are carrying out the calculation for the
$n$-city case discussed in Section III; the corresponding results for Section
II can be obtained simply by setting $n=2$.
The first point to mention is that there are apparently $n$ expansion
parameters: $\\{N_{1},\ldots,N_{n}\\}$. The method is valid if they are all
large and of the same order. More formally we can take, for instance,
$N_{1}\equiv N$ as the expansion parameter and express all the other $N_{j}$
in terms of it: $N_{j}=Nq_{j}$, where the $q_{j}=N_{j}/N$, $j=2,\ldots,n$ are
of order one. In practice the method seems to work well when the $q_{j}$ are
significantly different from one, but this has to be checked a posteriori, for
instance by comparing the analytic results with those obtained using computer
simulations. In what follows we will not introduce the $q_{j}$ explicitly; we
will simply take all the $N_{j}$’s to be of the same order in the expansion.
The van Kampen ansatz Eq. (8) replaces the discrete stochastic variables
$\sigma$ by the continuous stochastic variables $\mathbf{z}$ and so we write
the transformed probability distribution $P(\sigma,t)$ as $\Pi(\mathbf{z},t)$.
Since this transformation is time-dependent, substituting the ansatz into
$dP/dt$ on the left-hand side of Eq. (7) gives van Kampen (2007)
$\displaystyle\frac{dP(\sigma,t)}{dt}$ $\displaystyle=$
$\displaystyle\frac{\partial\Pi(\mathbf{z},t)}{\partial
t}-\sum^{n}_{j=1}\sqrt{N_{j}}\,\frac{\partial\Pi(\mathbf{z},t)}{\partial
x_{j}}\frac{ds_{j}}{dt}$ (26) $\displaystyle-$
$\displaystyle\sum^{n}_{j=1}\sqrt{N_{j}}\,\frac{\partial\Pi(\mathbf{z},t)}{\partial
y_{j}}\frac{di_{j}}{dt}\,.$
The right-hand side of the master equation (7) can be put into a form from
which it is simple to apply the expansion procedure. To do this one introduces
step-operators van Kampen (2007) defined by
$\displaystyle\epsilon^{\pm 1}_{S_{j}}$ $\displaystyle
f(S_{1},\ldots,S_{j},\ldots,S_{n},I_{1},\ldots,I_{n})$ $\displaystyle=$
$\displaystyle f(S_{1},\ldots,S_{j}\pm 1,\ldots,S_{n},I_{1},\ldots,I_{n}),$
$\displaystyle\epsilon^{\pm 1}_{I_{j}}$ $\displaystyle
f(S_{1},\ldots,S_{n},I_{1},\ldots,I_{j},\ldots,I_{n})$ $\displaystyle=$
$\displaystyle f(S_{1},\ldots,S_{n},I_{1},\ldots,I_{j}\pm 1,\ldots,I_{n}),$
(27)
for a general function $f$ and where $j=1,\ldots,n$. Using these operators the
master equation (7) may be written as
$\displaystyle\dfrac{dP(\sigma,t)}{dt}=\sum_{j=1}^{n}\left[\left(\epsilon_{I_{j}}-1\right)T_{j}+\left(\dfrac{\epsilon_{I_{j}}}{\epsilon_{S_{j}}}-1\right)T_{n+j}\right.$
$\displaystyle+\left.\left(\dfrac{1}{\epsilon_{S_{j}}}-1\right)T_{2n+j}+\left(\dfrac{\epsilon_{S_{j}}}{\epsilon_{I_{j}}}-1\right)T_{3n+j}\right]P(\sigma,t).$
(28)
Within the system-size expansion these operators have a simple structure:
$\epsilon_{S_{j}}=\sum\limits_{p=0}^{\infty}\dfrac{N_{j}^{-p/2}}{p!}\dfrac{\partial^{p}}{\partial
x_{j}^{p}},\ \
\epsilon_{I_{j}}=\sum\limits_{p=0}^{\infty}\dfrac{N_{j}^{-p/2}}{p!}\dfrac{\partial^{p}}{\partial
y_{j}^{p}},$ (29)
and so all the terms of the right-hand side of Eq. (28) may be
straightforwardly expanded. Comparing these with the left-hand side in Eq.
(26) the leading order ($\sim\sqrt{N_{j}}$) yields the deterministic equations
given by Eq. (18). The next-to-leading order (which is of order one) gives a
Fokker-Planck equation:
$\frac{\partial\Pi}{\partial t}=-\sum^{2n}_{J,K=1}\frac{\partial}{\partial
z_{J}}\left[A_{JK}z_{K}\Pi\right]+\frac{1}{2}\sum^{2n}_{J,K=1}B_{JK}\frac{\partial^{2}\Pi}{\partial
z_{J}\partial z_{K}}.$ (30)
The $2n\times 2n$ matrices $A$ and $B$ which appear in this equation have the
following form. Writing $A$ in blocks of four $n\times n$ submatrices:
$A=\left[\begin{array}[]{c|c}A^{(1)}&A^{(2)}\\\ \hline\cr
A^{(3)}&A^{(4)}\end{array}\right],$ (31)
the elements of these submatrices are
$\displaystyle A^{(1)}_{jk}$ $\displaystyle=$
$\displaystyle-\mu\delta_{jk}-\beta\delta_{jk}\sum_{\ell=1}^{n}c_{j\ell}i_{\ell},$
$\displaystyle A^{(2)}_{jk}$ $\displaystyle=$
$\displaystyle-\beta\left(\dfrac{N_{j}}{N_{k}}\right)^{1/2}s_{j}c_{jk},$
$\displaystyle A^{(3)}_{jk}$ $\displaystyle=$
$\displaystyle\beta\delta_{jk}\sum_{\ell=1}^{n}c_{j\ell}i_{\ell},$
$\displaystyle A^{(4)}_{jk}$ $\displaystyle=$
$\displaystyle-(\mu+\gamma)\delta_{jk}+\beta\left(\dfrac{N_{j}}{N_{k}}\right)^{1/2}s_{j}c_{jk}.$
(32)
Writing $B$ in a similar way to $A$ in Eq. (31), the elements of the
submatrices are
$\displaystyle B^{(1)}_{jk}$ $\displaystyle=$
$\displaystyle\mu\delta_{jk}\left(1-s_{j}\right)+\beta\delta_{jk}\sum_{\ell=1}^{n}s_{j}c_{j\ell}i_{\ell},$
$\displaystyle B^{(2)}_{jk}$ $\displaystyle=$ $\displaystyle
B^{(3)}_{jk}=-\mu\delta_{jk}i_{j}-\beta\delta_{jk}\sum_{\ell=1}^{n}s_{j}c_{j\ell}i_{\ell},$
$\displaystyle B^{(4)}_{jk}$ $\displaystyle=$
$\displaystyle\left(\gamma+\mu\right)\delta_{jk}i_{j}+\beta\delta_{jk}\sum_{\ell=1}^{n}s_{j}c_{j\ell}i_{\ell}.$
(33)
From Eqs. (32) and (33) it is clear that the matrices $A_{jk}$ and $B_{jk}$
depend on the solutions of the deterministic equations given in Eq. (18).
However, since we will be interested only in fluctuations about the stationary
state, these matrices are evaluated at the fixed point. Since the unique
stable fixed point is the symmetric one, the same for all cities, the entries
(32) and (33) are given by:
$\displaystyle A^{*(1)}_{jk}$ $\displaystyle=-\left[\mu+\beta
i^{*}\right]\delta_{jk},\ \
A^{*(2)}_{jk}=-\beta\left(\dfrac{N_{j}}{N_{k}}\right)^{1/2}s^{*}c_{jk},$
$\displaystyle A^{*(3)}_{jk}$ $\displaystyle=\beta i^{*}\delta_{jk},\ \
A^{*(4)}_{jk}=\beta\left(\dfrac{N_{j}}{N_{k}}\right)^{1/2}s^{*}c_{jk}-(\mu+\gamma)\delta_{jk},$
and
$\displaystyle B^{*(1)}_{jk}=2\mu\left(1-s^{*}\right)\delta_{jk},\ \
B^{*(4)}_{jk}=2\left(\gamma+\mu\right)i^{*}\delta_{jk},$ $\displaystyle
B^{*(2)}_{jk}=B^{*(3)}_{jk}=-i^{*}\left[\mu+\beta s^{*}\right]\delta_{jk},$
(35)
where we have used the fixed-point equation (21) to simplify some of the
entries in Eq. (35).
Finally, the Fokker-Planck equation (30) is equivalent to the stochastic
differential equation (10). We will work with the latter, since we wish to use
Fourier analysis to analyze the nature of the fluctuations, and since Eq. (10)
is linear, it can easily be Fourier transformed, as discussed in detail in
Section IV.
## Appendix B Some results for the $n$-city case
In this Appendix we give some of the derivations for the $n$-city case
discussed in Section III and Section IV which are too long and cumbersome to
be given in the main text.
### B.1 The coefficients $c_{jk}$
The coefficients $c_{jk}$ appearing in Eq. (16) may be read off from the four
types of term (i)-(iv) given in Section III:
$\displaystyle c_{jj}$ $\displaystyle=$
$\displaystyle\frac{\left(1-f_{j}\right)^{2}N_{j}}{\left[(1-f_{j})N_{j}+\sum_{m\neq
j}f_{jm}N_{m}\right]}$ (36) $\displaystyle+$ $\displaystyle\sum_{\ell\neq
j}\frac{f_{\ell
j}^{2}N_{j}}{\left[(1-f_{\ell})N_{\ell}+\sum_{m\neq\ell}f_{\ell
m}N_{m}\right]},$
for $j=1,\ldots,n$ and
$\displaystyle c_{jk}$ $\displaystyle=$
$\displaystyle\frac{\left(1-f_{j}\right)f_{jk}N_{k}}{\left[(1-f_{j})N_{j}+\sum_{m\neq
j}f_{jm}N_{m}\right]}$ (37) $\displaystyle+$
$\displaystyle\frac{f_{kj}\left(1-f_{k}\right)N_{k}}{\left[(1-f_{k})N_{k}+\sum_{m\neq
k}f_{km}N_{m}\right]}$ $\displaystyle+$ $\displaystyle\sum_{\ell\neq
j,k}\frac{f_{\ell j}f_{\ell
k}N_{k}}{\left[(1-f_{\ell})N_{\ell}+\sum_{m\neq\ell}f_{\ell m}N_{m}\right]},$
for $j,k=1,\ldots,n$ and $j\neq k$.
To prove the first relation given in Eq. (17), consider the sum
$c_{jj}+\sum_{k\neq j}c_{jk}$. The first term in Eq. (36) combines with the
first term in Eq. (37) to give $(1-f_{j})$. The last term in Eq. (36) combines
with the last term in Eq. (37) to give
$\displaystyle\sum_{\ell\neq j}\frac{f_{\ell j}\left[f_{\ell
j}N_{j}+\sum_{k\neq j,\ell}f_{\ell
k}N_{k}\right]}{\left[(1-f_{\ell})N_{\ell}+\sum_{m\neq\ell}f_{\ell
m}N_{m}\right]}$ $\displaystyle=\sum_{\ell\neq j}\frac{f_{\ell
j}\left[\sum_{k\neq\ell}f_{\ell
k}N_{k}\right]}{\left[(1-f_{\ell})N_{\ell}+\sum_{m\neq\ell}f_{\ell
m}N_{m}\right]}$ $\displaystyle=\sum_{k\neq j}\frac{f_{kj}\left[\sum_{m\neq
k}f_{km}N_{m}\right]}{\left[(1-f_{k})N_{k}+\sum_{m\neq k}f_{km}N_{m}\right]},$
(38)
where in the last line we have performed a relabeling. Combining the middle
term of Eq. (37) with the result in Eq. (38) gives
$\sum_{k\neq j}\frac{f_{kj}\left[\left(1-f_{k}\right)N_{k}+\sum_{m\neq
k}f_{km}N_{m}\right]}{\left[(1-f_{k})N_{k}+\sum_{m\neq
k}f_{km}N_{m}\right]}=f_{j},$ (39)
using Eq. (11). Adding this to the result $(1-f_{j})$ found earlier proves the
result $c_{jj}+\sum_{k\neq j}c_{jk}=1$.
We also note from Eq. (37) that $c_{jk}/N_{k}$ is symmetric under the
interchange of $j$ and $k$ Therefore
$\frac{c_{jk}}{N_{k}}=\frac{c_{kj}}{N_{j}},$ (40)
which is the second relation in Eq. (17).
### B.2 Uniqueness and stability of the fixed point
In Section III we asserted that the deterministic equations (18) have a unique
non-trivial fixed point, which was globally stable. Here we prove this by
giving a Liapunov function for the dynamical system in the invariant region
$R=\\{(s_{1},\ldots,s_{n},i_{1},\ldots,i_{n}):0\leq s_{j}\leq 1,0\leq
i_{j}\leq 1,s_{j}+i_{j}\leq 1,j=1,...,n\\}$ where the system is defined. This
is a modification of the function given in Ref. Guo et al. (2008) for the SEIR
model. The proof assumes that the matrix of the coupling coefficients $c_{jk}$
is irreducible, which means that any two cities have a direct or indirect
interaction. Otherwise the proof breaks down because the $n$ cities may be
split into non-interacting subsets, and several equilibria may be found by
combining disease extinction in some subsets with non-trivial equilibrium in
other subsets.
Let $\beta_{jk}\equiv\beta c_{jk}s^{*}_{j}i^{*}_{k}$, where
$(s^{*}_{1},\ldots,s^{*}_{n},i^{*}_{1},\ldots,i^{*}_{n})$ is a fixed point of
Eq. (18), and denote by $M$ the matrix defined by $M_{kj}=\beta_{jk},j\neq k$,
and $\sum_{k=1}^{n}M_{kj}=0,j=1,...,n$. It can be shown (Guo et al. (2008),
Lemma 2.1) that the solution space of the linear equation $Mv=0$ is spanned by
a single vector $(v_{1},\ldots,v_{n})$, $v_{j}>0,j=1,\ldots,n$. Let
$L(s_{1},\ldots,s_{n},i_{1},\ldots,i_{n})$ be defined as
$L=\sum_{j=1}^{n}v_{j}(s_{j}-s_{j}^{*}\log s_{j}+i_{j}-i_{j}^{*}\log i_{j}).$
$L$ has a global minimum in $R$ at the fixed point. Functions of this form
have been used in the literature as Liapunov functions for fixed points of
ecological and epidemiological models, whose variables take only positive
values Guo et al. (2008). Differentiating $L$ along the solutions of Eq. (18),
and following the proof of Theorem 1.1 in Ref. Guo et al. (2008), we obtain
$\dot{L}\leq\sum_{j,k=1}^{n}v_{j}M_{kj}\left(2-\frac{s_{j}^{*}}{s_{j}}-\frac{s_{j}}{s_{j}^{*}}\frac{i_{k}}{i_{k}^{*}}\frac{i_{j}^{*}}{i_{j}}\right).$
(41)
The properties of the coefficients $v_{j}$ in the definition of $L$ play a
crucial role in the derivation of the second term in this inequality. Use has
been made of the identity
$\sum_{j=1}^{n}v_{j}\sum_{k=1}^{n}\beta
c_{jk}s^{*}_{j}i_{k}=\sum_{j=1}^{n}v_{j}(\gamma+\mu)i_{j},$ (42)
which in turn uses the fact that $Mv=0$ can be written as
$\sum_{j=1}^{n}\beta c_{kj}s^{*}_{k}i^{*}_{j}v_{k}=\sum_{j=1}^{n}\beta
c_{jk}s^{*}_{j}i^{*}_{k}v_{j},\ ,k=1,\ldots,n.$ (43)
Following Ref. Guo et al. (2008), it can then be shown that the right-hand
side of Eq (41) is strictly negative except at
$(s^{*}_{1},\ldots,s^{*}_{n},i^{*}_{1},\ldots,i^{*}_{n})$. Therefore, $L$ is a
Liapunov function for this fixed point in $R$, and the fixed point is unique
and globally stable. Note that the result also holds when the disease
transmissibility $\beta$, the recovery rate $\gamma$ and the birth-death rate
$\mu$ are different in different cities, in which case the non-trivial
equilibrium is in general not symmetric.
### B.3 Nature of the eigenvalues of the matrix $A$
In this subsection we will give some results on the eigenvalues of $A$ which
are required for the discussion in Section IV.
We first recall that $A$ is closely related to the stability matrix of the
deterministic equations (18). In fact, in most applications of the system-size
expansion they are equal. In our case because we have $n$ expansion parameters
$\sqrt{N}_{j}$, they are not equal, but closely related. A simple calculation
of the Jacobian, $J$, from Eq. (18), shows that
$J=S^{-1}AS,\ \ {\rm where\ }S={\rm
diag}\left(\sqrt{N}_{1},\ldots,\sqrt{N}_{n}\right).$ (44)
The effect of the transformation is simply that one can obtain $J$ from $A$ by
omitting the terms $(N_{j}/N_{k})^{1/2}$ in $A^{(2)}_{jk}$ and $A^{(4)}_{jk}$
in Eq. (32) or in $A^{*(2)}_{jk}$ and $A^{*(4)}_{jk}$ in Eq.
(LABEL:A_entries*). This is useful, since it follows from the similarity
transformation (44) that the eigenvalues of $A$ are also the eigenvalues of
$J$. So we may study the simpler problem of finding the eigenvalues of the
Jacobian at the symmetric fixed point (22).
For orientation, let us explicitly calculate the characteristic polynomial of
the Jacobian for the cases of one city and two cities. These are
$n=1:\ \ \ R_{1}(\lambda)=Q^{-1}(d_{2}\lambda^{2}+d_{1}\lambda+d_{0}),$ (45)
where
$\displaystyle Q$ $\displaystyle=$ $\displaystyle\gamma+\mu,\ \
d_{2}=\gamma+\mu,\ \ d_{1}=\beta\mu,$ $\displaystyle d_{0}$ $\displaystyle=$
$\displaystyle\mu(\gamma+\mu)\left[\beta-(\gamma+\mu)\right],$ (46)
and
$n=2:\ \ \
R_{2}(\lambda)=Q^{-2}(d_{2}\lambda^{2}+d_{1}\lambda+d_{0})(g_{2}\lambda^{2}+g_{1}\lambda+g_{0}).$
Thus,
$R_{2}(\lambda)=R_{1}(\lambda)Q^{-1}(g_{2}\lambda^{2}+g_{1}\lambda+g_{0}),$
(47)
where
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle\gamma+\mu,\ \
g_{1}=\beta\mu+(c_{12}+c_{21})(\gamma+\mu)^{2},$ $\displaystyle g_{0}$
$\displaystyle=$
$\displaystyle\mu(\gamma+\mu)\left[\beta-(1-c_{12}-c_{21})(\gamma+\mu)\right].$
(48)
We see that the factor $R_{1}(\lambda)$ is common, which suggests that the
pair of eigenvalues found in the one city case might always be present in the
$n$ city case. This is easily proved by considering the vector
$\mathbf{v}=(v_{1},\ldots,v_{n},v_{n+1},\ldots,v_{2n})^{T}$ with components
satisfying $v_{i}=v$ and $v_{i+n}=v^{\prime}$ for $i=1,\ldots,n$. Then the
eigenvector equation $J^{*}{\bf v}=\lambda{\bf v}$ reduces to that for one
city as required.
A similar method can be used to find the characteristic polynomial for $n\geq
3$ cities with equal population sizes, where the couplings are equal, that is,
$c_{jk}=\begin{cases}1-(n-1)c,&j=k,\\\ c,&j\neq k,\end{cases}$ (49)
where $j,k=1,...,n$. We now take the components of the vector to be
$v_{1}=-v_{2}=v$, $v_{n+1}=-v_{n+2}=v^{\prime}$, and $v_{i}=v_{i+n}=0$ for
$i=3,\ldots,n$. The eigenvector equation $J^{*}{\bf v}=\lambda{\bf v}$ now
reduces to
$\displaystyle-\left(\dfrac{\beta\mu}{\gamma+\mu}+\lambda\right)v-(1-nc)(\gamma+\mu)v^{\prime}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(\dfrac{\beta\mu}{\gamma+\mu}-\mu\right)v-\left[nc(\gamma+\mu)+\lambda\right]v^{\prime}$
$\displaystyle=$ $\displaystyle 0.$ (50)
Therefore, both solutions of
$Q^{-1}\left(h_{2}\lambda^{2}+h_{1n}\lambda+h_{0n}\right)=0,$ (51)
where
$\displaystyle h_{2}$ $\displaystyle=$ $\displaystyle\gamma+\mu,\ \
h_{1n}=\beta\mu+nc(\gamma+\mu)^{2},$ $\displaystyle h_{0n}$ $\displaystyle=$
$\displaystyle\mu(\gamma+\mu)\left[\beta-(1-nc)(\gamma+\mu)\right],$ (52)
are eigenvalues of $J^{*}$. This procedure can be repeated for $n-1$
independent vectors with only four non-zero components and the same symmetry
as ${\bf v}$. Therefore, the characteristic polynomial of $J^{*}$,
$R_{n}(\lambda)$, factorizes as
$R_{n}(\lambda)=R_{1}(\lambda)\left[Q^{-1}\left(h_{2}\lambda^{2}+h_{1n}\lambda+h_{0n}\right)\right]^{n-1}.$
(53)
Finally let us consider 3 cities with arbitrary coupling and study the
eigenvalues of $J^{*}$ in the limit when the off-diagonal coefficients
$c_{jk}$ are small and of the same order. It will become clear that the
coupling range to explore corresponds to $c_{jk}$ of the order of $\sqrt{\mu}$
and it is convenient to introduce the notation
$c_{jk}(x)=\begin{cases}1-{\hat{c}}_{jj}\ x\ \sqrt{\mu},&j=k,\\\
{\hat{c}}_{jk}\ x\ \sqrt{\mu},&j\neq k,\end{cases}$ (54)
where $j,k=1,2,3$ and $x$ is a positive parameter. Eq. (54) represents, for
each choice of ${\hat{c}}_{jk}$, a family of systems with all the off-diagonal
coefficients $c_{jk}$ of the same order, that reaches the zero coupling limit
for $x=0$. The quantity $x$ measures the distance to zero coupling along each
particular family, scaled by $\sqrt{\mu}$. Taking into account the properties
of the matrix $c_{jk}$, given by Eq. (17), the characteristic polynomial of
$J^{*}$ is a polynomial of degree six that can be expressed in terms of this
distance $x\ \sqrt{\mu}$ and of three other independent parameters. We choose
these to be ${\hat{c}}_{jj}$, $j=1,2,3$. We know that this characteristic
polynomial factorizes as
$R_{1}(\lambda)(\lambda^{4}+p_{3}\lambda^{3}+p_{2}\lambda^{2}+p_{1}\lambda+p_{0})$,
where $p_{3}$, $p_{2}$, $p_{1}$ and $p_{0}$ are some coefficients. The roots
of $R_{1}(\lambda)$ are the pair of eigenvalues $\lambda_{1}^{\pm}$ shared by
all the characteristic equations of this family of systems. The polynomial of
degree four can be easily found by direct computation. For equal city sizes we
obtain for the coefficients
$\displaystyle p_{3}$ $\displaystyle=$ $\displaystyle\gamma\sigma\ x\
\sqrt{\mu}+{\cal O}(\mu),$ $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle 2(\beta-\gamma)\ \mu-3/4\ \gamma^{2}{\hat{p}}_{2}\ x^{2}\
\mu+{\cal O}(\mu^{3/2}),$ $\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\gamma(\beta-\gamma)\sigma\ x\ \mu^{3/2}+{\cal O}(\mu^{2}),$
$\displaystyle p_{0}$ $\displaystyle=$ $\displaystyle(\beta-\gamma)^{2}\
\mu^{2}+{\cal O}(\mu^{5/2}),$ (55)
where
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle{\hat{c}}_{11}+{\hat{c}}_{22}+{\hat{c}}_{33},$
$\displaystyle{\hat{p}}_{2}$ $\displaystyle=$
$\displaystyle{\hat{c}}_{11}^{2}+{\hat{c}}_{22}^{2}+{\hat{c}}_{33}^{2}-2({\hat{c}}_{11}{\hat{c}}_{22}+{\hat{c}}_{11}{\hat{c}}_{33}+{\hat{c}}_{22}{\hat{c}}_{33}).$
Keeping only the leading order terms in each of the coefficients given by Eq.
(55) we find a simple approximate expressions for the two additional
eigenvalue pairs $\lambda_{2}^{\pm}$, $\lambda_{3}^{\pm}$. In particular, we
find
$\displaystyle\operatorname{Re}(\lambda_{2}^{\pm})$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{4}(\sigma+k)\ x\ \sqrt{\mu}+{\cal O}(\mu),$
$\displaystyle\operatorname{Re}(\lambda_{3}^{\pm})$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{4}(\sigma-k)\ x\ \sqrt{\mu}+{\cal O}(\mu),$ (57)
where
$k^{2}=4({\hat{c}}_{11}^{2}+{\hat{c}}_{22}^{2}+{\hat{c}}_{33}^{2}-{\hat{c}}_{11}{\hat{c}}_{22}-{\hat{c}}_{11}{\hat{c}}_{33}-{\hat{c}}_{22}{\hat{c}}_{33}).$
(58)
Assuming without loss of generality that
${\hat{c}}_{33}\geq{\hat{c}}_{22}\geq{\hat{c}}_{11}$, $k^{2}$ is positive and
so $k$ is real. Note that $k=0$ in the symmetric case, and in that case (57)
coincide in the same order of approximation with the roots of Eq. (51) for
$n=3$. The quantities $\sigma\,x\,\sqrt{\mu}$ and $k\,x\,\sqrt{\mu}$ that
determine, in this approximation, the real parts of the two non-trivial
eigenvalue pairs can be interpreted as the overall coupling strength and the
coupling asymmetry for a system of family (54). We also find for the absolute
value of the eigenvalues
$|\lambda_{2,3}^{\pm}|=\sqrt{\beta-\gamma}\sqrt{\mu}+{\cal O}(\mu),$ (59)
which shows that, for all families of the form Eq. (54), the eigenvalues
$\lambda_{2,3}^{\pm}$ of $J^{*}$ move close to the circle ${\cal C}$ in the
complex plane centered at zero that goes through $\lambda_{1}^{\pm}$. For
arbitrary city sizes, the same calculation can be carried out to find that Eq.
(57) and Eq. (59) still hold, with Eq. (58) replaced by
$k^{2}=\sigma^{2}+\frac{1+q_{21}+q_{31}}{q_{21}q_{31}}{\tilde{k}}^{2},$ (60)
where $q_{jk}=N_{j}/N_{k}$ and
${\tilde{k}}^{2}={\hat{c}}_{11}^{2}+({\hat{c}}_{22}q_{21}-{\hat{c}}_{33}q_{31})^{2}-2{\hat{c}}_{11}({\hat{c}}_{22}q_{21}+{\hat{c}}_{33}q_{31}).$
(61)
The behavior of the two non-trivial eigenvalue pairs along a family (54) can
be described, in this approximation, in terms of the two parameters $\sigma$
and $k$ that characterize the family and of the scaled distance $x$. As $x$
increases away from zero, both eigenvalue pairs move along ${\cal C}$ with
speeds whose ratio is given by $(\sigma+k)/(\sigma-k)$. The parameter $k$ that
measures the asymmetry of the coupling causes the splitting of the two pairs
with respect to the degenerate, symmetric case. The first pair to reach the
real axis does so for $x=4\sqrt{\beta-\gamma}/(\gamma(\sigma+k))$, which lies
within the scope of the approximation. From then on the two real eigenvalues
keep changing with $x$ in such a way that the square root of their product
verifies the constraint Eq. (59) until for large $x$ the approximation breaks
down.
## References
* Pastor-Satorras and Vespignani (2001) R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001).
* Keeling and Eames (2005) M. J. Keeling and K. T. D. Eames, J. R. Soc. Interface 2, 295 (2005).
* Gross et al. (2006) T. Gross, C. J. D. D’Lima, and B. Blasius, Phys. Rev. Lett. 96, 208701 (2006).
* Volz and Meyers (2007) E. Volz and L. A. Meyers, Proc. R. Soc. B 274, 2925 (2007).
* Colizza et al. (2007) V. Colizza, R. Pastor-Satorras, and A. Vespignani, Nature Phys. 3, 276 (2007).
* Simoes et al. (2008) M. Simoes, M. M. Telo da Gama, and A. Nunes, J. R. Soc. Interface 5, 555 (2008).
* Anderson and May (1991) R. M. Anderson and R. M. May, _Infectious Diseases of Humans: Dynamics and Control_ (Oxford University Press, Oxford, 1991).
* Keeling and Rohani (2007) M. J. Keeling and P. Rohani, _Modelling Infectious Diseases in Humans and Animals_ (Princeton University Press, Princeton, 2007).
* Schenzle (1985) D. Schenzle, Math. Med. Biol. 1, 169 (1985).
* Keeling and Grenfell (2002) M. J. Keeling and B. T. Grenfell, Proc. R. Soc. London, Ser. B 269, 335 (2002).
* Eubank et al. (2004) S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai, and N. Want, Nature 429, 180 (2004).
* Meyers et al. (2005) L. A. Meyers, B. Pourbohloul, M. E. J. Newman, D. M. Skowronski, and R. C. Brunham, J. Theor. Biol. 232, 71 (2005).
* Christensen et al. (2010) C. Christensen, I. Albert, B. Grenfell, and R. Albert, Physica A 389, 2663 (2010).
* Aleman et al. (2011) D. M. Aleman, T. G. Wibisono, and B. Schwartz, Interfaces 41, 301 (2011).
* van Kampen (2007) N. G. van Kampen, _Stochastic Processes in Physics and Chemistry_ (Elsevier, Amsterdam, 2007).
* Risken (1989) H. Risken, _The Fokker-Planck Equation_ (Springer, Berlin, 1989).
* Gillespie (1976) D. T. Gillespie, J. Comp. Phys. 22, 403 (1976).
* McKane and Newman (2005) A. J. McKane and T. J. Newman, Phys. Rev. Lett. 94, 218102 (2005).
* Alonso et al. (2007) D. Alonso, A. J. McKane, and M. Pascual, J. R. Soc. Interface 4, 575 (2007).
* Rozhnova and Nunes (2010) G. Rozhnova and A. Nunes, Phys. Rev. E 82, 041906 (2010).
* Black and McKane (2010a) A. J. Black and A. J. McKane, J. Theor. Biol. 267, 85 (2010a).
* Black et al. (2009) A. J. Black, A. J. McKane, A. Nunes, and A. Parisi, Phys. Rev. E 80, 021922 (2009).
* Black and McKane (2010b) A. J. Black and A. J. McKane, J. R. Soc. Interface 7, 1219 (2010b).
* Rozhnova and Nunes (2009a) G. Rozhnova and A. Nunes, Phys. Rev. E 79, 041922 (2009a).
* Rozhnova and Nunes (2009b) G. Rozhnova and A. Nunes, Phys. Rev. E 80, 051915 (2009b).
* Lajmanovich and Yorke (1976) A. Lajmanovich and J. A. Yorke, Math. Biosc. 28, 221 (1976).
* Lloyd and May (1996) A. L. Lloyd and R. M. May, J. Theor. Biol. 179, 1 (1996).
* Keeling and Rohani (2002) M. J. Keeling and P. Rohani, Ecol. Lett. 5, 20 (2002).
* Arino and van den Driessche (2003) J. Arino and P. van den Driessche, Math. Populat. Studies 10, 175 (2003).
* Hagenaars et al. (2004) T. J. Hagenaars, C. A. Donnelly, and N. M. Ferguson, J. Theor. Biol. 229, 349 (2004).
* Colizza and Vespignani (2008) V. Colizza and A. Vespignani, J. Theor. Biol. 251, 450 (2008).
* Barthelemy et al. (2010) M. Barthelemy, C. Godreche, and J.-M. Luck, J. Theor. Biol. 267, 554 (2010).
* Guo et al. (2008) H. Guo, M. Y. Li, and Z. Shuai, Proc. Am. Math. Soc. 136, 2793 (2008).
* Bauch and Earn (2003) C. T. Bauch and D. J. D. Earn, Proc. R. Soc. London, Ser. B. 270, 1573 (2003).
|
arxiv-papers
| 2011-09-12T09:20:44 |
2024-09-04T02:49:22.190962
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Rozhnova, A. Nunes and A. J. McKane",
"submitter": "Ganna Rozhnova",
"url": "https://arxiv.org/abs/1109.3354"
}
|
1109.3398
|
# Studies with Onia at LHCb
L. Li Gioi - for the LHCb collaboration CNRS - Laboratoire de Physique
Corpusculaire de Clermont-Ferrand, France
###### Abstract
The production of $c\bar{c}$ and $b\bar{b}$ is studied in $pp$ collisions at
$\sqrt{s}=7$ TeV with the LHCb detector. The results of these measurements are
compared with different theoretical models. Results and prospects are also
shown for exotics: the mass measurement of the $X(3872)$ and the search of the
$X(4140)$. Most of the presented results are based on the 2010 LHCb dataset
(35 pb-1).
## I Introduction
Heavy quarkonium production remains a challenging problem for the
understanding of Quantum Chromodynamics (QCD). At the centre-of mass energies
for proton-proton collisions at the Large Hadron Collider, $c\bar{c}$ pairs
are expected to be produced predominantly via Leading Order gluon gluon
interactions, which can be computed using perturbative QCD, followed by the
formation of the bound charmonium states described by non perturbative models.
Recent approaches make use of non-relativistic QCD factorization (NRQCD) which
assumes a combination of the colour-singlet and colour-octet $c\bar{c}$ as it
evolves towards the final bound state via the exchange of soft gluons
ref:intr1 . Next-to-Leading Order (NLO) QCD corrections in charmonium and
bottomonium production are also essential for the description of the
experimental data ref:intr2 ; ref:intr3 . Studies of $\psi(2S)$,
$\Upsilon(1S)$ and $\chi_{c}$ production cross sections are presented.
The $X(3872)$ meson is an exotic meson discovered in 2003 by the Belle
collaboration in the $B^{\pm}\to X(3872)K^{\pm}$, $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay chain ref:intr4 . Its existence was confirmed by
the CDF ref:intr5 , D0 ref:intr6 and BaBar ref:intr7 collaborations. The
$X(3872)$ particle, together with several other new states subsequently
observed in the mass range 3.9-4.7 GeV/c2, has led to a resurgence of interest
in exotic meson spectroscopy ref:intr8 . Several properties of the $X(3872)$
have been measured; however, its nature is still uncertain and several models
have been proposed. First, it is not excluded that the $X(3872)$ is a
conventional charmonium state with one candidate being the $\eta_{c2}(1D)$
meson ref:intr8 . However, the mass of this state is predicted to be far below
the observed $X(3872)$ mass. Given the proximity of the $X(3872)$ mass to the
$D^{*0}\bar{D^{0}}$ threshold, one possibility is that the $X(3872)$ is a
loosely bound deuteron-like $D^{*0}\bar{D^{0}}$ molecule, i.e. a ((uc) - (cu))
system ref:intr8 . Another more exotic possibility is that the $X(3872)$ is a
tetraquark state ref:intr9 .
The CDF experiment has reported $>5\ \sigma$ evidence for $X(4140)$ state
(also referred to as $Y(4140)$ in the literature) in 6.0 fb-1 of $p\bar{p}$
data collected at Tevatron ref:intr10 . The relative rate was measured to be
${B(B^{+}\to X(4140)K^{+})\times B(X(4140)\to J/\psi\phi)}/{B(B^{+}\to
J/\psi\phi K^{+})}=0.149\pm 0.039\pm 0.024$. This observation has triggered
wide interest among model builders of exotic hadronic states. It has been
suggested that the $X(4140)$ could be a molecular state ref:intr11 ;
ref:intr12 , a tetraquark state ref:intr13 ; ref:intr14 , a hybrid state
ref:intr15 ; ref:intr16 or a re-scattering effect ref:intr17 ; ref:intr18 .
## II $\psi(2S)$ Production Cross Section
Two decay modes of the $\psi(2S)$ meson have been studied ref:confPsi2S :
$\psi(2S)\to\mu^{+}\mu^{-}$ and $\psi(2S)\to J/\psi(\mu\mu)\pi^{+}\pi^{-}$.
The differential cross-section for the inclusive $\psi(2S)$ production is
computed as
$\frac{d\sigma}{dp_{T}}(p_{T})=\frac{N_{\psi(2S)}(p_{T})}{L_{int}\epsilon(p_{T})B(\psi(2S)\to
f)\Delta p_{T}}$ (1)
where $N_{\psi(2S)}(p_{T})$ is the number of observed $\psi(2S)$ decays,
$L_{int}$ is the integrated luminosity, $\epsilon(p_{T})$ is the total
detection efficiency including acceptance effects, $B(\psi(2S)\to f)$ is
$B(\psi(2S)\to J/\psi\pi^{+}\pi^{-})B(J/\psi\to\mu^{+}\mu^{-})$ for
$\psi(2S)\to J/\psi(\mu\mu)\pi^{+}\pi^{-}$ and the dielectron branching ratio
($B(\psi(2S)\to e^{+}e^{-})$) for $\psi(2S)\to\mu^{+}\mu^{-}$ and $\Delta
p_{T}=$ 1 GeV/c is the bin size. The dielectron branching ratio is used,
assuming lepton universality, since it has a much smaller error than the
dimuon one (2.2% and 10% respectively). In order to estimate the number of
$\psi(2S)$ signal events, a fit is performed independently in each $(p_{T})$
bin.
Figure 1: Comparison of the LHCb results for the differential production
cross-section of $\psi(2S)$ with the predictions for prompt production by a
NLO NRQCD model ref:Psi2S_1 . LHCb data include also $\psi(2S)$ from b.
In Fig. 1 a comparison between the measurements presented of the differential
$\psi(2S)$ production cross-section multiplied by the
$\psi(2S)\to\mu^{+}\mu^{-}$ branching ratio and a recent theory prediction
ref:Psi2S_1 for prompt $\psi(2S)$ in the rapidity range $2\leq
y=\frac{1}{2}ln\frac{E+p_{z}}{E-p_{z}}\leq 4.5$ is shown. Here $E$ and $p_{z}$
are the $\psi(2S)$ energy and momentum in the $z$ direction measured in the pp
centre-of-mass frame; the z-axis is defined along the beam axis in the LHCb
frame, oriented from the VELO to the Muon Detector. The differential cross-
section in ref:Psi2S_1 has been evaluated for the $\psi(2S)$ prompt
production at the LHC at next-to-leading order in non-relativistic QCD,
including color-singlet and color-octet contributions. The integrated cross-
section in the full range of $p_{T}$ and $y$ respectively to
$\psi(2S)\to\mu^{+}\mu^{-}$ and $\psi(2S)\to J/\psi\pi^{+}\pi^{-}$ is found to
be:
$\displaystyle\sigma(0<p_{T}\leq 12\ \mbox{GeV/c},2<y\leq 4.5)$
$\displaystyle=$ $\displaystyle 1.88\pm 0.02\pm 0.31^{+0.25}_{-0.48}\
\mbox{$\mu$b}$ $\displaystyle\sigma(3<p_{T}\leq 16\ \mbox{GeV/c},2<y\leq 4.5)$
$\displaystyle=$ $\displaystyle 0.62\pm 0.04\pm 0.12^{+0.07}_{-0.14}\
\mbox{$\mu$b}$ (2)
where the first uncertainty is statistical, the second is systematic and the
third is the uncertainty due to the unknown polarization.
## III $\Upsilon(1S)$ Production Cross Section
$\Upsilon(1S)$ meson has been studied in the decay mode
$\Upsilon(1S)\to\mu^{+}\mu^{-}$ ref:confY1S . The double differential cross-
section for the inclusive $\Upsilon(1S)$ production is computed as
$\frac{d^{2}\sigma}{dp_{T}dy}(p_{T},y)=\frac{N_{\Upsilon(1S)}(p_{T},y)}{L_{int}\epsilon(p_{T},y)B(\Upsilon(1S)\to\mu^{+}\mu^{-})\Delta
p_{T}\Delta y}$ (3)
where $N_{\Upsilon(1S)}(p_{T},y)$ is the number of observed
$\Upsilon(1S)\to\mu^{+}\mu^{-}$ decays, $\epsilon(p_{T},y)$ is the total
detection efficiency including acceptance effects, $L_{int}$ is the integrated
luminosity, $B(\Upsilon(1S)\to\mu^{+}\mu^{-})$ is the branching fraction, and
$\Delta p_{T}\Delta y=1\times 0.5$ GeV/c is the rapidity and $p_{T}$ bin
sizes. In order to estimate the number of $\Upsilon(1S)$ signal events, a fit
is performed independently in each of the 15 $p_{T}$ times 5 $y$ bins. The
double differential cross-section as a function of $p_{T}$ and $y$ is shown in
Fig. 2. The integrated cross-section in the full range of $y$ and $p_{T}$ is
found to be
$\sigma(pp\to\Upsilon(1S)X;p_{T}(\Upsilon(1S))<15\
\mbox{GeV/c};2<y(\Upsilon(1S))<4.5)=108.3\pm 0.7^{+30.9}_{-25.8}\ \mbox{nb}$
(4)
where the first uncertainty is statistical, and the second systematic. The
latter includes ${}^{+18.8}_{-7.9}$ nb from the unknown polarization, $\pm
10.8$ nb from the luminosity determination and $\pm 22.0$ nb from other
sources. The integrated cross-section is about a factor 100 smaller than the
integrated $J/\psi$ cross-section in the identical $y$ and $p_{T}$ region
ref:Y1S_1 , and a factor three smaller than the integrated $\Upsilon(1S)$
cross-section in the central region $|y|<2$ as measured by CMS ref:Y1S_2 .
Figure 2: Differential $\Upsilon(1S)$ production cross-section as a function
of $p_{T}$ in bins of rapidity.
The CMS and the LHCb measurements agree; the difference in the integrated
cross section results is due to the different rapidity ranges. Fig. 3 shows
the comparison of the LHCb cross-section measurement in bins of $y$ integrated
over $p_{T}$ with the same measurement of CMS, in the $y$ ranges covered by
the two experiments.
Figure 3: Differential $\Upsilon(1S)$ production cross-section as a function
of $y$ integrated over $p_{T}$ as measured by the CMS ref:Y1S_2 and LHCb
experiments.
## IV $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ Production Cross Sections Ratio
$\chi_{c1}$ and $\chi_{c2}$ have been reconstructed in the final state
$J/\psi\gamma$. The production cross-section ratio of the $\chi_{c2}$ and
$\chi_{c1}$ states is measured ref:confChic using
$\frac{\sigma(\chi_{c2})}{\sigma(\chi_{c1})}=\frac{N_{\chi_{c2}}}{N_{\chi_{c1}}}\cdot\frac{\epsilon^{\chi_{c1}}_{J/\psi}\epsilon^{\chi_{c1}}_{\gamma}\epsilon^{\chi_{c1}}_{sel}}{\epsilon^{\chi_{c2}}_{J/\psi}\epsilon^{\chi_{c2}}_{\gamma}\epsilon^{\chi_{c2}}_{sel}}\cdot\frac{B(\chi_{c1}\rightarrow
J/\psi\gamma)}{B(\chi_{c2}\rightarrow J/\psi\gamma)}$ (5)
where $B(\chi_{c1}\rightarrow J/\psi\gamma)$ and $B(\chi_{c2}\rightarrow
J/\psi\gamma)$ are the $\chi_{c1}$ and $\chi_{c2}$ branching ratios to the
final state $J/\psi\gamma$, $\epsilon^{\chi_{c2}}_{J/\psi}$
($\epsilon^{\chi_{c2}}_{\gamma}$) is the efficiency to reconstruct and select
a $J/\psi$ ($\gamma$) from $\chi_{c}$ decay and $\epsilon^{\chi_{c}}_{sel}$ is
the efficiency to select the $\chi_{c}$ candidate. The measurement method
consists of extracting the two $N_{\chi_{ci}}$ yields (for $\chi_{c1}$ and
$\chi_{c2}$) from an unbinned maximum likelihood fit to $\Delta
M=M(\chi_{c})-M(J/\psi)$ mass difference distribution. With the ratio of the
resolution parameters and the mass differences fixed, a fit is then performed
to the data in the full $J/\psi\ p_{T}$ range $J/\psi\ p_{T}\in[3,15]$ GeV/c
in order to extract the resolution scale $\sigma_{res}(\chi_{c1})$. Here, the
sample is subdivided into candidates with converted (after the magnet) and
non-converted photons, in order to account for the different calorimeter
resolution in the two cases. Converted photons are identified by the presence
of activity in the scintillator pad detector at the entrance to the
calorimeter system. Photons that convert before the magnet have a low
probability to be reconstructed, because either one or both electrons are
swept out of the detector acceptance by the magnetic field, and are not
considered here. The resolution scales, $\sigma_{res}(\chi_{c1})$, are
measured to be $22.8\pm 1.1$ MeV/c2 and $18.4\pm 0.4$ MeV/c2 for converted and
non converted candidates, respectively. The fit is then performed in bins of
$J/\psi\ p_{T}$. For each bin the value of $\sigma_{res}(\chi_{c1})$ is fixed
to the value extracted from the fit to the full range $J/\psi\ p_{T}$. The
presence of $\chi_{c}$ polarized states would modify the efficiencies
calculated from the Monte Carlo. In order to take into account possible
polarization scenarios, the relevant combination of weights has to be taken
into account. The results from the not-converted and converted samples are
combined by sampling the combination of the statistical and uncorrelated
systematic uncertainties using a toy Monte Carlo. Finally, the correlated
systematic uncertainty from the branching ratio is calculated using the
combined central values for $\sigma(\chi_{c2})/\sigma(\chi_{c1})$. The
preliminary result for the ratio of the prompt $\chi_{c2}$ to $\chi_{c1}$
production cross-sections as a function of $J/\psi\ p_{T}$ is given in Fig. 4.
Comparisons to the theory predictions from the ChiGen MC generator ref:Chic_1
and from the NLO NRQCD calculations ref:Chic_2 , in the rapidity range [2;
4.5], are also shown in the figure. Fig. 4 also shows the maximum effect of
the unknown $\chi_{c}$ polarization on the result, shown as a black shaded
area around the data points. The upper limit of the shaded area corresponds to
the spin state ( $\chi_{c1}$ : $m_{J}$ = 1; $\chi_{c2}$ : $m_{J}$ = 2) and the
lower limit corresponds to the spin state ( $\chi_{c1}$ : $m_{J}$ = 0;
$\chi_{c2}$ : $m_{J}$ = 0). The results are broadly in agreement at high
$J/\psi\ p_{T}$ with the colour singlet model, however, they are not yet
precise enough to rule out sizeable colour octet terms. There are indications
of a discrepancy in the mid to low $J/\psi\ p_{T}$ region. This may be
explained by a more complete modeling of the transition from the high to low
$J/\psi\ p_{T}$ regions, which is sensitive to non-perturbative effects and/or
sizeable higher-order perturbative corrections.
Figure 4: The ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ in bins of in the
range of $J/\psi\ p_{T}$ : [3; 15] GeV/c. The internal error bars correspond
to the statistical error on the $\chi_{c1}$ and $\chi_{c2}$ yields; the
external error bars include the contribution from the systematic uncertainties
(apart from the polarization). The shaded area around the data points (black)
shows the maximum effect of the unknown $\chi_{c}$ polarization on the result.
The two other bands correspond to the ChiGen MC generator theoretical
prediction ref:Chic_1 (in blue) and NLO NRQCD ref:Chic_2 (in red).
## V $X(3872)$ Mass Measurement
Inclusively produced $X(3872)$ mesons are reconstructed and selected in the
$X(3872)\to J/\psi\pi^{+}\pi^{-}$, $J/\psi\to\mu^{+}\mu^{-}$ decay mode
ref:confXmass . A momentum scale calibration is performed to account for a
mixture of effects related to imperfections in the knowledge of the magnetic
field map and of the alignment of the tracking system.
### V.1 Momentum Scale Calibration
The momentum scale is calibrated using a large sample of
$J/\psi\to\mu^{+}\mu^{-}$ decays. This calibration, gives an overall scale
factor that is applied to all measurements of particle momenta. The
calibration is checked to be also valid for two-body decays of the
$\Upsilon(1S)$, $D^{0}$ and $K^{0}$. In addition, its effect on $\psi(2S)\to
J/\psi\pi^{+}\pi^{-}$, which have kinematics similar to the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$, is studied. The uncalibrated value of the $\psi(2S)$
mass in data is $3685.94\pm 0.06(stat)$ MeV/c2. After the momentum scale
calibration it becomes $3686.12\pm 0.06(stat)$ MeV/c2, in good agreement with
the PDG value of $3686.09\pm 0.04$ MeV/c2. The measured $J/\psi$ mass after
alignment and calibration is checked to be stable over the whole 2010 data-
taking period.
### V.2 Results
The masses of the $\psi(2S)$ and $X(3872)$ are determined from an extended
unbinned maximum likelihood fit of the reconstructed $J/\psi\pi^{+}\pi^{-}$
mass in the interval $3.6<M_{J/\psi\pi^{+}\pi^{-}}<3.95$ GeV/c2. The
functional form of the background is studied using the same-sign pion events.
The $\psi(2S)$ and $X(3872)$ signals are each described with a Voigt function
defined as the convolution of a non-relativistic Breit-Wigner with a Gaussian
function. The intrinsic width of the $\psi(2S)$ is fixed to the PDG value,
$\Gamma_{\psi(2S)}=0.317$ MeV/c2. The $X(3872)$ intrinsic width is poorly
known. The BaBar ref:Xmass_1 and Belle ref:intr4 collaborations have
published $90\%$ confidence level limits of $\Gamma_{X(3872)}<3.3$ MeV/c2 and
$\Gamma_{X(3872)}<2.3$ MeV/c2, respectively. The results of the Belle and
BaBar analyses have been combined by CDF to yield $\Gamma_{X(3872)}=1.3\pm
0.6$ MeV/c2 ref:Xmass_2 . For these studies, the procedure adopted in
ref:Xmass_2 has been adopted and the natural width has been fixed to this
value in the default fit. Fig. 5 shows the invariant $J/\psi\pi^{+}\pi^{-}$
mass distributions for opposite-sign (black points) and same-sign (blue filled
histogram) candidates. Clear signals for both the $\psi(2S)$ and the $X(3872)$
can be seen. From a comparison of fits with and without the $X(3872)$
component the statistical significance of the $X(3872)$ signal is estimated to
be $9\ \sigma$. The uncertainties on the parameters reported by the fit are in
good agreement with expectations based on toy Monte Carlo studies. The
preliminary LHCb result
$M_{X(3872)}=3871.96\pm 0.46\pm 0.10\ \mbox{MeV/c${}^{2}$}$ (6)
is in good agreement with the published measurements ref:intr4 ; ref:Xmass_1 ;
ref:Xmass_2 ; ref:intr5 and with their average, $3871.56\pm 0.22$ MeV/c2. The
new world average value including LHCb measurement, $3871.63\pm 0.20$ MeV/c2,
is consistent within uncertainties with the sum of the $D^{0}$ and $D^{*0}$
masses, $3871.79\pm 0.29$ MeV/c2, computed from the results of the global PDG
fit of the charm meson masses.
Figure 5: Invariant mass distribution of $J/\psi\pi^{+}\pi^{-}$ (black points
with statistical error bars) and same-sign $J/\psi\pi^{\pm}\pi^{\pm}$ (blue
filled histogram) candidates. The red curve is the result of the fit described
in the text. The insert shows a zoom of the region around the $X(3872)$ mass.
## VI $X(3872)$ Production Cross Section
The product of the inclusive production cross-section $\sigma(pp\to
X(3872)+...)$ by the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ branching fraction is
computed as ref:confXCrSec
$\sigma(pp\to X(3872)+...)\times B(X(3872)\to
J/\psi\pi^{+}\pi^{-})=\frac{N^{corr}_{X(3872)}}{\eta_{tot}\times L_{int}\times
B(J/\psi\to\mu^{+}\mu^{-})}$ (7)
where $N^{corr}_{X(3872)}$ is the efficiency-corrected yield of $X(3872)\to
J/\psi(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ signal decays, $\eta_{tot}$ is a
multiplicative factor to the efficiency that accounts for known differences
between the data and simulation, $L_{int}$ is the integrated luminosity, and
the notation $B(...)$ is used for branching fractions. The $X(3872)$ signal
yield is determined from an extended, unbinned maximum likelihood fit of the
reconstructed $J/\psi\pi^{+}\pi^{-}$ mass in the interval
$3.82<M_{J/\psi\pi^{+}\pi^{-}}<3.95$ GeV/c2. The fitting function is the same
used for the $X(3872)$ mass measurement. The mass resolution is fixed in this
analysis to
$\sigma(X(3872))=\sigma(\psi(2S))\frac{\sigma^{MC}_{X(3872)}}{\sigma^{MC}_{\psi(2S)}}=3.26\pm
0.10\ \mbox{MeV/c${}^{2}$}$ (8)
where $\sigma(\psi(2S))=2.48\pm 0.08$ MeV/c2 is the mass resolution fitted in
data for the kinematically similar decay $\psi(2S)\to J/\psi\pi^{+}\pi^{-}$,
and where $\sigma^{MC}_{X(3872)}$ and $\sigma^{MC}_{\psi(2S)}$ are the mass
resolutions determined from fully simulated Monte Carlo events. As in section
V.2, the natural width is fixed to 1.3 MeV/c2. The effect of fixing parameters
that are not perfectly known is investigated as part of the systematic
studies. The $X(3872)$ signal yield is approximately $68\%$ of that reported
in ref:confXmass due to the additional trigger requirements and fiducial cuts
on $y$ and $p_{T}$. Applying the procedure discussed above, and using the
$p_{T}-$ and $y-$dependent efficiency from simulation, the efficiency-
corrected signal yield is $N^{corr}_{X(3872)}=9597\pm 2217$, where the quoted
uncertainty is statistical. Two factors enter into the determination of the
correction factor $\eta_{tot}$. The first is a factor $1.024\pm 0.011$ which
accounts for differences in the efficiency of the muon identification in the
data and simulation. The second is a factor of $0.96\pm 0.02$ that accounts
for observed differences in the efficiency of global event cuts applied in the
trigger. Multiplying these values together gives $\eta_{tot}=0.983\pm 0.023$.
The uncertainty on this number is taken into account in the estimation of the
systematic uncertainty. The preliminary measurement is
$\sigma(pp\to X(3872)+...)\times B(X(3872)\to J/\psi\pi^{+}\pi^{-})=4.74\pm
1.10\pm 1.01\ \mbox{nb}$ (9)
where $\sigma(pp\to X(3872)+...)$ is the cross-section for producing an
$X(3872)$ particle in $pp$ collisions at $\sqrt{s}=7$ TeV (either promptly or
from the decay of other particles) with a transverse momentum between 5 and 20
GeV/c and a rapidity between 2.5 and 4.5.
## VII Search of the $X(4140)$
The $X(4140)$ has been studied using a data sample of approximately 0.376 fb-1
ref:confX4140 selecting $B^{+}\to J/\psi\phi K^{+}$, $\phi\to K^{+}K^{-}$. No
narrow structure has been seen near the threshold in $M(J/\psi\phi)-M(J/\psi)$
distribution as shown in Fig. 6. In the CDF analysis, they fit their data with
spin 0 relativistic Breit-Wigner function on top of three-body phase-space,
all smeared with the detector resolution ref:intr10 . To quantify the
disagreement with CDF the same function has been used. The efficiency
dependence is extracted from the MC simulations and applied as a correction to
the three-body phase-space function. Mass and width of $X(4140)$ peak are
fixed to the values obtained by the CDF collaboration. The mass difference
resolution is determined from the $B^{+}\to X(4140)K^{+}$ MC simulation. The
fit to $M(J/\psi\phi)-M(J/\psi)$ distribution gives a $X(4140)$ amplitude of
$6.9\pm 4.7$ events (Fig. 6a) and has a confidence level (CL) of $3\%$. When
using a quadratic polynomial instead of three-body phase-space function for
the background, the preferred value of the $X(4140)$ amplitude is zero (it is
restricted not to go below zero), with a positive error of 3 events. This fit
is shown in Fig. 6b and has a confidence level of $11\%$. Using the $B^{+}\to
J/\psi\phi K^{+}$ yield multiplied by this efficiency ratio ($B^{+}\to
X(4140)K^{+}$, $X(4140)\to J/\psi\phi$)/($B^{+}\to J/\psi\phi K^{+}$) and
multiplied by the CDF value for $B(B^{+}\to X(4140)K^{+})/B(B^{+}\to
J/\psi\phi K^{+})$ ref:intr10 , leads to a prediction of observed $35\pm 9\pm
6$ events, where the first uncertainty is statistical from the CDF data and
the second is systematic including both CDF and LHCb contributions. The
central value of this estimate is illustrated in Fig. 6. The CDF result
disagrees by three standard deviations with the fit to LHCb data using the
polynomial background. The disagreement is 2.4 $\sigma$ when the efficiency-
corrected three-body phase-space background shape is used.
Figure 6: Fit of $X(4140)$ signal on top of smooth background to
$M(J/\psi\phi)-M(J/\psi)$ for the $B^{+}\to J/\psi\phi K^{+}$ data. The solid
line represents the result of the fit to LHCb data. The dashed line on top
illustrates the expected signal amplitude from the CDF results as explained in
the text. The top and bottom plots differ by the type of the background
function used in the fit: a) efficiency-shaped three-body phase-spaced; b)
quadratic polynomial.
## VIII Exclusive Dimuon Production
Exclusive particle production in proton-proton collisions are elastic
processes in which the protons remain intact, $p+p\to p+X+p$, and the
additional particles are created through photon and/or gluon propagators. In
the case of purely photon propagators, this is an electromagnetic process
which can be theoretically calculated with high accuracy. When gluons are
involved, these processes constitute an important testing ground for QCD,
since the object that couples to the proton must be colourless. Thus the
pomeron (two gluon states) or possibly an odderon (three gluons), predicted in
QCD but never unambiguously observed, can be studied in a clean experimental
environment. The cleanest experimental manifestation of these occur in final
states containing two protons and two muons or in events containing two
protons, two muons and a photon. The former can be produced in the diphoton
process giving a continuous dimuon invariant mass spectrum, or in the photon-
pomeron process which can produce $\phi$, $J/\psi$, $\psi(2S)$,
$\Upsilon(1S)$, $\Upsilon(2S)$ or $\Upsilon(3S)$ which decay to two muons. The
latter is a signal for double pomeron exchange which produces
$\chi_{c}$($\chi_{b}$) that decay to $J/\psi$ ($\Upsilon$) plus a photon. The
final state protons are only marginally deflected, go down the beam-pipe, and
remain undetected. The experimental signal therefore in LHCb is a completely
empty event except for two muons and possibly a photon. However, because LHCb
is not hermetic, there will be sizeable backgrounds from non-elastic processes
where the other particles travel outside the detector acceptance.
The cross-section $\sigma$, is calculated ref:confExcl from the number, $N$,
of selected events having corrected for efficiency, $\epsilon$, and purity,
$p$, and dividing by the luminosity, $L$, via $\sigma=(pN)/(\epsilon L)$. The
efficiency for selecting the events has been determined from simulation. The
measured cross-sections are:
$\displaystyle\sigma_{J/\psi\to\mu^{+}\mu^{−}}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5)$
$\displaystyle=$ $\displaystyle 474\pm 12\pm 51\pm 92\ \mbox{pb}$
$\displaystyle\sigma_{\psi(2S)\to\mu^{+}\mu^{−}}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5)$
$\displaystyle=$ $\displaystyle 12.2\pm 1.8\pm 1.3\pm 2.4\ \mbox{pb}$
$\displaystyle\sigma_{\chi_{c0}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 9.3\pm 2.2\pm 3.5\pm 1.8\ \mbox{pb}$ (10)
$\displaystyle\sigma_{\chi_{c1}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 16.4\pm 5.3\pm 5.8\pm 3.2\ \mbox{pb}$
$\displaystyle\sigma_{\chi_{c2}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 28.0\pm 5.4\pm 9.7\pm 5.4\ \mbox{pb}$
$\displaystyle\sigma_{pp\to
p\mu^{+}\mu^{-}p}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5,M_{\mu^{+}\mu^{-}}>2.5\
\mbox{GeV/c${}^{2}$})$ $\displaystyle=$ $\displaystyle 67\pm 10\pm 7\pm 15\
\mbox{pb}$
where the first uncertainty is statistical, the second is systematic, and the
third comes from the estimate of the luminosity. Note that these numbers are
cross-section times the branching ratio into the final state of interest, and
all final state particles are required to be between pseudorapidities of 2 and
4.5. The cross sections are quoted in a limited pseudorapidity range as the
models under consideration have pseudorapidity dependence.
## IX Summary
LHCb performed many analysis of the quarkonium states using 2010 collected
data (35 pb-1). The measurement of the production cross sections of charmonium
and bottomonium states ($\psi(2S)$, $\Upsilon(1S)$, $\chi_{c}$) are useful to
test theoretical models. For the exotic states, the measurement of the
$X(3872)$ mass and cross section has been performed and the CDF narrow
$X(4140)$ has been studied using a dataset of 376 pb-1 and its existence not
confirmed. LHCb has a very high $J/\psi$ statistic in 2011 data that will
allow to have a lot of new results in the near future.
## References
* (1) G. T. Bodwin, et al. Phys.Rev., D51:1125–1171, 1995.
* (2) J. M. Campbell, et al. Phys.Rev.Lett., 98:252002, 2007.
* (3) Y.-Q. Ma, et al. Phys.Rev., D83:111503, 2011.
* (4) S.K. Choi et al. [Belle collaboration], Phys. Rev. Lett. 91 (2003) 262001, arXiv:hep-ex/0309032.
* (5) D. Acosta et al. [CDF collaboration], Phys. Rev. Lett. 93 (2004) 072001, arXiv:hep-ex/0312021.
* (6) V. M. Abazov et al. [D0 collaboration], Phys. Rev. Lett. 93 (2004) 162002, arXiv:hep-ex/0405004.
* (7) B. Aubert et al. [Babar collaboration], Phys. Rev. D 71 (2005) 071103, arXiv:hep-ex/0406022.
* (8) E. Swanson, Physics Reports 429 (2006) 243, arXiv:hep-ph/0601110.
* (9) L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D 71 (2005) 014028, arXiv:hep-ph/0412098.
* (10) The CDF Collaboration, T. Aaltonen et al., arXiv:1101.6058.
* (11) X. Liu and S.-L. Zhu, Phys.Rev. D80 (2009) 017502, [arXiv:0903.2529].
* (12) Z.-G. Wang, Z.-C. Liu, and X.-H. Zhang, Eur. Phys. J. C64 (2009) 373–386, [arXiv:0907.1467].
* (13) F. Stancu, J. Phys. G37 (2010) 075017, [arXiv:0906.2485].
* (14) N. V. Drenska, R. Faccini, and A. D. Polosa, Phys. Rev. D79 (2009) 077502, [arXiv:0902.2803].
* (15) N. 130 Mahajan, Phys. Lett. B679 (2009) 228–230, [arXiv:0903.3107].
* (16) Z.-G. Wang, Eur. Phys. J. C63 (2009) 115–122, [arXiv:0903.5200].
* (17) X. Liu, Phys. Lett. B680 (2009) 137–140, [arXiv:0904.0136].
* (18) D. V. Bugg, arXiv:1103.5363.
* (19) LHCb-CONF-2011-026.
* (20) Y.-Q.Ma et al., Phys. Rev. Lett. 106, 042002, (2011) and private communication.
* (21) LHCb-CONF-2011-016.
* (22) The LHCb Collaboration, R. Aaij et al., Eur.Phys.J.C71:1645,2011.
* (23) The CMS Collaboration, CMSBPH-10-003, CERN-PH-EP-2010-055; arXiv:1012.5545 [hep-ex].
* (24) LHCb-CONF-2011-020.
* (25) L. A. Harland-Lang andW. J. Stirling, http://projects.hepforge.org/superchic/chigen.html.
* (26) Y. Ma, K. Wang, K. Chao, arXiv:1002.3987 (hep-ph).
* (27) LHCb-CONF-2011-021.
* (28) B. Aubert et al. [Babar collaboration], Phys. Rev. D 77 (2008) 111101, arXiv:0803.2838 [hep-ex].
* (29) T. Aaltonen et al. [CDF collaboration], Phys. Rev. Lett. 103 (2009) 152001, arXiv:0906.5218 [hep-ex].
* (30) LHCb-CONF-2011-043.
* (31) M. Pivk and F.R. Le Diberder , Nucl. Instrum. Meth. A 555 (2005) 356–369, arXiv:physics/0402083v3.
* (32) S.-K. Choi et al. [Belle collaboration], arXiv:1107.0163v1 [hep-ex], submitted to Phys. Rev. D.
* (33) LHCb-CONF-2011-045.
* (34) LHCb-CONF-2011-022.
|
arxiv-papers
| 2011-09-15T16:20:39 |
2024-09-04T02:49:22.202445
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luigi Li Gioi",
"submitter": "Luigi Li Gioi",
"url": "https://arxiv.org/abs/1109.3398"
}
|
1109.3551
|
# SuSeFLAV 1.2: program for supersymmetric mass spectra with seesaw mechanism
and rare lepton flavor violating decays
Debtosh Chowdhury Centre for High Energy Physics,Indian Institute of Science,
Bangalore 560 012, India Raghuveer Garani Physikalisches Institut der
Universität Bonn, Nußalle 12, D-53115 Bonn, Germany Sudhir K. Vempati
###### Abstract
Accurate supersymmetric spectra are required to confront data from direct and
indirect searches of supersymmetry.
SuSeFLAV111http://cts.iisc.ernet.in/Suseflav/main.html is a numerical tool
which is capable of computing supersymmetric spectra precisely for various
supersymmetric breaking scenarios applicable even in the presence of flavor
violation. The program solves MSSM RGEs with complete $3\times 3$ flavor
mixing at 2-loop level and one loop finite threshold corrections to all MSSM
parameters by incorporating radiative electroweak symmetry breaking
conditions. The program also incorporates the Type-I seesaw mechanism with
three massive right handed neutrinos at user defined mass scales and mixing.
It also computes branching ratios of flavor violating processes such as
$l_{j}\,\rightarrow\,l_{i}\gamma$, $l_{j}\;\rightarrow\,3~{}l_{i}$,
$b\,\rightarrow\,s\gamma$ and supersymmetric contributions to flavor
conserving quantities such as $(g_{\mu}-2)$. A large choice of executables
suitable for various operations of the program are provided.
###### keywords:
MSSM , Right Handed Neutrinos , Lepton Flavor Violation.
††journal: Computer Physics Communications
PROGRAM SUMMARY
Manuscript Title: SuSeFLAV: program for supersymmetric mass spectra with
seesaw mechanism and rare lepton flavor violating decays
Authors: Debtosh Chowdhury, Raghuveer Garani, Sudhir K. Vempati
Program Title: SuSeFLAV
Journal Reference:
Catalogue identifier:
Licensing provisions: GNU Public License
Programming language: Fortran 95
Computer: Personal Computer, Work-Station
Operating system: Linux, Unix
RAM:
Number of processors used: Single
Supplementary material:
Keywords: MSSM, Right Handed Neutrinos, Lepton Flavor Violation.
Classification:
External routines/libraries:
Subprograms used:
Nature of problem: Determination of masses and mixing of supersymmetric
particles within the context of MSSM with conserved R-parity with and without
the presence of Type-I seesaw. Inter-generational mixing is considered while
calculating the mass spectrum. Supersymmetry breaking parameters are taken as
inputs at a high scales specified by the mechanism of supersymmetry breaking.
RG equations including full inter-generational mixing are then used to evolve
these parameters up to the electroweak breaking scale. The low energy
supersymmetric spectrum is calculated at the scale where successful radiative
electro-weak symmetry breaking occurs. At weak scale Standard Model fermion
masses, gauge couplings are determined ncluding the supersymmetric radiative
corrections. Once the spectrum is computed, the program proceeds to various
lepton flavor violating observables (e.g., BR$(\mu\ \rightarrow\ e\gamma)$,
BR$(\tau\ \rightarrow\ \mu\gamma)$ etc.) at the weak scale.
Solution method: Two loop RGEs with full $3\times 3$ flavor mixing for all
supersymmetry breaking parameters are used to compute the low energy
supersymmetric mass spectrum. Adaptive step size Runge-Kutta method is used to
solve the RGEs numerically between the high scale and the electroweak breaking
scale. Iterative procedure is employed to get the consistent radiative
electroweak symmetry breaking condition. The masses of the supersymmetric
particles are computed at 1-loop order. The third generation SM particles and
the gauge couplings are evaluated at the 1-loop order including supersymmetric
corrections. A further iteration of the full program is employed such that the
SM masses and couplings are consistent with the supersymmetric particle
spectrum.
Restrictions:
Unusual features:
Additional comments:
Several executables are presented for the user.
Running time: 0.2 seconds on a Intel(R) Core(TM) i5 CPU 650 with 3.20 GHz.
## 1 Introduction
Low energy supersymmetry [1] is currently being probed by the Large Hadron
Collider (LHC) at CERN and the Tevatron collider at the Fermilab222The
Tevatron collider has recently stopped functioning.. On the other hand, there
is already a huge amount of information which has been collected and is being
collected which gives information on low energy supersymmetric Lagrangian
indirectly. For example, the flavor experiments in the hadronic and the
leptonic sectors place strong constraints on the flavor off-diagonal entries
in the Lagrangian. Similarly, the astrophysical data on dark matter which has
been improved with the latest WMAP 7-year results [2] also strongly restricts
the parameter space where the mass and the couplings of the lightest
supersymmetric particle (LSP) correspond to the observed relic density. While
experimental evidence for supersymmetry is definitely far more superior
compared to the indirect detection of supersymmetry, the power of indirect
experimental data to constrain the parameter space cannot be underestimated.
Furthermore, as it is well known even if there is a positive experimental
signal at the LHC, it would be very hard to reconstruct the supersymmetric
breaking Lagrangian unambiguously due to the large number of degeneracies
present in the parameter space which can give similar signals at the colliders
[3]. It has also been noted that flavor violating observables and dark matter
could help to break these degeneracies [4].
Obviously, the flavor observables depend on the supersymmetric model in which
they are calculated. Unfortunately, most of the present supersymmetric mass
spectrum calculators do not take in to consideration the effect of flavor
violation in the running of soft mass parameters either in the hadronic sector
or the leptonic sector333The recent version of SPheno [38] is an exception..
In hadronic sector, typically the CKM is considered to be the only source of
flavor violation, while this works very well, as long as one restricts to the
scheme of Minimal Flavor Violation (MFV), in a more general scheme of
supersymmetric models, such an assumption cannot be supported. Recently
‘observed’ deviations from the Standard Model CKM paradigm [5, 6] might find
explanations in terms of a supersymmetric standard model with some amount of
flavor violation [7]. To study the associated phenomenology of such kinds of
models either for dark matter relic density, collider searches or other
theoretical aspects such as threshold corrections to fermion masses, gauge
coupling unification etc., would require precise computation of the mass
spectrum in these models.
In the leptonic sector, the case for flavor violation is even more stronger.
Firstly, neutrinos have non-zero masses and secondly their flavor mixing is
large as has been observed in the neutrino oscillation experiments. Most of
mechanisms of generating neutrino masses and mixing inevitably lead to large
flavor violation in supersymmetric theories. One of the simplest ways to
generate neutrino masses is the so called ‘seesaw mechanism’. In the present
work, we have restricted ourselves to Type-I seesaw mechanism, though the
program can be generalized to incorporate other seesaw mechanisms by adding
the corresponding RGEs. The Type-I seesaw mechanism444For a summary of the
seesaw mechanisms, please see [8]. has three singlet right handed neutrinos
added to the Standard Model which leads to two additional terms, the Dirac
mass term combining the left and right neutrino fields and the lepton number
violating Majorana mass term for the right handed singlet fields. The
interplay between these two terms leads to small Majorana masses to the left
handed neutrinos in the limit of heavy Majorana masses for the right
neutrinos. The supersymmetric version of the seesaw mechanism was proposed
long ago [9] and some of its consequences for leptonic flavor violations have
immediately been noticed. Over the years, other theoretical/phenomenological
consequences of having right handed neutrinos has been observed. In the
following we list some of them.
* 1.
$Y_{b}-Y_{\tau}$ Unification: The presence of right handed neutrinos could
significantly modify regions where $\tau-b$ Yukawa couplings unify at the GUT
scale in unified theories like SO(10) or SU(5) [10, 11, 12, 13]. This is due
to the fact that the neutrino Dirac Yukawa couplings enter the renormalization
group equations (RGE) of the $Y_{b}$ and $Y_{\tau}$ at 2-loop level and
$Y_{t}$ at the 1-loop. This is enough to change the $Y_{b}/Y_{\tau}$ ratio at
the weak scale, if the neutrino Yukawa couplings are large.
* 2.
Lepton Flavor Violation: As mentioned previously, one of the main consequences
of the seesaw mechanism in supersymmetric theories is the violation of lepton
numbers leading to rare flavor violating decays [9]. This flavor violation
will be generated through the RGE even if the supersymmetry breaking mechanism
at the high scale conserves flavor as in mSUGRA. In particular GUT models, the
generated flavor violations could be large enough to strongly constrain
observability of supersymmetry at the LHC [14, 16, 17, 18]. On going
experiments like MEG and future experiments like PRISM/PRIME and Super-B
factories have enhanced sensitivity to large amounts of parameter space even
with small mixing and small $\tan\beta$ [16, 18].
* 3.
Dark Matter: One of the most surprising phenomenological aspects of seesaw
mechanism and SUSY-GUT models has been the impact on Dark matter
phenomenology. The presence of a single right handed neutrino with a large
Yukawa coupling could significantly enhance the efficiency of the electroweak
symmetry breaking and thus making the focus point region unviable within
mSUGRA like models [19, 20]. Similarly, GUT effects can significantly effect
the stau co-annihilation region [19]. The co-annihilation region and the focus
point regions seem to be most vulnerable to these effects in other GUT models
and mSUGRA incorporating Type-II or Type-III seesaw mechanisms [21]. It has
been recently realized that even flavor effects could play a role in the relic
density calculations in the early universe [22].
* 4.
Hadronic flavor violation: Incorporating Type-I seesaw mechanism in Grand
Unified Theories (GUT) also has effects on the hadronic sector. For example,
CP violation in the neutrino sector could be transmitted to the quark sector
in SU(5) or SO(10) theories [23, 24]. The large phases of the neutrino mass
matrix can be transmitted to the hadronic sector with effects in $K$ and $B$
physics phenomenology. More generally in GUT theories, hadronic and leptonic
flavor violations are related to each other by the underlying GUT symmetry
[25].
* 5.
Collider Signals: Lepton flavor violation which might be inevitable due to the
presence of a seesaw mechanism, can lead to flavor violation in the sleptonic
sector as we have mentioned above. Such flavor violation can be studied at the
colliders by measuring the mass differences between the sleptons by observing
such as sleptonic oscillations etc. [26, 27, 28, 29].
* 6.
Gauge Coupling Unification: Finally, let us note that it has been pointed that
the presence of massive right handed neutrinos with large yukawa couplings in
the MSSM Lagrangian improves the accurate unification of the gauge couplings
because $\alpha_{3}$ gets contribution from the right handed neutrinos through
the RGE running at the two loop level [30].
The above points provide enough justification to determine the supersymmetric
mass spectrum in seesaw models at a high precision level including the effects
due to flavor violation555Programs like SuperIso[31] and SUSYFLAVOR[32]
compute the hadronic flavor violating processes for the given supersymmetric
spectrum at low energy with high precision. . Unfortunately, while there exist
very good spectrum calculators for supersymmetric theories like ISASUSY [33],
SuSpect [34] and SOFTSUSY [35], they do not consider full flavor violating
structure in the computation of the soft spectrum either in their RGE’s or
their mass matrices666 However there are programs which consider flavor
violation in their mass matrices. The program SPICE [36] computes mass
matrices with full flavor violation without considering intergenerational
mixing in the RGE. After this work has appeared we have been informed that
SOFTSUSY also has a new version where flavor violation is considered [37]..
For these reasons, present versions of these programs might not be suitable
for attacking problems listed above unless one significantly modifies them.
Our program was written to address this deficiency in publicly available
codes. We, however point out that the recent version of SPheno [39] is very
similar to the program we are presenting here. It has full flavor structure
for the soft masses as well as Yukawas at the 2-loop level and considers the
full $6\times 6$ mass matrices for the sparticles. The preliminary version of
our program was first presented at [40]. The present version is an expanded
and more structured version of the same. This paper explains the code in
detail. We attempted to link the physics discussion with the file structure of
the code wherever possible such that the user can modify the code with minimal
efforts.
SuSeFLAV is a program written in Fortran 95 in a fixed length format. The
recommended compiler for this program is gfortran available in various
distributions of Linux. The program can be made executable using other Fortran
compilers too, such as ifort, by modifying the Makefile in the main directory.
In addition to studying the spectrum of supersymmetric particles the code also
computes leptonic flavor violating decays and some hadronic decays. We have
implemented the SLHA 2.2 [41] format for dealing with the input parameters and
output data. This way, the output of the code can be fed in to other publicly
available programs either for computing Dark Matter relic density or for
calculating supersymmetric particle decays and production cross-sections at
LHC etc. While the main set of RGE’s are written for Type-I seesaw mechanism,
extending the program for other seesaw mechanisms or even other models would
require coding the RGE’s from respective models. However, other parts of the
code, like mass spectrum, one-loop corrections etc., which are given in
separate files in the source directory can be used to suit the user’s needs.
The rest of the paper is organized as follows. In section 2 we describe the
MSSM Lagrangian with Type-I seesaw mechanism. In section 3, we discuss the
various supersymmetry breaking scenarios considered in SuSeFLAV. In section 4
we describe the calculation of the low energy supersymmetric spectrum
implemented in the program. In section 5 we briefly describe about various
theoretical constraints we impose on the sparticle spectrum in SuSeFLAV. In
section 6 we discuss about the various low energy observables computed in the
program. In section 7 we show the instructions on how to install and execute
SuSeFLAV. In A we write down all the tree level mass matrices of the
sparticles and in B we outline all the one-loop radiative corrections to these
parameters SuSeFLAV. In C we graphically show the file structure in SuSeFLAV.
We close with a comparison chart of sparticle masses with other available
codes in D.
## 2 Minimal Supersymmetric Seesaw Model
Since 1998, ever increasing data from neutrino sector has firmly established
that neutrinos have non-zero masses and that their flavors mix with two of
their angles being close to maximal and the third angle is non-zero [42]. One
of the elegant mechanisms to generate non-zero neutrino masses is through the
seesaw mechanism [43], where right handed singlet fields are added to the
Standard Model particle spectrum. These singlet neutrinos break lepton number
typically at a scale much larger than the standard model scale through their
Majorana masses.
Supersymmetric version of the seesaw mechanism is straight forward extension
of the Minimal Supersymmetric Standard Model [1] by adding right handed
neutrino superfields. The field content of the MSSMRN(MSSM $+$ Right Handed
Neutrinos) and their transformation properties under the gauge group
${\mathcal{G}}_{{\rm SM}}\equiv{\rm SU}(3)_{c}\otimes{\rm
SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ is given as
$\displaystyle{L}$ $\displaystyle:$
$\displaystyle\left(1,2,-\frac{1}{2}\right);\quad{e^{c}}:\left(1,1,+1\right);\qquad{\nu^{c}}:\left(1,1,0\right);$
$\displaystyle Q$ $\displaystyle:$
$\displaystyle\left(3,2,+\frac{1}{6}\right);\quad{u^{c}}:\left({\bar{3}},1,-\frac{2}{3}\right);\quad{d^{c}}:\left({\bar{3}},1,\frac{1}{3}\right);$
(1) $\displaystyle H_{u}$ $\displaystyle:$
$\displaystyle\left(1,2,+\frac{1}{2}\right);\quad
H_{d}:\left(1,2,-\frac{1}{2}\right).$
where $Q$ and $L$ stand for the $SU(2)_{L}$ doublet quarks and leptons,
$u^{c}$, $d^{c}$, $e^{c}$ and $\nu^{c}$ stand for the $SU(2)_{L}$ singlet
quarks and leptons, and the two Higgs doublet chiral superfields are denoted
by $H_{u}$ and $H_{d}$. At the seesaw scale and above, $q^{2}\gtrsim
M_{R}^{2}$, the superpotential takes the form :
$\displaystyle\mathcal{W}\ =\ $ $\displaystyle{\bf
Y}^{d}_{ij}d^{c}_{i}Q_{j}H_{d}+{\bf Y}^{u}_{ij}u^{c}_{i}Q_{j}H_{u}+{\bf
Y}^{e}_{ij}e^{c}_{i}L_{j}H_{d}$ $\displaystyle+{\bf
Y}^{\nu}_{ij}{\nu}^{c}_{i}L_{j}H_{u}+\mu
H_{u}H_{d}-\frac{1}{2}{M_{R}}_{i}{\nu}^{c}_{i}{\nu}^{c}_{i},$ (2)
where $i,j=\\{1,2,3\\}$ are generation indices. Note that the right handed
neutrino Majorana mass matrix is diagonal. The SU(2) and SU(3) contractions
are suppressed in the above Lagrangian. In the program, the seesaw scale is
taken to be $q^{2}=M_{R_{3}}^{2}$. At this scale, the right handed neutrino
mass matrix can be diagonalized by a rotation of the right handed neutrino
fields. The Dirac Yukawa matrix ${\bf Y}^{\nu}$ is defined at this scale in
the basis where the right handed neutrino mass matrix is diagonal. SuSeFLAV
considers the inputs at the scale $M_{R_{3}}$ where right handed neutrino
masses as well as the neutrino Dirac Yukawa coupling matrix. Finally it should
be noted that in the present version of SuSeFLAV, the ordering of the right
handed neutrino mass eigenvalues is taken as $M_{R_{1}}\lesssim
M_{R_{2}}\lesssim M_{R_{3}}$. We have not included the option of inverted
hierarchy for the right handed neutrinos in the present version. The program
will abort if such a choice is made.
Below the seesaw scale, once the right handed neutrinos are integrated out, we
are left with the five dimensional operator defining the the light neutrino
mass matrix as
$\displaystyle\mathcal{W}\ =\ $ $\displaystyle{\bf
Y}^{d}_{ij}d^{c}_{i}Q_{j}H_{d}+{\bf Y}^{u}_{ij}u^{c}_{i}Q_{j}H_{u}+{\bf
Y}^{e}_{ij}e^{c}_{i}L_{j}H_{d}$ $\displaystyle+\mu
H_{u}H_{d}+\frac{\kappa_{ij}}{\Lambda}\;L_{i}H_{u}L_{j}H_{u}$ (3)
The light neutrino mass matrix is given at the weak scale after the ${\rm
SU(2)}_{L}\times{\rm U(1)}_{Y}$ breaking as
$\mathcal{M}_{\nu}=\frac{\kappa_{ij}}{\Lambda}\;\langle H_{u}^{0}\rangle^{2}$
(4)
where $\Lambda$ represents the right handed neutrino mass scale or the seesaw
scale and $\langle H_{u}^{0}\rangle$ is the vev of the $H_{u}$ superfield. The
five dimensional operator is renormalized from the seesaw scale to the
electroweak scale. These corrections can be significant for inverse hierarchal
and degenerate spectrum for neutrino masses [44]. The present version of the
program does not contain the renormalization for the light neutrinos, we refer
users to use the existing programs like REAP [44]. The main reason for not
including the RG effects for light neutrinos has been the famous ambiguity in
relating the light neutrino masses to the neutrino Yukawa couplings [45].
Furthermore, we are more interested in the effects of seesaw mechanism on the
soft supersymmetric masses and couplings. However, we do provide the option
for the users to define the Yukawa matrices in terms of the Casas-Ibarra
${\cal R}$-parameterization [45]. In this parametrization the neutrino Yukawa
matrix ${\bf Y}^{\nu}$ is defined in terms of light neutrino masses as well as
the right handed neutrino masses. In ${\cal R}$-parametrization, ${\bf
Y}^{\nu}$ at the seesaw scale defined as
${\bf Y}^{\nu}=\frac{1}{\langle H_{u}^{0}\rangle^{2}}\ {\cal
D}_{\sqrt{M_{R}}}\,{\cal R}\,{\cal D}_{\sqrt{\cal{M_{\nu}}}}\,{\bf
U}^{\dagger}_{PMNS}$ (5)
where ${\bf U}_{{\rm PMNS}}$ is the Pontecorvo-Maki-Nakagawa-Sakata matrix
[46] and ${\cal R}$ is any $3\times 3$ orthogonal matrix. The ${\cal
D}_{\sqrt{M_{R}}}$ and ${\cal D}_{\sqrt{\cal{M_{\nu}}}}$ are defined as
$\displaystyle{\cal D}_{\sqrt{M_{R}}}\ =$ $\displaystyle\ {\rm
diagonal}\left(\sqrt{M_{R_{1}}},\,\sqrt{M_{R_{2}}},\,\sqrt{M_{R_{3}}}\right)$
(6) $\displaystyle{\cal D}_{\sqrt{\cal{M_{\nu}}}}\ =$ $\displaystyle\ {\rm
diagonal}\left(\sqrt{\kappa_{1}},\,\sqrt{\kappa_{2}},\,\sqrt{\kappa_{3}}\right)$
(7)
where $\kappa_{1}$, $\kappa_{2}$ and $\kappa_{3}$ are light neutrino mass
eigenvalues. It is important to note that $\cal R$ can be complex in nature
also, but in the present version of SuSeFLAV we take $\cal R$ to be real
orthogonal matrix. One can parametrize the $\cal R$ matrix in terms of 3
angles but in SuSeFLAV we have not parametrized the $\cal R$ matrix and left
all the 9 elements of $\cal R$ as user defined input. Various other
parametrization of the ${\bf Y}_{\nu}$ matrix have also been carried out in
[47]. Users interested in using the R-parameterization can use sinputs-rpar.in
in the main directory for their computations.
In addition to the user defined ${\bf Y}^{\nu}$ at the seesaw scale, we also
provide two other choices for ${\bf Y}^{\nu}$ and $M_{R}$ based on Grand
Unified Models like SO(10). Both these cases consider hierarchal masses for
the light neutrinos (${\mathcal{M}}_{\nu}$). These are
1. 1.
CKM Case:
In this case the ${\bf Y}^{\nu}$ and $M_{R}$ are given as
$\displaystyle{\bf Y}^{\nu}=$ $\displaystyle\begin{pmatrix}h_{u}&0&0\\\
0&h_{c}&0\\\ 0&0&h_{t}\end{pmatrix}{\bf V}_{\text{CKM}}$ (8)
$\displaystyle\text{Diagonal}[M_{R}]=$
$\displaystyle\\{M_{R_{3}},M_{R_{2}},M_{R_{1}}\\}=\\{10^{14},10^{9},10^{6}\\}~{}\text{GeV}$
where $h_{u},h_{c},h_{t}$ are the Yukawa couplings of the up, charm and the
top quarks and ${\bf V}_{\text{CKM}}$ is the quark sector mixing matrix.
2. 2.
PMNS Case:
In this case the ${\bf Y}^{\nu}$ and $M_{R}$ are given as
$\displaystyle{\bf Y}^{\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}h_{u}&0&0\\\ 0&h_{c}&0\\\
0&0&h_{t}\end{array}\right){\bf U}_{\text{PMNS}}$ (12)
$\displaystyle\text{Diagonal}[M_{R}]$ $\displaystyle=$
$\displaystyle\\{M_{R_{3}},M_{R_{2}},M_{R_{1}}\\}=\\{10^{14},10^{9},10^{6}\\}~{}\text{GeV}$
where ${\bf U}_{{\rm PMNS}}$ is the leptonic mixing matrix. Both ${\bf
V}_{\text{CKM}}$ and ${\bf U}_{\text{PMNS}}$ are defined in the SuSemain.f
file in the src/ directory. Below the seesaw scale, $q^{2}\lesssim
M_{R_{3}}^{2}$, the right handed neutrinos (and sneutrinos) decouple from the
theory, and the model is defined by MSSM.
The soft supersymmetric breaking terms in the seesaw enhanced MSSM are same as
in the MSSM together with the additional terms involving the right handed
sneutrinos. These include the mass terms for the gauginos, mass squared terms
for all the scalar particles and also the bilinear terms and trilinear terms:
$\displaystyle-\mathcal{L}_{\text{soft}}\ \supset\ $
$\displaystyle\frac{1}{2}\left(M_{1}\widetilde{B}\widetilde{B}+M_{2}\widetilde{W}\widetilde{W}+M_{3}\tilde{g}\tilde{g}\right)$
$\displaystyle+m_{H_{u}}^{2}|H_{u}|^{2}+m_{H_{d}}^{2}|H_{d}|^{2}+{\bf
m}_{\tilde{L}_{ij}}^{2}\tilde{L}_{i}^{\star}\tilde{L}_{j}+{\bf
m}^{2}_{\widetilde{e}^{c}_{ij}}\widetilde{e^{c}}^{\star}_{i}\widetilde{e}^{c}_{j}+{\bf
m}^{2}_{\widetilde{\nu^{c}}_{ij}}\widetilde{\nu^{c}}^{\star}_{i}\widetilde{\nu^{c}}_{j}+\ldots$
$\displaystyle+B_{\mu}H_{u}H_{d}+{\bf
B}_{M_{ij}}\widetilde{\nu^{c}_{i}}\widetilde{\nu^{c}_{j}}+h.c.$
$\displaystyle+\tilde{{\bf
A}}_{ij}^{u}\widetilde{Q}_{i}\widetilde{u}^{c}_{j}H_{u}+\tilde{{\bf
A}}_{ij}^{d}\widetilde{Q}_{i}\widetilde{d}^{c}_{j}H_{d}+\tilde{{\bf
A}}_{ij}^{e}\widetilde{L}_{i}\widetilde{e}^{c}_{j}H_{d}+\tilde{{\bf
A}}_{ij}^{\nu}\widetilde{L}_{i}\widetilde{\nu}^{c}_{j}H_{u}$ (13)
We use the factorization $\tilde{{\bf A}}^{u}\equiv{\bf A}^{u}{\bf Y}^{u}$,
for all the trilinear couplings at the weak scale777The RGEs are however,
defined in terms of $\tilde{\bf A}^{f}$, which sets the format for inputs at
the high scale. In mSUGRA, we have $\tilde{\bf A}^{f}=A_{0}\textbf{1}$. As
noted before, at the energies $q^{2}\lesssim M_{R_{3}}^{2}$, the right handed
sneutrinos also decouple from the theory, along with the right handed
neutrinos. In the program, we decouple the right handed neutrinos sequentially
at different scales : the heaviest right handed neutrino at $M_{R_{3}}$ and
the second heaviest one at $M_{R_{2}}$ and the lightest one at$M_{R_{1}}$.
The right handed neutrinos can be easily removed from the model to recover
MSSM without right handed neutrinos, by choosing ${\bf Y}^{\nu}=\textbf{0}$.
This automatically decouples the right handed neutrinos in the theory. An
explicit option is also provided in the input files, where by turning on/off
the parameter rhn, one can either include/remove right handed neutrinos in the
theory. Finally, let us note that quantum effects above the scale of seesaw
$q^{2}\gg M_{R_{3}}^{2}$ will make the mass matrix $M_{R}$ non-diagonal. The
running of the Majorana mass matrix can also effect the ${\bf B}_{M}$ term in
the soft potential, described below, at the 1-loop level. The ${\bf B}_{M}$
can have implications for flavor physics and EDMs, if it is large through
finite terms. These effects [48] are not computed in SuSeFLAV.
The complete set of 2-loop RGE for a general superpotential and MSSM are
presented in [49]. For the supersymmetric seesaw model, we use the RGE’s from
[50].
## 3 SUSY Breaking Mechanisms
Supersymmetry is broken spontaneously in a hidden sector and is then
communicated to the MSSM sector through the ‘messenger sector’. The messengers
could be gauge interactions or gravitational interactions. The result of this
communication leads to soft supersymmetry breaking terms in the MSSM. While
the form of the soft Lagrangian Eq. (2) is itself not dependent on the
mediation mechanism, the physical quantities i.e., the masses, the couplings
etc., are determined in terms of few ‘fundamental’ parameters depending on the
mediation mechanism. Popular among such supersymmetry breaking schemes are (i)
minimal Supergravity (mSUGRA) (ii) Gauge Mediated Supersymmetry Breaking
(GMSB) (iii) Anomaly Mediated Supersymmetry Breaking (AMSB) (iv) Gaugino
mediation (v) Moduli mediation etc. among a host of other possibilities [51].
In addition there could be variations within each of the above schemes. In the
SuSeFLAV, we have in-built (i) mSUGRA and some of its variations: (a) Non-
Universal Higgs Models (NUHM), (b) Non-Universal Gaugino Models (NUGM) and (c)
Complete Non-Universal Model (CNUM) and (ii) Gauge Mediated Supersymmetry
Breaking (GMSB) models. The corresponding input files are given in the
examples/ directory. An input file with completely non-universal soft
parameters is also presented for supergravity mediation, where the users can
define the boundary conditions of their choice. For other supersymmetric
breaking models, the users can modify appropriately the slha.in file and run
the program accordingly. In the section below we describe the two
supersymmetric breaking scenarios considered in SuSeFLAV.
### 3.1 mSUGRA and its variations
Supersymmetry is broken spontaneously in a hidden sector and the communicated
to the visible sector through the gravitational interactions. If the
supergravity Kähler metric is canonical in matter fields, the soft terms
resultant after integrating out the supergravity multiplet (while keeping the
gravitino mass fixed), are universal in nature [52]. The property universal
refers to the flavor space i.e. all the soft terms take the same value
irrespective of the flavor at the mediation scale. In SuSeFLAV we have
considered the mediation scale to be $M_{\text{GUT}}$. At this scale, all the
soft terms are determined by four parameters and the sign of the
$\mu$-parameter.
1. 1.
At $M_{\text{GUT}}$ the gaugino masses are universal to a value $M_{1/2}$,
i.e.
$M_{1}(M_{\text{GUT}})=M_{2}(M_{\text{GUT}})=M_{3}(M_{\text{GUT}})\equiv
M_{1/2}$ (14)
2. 2.
The scalar and the Higgs masses are given by the parameter $m_{0}^{2}$ at
$M_{\text{GUT}}$.
$\displaystyle{\bf m}^{2}_{{\tilde{Q}}_{i}}(M_{GUT})={\bf
m}^{2}_{{\tilde{u}}_{R_{i}}}(M_{GUT})$ $\displaystyle={\bf
m}^{2}_{{\tilde{d}}_{R_{i}}}(M_{GUT})={\bf
m}^{2}_{{\tilde{L}}_{i}}(M_{GUT})={\bf m}^{2}_{{\tilde{l}}_{i}}(M_{GUT})\equiv
m^{2}_{0}\;{\bf 1}$ $\displaystyle m^{2}_{{H}_{u}}(M_{GUT})$
$\displaystyle=m^{2}_{{H}_{d}}(M_{GUT})\equiv m^{2}_{0}\;$ (15)
3. 3.
The trilinear couplings are given by the parameter $A_{0}$ at $M_{\text{GUT}}$
${\tilde{\bf A}}^{u}_{ij}(M_{GUT})={\tilde{\bf
A}}^{d}_{ij}(M_{GUT})={\tilde{\bf A}}^{l}_{ij}(M_{GUT})\equiv A_{0}\;{\bf
1}_{ij}$ (16)
To specify the spectrum at the weak scale, two more parameters need to be
fixed. First is the ratio of the vacuum expectation values (vevs) of the two
Higgs fields, $\tan\beta=v_{u}/v_{d}$. Second is a discrete parameter, the
sign of $\mu$ or the Higgsino mass parameter. The magnitude of $\mu$ is fixed
by the radiative electroweak symmetry breaking mechanism which has been
incorporated in the program.
#### 3.1.1 Non-Universal Models
In models based on Grand Unified theories, it has been proposed that the
strictly universal feature of the soft masses might not be valid and in fact
some amount of non-universalities can enter in a model-dependent fashion. For
example, in models where the hidden sector gauge kinetic function is no longer
singlet under the GUT group, gaugino masses would become non-universal at the
high scale. However the non-universalities enter in a predictive fashion when
a particular gauge group is chosen, as the ratios of the gaugino masses are
now fixed by the Clebsch-Gordan coefficients of respective decomposition.
These ratios are well known for the GUT models based on various gauge groups,
e.g., SU(5) [53, 54, 55, 56], SO(10) [57, 58, 59]. Without resorting to any
particular model, we have incorporated non-universal gaugino mass scenario in
SuSeFLAV, by considering
$M_{1}(M_{GUT})\;\neq\;M_{2}(M_{GUT})\;\neq\;M_{3}(M_{GUT}).$ (17)
The user has the freedom of choosing any ratios among these three parameters
at GUT scale. The corresponding input file is sinputs-nugm.in. A second class
of non-universality which has been incorporated in SuSeFLAV is for the Higgs
[60]. It has been argued that since in Grand Unified theories (GUTs) like
SO(10), all the matter sits in a single representation where as the Higgs sits
in separate representation, the universality of the soft masses need not
include Higgs, especially when supersymmetry breaking mediation happens close
to the GUT scale. It has also been realized that introducing such non-
universality makes the $\mu$ parameter free and thus leading to completely
different phenomenology at the weak scale especially for dark matter. Thus the
boundary conditions at the high scale in our notation are given by
$\displaystyle{\bf m}^{2}_{{\tilde{Q}}_{i}}(M_{GUT})=\;$ $\displaystyle{\bf
m}^{2}_{{\tilde{u}}_{R_{i}}}(M_{GUT})=\;{\bf
m}^{2}_{{\tilde{d}}_{R_{i}}}(M_{GUT})=\;{\bf
m}^{2}_{{\tilde{L}}_{i}}(M_{GUT})={\bf m}^{2}_{{\tilde{l}}_{i}}(M_{GUT})\equiv
m^{2}_{0}\;{\bf 1}$ $\displaystyle m^{2}_{{H}_{u}}(M_{GUT})\equiv
m^{2}_{10}\;\;\;;\;\;\;m^{2}_{{H}_{d}}(M_{GUT})\equiv m^{2}_{20}$ (18)
Note that we intermittently use the notation $m_{10}$ and $m_{20}$ for the
Higgs mass parameters as defined above in the non-universal Higgs mass model.
The input file for this case sinputs-nuhm.in. It should be noted that a
negative sign for any soft-mass as input would mean a negative sign for that
soft-mass squared in the program.
In addition to these non-universal input files, a completely generic input
file called sinputs-cnum.in is provided where the user can provide all the
supersymmetric breaking parameters at the high scale (mSUGRA) without any
assumptions on their structure. This input file is more suited for models with
flavor structure at the high scale.
### 3.2 GMSB
The second class of supersymmetric breaking models incorporated in SuSeFLAV is
Gauge Mediated Supersymmetric Breaking (GMSB). As before, supersymmetry is
broken spontaneously in the hidden sector, but now communicated to the MSSM
sector through gauge interactions. The minimal set of models under this
category goes under the name, Minimal Messenger Model (MMM) [61]. In this
model, a set of messenger superfields transforming as complete representations
of SU(5) ($\supset\mathcal{G}_{\text{SM}}$) gauge group and couple directly to
a singlet field which parameterized the supersymmetry breaking in the hidden
sector. Supersymmetry breaking is then transmitted to the MSSM through SM
gauge interactions. The following superpotential represents the messenger
sector coupling to the hidden sector
${\cal W}=\lambda X\Phi_{i}\bar{\Phi}_{i}$ (19)
where $\Phi$ and $\bar{\Phi}$ are messenger sector superfields transforming as
$5$ and $\bar{5}$ of SU(5) and $i$ runs for the number of messenger sector
superfields, typically $i=[1,5]$. The $X$ superfield representing the hidden
sector is parameterized by the vacuum expectation values for both its scalar
component $\langle X\rangle$ as well as its auxiliary component $\langle
F_{X}\rangle$. Gauge interactions with the messenger fields lead to gauginos
attaining masses at 1-loop which at the Messenger scale are given by
$M_{a}\left(M_{mess}\right)=\frac{\alpha_{a}\left(M_{mess}\right)}{4\pi}\;\Lambda\;g\left(\frac{\Lambda}{M_{mess}}\right)\sum_{i}n_{a}(i);$
(20)
where the Messenger scale $M_{mess}=\lambda\langle X\rangle$ and
$\Lambda=\langle F_{X}\rangle/\langle X\rangle$. Here $n_{a}(i)$ is the Dynkin
index for the messenger pair $\Phi,\overline{\Phi}$ and the sum runs over all
the messengers in each group.
The scalars attain their masses from the 2-loop diagrams. These are given as
$m^{2}_{s}(M_{mess})=2\Lambda^{2}\;f\left(\frac{\Lambda}{M_{mess}}\right)\;\sum_{a,i}n_{a}(i)\;C_{a}\;\left(\frac{\alpha_{a}(M_{mess})}{4\pi}\right)^{2}$
(21)
Where $C_{a}$ is the quadratic Casimir invariant of the MSSM fields and the
function $g(x)$ and $f(x)$ are defined in [62] and [63] respectively. The
leading order contribution to the tri-linear couplings comes from the 2-loop
diagrams but they are suppressed by an extra $\alpha/4\pi$ factor compared to
the gaugino masses. Thus to an very good approximation we can take
${\tilde{\bf A}}^{u,d,l}_{ij}(M_{Mess})\simeq 0$ (22)
Thus in minimal GMSB model considered in SuSeFLAV we have the following 5
parameters as inputs
$\tan\beta,\;{\rm sign}(\mu),\;M_{mess},\;\Lambda{\rm~{}and~{}}n.$ (23)
The input file sinput-gmsb.in and mgmsb1.1.in and so on, are built-in files to
specify the inputs of the GMSB model in the examples/ directory of SuSeFLAV.
## 4 Calculation of Supersymmetric Spectrum
Figure 1: Flowchart of the program
Once the user chooses the particular model of supersymmetry breaking by typing
in the various parameters in the relevant input file, the program computes the
spectrum at the weak scale, checks for the various direct and indirect search
limits and computes the various observables including the flavor violating
ones like $\mu\to e+\gamma$. The computation of the spectrum involves several
complicated intermediate steps which has been already explained in detail by
various existing programs [64, 65]. In SuSeFLAV we follow a similar approach
in computing the spectrum, however including flavor mixing as well as
couplings with right handed neutrinos. In Fig. (1), we have shown the
flowchart of the computation of the spectrum and the observables in the
program. We can summarize the computation in terms of three steps which
however, are not independent of each other as the procedure involves
significant number of iterations.
* 1.
RGE evolution: Using the MSSM RGE for the Yukawa and gauge couplings, run all
the known SM parameters like gauge couplings, Yukawa couplings up to the scale
of supersymmetry breaking. In the case of mSUGRA, run to the scale where the
gauge couplings corresponding to ${\rm SU}(2)_{L}$ ($g_{2}$) and ${\rm
U}(1)_{Y}$ $\left(g_{1}\equiv\sqrt{\frac{3}{5}}\;g_{Y}\right)$ meet, this
determines the GUT scale ($M_{{\rm GUT}}$). At the SUSY breaking scale
($M_{\text{GUT}}$ in case of mSUGRA and $M_{\text{mess}}$ in case of GMSB),
with the user defined input parameters and using the GUT scale SM parameters
as the boundary conditions, run all the MSSM RGEs including those for the soft
terms all the way down to $M_{{\rm SUSY}}$. For the initial run $M_{{\rm
SUSY}}$ is defined to be 1 TeV. If seesaw mechanism is switched on, the
program takes in to consideration the running of user defined neutrino Yukawa
couplings between the seesaw scale and the supersymmetric breaking scale in
both directions.
* 2.
Radiative Electroweak symmetry breaking: The resultant soft parameters at the
$M_{\text{SUSY}}$ are used to check if they satisfy the tree level electroweak
symmetry breaking conditions and compute the $\mu$ parameter. The full one
loop effective potential corrections are then computed using the ‘tree level’
$\mu$ parameter, which is then used to derive the 1-loop corrections to the
$\mu$ parameter. This is repeated iteratively until the $\mu$ parameter
converges.
* 3.
Convergence of the Spectrum: In the final step, we run all the soft terms to
the scale $M_{Z}$ where corrections to the SM parameters are added. We compute
the supersymmetric corrections to the SM gauge couplings and the third
generation fermion masses ($t$, $b$ and $\tau$). The resultant masses are fed
in to the RGE routine as shown in Fig. (1) and the soft spectrum is evaluated
and run to MZ scale. This full iteration is continued until the SM third
generation fermion masses converges to user defined precision (usually ${\cal
O}(10^{-3}-10^{-4})$). Once this masses get converged we calculate various low
energy observables (e.g. ${\rm BR}(\mu\to e\gamma)$, ${\rm BR}(b\to s\gamma)$
etc.). In section 6 we have discussed the low energy observables SuSeFLAV
calculates.
In the following we describe each of these steps in more detail.
### 4.1 RGE Evolution
The standard model fermion masses and gauge couplings are the inputs to the
program at the weak scale, taken to be equal to the Z-boson mass $M_{Z}$ in
the program. The parameters are divided in to two subsets depending on whether
radiative corrections are added or not. The parameters, masses of the first
two generations quarks and leptons, for which we do not add radiative
corrections are put in the file src/stdinputs.h. The values for these masses
are taken from PDG 2012 [66]. Most of the other parameters such as the
leptonic mixing matrix, ${\bf U}_{\text{PMNS}}$, the hadronic mixing matrix
(${\bf V}_{\text{CKM}}$) are defined in the file src/SuSemain.f. The
$\overline{\text{MS}}$ values of Z-boson mass, the pole masses of the top
quark $m_{t}^{pole}(m_{t})$ and tau lepton, $m_{\tau}^{pole}(m_{\tau})$ and
the $\overline{\text{MS}}$ mass of the bottom quark mass,
$m_{b}^{\overline{\text{MS}}}(m_{b})$, are left as user defined inputs. The
$\overline{\text{MS}}$ gauge couplings, electromagnetic, $\alpha_{em}(M_{Z})$
and the strong coupling, $\alpha_{s}(M_{Z})$ are also considered as inputs and
are contained in the input files. The $\overline{\text{MS}}$ inputs are
converted to $\overline{\text{DR}}$ as the RGEs are written in the
$\overline{\text{DR}}$ scheme. The conversion for the gauge couplings is given
by
$\alpha_{em}^{\overline{\text{DR}}}(M_{Z})=\left(\frac{1}{\alpha_{em}^{\overline{\text{MS}}}(M_{Z})}-\frac{1}{6\pi}\right)^{-1},\,\,\,\alpha_{s}^{\overline{\text{DR}}}(M_{Z})=\left({\frac{1}{\alpha_{s}^{\overline{\text{MS}}}(M_{Z})}-\frac{1}{4\pi}}\right)^{-1}$
(24)
The so defined $\alpha_{em}^{\overline{\text{DR}}}$ is in turn used to define
the $\overline{\text{DR}}$ values of the $\alpha_{1,2}$.
$\alpha_{1}(M_{Z})\equiv\frac{g_{1}^{2}}{4\pi}=\frac{5\alpha_{em}^{\overline{\text{DR}}}(M_{Z})}{3\cos^{2}\theta_{W}}\;,\quad\alpha_{2}(M_{Z})\equiv\frac{g_{2}^{2}}{4\pi}=\frac{\alpha_{em}^{\overline{\text{DR}}}(M_{Z})}{\sin^{2}\theta_{W}}$
(25)
In a similar fashion, the bottom mass is converted from the
$\overline{\text{MS}}$ to $\overline{\text{DR}}$ using [67, 68]
$m_{b}^{\overline{\text{DR}}}(M_{Z})=m_{b}^{\overline{\text{MS}}}(m_{b})\cdot\left[1-\frac{\alpha_{s}(M_{Z})}{3\pi}-\frac{23\alpha_{s}^{2}(M_{Z})}{72\pi^{2}}+\frac{3\alpha_{2}(M_{Z})}{32\pi}+\frac{13\alpha_{1}(M_{Z})}{288\pi}\right],$
(26)
where the coupling constants appearing in the parenthesis are their
$\overline{\text{DR}}$ values. The masses of the tau lepton and top quark are
converted from their pole masses using the following relations:
$\displaystyle m_{\tau}^{\overline{\text{DR}}}(M_{Z})$ $\displaystyle=$
$\displaystyle
m_{\tau}^{pole}\cdot\left[1-\frac{3}{8}\left(\alpha_{1}(M_{Z})-\frac{\alpha_{2}(M_{Z})}{4}\right)\right]$
$\displaystyle m_{t}^{\overline{\text{DR}}}(M_{Z})$ $\displaystyle=$
$\displaystyle m_{t}^{pole}\cdot\Delta_{m_{t}}^{QCD}$ (27)
where $\Delta_{m_{t}}^{QCD}$ is given by [69]
$\Delta_{m_{t}}^{QCD}=1-\frac{\alpha_{s}(m_{t})}{3\pi}\,\left(5-3\Delta_{tz}\right)-\alpha_{s}^{2}(m_{t})\,\left(0.538-\frac{43\Delta_{tz}}{24\pi}+\frac{3\Delta_{tz}^{2}}{8\pi^{2}}\right),$
(28)
with $\Delta_{tz}=2\ln\left(m_{t}^{pole}/M_{Z}\right)$ and
$\alpha_{s}(m_{t})=\frac{\alpha_{s}^{\overline{\text{DR}}}(M_{Z})}{1+\frac{3}{4\pi}{\alpha_{s}^{\overline{\text{DR}}}(M_{Z})\Delta_{tz}}}$
(29)
The $\overline{\text{DR}}$ corrected masses are used to define the $3\times 3$
Yukawa matrices at the $M_{Z}$ scale which form the inputs to the RGE.
$\displaystyle{\bf
Y}^{u}=\frac{\sqrt{2}}{v\sin\beta}\;\text{Diag}[m_{u},m_{c},m_{t}]\cdot{\bf
V}_{CKM}\;$ ; $\displaystyle\;{\bf
Y}^{d}=\frac{\sqrt{2}}{v\cos\beta}\;\text{Diag}[m_{d},m_{s},m_{b}]\;;$
$\displaystyle{\bf Y}^{e}=\frac{\sqrt{2}}{v\cos\beta}$
$\displaystyle\text{Diag}[m_{e},m_{\mu},m_{\tau}]$ (30)
We use the above defined Yukawas to run the full 2-loop RGEs from the weak
scale ($M_{Z}$) up to the scale at which the two gauge couplings ($g_{1}$ and
$g_{2}$) unify with an accuracy of 1%. For GMSB scenario this scale is set by
the user as messenger scale or $M_{\rm mess}$. In the case of right handed
neutrinos three intermediate scales get introduced in the theory. As mentioned
in the section 2, we consider the seesaw scale to be the mass of the heaviest
right handed neutrino, $M_{R_{3}}$. At this scale we set the neutrino Yukawa
(${\bf Y}^{\nu}$) and run the RGEs with this Yukawa up to the high scale. At
the high scale, depending on the model, we set the SUSY breaking boundary
conditions and then run the RGEs down to the heaviest right handed neutrino
mass scale i.e., $M_{R_{3}}$. Below this mass scale we decouple the heaviest
right handed neutrino by setting its couplings in the neutrino Yukawa matrix
to zero. From $M_{R_{3}}$ we run down the RGEs to the next heaviest right
handed neutrinos i.e., $M_{R_{2}}$ and then form $M_{R_{2}}$ to $M_{R_{1}}$.
Below each of these scale, i.e., $M_{R_{2}}$ and $M_{R_{1}}$, we decouple the
corresponding right handed neutrino by setting their couplings to zero. From
$M_{R_{1}}$ we run the RGEs down to the scale $M_{\rm SUSY}$. For the first
iteration we take a guess value for $M_{\rm SUSY}$ which is 1 TeV. From the
second iteration onwards the Electro-Weak Symmetry Breaking (EWSB) scale is
set to the geometric mean of the two stop masses or
$\sqrt{m_{\tilde{t}_{1}}\cdot m_{\tilde{t}_{2}}}$. At this scale we check for
the EWSB condition and then we calculate the supersymmetric spectrum. A
schematic picture of the integration of the RGE’s in mSUGRA with seesaw
mechanism is summarized in Fig. (2). In GMSB models, the high scale is the
messenger scale, $M_{\rm mess}$ instead of $M_{\rm GUT}$ and the procedure of
the integration is very similar. The seesaw mechanism can be incorporated in
this class of models as long as heaviest right handed neutrino is lighter than
the messenger scale ($M_{R_{3}}<M_{\rm mess}$).
Figure 2: Pictorial representation of evolution of RGEs
### 4.2 Radiative Electroweak Symmetry Breaking
The tree level EWSB conditions at the $M_{\rm SUSY}$ scale are defined as
below
$\displaystyle\left|\mu\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tan 2\beta(m_{H_{u}}^{2}\tan\beta-
m_{H_{d}}^{2}\cot\beta)-M_{Z}^{2}\right]$ $\displaystyle B_{\mu}$
$\displaystyle=$ $\displaystyle\frac{\sin
2\beta}{2}\left[m_{H_{u}}^{2}+m_{H_{d}}^{2}+2|\mu|^{2}\right]$ (31)
where $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are the RGE output at $M_{\rm
SUSY}$. For consistent electroweak symmetry breaking we require
$\left|\mu\right|^{2}>0$888Further, there should not be any tachyons in the
physical Higgs spectrum.. The tree level $|\mu|^{2}$ and RGE output of the
other SUSY soft masses is used to calculate the tree level spectrum as
described in A. Radiative corrections can however significantly modify the
tree level value of $\mu$. Using tree level sparticle spectrum, we calculate
the radiative corrections to the higgs potential up to one-loop order as given
by BPMZ [70]. The tadpoles modify the $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ as
$m_{H_{u}}^{2}\rightarrow m_{H_{u}}^{2}-\frac{t_{1}}{v_{1}}\quad;\quad
m_{H_{d}}^{2}\rightarrow m_{H_{d}}^{2}-\frac{t_{2}}{v_{2}}.$ (32)
With these radiatively corrected higges, using Eq. (4.2), we calculate the
radiatively corrected $|\mu|^{2}$. This is repeated iteratively until the
convergence of $|\mu|^{2}$ reaches the desired accuracy (default value is
$\mathcal{O}(10^{-4})$). This accuracy level can be changed by changing the
parameter tol in file src/ewsbiterate.f. There could be regions where the
$\mu$ parameter does not converge within a small number of iterations. In such
regions, the program considers the parameter point as $|\mu|$-non convergent.
Once $|\mu|$ has converged we calculate $B_{\mu}$ as
$B_{\mu}=\frac{\sin
2\beta}{2}\left[\overline{m}_{H_{u}}^{2}+\overline{m}_{H_{d}}^{2}+2|\mu|^{2}\right]$
(33)
It is important to note here that at $M_{\rm SUSY}$, SuSeFLAV checks for
D-flat directions in the potential as well as whether the potential is
unbounded from below. It also checks for charge and color breaking minima.
More details about these checks are discussed in the next section. Even if
these conditions are not satisfied the program still proceeds to compute the
spectrum however, a flag is raised and written in the output file.
Once $\mu$ is converged, the program uses it to compute complete one loop
corrections to the sparticle spectrum. We follow the work of BPMZ [70] in
computing these corrections999Current version of the program does not include
flavor violating contributions from sleptons to all the 1-loop corrections.
The sleptonic contributions are neglected in this version in the presence of
flavor violation.. Corrections to neutralinos and charginos are evaluated at
external momenta equal to $M_{\text{SUSY}}$ and the corrections to sfermion
masses are calculated at an external momenta equal to their pole mass as
prescribed by BPMZ. In B we have discussed more about these threshold
corrections to the sparticles. As mentioned before all the parameters of the
code are considered real, including the diagonalizing matrices. In determining
the neutral higgs masses the user has a choice to employ approximations for
two loop which are mostly top-stop enhanced [71] or full one loop tadpole
corrections as described in BPMZ or full one loop together with leading order
two loop corrections. We have implemented the two loop corrections due to
Slavich et. al [72] in our version 1.2. This is the default choice for the
Higgs spectrum in the present version.
### 4.3 Convergence of the Spectrum
In final step, the program evaluates the full one loop flavor conserving
supersymmetric threshold corrections to SM parameters. The parameters which
are corrected are $m_{t},\,m_{b},\,m_{\tau},\,\alpha_{s},\,\alpha_{em}$ and
$\sin^{2}\theta_{W}$. One loop corrected running masses are given by the
following,
$\Delta_{m_{t}}(M_{Z})=\Sigma_{t}^{BPMZ}+\Delta_{m_{t}}^{QCD},\quad
m_{t}(M_{Z})=m_{t}^{pole}\left[1+\Delta_{m_{t}}(M_{Z})\right]$ (34)
$m_{b}(M_{Z})^{\overline{DR}}_{MSSM}=\frac{m_{b}(M_{Z})^{\overline{DR}}_{SM}}{1+\Delta_{m_{b}}^{BPMZ}}$
(35)
$m_{\tau}(M_{Z})=m_{\tau}^{pole}\left[1+\Sigma_{\tau}^{BPMZ}\right]$ (36)
The quantity $\Sigma_{t}^{BPMZ}$, one loop correction to the top quark mass is
evaluated at external momenta equal to $m_{t}^{\overline{DR}}(M_{Z})$.
Whereas, $\Delta_{m_{b}}^{BPMZ}$ and $\Sigma_{\tau}^{BPMZ}$ are evaluated in
the limit of external momenta tending to zero. The expressions of these
parameters are described in the B. The three gauge couplings get corrected as
below
$\alpha_{1}(M_{Z})=\frac{5\alpha_{em}^{\overline{DR}}(M_{Z})}{3\left(1-\Delta_{\alpha_{em}}\right)\cos^{2}\theta_{W}}$
(37)
$\alpha_{2}(M_{Z})=\frac{\alpha_{em}^{\overline{DR}}(M_{Z})}{\left(1-\Delta_{\alpha_{em}}\right)\sin^{2}\theta_{W}}$
(38)
$\alpha_{3}(M_{Z})=\frac{\alpha_{s}(M_{Z})}{1-\Delta_{\alpha_{s}}}$ (39)
where $\Delta_{\alpha_{em}}$ and $\Delta_{\alpha_{s}}$ are one loop
corrections to the electromagnetic and strong coupling described in the B.
Note that $\sin^{2}\theta_{W}$ used in the above expressions is also
radiatively corrected. Iterative method is implemented to correctly evaluate
SUSY contributions to $\sin^{2}\theta_{W}$. The above corrected
$\alpha_{1}(M_{Z}),\alpha_{2}(M_{Z})$ and $\alpha_{3}(M_{Z})$ and also the
third generation SM fermions ($m_{t}$, $m_{b}$, $m_{\tau}$) masses are used as
the input for the next long iteration. The iteration continues until the SM
third generation fermions, namely top, bottom and tau mass are converged to
user defined precision (usually ${\cal O}(10^{-3})$). This precision can be
changed by the parameter named spectrum tolerance defined in the input files.
Once the SM fermion masses get converged the program proceeds to calculate the
various low energy observables. Finally, lets note that if both $LL$ and $RR$
type leptonic flavor violation is present, it could lead to corrections to
lepton self energies [73], which are not included in the present version of
the code.
## 5 Theoretical and Phenomenological Constraints
The requirement of consistent evaluation of supersymmetric spectrum involves a
check for theoretical constraints such as charge and color breaking minima
(CCB), scalar potential unbounded from below (UFB) and efficient electroweak
symmetry breaking at EWSB scale.
With every iterative step of the program SuSeFLAV checks for CCB and UFB
conditions at the tree level [74]. These conditions are governed by equations
40, 41 and are simultaneously implemented while computing the tree level $\mu$
parameter (checking for efficient EWSB, see section 4.2 for the complete
description).
$\displaystyle{\rm{\bf
CCB:}}\;\;\;\;3\left(m_{Q_{u}}^{2}+m_{f}^{2}+|\mu|^{2}+m_{H_{f}}^{2}\right)\geq\left|A_{f}\right|^{2}$
(40) $\displaystyle{\rm{\bf
UFB:}}\;\;\;\;\;m_{H_{u}}^{2}+m_{H_{d}}^{2}+2|\mu|^{2}\geq
2\left|B_{\mu}\right|^{2}{\;\rm at\;}Q>M_{EWSB}$ (41)
It is important to note that the complete MSSM spectrum is still calculated
even if CCB and UFB conditions are not satisfied. However, if the
supersymmetric scalar potential has a charge and color breaking minima which
is lower than electroweak minimum a warning flag CCB is generated. Similarly,
if the complete supersymmetric scalar is unbounded from below a warning flag
UFB is generated.
Apart form the above described theoretical constraints the program also
imposes additional phenomenological constraints on the obtained spectrum. From
the direct non-observation of charged dark matter in the Universe we require
$m_{\tilde{\tau}}>m_{\chi^{0}}$ or LSP being neutral. Regions for which this
condition is not true is excluded as $\tilde{\tau}$ LSP regions. Consequently
the flag is marked as LSPSTAU. Also, we require the the spectrum to be non-
tachyonic. If tachyonic spectrum is encountered it is flagged as TACSPEC. The
program also indicates the sector where the tachyon occurs, for example: a
tachyon in sleptonic sector is marked as TACSLP. Some lower bounds on various
sparticle masses that result from direct searches at colliders, e.g., the
lightest Higgs mass $m_{h}>114.5{\rm~{}GeV}$, and the Chargino mass
$m_{\chi^{\pm}}>103.5{\rm~{}GeV}$ [75] are also incorporated in the program.
Points failing to satisfy these bounds are flagged as LEPH and LEPC
respectively. The present version does not include recent direct search limits
from LHC [76].
## 6 Low Energy Observables
Once the complete supersymmetric spectrum is obtained we compute the following
low energy observables in SuSeFLAV.
* 1.
Fine Tuning: In MSSM the standard model masses and gauge couplings, e.g.
$M_{Z}$, $m_{t}$, $m_{b}$ etc. are function of the input parameters of the
model, i.e., for mSUGRA $m_{0},\ M_{1/2},\ A_{0},\ \tan\beta\ {\rm and}\ {\rm
sign}(\mu)$. Now, we can recast the EWSB Eq. (4.2) as below:
$M_{\bar{Z}}^{2}=-2|\mu|^{2}+\tan 2\beta\left(m_{{H}_{u}}^{2}\tan\beta-
m_{{H}_{d}}^{2}\cot\beta\right),$ (42)
where, all the parameters on the right side of Eq. (42) are at the weak scale.
Given that $M_{Z}$ is know at a few percent level, it can be seen that some
amount of ‘tuning’ between the parameters in the right side is needed. Various
measures [77, 78, 79, 80] have been proposed to quantify the ‘tuning’ needed
in some input parameter. In SuSeFLAV, we have followed Barbieri and Giudice
[77, 80], in evaluating the fine tuning for a given parameter $\lambda_{i}$ of
the model as
$\frac{\delta
M_{Z}^{2}}{M_{Z}^{2}}=f(M_{Z}^{2},\lambda_{i})\;\frac{\delta\lambda_{i}}{\lambda_{i}}$
(43)
Where Barbieri-Giudice function or $f(M_{Z}^{2},\lambda_{i})$ is defined as
$f(M_{Z}^{2},\lambda_{i})=\frac{\lambda_{i}}{M_{Z}^{2}}\;\frac{\partial
M_{Z}^{2}}{\partial\lambda_{i}}$ (44)
We derive the fine tuning in $M_{Z}^{2}$ with respect to $\mu^{2}$ for which
$f(M_{Z}^{2},\mu^{2})$ takes the following form
$f(M_{Z}^{2},\mu^{2})=\frac{2|\mu|^{2}}{M_{Z}^{2}}\left[1+\frac{(\tan^{2}\beta+1)}{(\tan^{2}\beta-1)^{3}}\;\frac{4\tan^{2}\beta\;\left(m_{\overline{H}_{d}}^{2}-m_{\overline{H}_{u}}^{2}\right)}{\left(m_{\overline{H}_{d}}^{2}+m_{\overline{H}_{u}}^{2}\right)}\right].$
(45)
The fine tuning in $m_{t}$ with respect to $\mu^{2}$ or $f(M_{Z}^{2},m_{t})$
is expressed as follows
$f(m_{t},\mu^{2})=\frac{1}{2}f(M_{Z}^{2},\mu^{2})+\frac{1}{\tan^{2}\beta-1}\;\frac{2|\mu|^{2}}{m_{\overline{H}_{u}}^{2}+m_{\overline{H}_{d}}^{2}}$
(46)
* 2.
Electro-Weak Precision Measurement: The electroweak observables $\rho$
parameter and $\sin^{2}\theta_{W}$ are defined in 47 and 48.
$\displaystyle\rho=\frac{M_{W}}{M_{Z}\cos\theta_{W}}$ (47)
$\displaystyle\sin^{2}\theta_{W}=1-\frac{M^{2}_{W}}{M^{2}_{Z}}$ (48)
At the tree level $\rho$ parameter is unity and $\sin^{2}\theta_{W}=0.2286$. A
deviation from these values are observed at one-loop level stemming from SM
and supersymmetric corrections to $W$ and $Z$ boson masses. A measure of this
deviation to $\rho$ parameter is given below
$\displaystyle\Delta\rho$
$\displaystyle=\frac{\Pi_{ZZ}(0)}{M_{Z}^{2}}-\frac{\Pi_{WW}(0)}{M_{W}^{2}}$
(49) $\displaystyle{\rm and\;}\rho$ $\displaystyle=\frac{1}{(1-\Delta\rho)}$
(50)
The correction to $\sin^{2}\theta_{W}$ is described in B. A precise
measurement of these parameters by LEP, SLC and Tevetron imposes stringent
constraints on $\Delta\rho$, requiring $\Delta\rho\lesssim 10^{-3}$ for a
physically viable spectrum.
* 3.
BR($b\rightarrow s\gamma$) Constraint: Another sector where the effect of SUSY
particles can be seen is the radiative flavor changing decay of bottom quark,
$b\rightarrow s\gamma$ [81]. In the Standard model, this decay is mediated by
loops containing up sector quarks and $W$ bosons. In supersymmetric theories,
if CKM is the only source of hadronic flavor violation, additional
contributions to $b\rightarrow s\gamma$ process come from chargino loops
($\widetilde{\chi}^{\pm}$), stop ($\tilde{t}$) squarks loops and charged Higgs
bosons ($H^{\pm}$) loops. As in the perturbation theory the SM and
supersymmetric correction appear at the same order, the measurement of
BR($b\rightarrow s\gamma$) is a very powerful tool for constraining the SUSY
parameter space. In SuSeFLAV we have followed Bartl et al. [82], which
includes Standard model NLO as well as MSSM LO contributions. Like the other
observables, this branching ratio is calculated for all the valid points and
written in the SLHA file. More precise evaluations of this process are
available publicly in the [31, 32, 38, 83]. However, given that we are
concentrating on evaluation of flavor violating observables in the presence of
flavor violation in the soft sector, we would like upgrade our computation of
this processes in a future version to include gluino contributions which could
become important in the presence of squark flavor violating terms. Currently,
we refer our users to couple the output of our program to one of the above
existing codes depending on the accuracy required and input parameters they
are interested in. For example SuperIso [31] computes the process at NNLO. It
has an added advantage that it can compute other B-physics observables also.
As of now our program does not have an interface with SuperIso, but one can
link to it using one of its programs (cmssm.c etc ) to our code.
* 4.
Anomalous Magnetic Moment of Muon $(g_{\mu}-2)$: The muon anomalous magnetic
moment ($g_{\mu}-2$) has been precisely measured by Muon $(g-2)$ collaboration
[84, 85]. Due to supersymmetric particles in the loop, ($g_{\mu}-2$) gets non-
negligible correction apart from the SM contribution. The experimentally
measured value of anomalous magnetic moment of muon is:
$a_{\mu}^{exp}\equiv\frac{(g_{\mu}-2)}{2}=(11659208\pm 6)\times 10^{-10}.$
(51)
Whereas, the difference in theoretical prediction by SM and experimental
value, i.e $a_{\mu}^{exp}-a_{\mu}^{SM}=(28.7\pm 8.0)\times 10^{-10}$ [86].
Here the difference arises because of the fact of different estimates of
hadronic vacuum polarization contribution. The contribution from the
supersymmetric particles to $(g_{\mu}-2)$ is through
$(\widetilde{\chi}^{0}-\tilde{\mu}/\tilde{\tau}/\tilde{e})$ loop or through
$(\widetilde{\chi}^{\pm}-\tilde{\nu})$ loop. Supersymmetric parameter space
can be severely constrained [87] as a supersymmetric explanation for the
discrepancy with the experiment prefers a ‘light’ supersymmetric spectrum.
Moreover, both these contributions are $\tan\beta$ enhanced, so large values
of $\tan\beta\,(\gtrsim 30)$ are more severely constrained. We have taken the
expression for one-loop contribution, due to supersymmetric particles, to the
$(g_{\mu}-2)$ from Hisano et al. [14]. For a given set of input parameter
SuSeFLAV calculates the $(g_{\mu}-2)$ and writes it into the SLHA file. Two
loop contributions are not added in the present version which could be
important in the very large tan$\beta$ regime [88]. One of the main reasons
for not including these contributions is the absence of two loop computations
including flavor violation in the sleptonic sector. The present version
includes flavor violating contributions at the 1-loop level which become
important when the seesaw is switched on. We hope to generalize the two loop
contributions including flavor violations in a future version.
* 5.
Lepton Flavor Violating Decays: Subject to satisfying phenomenological
constraints at $M_{\rm SUSY}$, we evaluate decay rates of rare lepton flavor
violating processes. The decay rates and branching ratios for the following
processes are computed $\mu\,\rightarrow\,e\gamma$,
$\tau\,\rightarrow\,e\gamma$, $\tau\,\rightarrow\,\mu\gamma$,
$\mu\,\rightarrow\,e\gamma$, $\mu^{-}\,\rightarrow\,e^{+}e^{-}e^{-}$,
$\tau^{-}\,\rightarrow\,\mu^{+}\mu^{-}\mu^{-}$,
$\tau^{-}\,\rightarrow\,e^{+}e^{-}e^{-}$ and $\mu-e$ conversion rate in the
nuclei. The search for charged lepton flavor violating decays can play a
pivotal role in studying and discovering new physics beyond the standard model
at TeV scale and above.
Table 1: Present Experimental Bounds on LFV Processes Process | Experiment | Present upper bound
---|---|---
BR$(\mu\rightarrow e\,\gamma)$ | MEG [89] | $2.4\times 10^{-12}$
BR$(\mu\rightarrow e\,e\,e)$ | SINDRUM [90] | $1.0\times 10^{-12}$
CR$(\mu\rightarrow e\,{\rm in}\,{\bf Au})$ | SINDRUM-II [91] | $7\times 10^{-13}$
CR$(\mu\rightarrow e\,{\rm in}\,{\bf Ti})$ | SINDRUM-II [92] | $4.3\times 10^{-12}$
BR$(\tau\rightarrow\mu\,\gamma)$ | BABAR/Belle [93] | $4.4\times 10^{-8}$
BR$(\tau\rightarrow e\,\gamma)$ | BABAR/Belle [93] | $3.3\times 10^{-8}$
BR$(\tau\rightarrow\mu\,\mu\,\mu)$ | BABAR/Belle [93] | $2.0\times 10^{-8}$
BR$(\tau\rightarrow e\,e\,e)$ | BABAR/Belle [93] | $2.6\times 10^{-8}$
The amplitudes of these processes are also sensitive to the left-right mixing
in the sleptonic mass matrices in addition to the intergenerational mixing
[94]. In calculating the amplitudes we closely follow the notations and
expressions provided in [14, 15]. The contribution to $\mu-e$ conversion rate
in the nuclei is through penguin diagram where $\gamma$, $Z$ is exchanged and
through box diagrams containing $\tilde{\chi}^{0}-\tilde{l}_{i}-q_{u,d}$ loops
or $\tilde{\chi}^{-}-\tilde{\nu}_{i}-q_{u,d}$ loops. The current experimental
upper bound on various LFV processes is tabulated in Table 1.
In the current version of SuSeFLAV we use the tree level masses and full
$6\times 6$ slepton mixing matrix to calculate LFV observables. The output is
written in the BLOCK SuSeFLAVLOWENERGY in SLHA format.
* 6.
Light Neutral Higgs Boson Mass: Recently, the ATLAS and CMS experiments at the
Large Hadron Collider (LHC) at CERN have reported the discovery of a Higgs
like boson with a statistical significance of more than $5$ sigma. Both the
experiments report the mass of this particle to be close to 125 GeV with the
exact numbers given by :
$\displaystyle m_{h}=125.3\pm 0.4~{}(\text{stat})\pm
0.5~{}(\text{sys})~{}~{}\text{GeV}~{}~{}~{}~{}~{}~{}\text{CMS
\cite[cite]{[\@@bibref{Number}{cmshiggs}{}{}]}}~{}~{}$ $\displaystyle
m_{h}=126.0\pm 0.4~{}(\text{stat})\pm
0.4~{}(\text{sys})~{}~{}\text{GeV}~{}~{}~{}~{}~{}~{}\text{ATLAS
\cite[cite]{[\@@bibref{Number}{atlashiggs}{}{}]}}~{}~{}$
In MSSM, this would correspond to the mass of lightest neutral CP-even Higgs
boson provided it has SM-like couplings [97]. Both these requirements put a
strong constraints on most models of supersymmetry breaking especially the
ones with universal boundary conditions[98]. In the universal class of models,
large values of the trilinear couplings are required to generate a light higgs
mass of that order. Two loop contributions to the higgs mass matrices are
important and cannot be neglected while studying the implications on model
parameter space. As mentioned earlier, SuSeFLAVcomputes the light higgs mass
at the two loop level making it appropriate for analysis of this constraint
over a wide range of supersymmetry breaking models. In particular,
implications on the supersymmetric seesaw models and the complementarity with
flavour processes has already been studied [99] assuming universal and non-
universal higgs boundary conditions. As of now the light higgs mass constraint
is not imposed automatically in the program. Since there are still some
theoretical and experimental uncertainties which are subject to modification
as more data pours in from LHC experiments, the exact range of the light higgs
mass is left to the user.
## 7 Executing SuSeFLAV
Instruction to compile and install the package is provided in README file.
SuSeFLAV uses the Diag package by T. Hanh to diagonalize the mass
matrices101010Note that LAPACK [100] library is a another option for an
earlier version of SuSeFLAV for diagonalizing mass matrices. It is available
on the website for downloading. We do not recommend the old version.. SuSeFLAV
package produces three executables when compiled, namely suseflav,
suseflavslha and suseflavscan in the bin/ sub-directory. To compute the
spectrum for a single point the usage of executables suseflav and suseflavslha
is recommended. The user must modify the corresponding input files sinputs.in
and slha.in where all the input parameters are specified. To scan the
parameter space the usage of the executable suseflavscan is recommended. The
corresponding input parameters are in the input file sinputs_scan.in.
### 7.1 Sample Input/Output
In this section we provide examples of input files which can be used to run
the program. SuSeFLAV has two input/output modes. This can be broadly
classified into SLHA I/O interface and non-SLHA or traditional SuSeFLAV I/O
interface. The directory examples/ contains a variety of input files with
which the user can run different models.
#### 7.1.1 SLHA Interface
The main source file is runslha.f. To use the SLHA interface the user should
execute suseflavslha by modifying the corresponding input file slha.in. Note
that the user must rename the required SLHA file as slha.in to use that
particular file as input. To run the program type the following command at the
terminal
./suseflavslha
This sample input/output corresponds to point 10.1.5 [101] with Type-I see-saw
mechanism with maximal mixing in mSUGRA/CMSSM scenario.
Block MODSEL # Select model
1 1 # sugra
6 1 # Quark Flavor Violation
Block SMINPUTS # Standard Model inputs
1 1.279340000E+02 # alpha^(-1) SM MSbar(MZ)
2 1.166390000E-05 # G_Fermi
3 1.172000000E-01 # alpha_s(MZ) SM MSbar
4 9.118760000E+01 # MZ(pole)
5 4.230000000E+00 # mb(mb) SM MSbar
6 1.727000000E+02 # mtop(pole)
7 1.770000000E+00 # mtau(pole)
Block MINPAR # Input parameters
1 1.750000000E+02 # m0
2 7.000000000E+02 # m12
3 1.000000000E+01 # tanb
4 1.000000000E+00 # sign(mu)
5 0.000000000E+00 # A0
Block SUSEFLAV # Algorithm specific inputs
1 1.000000000E-03 # spectrum tolerance
5 2.000000000E+00 # 2-loop running (1 = 1loop; 2 = 2loop)
3 1.000000000E+00 # print Control (0 = do not print output; 1 = print output)
2 1.000000000E+00 # Right handed neutrino (1 = yes; 0 = no)
4 1.000000000E+00 # RHN mixing : 1 = ckm; 2 =PMNS; 3 = user defined
7 1.000000000E+06 # MR1, Lightest rhn decoupling scale
8 1.000000000E+09 # MR2, second lightest rhn decoupling scale
9 1.000000000E+14 # MR3, Heaviest rhn decoupling scale
10 0.000000000E+00 # Dirac neutrino mixing matrix 1,1
11 0.000000000E+00 # Dirac neutrino mixing matrix 1,2
12 0.000000000E+00 # Dirac neutrino mixing matrix 1,3
13 0.000000000E+00 # Dirac neutrino mixing matrix 2,1
14 0.000000000E+00 # Dirac neutrino mixing matrix 2,2
15 0.000000000E+00 # Dirac neutrino mixing matrix 2,3
16 0.000000000E+00 # Dirac neutrino mixing matrix 3,1
17 0.000000000E+00 # Dirac neutrino mixing matrix 3,2
18 0.000000000E+00 # Dirac neutrino mixing matrix 3,3
The corresponding SLHA output is generated in slha.out
# Spectrum Output in SUSY Les Houches Accord 2
# SuSeFLAV v1.2.0
# D. Chowdhury, R. Garani and S. K. Vempati, hep-ph/1109.3551
# For bug reports or any other queries please send email to suseflav@cts.iisc.ernet.in
# Created on 12.06.2012 at 17:30 Hrs
#
BLOCK SPINFO # Program information
1 SuSeFLAV # Spectrum calculator
2 1.2.0 # Version number
#
BLOCK MODSEL # MODEL NAME
1 1 # mSUGRA
3 4 # mSUGRA + RHN
6 3 # Lepton and quark flavor is violated
#
BLOCK MINPAR # Input parameters
1 1.75000000E+02 # m0
2 7.00000000E+02 # m_1/2
3 1.00000000E+01 # tanbeta(mz)
4 1.00000000E+00 # sign(mu)
5 0.00000000E+00 # A0
#
BLOCK SMINPUTS # Standard Model inputs
1 1.27934000E+02 # alpha_em (M_Z)^MSbar
2 1.16639000E-05 # G_F [GeV^-2]
3 1.17200000E-01 # alpha_S(M_Z)^MSbar
4 9.11876000E+01 # M_Z pole mass
5 4.23000000E+00 # mb(mb)^MSbar
6 1.72700000E+02 # mt pole mass
7 1.77000000E+00 # mtau pole mass
#
BLOCK EXTPAR # Extra Input parameters
0 1.97682880E+16 # Unification Scale
#
BLOCK MASS # Mass Spectrum
# PDG code mass particle
24 8.05095481E+01 # W+
25 1.16435226E+02 # h
35 9.79136838E+02 # H
36 9.78810446E+02 # A
37 9.82557324E+02 # H+
1000001 1.29049183E+03 # ~d_1
1000003 1.33317083E+03 # ~d_2
1000005 1.33782720E+03 # ~d_3
2000001 1.33783952E+03 # ~d_4
2000003 1.39947428E+03 # ~d_5
2000005 1.39948751E+03 # ~d_6
1000002 1.08957331E+03 # ~u_1
1000004 1.32515013E+03 # ~u_2
1000006 1.34374200E+03 # ~u_3
2000002 1.34410204E+03 # ~u_4
2000004 1.39733996E+03 # ~u_5
2000006 1.39738676E+03 # ~u_6
1000011 3.12007210E+02 # ~l_1
1000013 3.18856813E+02 # ~l_2
1000015 3.18880311E+02 # ~l_3
2000011 4.94924639E+02 # ~l_4
2000013 4.96469065E+02 # ~l_5
2000015 4.96470193E+02 # ~l_6
1000012 4.87191917E+02 # ~nu_1
1000014 4.90375557E+02 # ~nu_2
1000016 4.90381909E+02 # ~nu_3
1000021 1.57599127E+03 # ~g
1000022 2.89580143E+02 # ~chi_10
1000023 5.44321259E+02 # ~chi_20
1000025 8.58605742E+02 # ~chi_30
1000035 8.69437471E+02 # ~chi_40
1000024 5.33851586E+02 # ~chi_1+
1000037 8.62050764E+02 # ~chi_2+
#
BLOCK NMIX # Neutralino Mixing Matrix
1 1 9.97666897E-01 # N_11
1 2 -1.03658855E-02 # N_12
1 3 6.20427374E-02 # N_13
1 4 -2.65332008E-02 # N_14
2 1 2.32098394E-02 # N_21
2 2 9.80197789E-01 # N_22
2 3 -1.61409861E-01 # N_23
2 4 1.12340793E-01 # N_24
3 1 2.46412419E-02 # N_31
3 2 -3.55274833E-02 # N_32
3 3 -7.04937074E-01 # N_33
3 4 -7.07950795E-01 # N_34
4 1 -5.92863909E-02 # N_41
4 2 1.94531848E-01 # N_42
4 3 6.87867194E-01 # N_43
4 4 -6.96764815E-01 # N_44
#
BLOCK UMIX # Chargino Mixing Matrix U
1 1 9.74989137E-01 # U_11
1 2 -2.22252520E-01 # U_12
2 1 2.22252520E-01 # U_21
2 2 9.74989137E-01 # U_22
#
BLOCK VMIX # Chargino Mixing Matrix V
1 1 9.88224902E-01 # V_11
1 2 -1.53008313E-01 # V_12
2 1 1.53008313E-01 # V_21
2 2 9.88224902E-01 # V_22
#
BLOCK USQMIX # squark Mixing Matrix
1 1 1.10996359E-03 # USQMIX_11
1 2 -5.26653686E-03 # USQMIX_12
1 3 3.55285163E-01 # USQMIX_13
1 4 8.74704256E-04 # USQMIX_14
1 5 -4.21826454E-03 # USQMIX_15
1 6 9.34732542E-01 # USQMIX_16
2 1 1.12595131E-02 # USQMIX_21
2 2 -5.57113248E-02 # USQMIX_22
2 3 9.23559656E-01 # USQMIX_23
2 4 2.86784533E-02 # USQMIX_24
2 5 -1.38092429E-01 # USQMIX_25
2 6 -3.52015712E-01 # USQMIX_26
3 1 2.97503603E-05 # USQMIX_31
3 2 -9.98459110E-05 # USQMIX_32
3 3 1.03612223E-03 # USQMIX_33
3 4 9.77509497E-01 # USQMIX_34
3 5 2.10888527E-01 # USQMIX_35
3 6 -3.57456096E-04 # USQMIX_36
4 1 2.18713035E-03 # USQMIX_41
4 2 2.22241357E-03 # USQMIX_42
4 3 1.33658497E-01 # USQMIX_43
4 4 -2.08919254E-01 # USQMIX_44
4 5 9.67647238E-01 # USQMIX_45
4 6 -4.62304107E-02 # USQMIX_46
5 1 9.82212602E-01 # USQMIX_51
5 2 1.87753700E-01 # USQMIX_52
5 3 -9.34002684E-04 # USQMIX_53
5 4 5.08408180E-04 # USQMIX_54
5 5 -2.40125319E-03 # USQMIX_55
5 6 2.35206342E-04 # USQMIX_56
6 1 -1.87418278E-01 # USQMIX_61
6 2 9.80618229E-01 # USQMIX_62
6 3 5.42538107E-02 # USQMIX_63
6 4 2.10966003E-03 # USQMIX_64
6 5 -9.57981337E-03 # USQMIX_65
6 6 -1.49190697E-02 # USQMIX_66
#
BLOCK DSQMIX # squark Mixing Matrix
1 1 7.97027235E-03 # DSQMIX_11
1 2 -3.83864371E-02 # DSQMIX_12
1 3 9.76209913E-01 # DSQMIX_13
1 4 -1.04373101E-05 # DSQMIX_14
1 5 -2.41872348E-04 # DSQMIX_15
1 6 2.13253615E-01 # DSQMIX_16
2 1 -2.81490374E-03 # DSQMIX_21
2 2 1.35643992E-02 # DSQMIX_22
2 3 -2.12865830E-01 # DSQMIX_23
2 4 1.79277418E-05 # DSQMIX_24
2 5 9.89907963E-04 # DSQMIX_25
2 6 9.76982723E-01 # DSQMIX_26
3 1 6.10926291E-06 # DSQMIX_31
3 2 5.89431513E-03 # DSQMIX_32
3 3 6.86044622E-04 # DSQMIX_33
3 4 -2.59132138E-05 # DSQMIX_34
3 5 9.99981946E-01 # DSQMIX_35
3 6 -9.45553891E-04 # DSQMIX_36
4 1 2.96437790E-04 # DSQMIX_41
4 2 -7.98684964E-07 # DSQMIX_42
4 3 1.15232653E-05 # DSQMIX_43
4 4 9.99999956E-01 # DSQMIX_44
4 5 2.58944876E-05 # DSQMIX_45
4 6 -1.50004569E-05 # DSQMIX_46
5 1 -1.75678721E-01 # DSQMIX_51
5 2 9.83552600E-01 # DSQMIX_52
5 3 4.12376560E-02 # DSQMIX_53
5 4 5.24615573E-05 # DSQMIX_54
5 5 -5.82957991E-03 # DSQMIX_55
5 6 -5.17098306E-03 # DSQMIX_56
6 1 9.84411220E-01 # DSQMIX_61
6 2 1.75875037E-01 # DSQMIX_62
6 3 -1.15326259E-03 # DSQMIX_63
6 4 -2.91633802E-04 # DSQMIX_64
6 5 -1.04177570E-03 # DSQMIX_65
6 6 1.44250146E-04 # DSQMIX_66
#
BLOCK SELMIX # Slepton Mixing Matrix
1 1 1.05172917E-05 # SELMIX_11
1 2 -5.07243990E-05 # SELMIX_12
1 3 1.03659288E-01 # SELMIX_13
1 4 6.00795431E-08 # SELMIX_14
1 5 -1.11136067E-05 # SELMIX_15
1 6 9.94612864E-01 # SELMIX_16
2 1 -3.08963419E-08 # SELMIX_21
2 2 6.24290695E-03 # SELMIX_22
2 3 -2.02732368E-06 # SELMIX_23
2 4 -6.78386431E-09 # SELMIX_24
2 5 9.99980513E-01 # SELMIX_25
2 6 1.17032561E-05 # SELMIX_26
3 1 2.98889602E-05 # SELMIX_31
3 2 -6.33346635E-11 # SELMIX_32
3 3 1.18627921E-08 # SELMIX_33
3 4 1.00000000E+00 # SELMIX_34
3 5 6.78606456E-09 # SELMIX_35
3 6 -6.19572827E-08 # SELMIX_36
4 1 9.84024127E-03 # SELMIX_41
4 2 -4.73545901E-02 # SELMIX_42
4 3 9.93448538E-01 # SELMIX_43
4 4 -3.12319765E-07 # SELMIX_44
4 5 2.98862230E-04 # SELMIX_45
4 6 -1.03540457E-01 # SELMIX_46
5 1 9.99950259E-01 # SELMIX_51
5 2 2.09161645E-03 # SELMIX_52
5 3 -9.70065714E-03 # SELMIX_53
5 4 -2.98872962E-05 # SELMIX_54
5 5 -1.30585027E-05 # SELMIX_55
5 6 1.00054245E-03 # SELMIX_56
6 1 -1.62739211E-03 # SELMIX_61
6 2 9.98856442E-01 # SELMIX_62
6 3 4.71237978E-02 # SELMIX_63
6 4 4.78864799E-08 # SELMIX_64
6 5 -6.23573698E-03 # SELMIX_65
6 6 -4.86038870E-03 # SELMIX_66
#
BLOCK SNUMIX # Sneutrino Mixing Matrix
1 1 4.85609162E-03 # SNUMIX_11
1 2 -2.34212181E-02 # SNUMIX_12
1 3 9.99713892E-01 # SNUMIX_13
2 1 -5.74301350E-02 # SNUMIX_21
2 2 9.98069089E-01 # SNUMIX_22
2 3 2.36616496E-02 # SNUMIX_23
3 1 9.98337717E-01 # SNUMIX_31
3 2 5.75286069E-02 # SNUMIX_32
3 3 -3.50163123E-03 # SNUMIX_33
#
BLOCK ALPHA # Higgs mixing
-1.05473355E-01 # Mixing angle in the neutral Higgs boson sector
#
BLOCK HMIX Q= 1.20195104E+03 # DRbar Higgs Parameters
1 8.50892351E+02 # mu(Q)
2 9.63860969E+00 # tanbeta(Q)
3 2.43818892E+02 # vev(Q)
4 9.93075833E+05 # MA^2(Q)
#
BLOCK GAUGE Q= 1.20195104E+03 # The gauge couplings
1 3.62957251E-01 # gprime(Q) DRbar
2 6.42610014E-01 # g(Q) DRbar
3 1.04880243E+00 # g_3(Q) DRbar
#
BLOCK TU Q= 1.20195104E+03 # The trilinear couplings
1 1 -1.26688884E-01 # TURG_11
1 2 -2.93067837E-02 # TURG_12
1 3 -3.97143909E-04 # TURG_13
2 1 2.93036897E+01 # TURG_21
2 2 -1.26686864E+02 # TURG_22
2 3 -5.37734251E+00 # TURG_23
3 1 -1.11137936E+02 # TURG_31
3 2 5.35228445E+02 # TURG_32
3 3 -1.29451330E+04 # TURG_33
#
BLOCK TD Q= 1.20195104E+03 # The trilinear couplings
1 1 -6.05492867E+00 # TDRG_11
1 2 -1.02056677E-04 # TDRG_12
1 3 2.38753240E-03 # TDRG_13
2 1 -2.04113446E-03 # TDRG_21
2 2 -1.21088577E+02 # TDRG_22
2 3 -2.29965104E-01 # TDRG_23
3 1 1.22702682E+00 # TDRG_31
3 2 -5.90930668E+00 # TDRG_32
3 3 -2.95665819E+03 # TDRG_33
#
BLOCK TE Q= 1.20195104E+03 # The trilinear couplings
1 1 -1.44939639E-01 # TERG_11
1 2 -2.31067898E-08 # TERG_12
1 3 -4.08400006E-07 # TERG_13
2 1 -4.82631701E-06 # TERG_21
2 2 -3.02751406E+01 # TERG_22
2 3 4.10638455E-04 # TERG_23
3 1 -1.53360878E-03 # TERG_31
3 2 7.38111171E-03 # TERG_32
3 3 -5.16767721E+02 # TERG_33
#
BLOCK YU Q= 1.20195104E+03 # The top Yukawa coupling
1 1 6.44039888E-06 # YU_11
1 2 1.48968552E-06 # YU_12
1 3 2.41182558E-08 # YU_13
2 1 -1.48971566E-03 # YU_21
2 2 6.44041987E-03 # YU_22
2 3 2.72567759E-04 # YU_23
3 1 7.22768067E-03 # YU_31
3 2 -3.48078574E-02 # YU_32
3 3 8.43790071E-01 # YU_33
#
BLOCK YD Q= 1.20195104E+03 # The down Yukawa coupling
1 1 2.53400760E-04 # YD_11
1 2 -1.10093360E-09 # YD_12
1 3 2.56831364E-08 # YD_13
2 1 -2.20186781E-08 # YD_21
2 2 5.06812345E-03 # YD_22
2 3 -2.47377375E-06 # YD_23
3 1 1.31978305E-05 # YD_31
3 2 -6.35600675E-05 # YD_32
3 3 1.31856452E-01 # YD_33
#
BLOCK YE Q= 1.20195104E+03 # The tau Yukawa coupling
1 1 2.77703519E-05 # YE_11
1 2 -1.17975032E-11 # YE_12
1 3 4.03483137E-10 # YE_13
2 1 -2.46439380E-09 # YE_21
2 2 5.80081939E-03 # YE_22
2 3 -4.05872732E-07 # YE_23
3 1 1.45732796E-06 # YE_31
3 2 -7.01783388E-06 # YE_32
3 3 9.96765043E-02 # YE_33
#
BLOCK MSOFT Q= 1.20195104E+03 # soft SUSY breaking masses at scale Q
1 2.95504689E+02 # M_1
2 5.45936065E+02 # M_2
3 1.53262491E+03 # M_3
21 2.14879878E+05 # M^2_Hd
22 -6.96254426E+05 # M^2_Hu
31 4.94404728E+02 # M_eL
32 4.94396817E+02 # M_muL
33 4.91242768E+02 # M_tauL
34 3.15861187E+02 # M_eR
35 3.15846380E+02 # M_muR
36 3.11472426E+02 # M_tauR
41 1.39829222E+03 # M_q1L
42 1.39811849E+03 # M_q2L
43 1.29136113E+03 # M_q3L
44 1.34421791E+03 # M_uR
45 1.34420302E+03 # M_cR
46 1.11340022E+03 # M_tR
47 1.33760077E+03 # M_dR
48 1.33759064E+03 # M_sR
49 1.33101944E+03 # M_bR
#
BLOCK MSQ2 Q= 1.20195104E+03 # M^2_Q soft SUSY breaking masses
1 1 1.95522114E+06 # mSQRG_11
1 2 1.03013845E+02 # mSQRG_12
1 3 -2.38957910E+03 # mSQRG_13
2 1 1.03013845E+02 # mSQRG_21
2 2 1.95473530E+06 # mSQRG_22
2 3 1.15080756E+04 # mSQRG_23
3 1 -2.38957910E+03 # mSQRG_31
3 2 1.15080756E+04 # mSQRG_32
3 3 1.66761356E+06 # mSQRG_33
#
BLOCK MSU2 Q= 1.20195104E+03 # M^2_U soft SUSY breaking masses
1 1 1.80692178E+06 # mSURG_11
1 2 -5.84935042E-06 # mSURG_12
1 3 -1.19615520E-02 # mSURG_13
2 1 -5.84935042E-06 # mSURG_21
2 2 1.80688177E+06 # mSURG_22
2 3 4.19949204E+00 # mSURG_23
3 1 -1.19615520E-02 # mSURG_31
3 2 4.19949204E+00 # mSURG_32
3 3 1.23966005E+06 # mSURG_33
#
BLOCK MSD2 Q= 1.20195104E+03 # M^2_D soft SUSY breaking masses
1 1 1.78917582E+06 # mSDRG_11
1 2 -9.47987536E-06 # mSDRG_12
1 3 5.70618349E-03 # mSDRG_13
2 1 -9.47987536E-06 # mSDRG_21
2 2 1.78914873E+06 # mSDRG_22
2 3 -5.49615942E-01 # mSDRG_23
3 1 5.70618349E-03 # mSDRG_31
3 2 -5.49615942E-01 # mSDRG_32
3 3 1.77161275E+06 # mSDRG_33
#
BLOCK MSL2 Q= 1.20195104E+03 # M^2_L soft SUSY breaking masses
1 1 2.44436035E+05 # mSLRG_11
1 2 7.11764708E-01 # mSLRG_12
1 3 -1.51306954E+01 # mSLRG_13
2 1 7.11764708E-01 # mSLRG_21
2 2 2.44428213E+05 # mSLRG_22
2 3 7.28699498E+01 # mSLRG_23
3 1 -1.51306954E+01 # mSLRG_31
3 2 7.28699498E+01 # mSLRG_32
3 3 2.41319457E+05 # mSLRG_33
#
BLOCK MSE2 Q= 1.20195104E+03 # M^2_E soft SUSY breaking masses
1 1 9.97682891E+04 # mSERG_11
1 2 6.08801261E-09 # mSERG_12
1 3 1.01423906E-05 # mSERG_13
2 1 6.08801261E-09 # mSERG_21
2 2 9.97589360E+04 # mSERG_22
2 3 1.15378509E-03 # mSERG_23
3 1 1.01423906E-05 # mSERG_31
3 2 1.15378509E-03 # mSERG_32
3 3 9.70150720E+04 # mSERG_33
#
BLOCK SuSeFLAVLOWENERGY # PARAMETERS
1 3.39444376E-03 # Delta rho parameter
2 5.11182018E-10 # g_mu - 2
3 3.73074715E-04 # Br(b -> s gamma)
4 1.70603079E-13 # Br(tau -> mu gamma)
5 7.32782350E-15 # Br(tau -> e gamma)
6 9.20566951E-17 # Br(mu -> e gamma)
7 5.43318338E-16 # Br(tau -> mu mu mu)
8 8.82689326E-17 # Br(tau -> e e e)
9 6.80189300E-19 # Br(mu -> e e e)
#
BLOCK FINETUNE #
1 1.75230800E+02 # delta mZ^2/mZ^2 (mu^2)
2 8.76436259E+01 # delta mt/mt (mu^2)
The program prints BLOCK SELMIX in the SLHA output, which contains the
elements of the tree level $6\times 6$ slepton mixing matrix. The notation and
the method of diagonalization is explained in more detail in the A.
#### 7.1.2 Non-SLHA Interface
The source file for non-SLHA input is runonce.f. To use the traditional
SuSeFLAV interface the user should execute suseflav by modifying the
corresponding input file. This executable takes the following input files,
sinputs.in for mSUGRA/CMSSM models, sinputs-gmsb.in for GMSB models, sinputs-
nuhm.in for non-universal higgs model (NUHM2) models and sinputs-cnum.in for
complete non-universal models. For example, to run the program with complete
non-universal boundary conditions type the following command in the terminal
./suseflav <sinputs-cnum.in
The sample input/output below corresponds to 10.4.4 of [101] with Type-I see-
saw mechanism with minimal mixing in the context of mSUGRA/CMSSM scenario. To
execute the program for the sample point type the following command in the
terminal
./suseflav <sinputs.in
mSUG # MODEL name
1 # prinstat (1 for printing output on terminal, 0 otherwise)
1.00000E+01 # tanbeta
5.00000E+02 # m0
0.00000E+00 # a0
1.05000E+03 # M1/2
1.00000E+00 # sgn(mu)
1.16639E-05 # G_fermi
1.27934E+02 # alpha_em^(-1)
1.17200E-01 # alpha_strong
9.11876E+01 # mz_pole
2 # one loop or two loops
1 # quark mixing
1 # 1= rhn on; 0 = rhn off
CKM # case: CKM/MNS/USD
1.00000E-03 # spectrum tolerance
1.72700E+02 # Mtpole
4.23000E+00 # Mbpole
1.77000E+00 # Mtaupole
1.00000E+06 # MR1
1.00000E+09 # MR2
1.00000E+14 # MR3
0.00000E+00 # Ynu(1,1)
0.00000E+00 # Ynu(1,2)
0.00000E+00 # Ynu(1,3)
0.00000E+00 # Ynu(2,1)
0.00000E+00 # Ynu(2,2)
0.00000E+00 # Ynu(2,3)
0.00000E+00 # Ynu(3,1)
0.00000E+00 # Ynu(3,2)
0.00000E+00 # Ynu(3,3)
------------------------------------------------
The Output in SuSeFLAV format is contained in the file suseflav.out.
******** Begin Program SuSeFLAV ************
model = mSUG
loop = 2
tanbeta = 1.0000E+01
m0 = 5.0000E+02
a0 = 0.0000E+00
M12 = 1.0500E+03
m10 = 5.0000E+02
m20 = 5.0000E+02
sign mu = 1.0000E+00
qmix = 1
top pole mass = 1.7270E+02
rhn = 1
case = CKM
MR1 = 1.0000E+06
MR2 = 1.0000E+09
MR3 = 1.0000E+14
********************************************
up-type yukawa at high energy:
2.6401E-06 6.1056E-07 1.2205E-08
-6.1068E-04 2.6401E-03 1.1119E-04
3.8672E-03 -1.8624E-02 4.5212E-01
down-type yukawa at high energy:
7.9987E-05 -3.1319E-09 7.4082E-08
-6.2638E-08 1.6000E-03 -7.1355E-06
3.7936E-05 -1.8270E-04 4.5575E-02
lepton-type yukawa at high energy:
1.7539E-05 -1.3061E-10 3.1014E-09
-2.7282E-08 3.6638E-03 -3.1200E-06
1.1175E-05 -5.3817E-05 6.4558E-02
neutrino yukawa at high energy:
3.1809E-06 7.3564E-07 1.4835E-08
-7.3579E-04 3.1810E-03 1.3396E-04
4.7106E-03 -2.2686E-02 5.5081E-01
up-type yukawa at msusy :
6.3637E-06 1.4719E-06 2.3911E-08
-1.4720E-03 6.3637E-03 2.6931E-04
7.1199E-03 -3.4289E-02 8.3124E-01
down-type yukawa at msusy :
2.4935E-04 -1.2210E-09 2.8477E-08
-2.4420E-08 4.9870E-03 -2.7429E-06
1.4571E-05 -7.0171E-05 1.2939E-01
lepton-type yukawa at msusy :
2.7626E-05 -1.0563E-11 3.7615E-10
-2.2066E-09 5.7706E-03 -3.7838E-07
1.3609E-06 -6.5535E-06 9.9328E-02
Spectrum at msusy, q = 1.7679E+03
alpha_1 = 3.6366E-01
alpha_2 = 6.4050E-01
alpha_3 = 1.0297E+00
vev1 = 2.5253E+01
vev2 = 2.4230E+02
newtbeta = 9.5948E+00
\mu = 1.2105E+03
~gluino = 2.3009E+03
~stop_1 = 1.6702E+03
~stop_2 = 1.9889E+03
~scharm_R = 2.0550E+03
~scharm_L = 2.1380E+03
~sup_R = 2.0550E+03
~sup_L = 2.1383E+03
~sbottom_1 = 1.9606E+03
~sbottom_2 = 2.0368E+03
~sstrange_R = 2.0460E+03
~sstrange_L = 2.1394E+03
~sdown_R = 2.0461E+03
~sdown_L = 2.1396E+03
~stau_1 = 6.3321E+02
~stau_2 = 8.5289E+02
~smu_R = 6.4202E+02
~smu_L = 8.6003E+02
~sel_R = 6.4206E+02
~sel_L = 8.6012E+02
~tausnu = 8.4727E+02
~musnu = 8.5526E+02
~elsnu = 8.5520E+02
Higgs Spectrum
mA0 = 1.4726E+03
mh_charged = 1.4754E+03
mh0 = 1.1866E+02
mH = 1.4730E+03
\alpha_h = -1.0481E-01
Neutralino spectrum
N1 = 4.4421E+02
N2 = 8.3081E+02
N3 = 1.2193E+03
N4 = 1.2280E+03
Chargino spectrum
C1 = 8.1458E+02
C2 = 1.2179E+03
Low Energy Observables
Fine tuning, Cmz^2mu^2 = 3.5356E+02
Fine tuning, Cmt^2mu^2 = 1.7680E+02
Br(B => s,gamma) = 3.8430E-04
Br(mu => e,gamma) = 3.1044E-06 X 10^(-11)
Br(tau => mu,gamma) = 5.9654E-06 X 10^(-08)
Br(tau => e,gamma) = 2.5629E-08 X 10^(-07)
Br(tau => e,e,e) = 3.0580E-10 X 10^(-07)
Br(tau => mu,mu,mu) = 1.8321E-09 X 10^(-07)
Br(mu => e,e,e) = 2.2577E-07 X 10^(-12)
Br(mu => e in Ti) = 2.0425E-06 X 10^(-12)
(g_mu - 2) = 1.8491E+00 X 10^(-10)
-----------------------------
flag for the point is AOK
--------------------------------------------------
FLAGS AND THEIR MEANINGS
AOK = Everything is fine.
BMUNEG = B_mu is negative at Msusy.
REWSB = |\mu|^2 < 0 at Msusy.
MUNOC = Non-convergent |\mu| at Msusy.
SW2NOC = Non-convergent Sin^2_thetaw at Mz.
TACSPEC = Spectrum is tachyonic at Msusy.
NPERTYUK = Non-perturbative yukawa.
TACSPECMZ = Spectrum is tachyonic at Mz.
FSNC = Final spectrum non-convergent.
TACMh = Lightest CP-even neutral higgs tachyonic.
TACMH = Heaviest CP-even neutral higgs tachyonic.
TACMA = CP-odd neutral higgs tachyonic.
TACMHCH = Charged higgs tachyonic.
VARUNDER = Stepsize is zero while integrating the RGEs.
TACSUP = SUP sector tachyonic.
TACSDN = SDOWN sector tachyonic.
TACSLP = SLEPTON sector tachyonic.
TACSNU = SNEUTRINO sector tachyonic.
LEPH = Lightest higgs mass below LEP limit.
LEPC = Lightest chargino mass < 103.5 GeV.
LSPSTAU = Lightest stau is LSP.
--------------------------------------------------
## 8 Outlook
The MEG experiment has recently reported the latest limits on the rare
leptonic decay $\mu\to e+\gamma$ [89]. Simultaneously, there are results from
LHC as well as direct detection dark matter experiments. SuSeFLAV is designed
to compute supersymmetric spectrum with full flavor violation and further to
compute most of the leptonic flavor violating observables. The program can be
coupled to Dark Matter routines such as MicrOMEGAs [102], DarkSUSY [103] and
SuperIso [31] to compute the relic density and direct detection rates. The
program is free under the GNU Public License and can be downloaded from the
following websites:
* 1.
http://cts.iisc.ernet.in/Suseflav/main.html
* 2.
http://projects.hepforge.org/suseflav/
In the present version of the program only Type-I seesaw mechanism is
implemented. Future versions will include other seesaw mechanisms as well as
other improvements like inclusion new flavor violating decays rates in the
quark sector.
## Acknowledgments
We appreciate discussions with and acknowledge suggestions from L. Calibbi, S.
Kraml, W. Porod and O. Vives. We thank D. Grellscheid, T. Hahn and P. Slavich
for important inputs. We thank U. Chattopadhyaya, Aseshkrishna Dutta and M.
Guchait for encouragement. SKV thanks M. Ciuchini, A. Faccia, A. Masiero, P.
Paradisi and L. Silvestrini for collaborations during which parts of this
program were written. We thank Ketan Patel, Brandon Murakami for reporting
bugs. SKV acknowledges support from DST project “Complementarity between
direct and indirect searches for Supersymmetry” and also support from DST
Ramanujan Fellowship SR/S2/RJN-25/2008. RG acknowledges support from
SR/S2/RJN-25/2008. DC acknowledges partial support from SR/S2/RJN-25/2008.
## Appendix A Tree-Level Masses
Here we suppress any gauge indices and follow the notation of BPMZ111111Except
in our case the sign of $\mu$ parameter is opposite of BPMZ. [70] closely. The
Lagrangian contains the neutralino mass matrix as
${-\frac{1}{2}{\tilde{\psi}^{0\,T}}{\cal
M}_{\tilde{\psi}^{0}}\tilde{\psi}^{0}+h.c.\,,}$ (52)
where $\tilde{\psi}^{0}=$ $(\widetilde{B},$ $\widetilde{W}^{0},$
$\widetilde{H}_{d},$ $\widetilde{H}_{u})^{T}$ and
${\cal M}_{\tilde{\psi}^{0}}\ =\
\left(\begin{array}[]{cccc}M_{1}&0&-M_{Z}c_{\beta}s_{W}&M_{Z}s_{\beta}s_{W}\\\
0&M_{2}&M_{Z}c_{\beta}c_{W}&-M_{Z}s_{\beta}c_{W}\\\
-M_{Z}c_{\beta}s_{W}&M_{Z}c_{\beta}c_{W}&0&-\mu\\\
M_{Z}s_{\beta}s_{W}&-M_{Z}s_{\beta}c_{W}&-\mu&0\end{array}\right).$ (53)
We use the letter $s$ and $c$ for sine and cosine, so that
$s_{\beta}\equiv\sin\beta,\ c_{\beta}\equiv\cos\beta$ and $s_{W}(c_{W})$ is
the sine (cosine) of the weak mixing angle. The $4\times 4$ neutralino mixing
matrix is an orthogonal matrix $O$ with real entries, such that $O^{T}{\cal
M}_{\widetilde{\psi}^{0}}O$ is diagonal. The neutralinos $\chi^{0}_{i}$ are
defined such that their absolute masses increase with increasing $i$. Some of
their mass values can be negative.
We make the identification ${\widetilde{W}}^{\pm}=({\widetilde{W}^{1}}\mp
i{\widetilde{W}^{2}})/\sqrt{2}$ for the charged winos and
${\widetilde{H}_{u}^{+}},{\widetilde{H}_{d}^{-}}$ for the charged higgsinos.
The Lagrangian contains the chargino mass matrix as
$-\frac{1}{2}{\widetilde{\psi}^{-\,T}}{\cal
M}_{\widetilde{\psi}^{+}}\widetilde{\psi}^{+}+h.c.,$ (54)
where $\widetilde{\psi}^{+}=(\widetilde{W}^{+},\ \widetilde{H}_{u}^{+})^{T},\
\widetilde{\psi}^{-}=(\widetilde{W}^{-},\ \widetilde{H}_{d}^{-})^{T}$ and
${\cal M}_{\tilde{\psi}^{+}}\ =\
\left(\begin{array}[]{cc}M_{2}&\sqrt{2}\,M_{W}s_{\beta}\\\
\sqrt{2}\,M_{W}c_{\beta}&\mu\end{array}\right).$ (55)
The chargino masses are found by acting on the matrix ${\cal
M}_{\tilde{\psi}^{+}}$ with a bi-unitary transformation, so that $U^{*}{\cal
M}_{\tilde{\psi}^{+}}V^{\dagger}$ is a diagonal matrix containing the two
chargino mass eigenvalues, $m_{\tilde{\chi}_{i}^{+}}$. The matrices $U$ and
$V$ are easily found, as they diagonalize, respectively, the matrices ${\cal
M}_{\tilde{\psi}^{+}}^{*}{\cal M}_{\tilde{\psi}^{+}}^{\rm T}$ and ${\cal
M}_{\tilde{\psi}^{+}}^{\dagger}{\cal M}_{\tilde{\psi}^{+}}$. And we have taken
the $U$ and $V$ matrices to be real.
The tree-level squark and slepton mass squared values for the family $i$ are
found by diagonalizing the following mass matrices
${\mathcal{M}}_{\tilde{f}}^{2}$ defined in the
$({\tilde{f}}_{iL},{\tilde{f}}_{iR})^{T}$ basis:
$\left(\begin{array}[]{cc}(m_{\tilde{Q}}^{2})_{ii}+m_{u_{i}}^{2}+\left(\frac{1}{2}-\frac{2}{3}s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}&m_{u_{i}}\left(({\tilde{A}}_{\tilde{u}})_{ii}-\mu\cot\beta\right)\\\
m_{u_{i}}\left(({\tilde{A}}_{\tilde{u}})_{ii}-\mu\cot\beta\right)&(m^{2}_{\tilde{u}})_{ii}+m_{u_{i}}^{2}+\frac{2}{3}s_{W}^{2}M_{Z}^{2}c_{2\beta}\end{array}\right)\
,$ (56)
$\left(\begin{array}[]{cc}(m_{\tilde{Q}}^{2})_{ii}+m_{d_{i}}^{2}-\left(\frac{1}{2}-\frac{1}{3}s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}&m_{d_{i}}\left(({\tilde{A}}_{\tilde{d}})_{ii}-\mu\tan\beta\right)\\\
m_{d_{i}}\left(({\tilde{A}}_{\tilde{d}})_{ii}-\mu\tan\beta\right)&(m^{2}_{\tilde{d}})_{ii}+m_{d_{i}}^{2}-\frac{1}{3}s_{W}^{2}M_{Z}^{2}c_{2\beta}\end{array}\right)\
,$ (57)
$\Biggl{(}\begin{array}[]{cc}(m_{\tilde{L}}^{2})_{ii}+m_{e_{i}}^{2}-\left(\frac{1}{2}-s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}&m_{e_{i}}\left(({\tilde{A}}_{\tilde{e}})_{ii}-\mu\tan\beta\right)\\\
m_{e_{i}}\left(({\tilde{A}}_{\tilde{e}})_{ii}-\mu\tan\beta\right)&(m^{2}_{\tilde{e}})_{ii}+m_{e_{i}}^{2}-s_{W}^{2}M_{Z}^{2}c_{2\beta}\end{array}\Biggr{)}\
,$ (58)
where, $m_{f},e_{f}$ are the mass and electric charge of fermion $f$
respectively. The mixing of the first two families is suppressed by a small
fermion mass, which we approximate to zero. The sfermion mass eigenstates are
given by
$\left(\begin{array}[]{cc}m^{2}_{\tilde{f}_{1}}&0\\\
0&m^{2}_{\tilde{f}_{2}}\end{array}\right)=\left(\begin{array}[]{cc}c_{f}&s_{f}\\\
-s_{f}&c_{f}\end{array}\right){\mathcal{M}}_{\tilde{f}}^{2}\left(\begin{array}[]{cc}c_{f}&-s_{f}\\\
s_{f}&c_{f}\end{array}\right)$ (59)
where $c_{f}$ is the cosine of the sfermion mixing angle, $\cos\theta_{f}$,
and $s_{f}$ is $\sin\theta_{f}$. These angles are given by
$\displaystyle\tan(2\theta_{u})$ $\displaystyle=$
$\displaystyle\frac{2\,m_{u}\left({\tilde{A}}_{\tilde{u}}-\mu\cot\beta\right)}{m_{\tilde{Q}}^{2}-m_{\tilde{u}}^{2}+\left({\frac{1}{2}}-2\,e_{u}\,s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}}\
,$ (60) $\displaystyle\tan(2\theta_{d})$ $\displaystyle=$
$\displaystyle\frac{2\,m_{d}\left({\tilde{A}}_{\tilde{d}}-\mu\tan\beta\right)}{m_{\tilde{Q}}^{2}-m_{\tilde{d}}^{2}+\left(-{\frac{1}{2}}-2\,e_{d}\,s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}}\
.$ (61)
To calculate the lepton flavor violating decays, we diagonalize the full
$6\times 6$ sleptonic mass matrix (${\cal M}^{2}_{\tilde{l}}$) by $U^{T}\
{\cal M}^{2}_{\tilde{l}}\ U$, where $U$ is the sleptonic mixing matrix with
real entries. In the gauge basis of
$\left\\{\tilde{e}_{L},\tilde{\mu}_{L},\tilde{\tau}_{L},\tilde{e}_{R},\tilde{\mu}_{R},\tilde{\tau}_{R}\right\\}^{T}$
the sleptonic mass matrix is defined as
${\cal M}^{2}_{\tilde{l}}\ =\ \begin{pmatrix}{\bf
m}_{\tilde{L}}^{2}+\left[m_{e_{i}}^{2}-\left(\frac{1}{2}-s_{W}^{2}\right)M_{Z}^{2}c_{2\beta}\right]{\bf
1}&m_{e_{i}}\left({\tilde{\bf A}}_{\tilde{e}}-\mu\tan\beta\;{\bf 1}\right)\\\
m_{e_{i}}\left({\tilde{\bf A}}^{T}_{\tilde{e}}-\mu\tan\beta\ {\bf
1}\right)&{\bf
m}^{2}_{\tilde{e}}+\left[m_{e_{i}}^{2}-s_{W}^{2}M_{Z}^{2}c_{2\beta}\right]{\bf
1}\end{pmatrix}\ ,$ (62)
where ${\bf m}_{\tilde{L}}^{2}$, ${\bf m}^{2}_{\tilde{e}}$ and ${\tilde{\bf
A}}_{\tilde{e}}$ are the scalar mass matrices and tri-linear coupling matrix
respectively. Using the full $6\times 6$ sleptonic mixing matrix we define the
lepton flavor violating couplings following Hisano et al. [14]. With these
couplings we calculate the rare lepton flavor violating decays as described in
section 6.
Given values for $\tan\beta$ one can write the CP-odd Higgs-boson mass,
$m_{A}$, and the other Higgs masses at tree level by
$m_{A}^{2}=\frac{2B_{\mu}}{\sin
2\beta}=2|\mu|^{2}+m^{2}_{H_{u}}+m^{2}_{H_{d}}\ ,$ (63) $m_{H,h}^{2}\ =\
\frac{1}{2}\Biggl{(}m_{A}^{2}+M_{Z}^{2}\pm\sqrt{\left(m_{A}^{2}+M_{Z}^{2}\right)^{2}-4m_{A}^{2}M_{Z}^{2}c_{2\beta}^{2}}\Biggr{)}\
,$ (64)
and
$m_{H^{\pm}}^{2}\ =\ m_{A}^{2}\ +\ M_{W}^{2}\ .$ (65)
The CP-even gauge eigenstates $(H^{0}_{d},\,H^{0}_{u})$ are rotated by the
angle $\alpha$ into the mass eigenstates $(H,\,h)$ as follows,
$\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}H\\\\[5.69054pt]
h\end{array}\right)\ =\
\left(\begin{array}[]{cc}c_{\alpha}&s_{\alpha}\\\\[5.69054pt]
-s_{\alpha}&c_{\alpha}\end{array}\right)\left(\begin{array}[]{c}\mathfrak{Re}H^{0}_{d}\\\\[5.69054pt]
\mathfrak{Re}H^{0}_{u}\end{array}\right)~{}.$ (66)
At tree level, the angle $\alpha$ is given by
$\tan 2\alpha\ =\ \frac{m_{A}^{2}+M_{Z}^{2}}{m_{A}^{2}-M_{Z}^{2}}\tan 2\beta\
.$ (67)
## Appendix B One Loop Threshold Corrections
We compute flavor conserving complete one-loop corrections to masses and
couplings in MSSM following BPMZ [70]. In this appendix we present the summary
of one-loop corrections implemented in SuSeFLAV.
* 1.
Quarks and Leptons
The corrections to fermions are evaluated at the weak scale. The running
masses $m_{\hat{f}}$ are related to the corresponding $\overline{DR}$ masses
by the expression
$m_{f}=m_{\hat{f}}-\Sigma_{f}^{BPMZ}(m^{2}_{f})$ (68)
Where, $\Sigma_{f}^{BPMZ}(m^{2}_{f})$ is the one loop self-energy of the
fermion $f$. We follow equation D.18 of BPMZ [70]. At the present version of
SuSeFLAV we add the correction only to the third generation fermions.
* 2.
W and Z Bosons
Corrections to W and Z bosons are evaluated at the weak scale as well as EWSB
scale. The $\overline{DR}$ running W and Z boson mass squared at the scale $Q$
($\hat{M}_{Z}^{2}(Q)$ and $\hat{M}_{W}^{2}(Q)$) are related to the physical
pole mass of gauge bosons as follows
$\displaystyle M_{Z}^{2}$ $\displaystyle=$ $\displaystyle\hat{M}_{Z}^{2}(Q)\
-\ \Pi^{T}_{ZZ}(M_{Z}^{2})$ (69) $\displaystyle M_{W}^{2}$ $\displaystyle=$
$\displaystyle\hat{M}_{W}^{2}(Q)\ -\ \Pi^{T}_{WW}(M_{W}^{2})$ (70)
Where, $\Pi^{T}_{ZZ}(M_{Z}^{2})$ and $\Pi^{T}_{WW}(M_{W}^{2})$ are the
transverse part of self energy terms. Consult appendix D of BPMZ [70] for
detailed discussion and the complete expression of the self energy terms.
* 3.
sQuarks and sLeptons
Flavor conserving one-loop masses and mixings for squarks and sleptons are
evaluated by diagonalizing ${\cal M}_{\tilde{f}}^{2}(p^{2})$ for eigenvalues
at the EWSB scale.
${\cal M}_{\tilde{f}}^{2}(p^{2})\ =\
\left(\begin{array}[]{cc}M_{\tilde{f}_{L}\tilde{f}_{L}}^{2}-\Pi_{\tilde{f}_{L}\tilde{f}_{L}}(p^{2})&\qquad
M_{\tilde{f}_{L}\tilde{f}_{R}}^{2}-\Pi_{\tilde{f}_{L}\tilde{f}_{R}}(p^{2})\\\\[5.69054pt]
M_{\tilde{f}_{R}\tilde{f}_{L}}^{2}-\Pi_{\tilde{f}_{R}\tilde{f}_{L}}(p^{2})&\qquad
M_{\tilde{f}_{R}\tilde{f}_{R}}^{2}-\Pi_{\tilde{f}_{R}\tilde{f}_{R}}(p^{2})\end{array}\right)~{}$
(71)
$M^{2}_{\tilde{f}\tilde{f}}$ is the $2\times 2$ tree level mass matrix defined
in equations 56, 57 and 58. The self-energy terms of the above matrix
($\Pi_{\tilde{f}_{L}\tilde{f}_{L}}$, $\Pi_{\tilde{f}_{L}\tilde{f}_{R}}$,
$\Pi_{\tilde{f}_{R}\tilde{f}_{L}}$, $\Pi_{\tilde{f}_{R}\tilde{f}_{R}}$) are
defined in appendix D of BPMZ [70].
* 4.
Higgs Bosons
The mass of the two loop CP-even higges are obtained by diagonalizing ${\cal
M}^{2}_{s}(p^{2})$ at the EWSB scale. Where the matrix ${\cal
M}^{2}_{s}(p^{2})$ is given by,
${\cal M}^{2}_{s}(p^{2})\ =\
\left(\begin{array}[]{cc}\hat{M}_{Z}^{2}c_{\beta}^{2}+\hat{m}_{A}^{2}s_{\beta}^{2}-\,\Pi_{s_{1}s_{1}}(p^{2})+t_{1}/v_{1}&-(\hat{M}_{Z}^{2}+\hat{m}_{A}^{2})s_{\beta}c_{\beta}-\,\Pi_{s_{1}s_{2}}(p^{2})\\\\[5.69054pt]
-(\hat{M}_{Z}^{2}+\hat{m}_{A}^{2})s_{\beta}c_{\beta}-\,\Pi_{s_{2}s_{1}}(p^{2})&\hat{M}_{Z}^{2}s_{\beta}^{2}+\hat{m}_{A}^{2}c_{\beta}^{2}-\,\Pi_{s_{2}s_{2}}(p^{2})+t_{2}/v_{2}\end{array}\right)\
$ (72)
The self energy terms $\Pi_{s_{1}s_{1}},\,\Pi_{s_{1}s_{2}},\,\Pi_{s_{2}s_{1}}$
and $\Pi_{s_{2}s_{2}}$ are presented in appendix D of BPMZ [70]. Whereas, one
loop tadpole contributions $t_{1}/v_{1}$ and $t_{2}/v_{2}$ are presented in
appendix E of BPMZ [70]. Besides the complete one loop correction to $m_{h}$
and $m_{H}$, we evaluate the top mass enhanced dominant two loop corrections
provided in [71]. From version 1.2 onwards, we have implemented the full two
loop Higgs corrections due to Slavich et. al [72].
The tree level mass of CP-odd higgs boson, $m_{A}$ and charged higgs boson
$m_{H^{\pm}}$ is given by equations 63 and 65 respectively. The one loop
correction to masses is evaluated at EWSB scale and given by
$\displaystyle\hat{m}^{2}_{A}$ $\displaystyle=$ $\displaystyle
m^{2}_{A}+\Pi_{AA}(m_{A}^{2})-b_{A}$ (73) $\displaystyle\hat{m}_{H^{\pm}}$
$\displaystyle=$ $\displaystyle
m^{2}_{A}+M^{2}_{W}+\Pi_{AA}(m_{A}^{2})+\Pi^{T}_{WW}(M_{W}^{2})-\Pi_{H^{+}H^{-}}(m^{2}_{H^{\pm}})$
(74)
Where, $b_{A}=s^{2}_{\beta}\ t_{1}/v_{1}+c^{2}_{\beta}\ t_{2}/v_{2}$.
* 5.
Neutralino and Chargino
The one loop corrected neutralino mass matrix has the following form,
${\cal M}_{\tilde{\psi}^{0}}\ +\ \frac{1}{2}\left(\delta{\cal
M}_{\tilde{\psi}^{0}}(p^{2})+\delta{\cal
M}_{\tilde{\psi}^{0}}^{T}(p^{2})\right)~{}$ (75)
where
$\delta{\cal M}_{\tilde{\psi}^{0}}(p^{2})\ =\ -\ \Sigma_{R}^{0}(p^{2}){\cal
M}_{\tilde{\psi}^{0}}\ -\ {\cal M}_{\tilde{\psi}^{0}}\Sigma_{L}^{0}(p^{2})\ -\
\Sigma_{S}^{0}(p^{2})\ $ (76)
Where, ${\cal M}_{\tilde{\psi}^{0}}$ is the tree level neutralino mass matrix
defined in 53. And the matrix $\delta{\cal M}_{\tilde{\psi}^{0}}(p^{2})$ is
the one loop correction to the neutralino mass matrix.
The one-loop chargino mass matrix is as follows,
${\cal M}_{\tilde{\psi}^{+}}\ -\ \Sigma_{R}^{+}(p^{2})\,{\cal
M}_{\tilde{\psi}^{+}}\ -\ {\cal M}_{\tilde{\psi}^{+}}\,\Sigma_{L}^{+}(p^{2})\
-\ \Sigma_{S}^{+}(p^{2})~{}$ (77)
Where, ${\cal M}_{\tilde{\psi}^{+}}$ is the tree level chargino mass matrix
defined in 55. We evaluate the self energies $\Sigma^{+,0}_{L,R,S}$ with
$p^{2}=Q$(EWSB scale). See appendix D of BPMZ for the complete expressions of
these self-energies. One loop neutralino and chargino mass matrices are then
diagonalized to obtain the eigenvalues and eigenvectors at EWSB scale.
* 6.
Gauge Couplings and $\sin^{2}\theta_{W}$
The correction to $\overline{DR}$ electromagnetic coupling at the weak scale
is given by,
$\hat{\alpha}=\frac{\alpha_{em}}{1-\Delta\hat{\alpha}}\qquad{\rm
and}\qquad\alpha_{em}=\frac{1}{127.934}$ (78)
Where $\Delta\hat{\alpha}$ contains contribution from SM particles as well as
SUSY particles. Similarly, the correction to strong coupling $\alpha_{s}$ at
the weak scale is given by,
$\hat{\alpha}_{s}=\frac{\alpha_{s}}{1-\Delta\alpha_{s}}$ (79)
$\Delta\alpha_{s}$ receives contribution from top quark, gluino and squarks.
The exact expression given by equation 3 of BPMZ [70]. The correction to
$\overline{DR}$ weak mixing angle is given by,
$\sin^{2}2\hat{\theta}_{W}=\frac{4\pi\hat{\alpha}}{\sqrt{2}M_{Z}^{2}G_{\mu}(1-\Delta\hat{r})}$
(80)
The exact expression of $\Delta\hat{\alpha}$ and $\Delta\hat{r}$ is presented
in the appendix C of BPMZ [70].
* 7.
Gluino
One loop corrected gluino mass $m_{\tilde{g}}$ at scale Q is given by,
$m_{\tilde{g}}=M_{3}(Q)-\Sigma_{\tilde{g}}(m_{\tilde{g}})$ (81)
where $M_{3}(Q)$ is the tree level gluino mass generated by the RGEs and
$\Sigma_{\tilde{g}}(m_{\tilde{g}})$ is the one loop self-energy contribution
to the gluino mass. See equation D.44 of BPMZ [70] for the complete
expression.
## Appendix C Program File Structure
Figure 3: File structure and program flow
In this appendix we briefly list the subroutines of interest in the program
and its purpose (for more details see the technical manual provided with the
program). Fig. (3) depicts the program structure and the flow of variables in
the program. The subroutine name is printed first followed by the respective
file which contains the subroutine.
The program begins by reading inputs from the main program contained in the
files ‘runonce.f’, ‘runslha.f’ (for SLHA input format), ‘scanning.f’. The
convention followed in the program is such that all the input parameters which
do not receive one-loop susy threshold correction such as the masses of the
first two generation of quarks and leptons are contained in the file
‘stdinputs.h’. Whereas, parameters such as $\sin^{2}\theta_{W}$, gauge
couplings and the third generation quark and lepton masses which receive one-
loop SUSY threshold correction are contained in the file ‘SuSemain.f’.
Moreover, the complete $3\times 3$ structure of the Yukawas and soft terms are
defined in ‘SuSemain.f’.
* 1.
rgeiterate.f: The heart of the program. The iterative algorithm is contained
here.
* 2.
softspectrum.f: Given the RGE output at low scale, this routine computes the
complete tree level SUSY spectrum by diagonalizing the mass matrices.
* 3.
slha.f: Contains the ingredients for SLHA input/output interface.
* 4.
mueconvernew.f: Contains routines which calculate the decay rates and
branching fractions of rare lepton flavor violating decays.
* 5.
mssmrge.f: The complete MSSM two loop RGEs are contained in this file.
* 6.
oneloopparticle.f: A set of several files which contain the one loop
corrections to the corresponding particle, which could be gauge, neutralino,
charging, sfermion etc.
* 7.
oneloopPV.f: The analytical form of all scalar one loop Passarino-Veltman
functions are contained in this file.
* 8.
math.f: Routines for matrix manipulations, integration routine for ordinary
differential equations using Runge-Kutta with adaptive step size and other
general purpose functions such as random number generator.
## Appendix D Spectrum Comparison with Other Codes
In this section we compare the spectrum of SuSeFLAV with other codes for two
different parameter space points form [101]. For both the points we have set
RHN equals to zero.
Table 2: Spectrum comparison for point 10.1.5 of [101] 10.1.5 | $m_{0}=175$, $M_{\frac{1}{2}}=700$, $\tan\beta=10$, $A_{0}=0$, sign$(\mu)>0$
---|---
Parameters | SuSeFLAV | SOFTSUSY | SuSpect | SPheno
$W^{+}$ | 8.05096462E+01 | 8.03860774e+01 | 8.04813477E+01 | 8.03115948E+01
$h$ | 1.16294107E+02 | 1.16694326e+02 | 1.16799828E+02 | 1.16829168E+02
$H$ | 9.66133037E+02 | 9.74430227e+02 | 9.68302501E+02 | 9.79779724E+02
$A$ | 9.65805171E+02 | 9.74206243e+02 | 9.68101504E+02 | 9.79577186E+02
$H^{+}$ | 9.69592528E+02 | 9.77728714e+02 | 9.71660981E+02 | 9.83266787E+02
$\tilde{q}_{L}$ | 1.43474627E+03 | 1.43580606e+03 | 1.43214020E+03 | 1.43992830E+03
$\tilde{q}_{R}$ | 1.37749634E+03 | 1.37825389e+03 | 1.37724979E+03 | 1.38138749E+03
$\tilde{t}_{1}$ | 1.11659996E+03 | 1.11105612e+03 | 1.11462082E+03 | 1.10949455E+03
$\tilde{t}_{2}$ | 1.35668185E+03 | 1.35225046e+03 | 1.35767238E+03 | 1.35426026E+03
$\tilde{b}_{1}$ | 1.31765879E+03 | 1.31707770e+03 | 1.32640786E+03 | 1.31861062E+03
$\tilde{b}_{2}$ | 1.36683066E+03 | 1.36729669e+03 | 1.36701474E+03 | 1.37028335E+03
$\tilde{l}_{L}$ | 5.01303257E+02 | 5.01757612e+02 | 4.94643201E+02 | 5.01101877E+02
$\tilde{l}_{R}$ | 3.23056643E+02 | 3.18458635e+02 | 3.13310890E+02 | 3.18342964E+02
$\tilde{\tau}_{1}$ | 3.15781781E+02 | 3.11269901e+02 | 3.06660771E+02 | 3.11160940E+02
$\tilde{\tau}_{2}$ | 5.01464374E+02 | 5.01745556e+02 | 4.94979107E+02 | 5.01126010E+02
$\tilde{\nu}_{l}$ | 4.94057805E+02 | 4.95324112e+02 | 4.88544346E+02 | 4.94578335E+02
$\tilde{\nu}_{\tau}$ | 4.92669260E+02 | 4.93751173e+02 | 4.87265068E+02 | 4.92997226E+02
$\tilde{g}$ | 1.56391437E+03 | 1.56476030e+03 | 1.56495608E+03 | 1.57232851E+03
$\tilde{\chi}^{0}_{1}$ | 2.95219584E+02 | 2.91289971e+02 | 2.92926561E+02 | 2.92509855E+02
$\tilde{\chi}^{0}_{2}$ | 5.45858411E+02 | 5.51079728e+02 | 5.52119478E+02 | 5.52386977E+02
$\tilde{\chi}^{0}_{3}$ | 8.44975403E+02 | 8.50445968e+02 | 8.60389258E+02 | 8.56133860E+02
$\tilde{\chi}^{0}_{4}$ | 8.56487466E+02 | 8.62039492e+02 | 8.72046145E+02 | 8.67579765E+02
$\tilde{\chi}^{\pm}_{1}$ | 5.35344839E+02 | 5.51280182e+02 | 5.52043599E+02 | 5.52530182E+02
$\tilde{\chi}^{\pm}_{2}$ | 8.49067665E+02 | 8.61668017e+02 | 8.71711677E+02 | 8.67856046E+02
$\mu$ | 8.37298648E+02 | 8.44864881e+02 | 8.55544693E+02 | 8.50798325E+02
$M_{\rm SUSY}$ | 1.19242602E+03 | 1.19031414e+03 | 1.19174751E+03 | 1.22578828E+03
$M_{\rm GUT}$ | 1.26703749E+16 | 1.67323027e+16 | $-$ | 1.53713886E+16
Table 3: Spectrum comparison for point 40.2.3 of [101] 40.2.3 | $m_{0}=650$, $M_{\frac{1}{2}}=550$, $\tan\beta=40$, $A_{0}=-500$, sign$(\mu)>0$
---|---
Parameters | SuSeFLAV | SOFTSUSY | SuSpect | SPheno
$W^{+}$ | 8.05066107E+01 | 8.03827802e+01 | 8.04773725E+01 | 8.03087734E+01
$h$ | 1.16958052E+02 | 1.17297317e+02 | 1.17322254E+02 | 1.17682268E+02
$H$ | 7.30962046E+02 | 7.47689856e+02 | 7.18511292E+02 | 7.72497154E+02
$A$ | 7.31051866E+02 | 7.47643928e+02 | 7.18551942E+02 | 7.72499188E+02
$H^{+}$ | 7.35889065E+02 | 7.52307728e+02 | 7.23362920E+02 | 7.77278228E+02
$\tilde{u}_{L}$ | 1.30348659E+03 | 1.28439273e+03 | 1.30388612E+03 | 1.31020706E+03
$\tilde{u}_{R}$ | 1.26813307E+03 | 1.24781930e+03 | 1.27013696E+03 | 1.27389691E+03
$\tilde{d}_{L}$ | 1.30579237E+03 | 1.28668811e+03 | 1.30621620E+03 | 1.31246645E+03
$\tilde{d}_{R}$ | 1.26498184E+03 | 1.24450694e+03 | 1.26672885E+03 | 1.27054542E+03
$\tilde{t}_{1}$ | 9.24983971E+02 | 9.21498284e+02 | 9.69783691E+02 | 9.19586596E+02
$\tilde{t}_{2}$ | 1.14470194E+03 | 1.14136271e+03 | 1.17704096E+03 | 1.14412090E+03
$\tilde{b}_{1}$ | 1.08597531E+03 | 1.08516841e+03 | 1.11715522E+03 | 1.09021482E+03
$\tilde{b}_{2}$ | 1.15972419E+03 | 1.16046164e+03 | 1.19253887E+03 | 1.16721088E+03
$\tilde{l}_{L}$ | 7.46153736E+02 | 7.46218781e+02 | 7.43808878E+02 | 7.46077658E+02
$\tilde{l}_{R}$ | 6.83925820E+02 | 6.82393714e+02 | 6.80821036E+02 | 6.82370617E+02
$\tilde{\tau}_{1}$ | 5.23418984E+02 | 5.18531351e+02 | 5.54473085E+02 | 5.18371511E+02
$\tilde{\tau}_{2}$ | 6.95666250E+02 | 6.94111081e+02 | 7.16257580E+02 | 6.94363534E+02
$\tilde{\nu}_{l}$ | 7.40378630E+02 | 7.41675551e+02 | 7.39679081E+02 | 7.41481196E+02
$\tilde{\nu}_{\tau}$ | 6.79971865E+02 | 6.77798789e+02 | 6.94510135E+02 | 6.77596601E+02
$\tilde{g}$ | 1.27460200E+03 | 1.27746572e+03 | 1.28080280E+03 | 1.28539752E+03
$\tilde{\chi}^{0}_{1}$ | 2.33065416E+02 | 2.29995196e+02 | 2.31351813E+02 | 2.30719919E+02
$\tilde{\chi}^{0}_{2}$ | 4.32028446E+02 | 4.38269965e+02 | 4.43044390E+02 | 4.39375323E+02
$\tilde{\chi}^{0}_{3}$ | 7.36025983E+02 | 7.45547839e+02 | 9.13965835E+02 | 7.54411085E+02
$\tilde{\chi}^{0}_{4}$ | 7.44572370E+02 | 7.54116707e+02 | 9.19380416E+02 | 7.62689372E+02
$\tilde{\chi}^{\pm}_{1}$ | 4.22498882E+02 | 4.38347771e+02 | 4.43031333E+02 | 4.39557714E+02
$\tilde{\chi}^{\pm}_{2}$ | 7.40879408E+02 | 7.54555638e+02 | 9.20007967E+02 | 7.63794652E+02
$\mu$ | 7.29597776E+02 | 7.41004045e+02 | 9.12311446E+02 | 7.50392113E+02
$M_{\rm SUSY}$ | 9.97283261E+02 | 9.96175665e+02 | 1.03350532E+03 | 1.02572777E+03
$M_{\rm GUT}$ | 1.23171917E+16 | 1.88768830e+16 | $-$ | 1.75795520E+16
## References
## References
* [1] For a comprehensive review, please see S. P. Martin, “A Supersymmetry primer,” arXiv:hep-ph/9709356.
* [2] D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, M. R. Nolta, C. L. Bennett, B. Gold, M. Halpern et al., Astrophys. J. Suppl. 192, 16 (2011). [arXiv:1001.4635 [astro-ph.CO]].
* [3] See for example, the discussion in N. Arkani-Hamed, G. L. Kane, J. Thaler and L. T. Wang, JHEP 0608, 070 (2006) [arXiv:hep-ph/0512190].
* [4] See for example, discussion in C. F. Berger, J. S. Gainer, J. L. Hewett and T. G. Rizzo, JHEP 0902, 023 (2009) [arXiv:0812.0980 [hep-ph]].
* [5] M. Bona et al. [UTfit Collaboration], PMC Phys. A 3, 6 (2009) [arXiv:0803.0659 [hep-ph]]; E. Lunghi and A. Soni, Phys. Lett. B 697, 323 (2011) [arXiv:1010.6069 [hep-ph]].
* [6] For a review, please see A. J. Buras, Acta Phys. Polon. B 41, 2487 (2010) [arXiv:1012.1447 [hep-ph]].
* [7] L. Silvestrini, Ann. Rev. Nucl. Part. Sci. 57, 405 (2007) [arXiv:0705.1624 [hep-ph]].
* [8] S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008) [arXiv:0802.2962 [hep-ph]].
* [9] F. Borzumati and A. Masiero, Phys. Rev. Lett. 57, 961 (1986).
* [10] A. Brignole, H. Murayama and R. Rattazzi, Phys. Lett. B 335, 345 (1994) [arXiv:hep-ph/9406397].
* [11] F. Vissani and A. Y. Smirnov, Phys. Lett. B 341, 173 (1994) [arXiv:hep-ph/9405399]. M. S. Carena, J. R. Ellis, S. Lola and C. E. M. Wagner, Eur. Phys. J. C 12, 507 (2000) [arXiv:hep-ph/9906362].
* [12] J. Sato, K. Tobe and T. Yanagida, Phys. Lett. B 498, 189 (2001) [arXiv:hep-ph/0010348].
* [13] M. E. Gomez, S. Lola, P. Naranjo and J. Rodriguez-Quintero, JHEP 0904, 043 (2009) [arXiv:0901.4013 [hep-ph]].
* [14] J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Phys. Rev. D 53 (1996) 2442 [arXiv:hep-ph/9510309]. For a review in this direction, please see, A. Masiero, S. K. Vempati and O. Vives, New J. Phys. 6 (2004) 202 [arXiv:hep-ph/0407325].
* [15] J. Hisano, D. Nomura, Phys. Rev. D59, 116005 (1999). [hep-ph/9810479].
* [16] L. Calibbi, A. Faccia, A. Masiero and S. K. Vempati, Phys. Rev. D 74, 116002 (2006) [arXiv:hep-ph/0605139].
* [17] A. Ibarra and C. Simonetto, JHEP 0804, 102 (2008) [arXiv:0802.3858 [hep-ph]].
* [18] J. Hisano, M. Nagai, P. Paradisi and Y. Shimizu, JHEP 0912 (2009) 030 [arXiv:0904.2080 [hep-ph]].
* [19] L. Calibbi, Y. Mambrini and S. K. Vempati, JHEP 0709, 081 (2007) [arXiv:0704.3518 [hep-ph]].
* [20] V. Barger, D. Marfatia and A. Mustafayev, Phys. Lett. B 665, 242 (2008) [arXiv:0804.3601 [hep-ph]].
* [21] C. Biggio and L. Calibbi, JHEP 1010, 037 (2010) [arXiv:1007.3750 [hep-ph]]; M. E. Gomez, S. Lola, P. Naranjo and J. Rodriguez-Quintero, JHEP 0904, 043 (2009) [arXiv:0901.4013 [hep-ph]]; K. Kadota and K. A. Olive, Phys. Rev. D 80, 095015 (2009) [arXiv:0909.3075 [hep-ph]]; M. Drees and J. M. Kim, JHEP 0812, 095 (2008) [arXiv:0810.1875 [hep-ph]]; J. N. Esteves, J. C. Romao, M. Hirsch, F. Staub and W. Porod, Phys. Rev. D 83, 013003 (2011) [arXiv:1010.6000 [hep-ph]].
* [22] D. Choudhury, R. Garani and S. K. Vempati, JHEP 1206, 014 (2012) [arXiv:1104.4467 [hep-ph]].
* [23] T. Moroi, Phys. Lett. B 493, 366 (2000) [arXiv:hep-ph/0007328].
* [24] D. Chang, A. Masiero and H. Murayama, Phys. Rev. D 67 (2003) 075013 [arXiv:hep-ph/0205111].
* [25] M. Ciuchini, A. Masiero, L. Silvestrini, S. K. Vempati, O. Vives, Phys. Rev. Lett. 92, 071801 (2004). [hep-ph/0307191] ; M. Ciuchini, A. Masiero, P. Paradisi, L. Silvestrini, S. K. Vempati, O. Vives, Nucl. Phys. B783, 112-142 (2007). [hep-ph/0702144 [HEP-PH]].
* [26] N. Arkani-Hamed, H. C. Cheng, J. L. Feng and L. J. Hall, Phys. Rev. Lett. 77, 1937 (1996) [arXiv:hep-ph/9603431].
* [27] J. Hisano, M. M. Nojiri, Y. Shimizu and M. Tanaka, Phys. Rev. D 60, 055008 (1999) [arXiv:hep-ph/9808410]. J. Hisano, R. Kitano and M. M. Nojiri, Phys. Rev. D 65, 116002 (2002) [arXiv:hep-ph/0202129].
* [28] A. J. Buras, L. Calibbi and P. Paradisi, JHEP 1006, 042 (2010) [arXiv:0912.1309 [hep-ph]].
* [29] M. Hirsch, J. W. F. Valle, W. Porod, J. C. Romao and A. Villanova del Moral, Phys. Rev. D 78, 013006 (2008) [arXiv:0804.4072 [hep-ph]].
* [30] J. A. Casas, J. R. Espinosa, A. Ibarra and I. Navarro, Phys. Rev. D 63, 097302 (2001) [arXiv:hep-ph/0004166].
* [31] F. Mahmoudi, Comput. Phys. Commun. 180, 1718 (2009).
* [32] A. Crivellin, J. Rosiek, P. H. Chankowski, A. Dedes, S. Jaeger and P. Tanedo, arXiv:1203.5023 [hep-ph].
* [33] F. E. Paige, S. D. Protopopescu, H. Baer and X. Tata, arXiv:hep-ph/0312045.
* [34] A. Djouadi, J. L. Kneur and G. Moultaka, Comput. Phys. Commun. 176, 426 (2007) [arXiv:hep-ph/0211331].
* [35] B. C. Allanach, Comput. Phys. Commun. 143, 305 (2002) [arXiv:hep-ph/0104145].
* [36] G. Engelhard, J. L. Feng, I. Galon, D. Sanford and F. Yu, Comput. Phys. Commun. 181, 213 (2010) [arXiv:0904.1415 [hep-ph]].
* [37] http://projects.hepforge.org/softsusy/
* [38] W. Porod and F. Staub, arXiv:1104.1573 [hep-ph].
* [39] W. Porod, Comput. Phys. Commun. 153, 275 (2003) [arXiv:hep-ph/0301101].
* [40] D. Chowdhury, R. Garani and S. K. Vempati, SuSeFlav: A supersymmetric Spectrum Calculator, DAE-BRNS Symposium on High Energy Physics, Jaipur, India, Dec 13-18, 2010.
* [41] B. C. Allanach, C. Balazs, G. Belanger, M. Bernhardt, F. Boudjema, D. Choudhury, K. Desch, U. Ellwanger et al., Comput. Phys. Commun. 180, 8-25 (2009). [arXiv:0801.0045 [hep-ph]].
* [42] T. Schwetz, M. A. Tortola and J. W. F. Valle, New J. Phys. 10, 113011 (2008) [arXiv:0808.2016 [hep-ph]]; M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rept. 460, 1 (2008) [arXiv:0704.1800 [hep-ph]].
* [43] P. Minkowski, Phys. Lett. B67 (1977) 421; M. Gell–Mann, P. Ramond, and R. Slansky in Supergravity, p. 315, edited by F. Nieuwenhuizen and D. Friedman, North Holland, Amsterdam, 1979; T. Yanagida, Proc. of the Workshop on Unified Theories and the Baryon Number of the Universe, edited by O. Sawada and A. Sugamoto, KEK, Japan 1979; R. N. Mohapatra, G. Senjanović, Phys. Rev. Lett. 44 (1980) 912; J. Schechter, J. W. F. Valle Phys.Rev. D22 (1980) 2227; J. Schechter, J. W. F. Valle Phys.Rev. D25 (1982) 774.
* [44] For a review and up to date analysis, please see S. Antusch, J. Kersten, M. Lindner, M. Ratz, M. A. Schmidt, JHEP 0503, 024 (2005). [hep-ph/0501272].
* [45] J. A. Casas, A. Ibarra, Nucl. Phys. B618, 171-204 (2001). [hep-ph/0103065].
* [46] Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28, 870-880 (1962).
* [47] S. Davidson, A. Ibarra, Nucl. Phys. B648, 345-375 (2003). [hep-ph/0206304]. S. Davidson, JHEP 0303, 037 (2003). [hep-ph/0302075]. G. C. Branco, R. Gonzalez Felipe, F. R. Joaquim, I. Masina, M. N. Rebelo, C. A. Savoy, Phys. Rev. D67, 073025 (2003). [hep-ph/0211001]. S. Pascoli, S. T. Petcov, W. Rodejohann, Phys. Rev. D68, 093007 (2003). [hep-ph/0302054]. S. Pascoli, S. T. Petcov, C. E. Yaguna, Phys. Lett. B564, 241-254 (2003). [hep-ph/0301095]. J. R. Ellis, M. Raidal, T. Yanagida, Phys. Lett. B581, 9-18 (2004). [hep-ph/0303242].
* [48] Y. Farzan and M. E. Peskin, Phys. Rev. D 70, 095001 (2004) [arXiv:hep-ph/0405214].
* [49] S. P. Martin and M. T. Vaughn, Phys. Rev. D 50, 2282 (1994) [Erratum-ibid. D 78, 039903 (2008)].
* [50] A. Ibarra and C. Simonetto, JHEP 0804 (2008) 102 [arXiv:0802.3858 [hep-ph]].
* [51] M. A. Luty, arXiv:hep-th/0509029.
* [52] H. P. Nilles, Phys. Rept. 110, 1 (1984).
* [53] J. R. Ellis, C. Kounnas, D. V. Nanopoulos, Nucl. Phys. B247, 373-395 (1984).
* [54] J. R. Ellis, K. Enqvist, D. V. Nanopoulos, K. Tamvakis, Phys. Lett. B155, 381 (1985).
* [55] M. Drees, Phys. Lett. B158, 409 (1985).
* [56] A. Corsetti, P. Nath, Phys. Rev. D64, 125010 (2001). [hep-ph/0003186].
* [57] N. Chamoun, C. -S. Huang, C. Liu, X. -H. Wu, Nucl. Phys. B624, 81-94 (2002). [hep-ph/0110332].
* [58] S. Bhattacharya, J. Chakrabortty, Phys. Rev. D81, 015007 (2010). [arXiv:0903.4196 [hep-ph]].
* [59] S. P. Martin, Phys. Rev. D79, 095019 (2009). [arXiv:0903.3568 [hep-ph]].
* [60] J. R. Ellis, T. Falk, K. A. Olive, Y. Santoso, Nucl. Phys. B652, 259-347 (2003). [hep-ph/0210205].
* [61] M. Dine, A. E. Nelson, Phys. Rev. D48, 1277-1287 (1993), [hep-ph/9303230]. M. Dine, A. E. Nelson, Y. Shirman, Phys. Rev. D51, 1362-1370 (1995), [hep-ph/9408384]. M. Dine, A. E. Nelson, Y. Nir and Y. Shirman, Phys. Rev. D 53, 2658 (1996), [arXiv:hep-ph/9507378].
* [62] S. P. Martin, Phys. Rev. D55, 3177-3187 (1997). [hep-ph/9608224].
* [63] S. Dimopoulos, G. F. Giudice, A. Pomarol, Phys. Lett. B389, 37-42 (1996). [hep-ph/9607225].
* [64] B. C. Allanach, Comput. Phys. Commun. 143, 305-331 (2002). [hep-ph/0104145].
* [65] A. Djouadi, J. -L. Kneur, G. Moultaka, Comput. Phys. Commun. 176, 426-455 (2007). [hep-ph/0211331].
* [66] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010)
* [67] L. V. Avdeev and M. Y. .Kalmykov, Nucl. Phys. B 502, 419 (1997) [hep-ph/9701308].
* [68] H. Baer, J. Ferrandis, K. Melnikov, X. Tata, Phys. Rev. D66, 074007 (2002). [hep-ph/0207126].
* [69] A. Bednyakov, A. Onishchenko, V. Velizhanin, O. Veretin, Eur. Phys. J. C29, 87-101 (2003). [hep-ph/0210258].
* [70] D. M. Pierce, J. A. Bagger, K. T. Matchev and R. J. Zhang, Nucl. Phys. B 491, 3 (1997) [hep-ph/9606211].
* [71] S. Heinemeyer, W. Hollik, G. Weiglein, Phys. Lett. B455, 179-191 (1999). [hep-ph/9903404].
* [72] A. Brignole, G. Degrassi, P. Slavich and F. Zwirner, Nucl. Phys. B 631, 195 (2002) [hep-ph/0112177]; A. Dedes and P. Slavich, Nucl. Phys. B 657, 333 (2003) [hep-ph/0212132]; A. Dedes, G. Degrassi and P. Slavich, Nucl. Phys. B 672, 144 (2003) [hep-ph/0305127].
* [73] J. Girrbach, S. Mertens, U. Nierste, S. Wiesenfeldt, JHEP 1005, 026 (2010). [arXiv:0910.2663 [hep-ph]].
* [74] J. M. Frere, D. R. T. Jones, S. Raby, Nucl. Phys. B222, 11 (1983). L. Alvarez-Gaume, J. Polchinski, M. B. Wise, Nucl. Phys. B221, 495 (1983). M. Claudson, L. J. Hall, I. Hinchliffe, Nucl. Phys. B228, 501 (1983) L. J. Hall, J. D. Lykken, S. Weinberg, Phys. Rev. D27, 2359-2378 (1983). J. A. Casas, J. R. Espinosa, M. Quiros, Phys. Lett. B342, 171-179 (1995). [hep-ph/9409458].
* [75] R. Barate et al. [ LEP Working Group for Higgs boson searches and ALEPH and DELPHI and L3 and OPAL Collaborations ], Phys. Lett. B565, 61-75 (2003). [hep-ex/0306033].
* [76] https://twiki.cern.ch/twiki/bin/view/AtlasPublic.
https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsHIG.
* [77] R. Barbieri, G. F. Giudice, Nucl. Phys. B306, 63 (1988).
* [78] B. de Carlos, J. A. Casas, Phys. Lett. B309, 320-328 (1993). [hep-ph/9303291].
* [79] G. W. Anderson, D. J. Castano, Phys. Lett. B347, 300-308 (1995). [hep-ph/9409419].
* [80] K. Agashe, M. Graesser, Nucl. Phys. B507, 3-34 (1997).[hep-ph/9704206].
* [81] S. Bertolini, F. Borzumati, A. Masiero, G. Ridolfi, Nucl. Phys. B353, 591-649 (1991).
* [82] A. Bartl, T. Gajdosik, E. Lunghi, A. Masiero, W. Porod, H. Stremnitzer, O. Vives, Phys. Rev. D64, 076009 (2001). [hep-ph/0103324].
* [83] G. Degrassi, P. Gambino, P. Slavich, Comput. Phys. Commun. 179, 759-771 (2008). [arXiv:0712.3265 [hep-ph]]. G. Belanger, F. Boudjema, P. Brun, A. Pukhov, S. Rosier-Lees, P. Salati, A. Semenov, Comput. Phys. Commun. 182, 842-856 (2011). [arXiv:1004.1092 [hep-ph]].
* [84] G. W. Bennett et al. [ Muon g-2 Collaboration ], Phys. Rev. Lett. 89, 101804 (2002). [hep-ex/0208001].
* [85] G. W. Bennett et al. [ Muon g-2 Collaboration ], Phys. Rev. Lett. 92, 161802 (2004). [hep-ex/0401008].
* [86] M. Davier, A. Hoecker, B. Malaescu, Z. Zhang, Eur. Phys. J. C71, 1515 (2011). [arXiv:1010.4180 [hep-ph]]; M. Passera, W. J. Marciano and A. Sirlin, Phys. Rev. D 78, 013009 (2008) [arXiv:0804.1142 [hep-ph]]. M. Passera, W. J. Marciano, A. Sirlin, [arXiv:1001.4528 [hep-ph]].
* [87] U. Chattopadhyay, P. Nath, Phys. Rev. D53, 1648-1657 (1996). [hep-ph/9507386].
* [88] S. Marchetti, S. Mertens, U. Nierste, D. Stockinger, Phys. Rev. D79, 013010 (2009). [arXiv:0808.1530 [hep-ph]].
* [89] J. Adam et al. [ MEG Collaboration ], [arXiv:1107.5547 [hep-ex]].
* [90] U. Bellgardt et al. [SINDRUM Collaboration], Nucl. Phys. B 299, 1 (1988).
* [91] W. H. Bertl et al. [ SINDRUM II Collaboration ], Eur. Phys. J. C47, 337-346 (2006).
* [92] C. Dohmen et al. [SINDRUM II. Collaboration], Phys. Lett. B 317, 631 (1993).
* [93] D. Asner et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1010.1589 [hep-ex].
* [94] I-H. Lee, Phys. Lett. B 138, 121 (1984); I-H. Lee, Nucl. Phys. B 246, 120 (1984).
* [95] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012) [arXiv:1207.7235 [hep-ex]].
* [96] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214 [hep-ex]].
* [97] P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, arXiv:1207.1347 [hep-ph] ; J. Ellis and T. You, arXiv:1207.1693 [hep-ph]; J. R. Espinosa, C. Grojean, M. Muhlleitner and M. Trott, arXiv:1207.1717 [hep-ph]; D. Carmi, A. Falkowski, E. Kuflik, T. Volansky and J. Zupan, arXiv:1207.1718 [hep-ph].
* [98] L. J. Hall, D. Pinner and J. T. Ruderman, JHEP 1204, 131 (2012) [arXiv:1112.2703 [hep-ph]] ; S. Heinemeyer, O. Stal and G. Weiglein, Phys. Lett. B 710, 201 (2012) [arXiv:1112.3026 [hep-ph]]; A. Arbey, M. Battaglia and F. Mahmoudi, Eur. Phys. J. C 72, 1906 (2012) [arXiv:1112.3032 [hep-ph]]; A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, Phys. Lett. B 708, 162 (2012) [arXiv:1112.3028 [hep-ph]]; P. Draper, P. Meade, M. Reece and D. Shih, Phys. Rev. D 85, 095007 (2012) [arXiv:1112.3068 [hep-ph]]; M. Carena, S. Gori, N. R. Shah and C. E. M. Wagner, JHEP 1203, 014 (2012) [arXiv:1112.3336 [hep-ph]]; N. D. Christensen, T. Han and S. Su, Phys. Rev. D 85, 115018 (2012) [arXiv:1203.3207 [hep-ph]]; H. Baer, V. Barger and A. Mustafayev, Phys. Rev. D 85, 075010 (2012) [arXiv:1112.3017 [hep-ph]]; M. Kadastik, K. Kannike, A. Racioppi and M. Raidal, JHEP 1205, 061 (2012) [arXiv:1112.3647 [hep-ph]]; O. Buchmueller, R. Cavanaugh, A. De Roeck, M. J. Dolan, J. R. Ellis, H. Flacher, S. Heinemeyer and G. Isidori et al., Eur. Phys. J. C 72, 2020 (2012) [arXiv:1112.3564 [hep-ph]]; L. Aparicio, D. G. Cerdeno and L. E. Ibanez, JHEP 1204, 126 (2012) [arXiv:1202.0822 [hep-ph]]; J. Ellis and K. A. Olive, Eur. Phys. J. C 72, 2005 (2012) [arXiv:1202.3262 [hep-ph]]; H. Baer, V. Barger and A. Mustafayev, JHEP 1205, 091 (2012) [arXiv:1202.4038 [hep-ph]]; M. Hirsch, F. R. Joaquim and A. Vicente, arXiv:1207.6635 [hep-ph].
* [99] L. Calibbi, D. Chowdhury, A. Masiero, K. M. Patel and S. K. Vempati, arXiv:1207.7227 [hep-ph].
* [100] http://www.netlib.org/lapack/.
* [101] S. S. AbdusSalam et al., “Benchmark Models, Planes, Lines and Points for Future SUSY Searches at the Lhc,” Eur. Phys. J. C 71 (2011) 1835 [arXiv:1109.3859 [hep-ph]].
* [102] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 180, 747 (2009) [arXiv:0803.2360 [hep-ph]].
* [103] P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, JCAP 0407, 008 (2004) [astro-ph/0406204].
|
arxiv-papers
| 2011-09-16T08:54:44 |
2024-09-04T02:49:22.220302
|
{
"license": "Public Domain",
"authors": "Debtosh Chowdhury, Raghuveer Garani and Sudhir K. Vempati",
"submitter": "Sudhir Vempati",
"url": "https://arxiv.org/abs/1109.3551"
}
|
1109.3631
|
# Lovelock gravity from entropic force
A. Sheykhi1,2, 111 sheykhi@uk.ac.ir H. Moradpour 1 and N. Riazi1 222
riazi@susc.ac.ir 1 hysics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
###### Abstract
In this paper, we first generalize the formulation of entropic gravity to
($n+1$)-dimensional spacetime. Then, we propose an entropic origin for Gauss-
Bonnet gravity and more general Lovelock gravity in arbitrary dimensions. As a
result, we are able to derive Newton’s law of gravitation as well as the
corresponding Friedmann equations in these gravity theories. This procedure
naturally leads to a derivation of the higher dimensional gravitational
coupling constant of Friedmann/Einstein equation which is in complete
agreement with the results obtained by comparing the weak field limit of
Einstein equation with Poisson equation in higher dimensions. Our study shows
that the approach presented here is powerful enough to derive the
gravitational field equations in any gravity theory.
keywords: entropic; gravity; thermodynamics.
## I Introduction
Nowadays, it is a general belief that there should be some deep connection
between gravity and thermodynamics. Indeed, this connection has a long history
since the discovery of black holes thermodynamics in 1970’s by Bekenstein and
Hawking HB . The studies on the profound connection between gravity and
thermodynamics have been continued B ; D until in 1995 Jacobson Jac
disclosed that the Einstein field equation is just an equation of state for
spacetime and in particular it can be derived from the the first law of
thermodynamics together with the relation between the horizon area and
entropy. Inspired by Jacobson’s arguments, an overwhelming flood of papers has
appeared which attempt to show that there is indeed a deeper connection
between gravitational field equations and horizon thermodynamics. It has been
shown that the gravitational field equations in a wide variety of theories,
when evaluated on a horizon, reduce to the first law of thermodynamics and
vice versa. This result, first pointed out in Pad , has now been demonstrated
in various theories including f(R) gravity Elin , cosmological setups Cai2 ;
Cai3 ; CaiKim ; Wang ; Cai33 ; Shey0 , and in braneworld scenarios Shey1 ;
Shey2 . For a recent review on the thermodynamical aspects of gravity and
complete list of references see Pad0 . The deep connection between horizon
thermodynamics and gravitational field equations, help to understand why the
field equations should encode information about horizon thermodynamics. These
results prompt people to take a statistical physics point of view on gravity.
A remarkable new perspective was recently suggested by Verlinde Ver who
claimed that the laws of gravitation are no longer fundamental, but rather
emerge naturally from the second law of thermodynamics as an “entropic force”.
Similar discoveries are also made by Padmanabhan Padm who observed that the
equipartition law for horizon degrees of freedom combined with the Smarr
formula leads to the Newton’s law of gravity. This may imply that the entropy
links general relativity with the statistical description of unknown spacetime
microscopic structure when a horizon is present. The investigations on the
entropic gravity has attracted a lot of interest recently Cai4 ; Smolin ; Li ;
Tian ; Vancea ; Modesto ; Sheykhi1 ; BLi ; Sheykhi2 ; Gu ; other ; mann .
On the other hand, the effect of string theory on classical gravitational
physics is usually investigated by means of a low energy effective action
which describes gravity at the classical level. This effective action consists
of the Einstein-Hilbert action plus curvature-squared (Gauss-Bonnet) term and
also higher order derivatives curvature terms. Lovelock gravity Lov ; Der
which is a natural generalization of Einstein gravity in higher dimensional
spacetimes contains higher order derivatives curvature terms, however there
are no terms with more than second order derivatives of metric in equations of
motion just as in Gauss-Bonnet gravity. Since the Lovelock tensor contains
metric derivatives no higher than second order, the quantization of the
linearized Lovelock theory is ghost-free Zwi .
Since the entropic gravity is fundamentally based on the holographic
principle, one expects that entropic gravity can be generalized to any
arbitrary dimension Ver . The motivation for studying higher dimensional
gravity originates from string theory, which is a promising approach to
quantum gravity. String theory predicts that spacetime has more than four
dimensions. Another striking motivation for studying higher dimensional
gravity comes from AdS/CFT correspondence conjecture Mad , which associates an
$n$-dimensional conformal field theory with a gravitational theory in $(n+1)$
dimension. The generalization of this duality is embodied by the holographic
principle Sus , which posits that the entropy content of any region of space
is defined by the bounding area of the region. These considerations have
provided us enough motivation to study the formulation of the entropic gravity
in $(n+1)$-dimensional spacetime. In this paper, we consider the problem of
formulating entropic gravity in all higher dimensions. We also show that in an
string inspired model of gravity the formalism of entropic force works well
and can be employed to derive the Newton’s law of gravity as well as the
$(n+1)$-dimensional Friedmann equation in Gauss-Bonnet theory and more general
Lovelock gravity.
This paper is organized as follow. In the next section we generalize the
entropic gravity to arbitrary dimensions and will derive successfully Newton’s
law of gravitation as well as Friedmann equation in $(n+1)$-dimensions. In
section III, we derive Newton’s law of gravity and the $(n+1)$-dimensional
Friedmann equation in Gauss-Bonnet theory from the entropic gravity
perspective. In section IV, we generalize our study to the more general
Lovelock gravity. The last section is devoted to conclusions and discussions.
## II Entropic Gravity in ($n+1$)-dimensions
According to Verlinde, when a particle is on one side of screen and the screen
carries a temperature, it will experience an entropic force equal to
$F=-T\frac{\triangle S}{\triangle x}.$ (1)
By definition, $F$ is a force resulting from the tendency of a system to
increase its entropy. Note that $\triangle S>0$ and hence the sign of the
force is determined by how one chooses the definition of $\triangle x$ as it
relates to the proposed system. Here $\triangle x$ is the displacement of the
particle from the holographic screen, while $T$ and $\triangle S$ are the
temperature and the entropy change on the screen, respectively. Suppose we
have a mass distribution $M$ which is distributed uniformly inside an screen
$\Sigma$. We have also a test mass $m$ which is located outside the screen.
The surface $\Sigma$ surrounds the mass distribution $M$ has a spherically
symmetric property, while the test mass $m$ is assumed to be very close to
$\Sigma$ comparing to its reduced Compton wavelength
$\lambda_{m}=\frac{\hbar}{mc}$. Now, consider an ($n+1$)-dimensional spacetime
with $n$ spacial dimensions. The mass $M$ induces a holographic screen
$\Sigma_{n}$ at distance $R$ that has encoded on it gravitational information.
The volume and area of this $n$-sphere are
$V_{n}=\Omega_{n}R^{n},\ \ \ \Sigma_{n}=n\Omega_{n}R^{n-1},$ (2)
where
$\Omega_{n}=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)},\ \ \ \
\Gamma(\frac{n}{2}+1)=\frac{n}{2}(\frac{n}{2}-1)!.$ (3)
According to the holographic principle, the screen encodes all physical
information contained within its volume in bits on the screen. The maximal
storage space, or total number of bits, is proportional to the area
$\Sigma_{n}$. Let us denote the number of used bits by $N$. It is natural to
assume that this number is proportional to the area $\Sigma_{n}$, namely
$\Sigma_{n}=NQ,$ (4)
where $Q$ is a constant which should be specified later. Since $N$ denotes the
number of bits, thus for one unit change we find $\triangle N=1$. Therefore,
from relation (4) one gets $\triangle\Sigma_{n}=Q$. Motivated by Bekenstein’s
area law of black hole entropy, we assume the entropy of the
($n-1$)-dimensional holographic screen obeys the area law, namely
$S=\frac{k_{B}c^{3}\Sigma_{n}}{4\hbar G_{n+1}},$ (5)
where
$G_{n+1}=2\pi^{1-n/2}\Gamma(\frac{n}{2})\frac{c^{3}\ell_{p}^{n-1}}{\hbar},$
(6)
is the ($n+1$)-dimensional gravitational constant mann . We also assume the
entropy change
$\triangle S=\frac{k_{B}c^{3}\triangle\Sigma_{n}}{4\hbar
G_{n+1}}=\frac{k_{B}c^{3}Q}{4\hbar G_{n+1}}.$ (7)
is one fundamental unit of entropy when $\triangle x=\frac{\hbar}{mc}$, and
the entropy gradient points radially from the outside of the surface to
inside. Assuming that the total energy of the system,
$E=Mc^{2},$ (8)
is evenly distributed over the bits. Then according to the equipartition law
of energy Pad1 , the total energy on the screen is
$E=\frac{1}{2}Nk_{B}T.$ (9)
Combining Eqs. (4), (8) and (9), we find
$T=\frac{2Mc^{2}Q}{\Sigma_{n}k_{B}}.$ (10)
Finally, inserting Eqs. (7) and (10) as well as relation $\triangle
x=\frac{\hbar}{mc}$ in Eq. (1), after using relation
$\Sigma_{n}=n\Omega_{n}R^{n-1}$, it is straightforward to show that the
entropic force yields the ($n+1$)-dimensional Newton’s law of gravitation
$F=-\frac{Mm}{R^{n-1}}\left[\frac{Q^{2}c^{6}}{2n\Omega_{n}\hbar^{2}G_{n+1}}\right],$
(11)
This is nothing but the Newton’s law of gravitation in arbitrary dimensions
provided we define
$Q^{2}\equiv\frac{2\hbar^{2}}{c^{6}}n\Omega_{n}G^{2}_{n+1}.$ (12)
For $n=3$ we have $G_{4}=G=\ell_{p}^{2}c^{3}/\hbar$ and the above expression
reduces to $Q^{2}=8\pi\ell_{p}^{4}$ Sheykhi1 . It is important to note that
the relations $N=\frac{Ac^{3}}{G\hbar}$ and $\triangle S=2\pi k_{B}$
postulated by Verlinde Ver . Our assumption here differs a bit from Verlinde
postulates. For example, we have taken $\triangle S=k_{B}\frac{c^{3}Q}{4\hbar
G}$ instead of $\triangle S=2\pi k_{B}$. Indeed, $\triangle S$ in our paper is
$\frac{1}{\sqrt{8\pi}}\times\triangle S$ Verlinde. So we do not expect to have
exactly $Q=\frac{G\hbar}{c^{3}}$. Although our main assumptions
$\Sigma_{n}\propto N$ and $\triangle S\propto k_{B}$ are similar but the
constants of proportionality are just assumption for later convenience [16].
Combining Eq. (12) with (11) we reach
$F=-G_{n+1}\frac{Mm}{R^{n-1}}.$ (13)
As the next step, we generalize the study to the cosmological setup. Assuming
a homogeneous and isotropic Friedmann-Robertson-Walker (FRW) spacetime which
is described by the line element
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+R^{2}d\Omega_{n-1}^{2}.$ (14)
Here $R=a(t)r$, $x^{0}=t,x^{1}=r$, and $h_{\mu\nu}$=diag
$(-1,a^{2}/(1-kr^{2}))$ is the two dimensional metric, while
$d\Omega^{2}_{n-1}$ is the metric of $(n-1)$-dimensional unit sphere. The
dynamical apparent horizon can be determined using relation
$h^{\mu\nu}\partial_{\mu}R\partial_{\nu}R=0$. It is a straightforward
calculation to show that the radius of the apparent horizon for the FRW
universe becomes
$R=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (15)
We also assume the matter source in the FRW universe is a perfect fluid with
stress-energy tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (16)
Conservation of energy-momentum in $(n+1)$-dimensions leads to the following
continuity equation
$\dot{\rho}+nH(\rho+p)=0,$ (17)
where $H=\dot{a}/a$ is the Hubble parameter. First of all, we derive the
dynamical equation for Newtonian cosmology. Consider a compact spatial region
$V_{n}$ with a compact boundary $\Sigma_{n}$, which is a sphere with physical
radius $R=a(t)r$. If we combine the gravitational force (13) with the second
law of Newton for the test particle $m$ near the screen $\Sigma_{n}$, then we
obtain
$F=m\ddot{R}=m\ddot{a}r=-G_{n+1}\frac{Mm}{R^{n-1}}.$ (18)
The total physical mass $M$ in the spatial region $V_{n}$ is defined as Cai4
$M=\int{dV\left(T_{\mu\nu}u^{\mu}u^{\nu}\right)}=\Omega_{n}R^{n}\rho,$ (19)
where $\rho=M/V_{n}$ is the energy density of the matter inside the the volume
$V_{n}=\Omega_{n}R^{n}$. Combining Eqs. (18) and (19) we reach
$\frac{\ddot{a}}{a}=-G_{n+1}\Omega_{n}\rho=-\frac{2G_{n+1}\pi^{n/2}}{n(\frac{n}{2}-1)!}\rho.$
(20)
This is the dynamical equation for ($n+1$)-dimensional Newtonian cosmology. In
four dimensional spacetime where $n=3$, we recover the well-known formula,
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho.$ (21)
In order to derive the ($n+1$)-dimensional Friedmann equations of the FRW
universe, let us notice that the quantity which produces the acceleration in a
dynamical background is the active gravitational mass $\mathcal{M}$ Pad3
rather than the total mass $M$. To determine the active gravitational mass, we
should express $\mathcal{M}$ in terms of energy-momentum tensor $T_{\mu\nu}$.
The key point here is to connect the energy momentum $T_{\mu\nu}$ with the
spacetime curvature with the use of the Tolman-Komar’s definition of active
gravitational mass. The active gravitational mass in $(n+1)$-dimension is
defined as Cai4
$\mathcal{M}=\frac{n-1}{n-2}\int_{V}{dV\left(T_{\mu\nu}-\frac{1}{n-1}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}}.$
(22)
It is a matter of calculation to show that
$\mathcal{M}=\frac{\Omega_{n}R^{n}}{n-2}[(n-2)\rho+np]=\frac{2\pi^{n/2}R^{n}}{n(n-2)(\frac{n}{2}-1)!}[(n-2)\rho+np].$
(23)
Now, we can combine Eq. (23) with (18) provided we replace $M$ in Eq. (18)
with induced active gravitational mass $\mathcal{M}$. This can be done because
according to the weak equivalence principle of general relativity, the active
gravitational mass of a system (here the universe) in general relativity is
equal to its total mass in Newtonian gravity. We find
$\frac{\ddot{a}}{a}=-\frac{G_{n+1}}{n-2}\Omega_{n}[(n-2)\rho+np]=-\frac{2G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}[(n-2)\rho+np].$
(24)
This is the acceleration equation for the dynamical evolution of the FRW
universe in $(n+1)$-dimensional spacetime. Multiplying $\dot{a}a$ on both
sides of Eq. (24), and using the continuity equation (17), after integrating
we find
$H^{2}+\frac{k}{a^{2}}=\frac{2G_{n+1}}{n-2}\Omega_{n}\rho=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho,$
(25)
where $k$ is an integration constant. When $n=3$, we have $\Omega_{3}=4\pi/3$
and one recovers the standard Friedmann equation
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho.$ (26)
Is it worth noting that in the literature the $(n+1)$-dimensional Friedmann
equation, in Einstein gravity, usually is written as
$H^{2}+\frac{k}{a^{2}}=\frac{2\kappa_{n}}{n(n-1)}\rho,$ (27)
with $\kappa_{n}=8\pi G_{n+1}$ (sometimes it is also written $\kappa_{n}=8\pi
G$). Let us note that the coupling in the r.h.s of this equation differs from
that we derived in Eq. (25) for $n\geq 4$. A question then arises, which one
is the correct Einstein gravitational constant? We believe that the coupling
constant we derived from the entropic force approach is the correct one. To
show this, let us note that the root of the factor $8\pi$ in Eq. (26) and also
(27) is the relation
$R_{00}=\nabla^{2}\phi,$ (28)
where $\phi$ is the Newtonian gravitational potential and $R_{00}$ is the (00)
component of the Ricci tensor. Now, the coefficient in the Poisson equation,
i.e. $4\pi$ has been obtained using the Gauss law for 3-dimensional space.
Thus we should first derive the correct coefficient for $n$-dimensional space.
Applying Guass’s law for an $n$-dimensional volume, one finds the Poisson
equation for arbitrary fixed dimension mansori
$\nabla^{2}\phi=\frac{2G_{n+1}\pi^{n/2}}{(\frac{n}{2}-1)!}\rho,$ (29)
On the other hand for $n\geq 3$, one finds mansori
$R_{00}=\left(\frac{n-2}{n-1}\right)\kappa_{n}\rho,$ (30)
Comparing Eqs. (28), (29) and (30) gives us the following modified Einstein
gravitational constant for arbitrary $n\geq 3$ dimensions
$\kappa_{n}=\frac{2(n-1)\pi^{n/2}G_{n+1}}{(n-2)(\frac{n}{2}-1)!}.$ (31)
Substituting relation (31) in (27), immediately shows that the correct form of
the Friedmann equation in $n\geq 3$ dimension is the expression we derived in
Eq. (25). This is a remarkable result and shows that the approach presented
here is powerful enough to derive the correct form of the gravitational field
equations.
## III Gauss-Bonnet Entropic Gravity
Next we study the entropic force idea in Gauss-Bonnet gravity. This theory
contains a special combination of curvature-squared term, added to the
Einstein-Hilbert action. The key point which should be noticed here is that in
Gauss-Bonnet gravity the entropy of the holographic screen does not obey the
area law. The lagrangian of the Gauss-Bonnet correction term is given by
${\mathcal{L}}_{GB}=R^{2}-4R^{ab}R_{ab}+R^{abcd}R_{abcd}.$ (32)
The low energy effective action of heterotic string theory naturally produces
the Gauss-Bonnet correction term. The Gauss-Bonnet term does not have any
dynamical effect in four dimensions since it is just a topological term in
four dimensions. Static black hole solutions of Gauss-Bonnet gravity have been
found and their thermodynamics have been investigated in ample details Bou ;
caigb . The entropy of the static spherically symmetric black hole in Gauss-
Bonnet theory has the following expression caigb
$S=\frac{k_{B}c^{3}\Sigma_{n}}{4\hbar
G_{n+1}}\left[1+\frac{n-1}{n-3}\frac{2\tilde{\alpha}}{r_{+}^{2}}\right],$ (33)
where $\Sigma_{n}$ is the horizon area and $r_{+}$ is the horizon radius. In
the above expression $\tilde{\alpha}=(n-2)(n-3)\alpha$, where $\alpha$ is the
Gauss-Bonnet coefficient which is positive Bou , namely $\alpha>0$. We assume
the entropy expression (33) also holds for the apparent horizon of the FRW
universe in Gauss-Bonnet gravity Cai2 . The only change we need to apply is
the replacement of the horizon radius $r_{+}$ with the apparent horizon radius
$R$, namely
$S=\frac{k_{B}c^{3}\Sigma_{n}}{4\hbar
G_{n+1}}\left[1+\frac{n-1}{n-3}\frac{2\tilde{\alpha}}{R^{2}}\right].$ (34)
For $n=3$ we have $\tilde{\alpha}=0$, thus the Gauss-Bonnet correction term
contributes only for $n\geq 4$ as we mentioned. In this case, the change in
the entropy becomes
$\triangle S=\frac{k_{B}c^{3}\triangle\Sigma_{n}}{4\hbar
G_{n+1}}+\frac{n-1}{n-3}\frac{k_{B}c^{3}\tilde{\alpha}}{2\hbar
G_{n+1}}\triangle\left(\frac{\Sigma_{n}}{R^{2}}\right).$ (35)
Using the relation $\Sigma_{n}=n\Omega_{n}R^{n-1}$ we have
$\triangle\Sigma_{n}R=(n-1)\Sigma_{n}\triangle R$. Combining this expression
with Eq. (35) after using relation $\triangle\Sigma_{n}=Q$, we obtain
$\triangle
S=\frac{Qk_{B}c^{3}}{4G_{n+1}\hbar}\left[1+\frac{2\tilde{\alpha}}{R^{2}}\right].$
(36)
Inserting Eqs. (10), (12) and (36) in Eq. (1) we find
$F=-G_{n+1}\frac{Mm}{R^{n-1}}\left[1+\frac{2\tilde{\alpha}}{R^{2}}\right].$
(37)
This is the Newton’s law of gravitation in Gauss-Bonnet gravity resulting from
the entropic force approach. In the absence of Gauss-Bonnet term
($\tilde{\alpha}=0$) one recovers Eq. (13). It is worth mentioning that the
correction term in Eq. (37) can be comparable to the first term only when $R$
is very small, namely for strong gravity. This implies that the correction
make sense only at the very small distances. When $R$ becomes large, i.e. for
weak gravity, the modified Newton’s law reduces to the usual Newton’s law of
gravitation.
Finally, we derive the $(n+1)$-dimensional Friedmann equation of FRW universe
in Gauss-Bonnet gravity using the approach we developed in the previous
section. In the presence of Gauss-Bonnet term Eq. (20) is modified as
$\frac{\ddot{a}}{a}=-\frac{2G_{n+1}\pi^{n/2}}{n(\frac{n}{2}-1)!}\rho\left[1+\frac{2\tilde{\alpha}}{R^{2}}\right].$
(38)
Note that $R=a(t)r$ is a function of time. Eq. (38) is the dynamical equation
for ($n+1$)-dimensional Newtonian cosmology in Gauss-Bonnet gravity. The main
difference between this equation and Eq. (20) is that the correction term
depends explicitly on the radius $R$. In order to remove this confusion, we
suppose that for Newtonian cosmology the spacetime is Minkowskian with $k=0$.
In this case we have $R=1/H$, and thus we can rewrite Eq. (38) as
$\frac{\ddot{a}}{a}=-\frac{2G_{n+1}\pi^{n/2}}{n(\frac{n}{2}-1)!}\rho\left[1+2\tilde{\alpha}\left(\frac{\dot{a}}{a}\right)^{2}\right].$
(39)
Combining Eq. (38) with (23), after replacing $M$ by $\mathcal{M}$, we get
$\frac{\ddot{a}}{a}=-\frac{2G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}[(n-2)\rho+np]\left[1+\frac{2\tilde{\alpha}}{R^{2}}\right].$
(40)
Thus we have derived the acceleration equation for the dynamical evolution of
the FRW universe in Gauss-Bonnet theory. Multiplying $\dot{a}a$ on both sides
of Eq. (40), and using the continuity equation (17), we get
$d(\dot{a}^{2})=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\left[d(\rho
a^{2})+\frac{2\tilde{\alpha}}{r^{2}}\frac{d(\rho a^{2})}{a^{2}}\right].$ (41)
Integrating yields
$H^{2}+\frac{k}{a^{2}}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho\left[1+\frac{2\tilde{\alpha}}{\rho
R^{2}}\int{\frac{d(\rho a^{2})}{a^{2}}}\right].$ (42)
Now, in order to calculate the correction term we need to find $\rho=\rho(a)$.
Suppose a constant equation of state parameter $w=p/\rho$, integrating the
continuity equation (17) immediately yields
$\rho=\rho_{0}a^{-n(1+w)},$ (43)
where $\rho_{0}$, an integration constant, is the present value of the energy
density. Inserting relation (43) in Eq. (42), after integration, we obtain
$H^{2}+\frac{k}{a^{2}}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho\left[1+\frac{2\tilde{\alpha}}{R^{2}}\left(\frac{n(1+w)-2}{n(1+w)}\right)\right].$
(44)
Using Eq. (15) we can further rewrite the above equation as
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)\left[1+2\tilde{\alpha}\left(H^{2}+\frac{k}{a^{2}}\right)\frac{n(1+w)-2}{n(1+w)}\right]^{-1}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho$
(45)
Next, we expand the above equation up to the linear order of $\tilde{\alpha}$.
We find
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)+{\alpha}^{\prime}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho,$
(46)
where we have defined
$\displaystyle\alpha^{\prime}\equiv\frac{2\tilde{\alpha}[2-n(1+w)]}{n(1+w)},$
(47)
and we have neglected $O(\tilde{\alpha}^{2})$ terms and higher powers of
$\tilde{\alpha}$. This is due to the fact that at the present time $R\gg 1$
and hence $H^{2}+k/a^{2}\ll 1$. Indeed for the present time where the apparent
horizon radius becomes large, the correction term is relatively small and the
usual Friedman equation is recovered. Thus, the correction make sense only at
the early stage of the universe where $a\rightarrow 0$. When $a\rightarrow 0$,
even the higher powers of $\tilde{\alpha}$ should be considered. With
expansion of the universe, the modified Friedmann equation reduces to the
usual Friedman equation. From Eq. (46) we see that the correction term
explicitly depends on the matter content through the equation of state
parameter, $w$, where we have assumed to be a constant.
Eq. (46) is the $(n+1)$-dimensional Friedmann equation in Gauss-Bonnet
Gravity. The Friedmann equation obtained here from entropic force approach is
in good agreement with that obtained from the gravitational field equation in
Gauss-Bonnet gravity Caimy . This fact further supports the viability of
Verlinde formalism.
## IV LOVELOCK Entropic Gravity
Finally we generalize our discussion to a more general case, the so-called
Lovelock gravity, which is a generalization of the Gauss-Bonnet gravity. The
most general lagrangian which keeps the field equations of motion for the
metric of second order, as the pure Einstein-Hilbert action, is Lovelock
lagrangian Lov . This lagrangian is constructed from the dimensionally
extended Euler densities and can be written as
$L=\sum_{p=0}^{m}c_{p}L_{p},$ (48)
where $c_{p}$ and $L_{p}$ are arbitrary constant and Euler density,
respectively. $L_{0}$ set to be one, so $c_{0}$ plays the role of the
cosmological constant, $L_{1}$ and $L_{2}$ are, respectively, the usual
curvature scalar and Gauss-Bonnet term. In an $(n+1)$-dimensional spacetime
$m=[n/2]$. The entropy of the spherically symmetric black hole solutions in
Lovelock theory can be expressed as Cai6
$S=\frac{k_{B}c^{3}\Sigma_{n}}{4\hbar
G_{n+1}}\sum_{i=1}^{m}\frac{i(n-1)}{n-2i+1}{\hat{c}_{i}}{{r}_{+}}^{2-2i}.$
(49)
where $\Sigma_{n}=n\Omega_{n}r^{n-1}_{+}$ is the horizon area. In the above
expression the coefficients ${\hat{c}_{i}}$ are given by
${\hat{c}_{0}}=\frac{{c_{0}}}{n(n-1)},\ \ {\hat{c}_{1}}=1,\ \
{\hat{c}_{i}}=c_{i}\prod_{j=3}^{2m}(n+1-j)\ \ i>1.$ (50)
Note that in expression (49) for entropy, the cosmological constant term
${\hat{c}_{0}}$ doesn’t appear. This is a reasonable result, and due to the
fact that the black hole entropy depends only on its horizon geometry. We
further assume the entropy expression (49) are valid for a FRW universe
bounded by the apparent horizon in the Lovelock gravity provided we replace
the horizon radius $r_{+}$ with the apparent horizon radius $R$, namely
$S=\frac{k_{B}c^{3}\Sigma_{n}}{4\hbar
G_{n+1}}\sum_{i=1}^{m}\frac{i(n-1)}{n-2i+1}{\hat{c}_{i}}R^{2-2i}.$ (51)
It is easy to show that, the first term in the above expression leads to the
well-known area law. The second term yields the apparent horizon entropy in
Gauss-Bonnet gravity. The change in the general entropy expression of Lovelock
gravity is obtained as
$\triangle S=\frac{k_{B}c^{3}Q}{4\hbar
G_{n+1}}\sum_{i=1}^{m}i{\hat{c}_{i}}R^{2-2i}.$ (52)
where we have used Eq. (4). Inserting Eq. (10), (12) and (52) in Eq. (1) one
finds
$F=-G_{n+1}\frac{Mm}{R^{n-1}}\sum_{i=1}^{m}i{\hat{c}_{i}}R^{2-2i}.$ (53)
Thus we have derived the Newton’s law of gravitation in Lovelock gravity
resulting from the entropic force. It is obvious that the first term of the
above expression yields the famous Newton’s law of gravity, and the others
terms will be important only for strong gravity or small distances. In this
manner, the dynamical equation for $(n+1)$-dimensional Newtonian cosmology
takes the following form
$\frac{\ddot{a}}{a}=-\frac{2G_{n+1}\pi^{n/2}}{n(\frac{n}{2}-1)!}\rho\sum_{i=1}^{m}i{\hat{c}_{i}}\left(\frac{\dot{a}}{a}\right)^{2i-2}.$
(54)
The acceleration equation for the dynamical evolution of the FRW universe in
$(n+1)$-dimensional Lovelock gravity is obtained following the method
developed in the previous section. The result is
$\frac{\ddot{a}}{a}=-\frac{2G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}[(n-2)\rho+np]\sum_{i=1}^{m}i{\hat{c}_{i}}R^{2-2i}.$
(55)
Multiplying $\dot{a}a$ on both sides of Eq. (55), and using the continuity
equation (17), after integrating, we get
$H^{2}+\frac{k}{a^{2}}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho\left[1+\sum_{i=2}^{m}\frac{i{\hat{c}_{i}}}{\rho
a^{2}r^{2i-2}}\int{\frac{d(\rho a^{2})}{a^{2i-2}}}\right].$ (56)
Using Eq. (43), we can perform the integration. We obtain
$H^{2}+\frac{k}{a^{2}}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho\left[1+\sum_{i=2}^{m}\frac{[2-n(1+w)]i{\hat{c}_{i}}}{[2(2-i)-n(1+w)]}\times\frac{1}{R^{2(i-1)}}\right].$
(57)
Eq. (57) can be rewritten in the following form
$\left(H^{2}+\frac{k}{a^{2}}\right)\left[1+\sum_{i=2}^{m}\frac{[2-n(1+w)]i{\hat{c}_{i}}}{[2(2-i)-n(1+w)]}\times\frac{1}{R^{2(i-1)}}\right]^{-1}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho.$
(58)
At the present time where $R\gg 1$, we can expand the l.h.s of the above
equation. Using Eq. (15), we reach
$\left(H^{2}+\frac{k}{a^{2}}\right)\left[1-\sum_{i=2}^{m}\frac{(2-n(1+w))i{\hat{c}_{i}}}{[2(2-i)-n(1+w)]}\left(H^{2}+\frac{k}{a^{2}}\right)^{i-1}\right]=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho.$
(59)
If we define
$\beta_{i}\equiv\frac{i[2-n(1+w)]{\hat{c}_{i}}}{n(1+w)-2(2-i)},$ (60)
then we can write Eq. (59) in the following form
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)+\sum_{i=2}^{m}\beta_{i}\left(H^{2}+\frac{k}{a^{2}}\right)^{i}=\frac{4G_{n+1}\pi^{n/2}}{n(n-2)(\frac{n}{2}-1)!}\rho.$
(61)
In this way we derive the $(n+1)$-dimensional Friedmann equations in Lovelock
gravity from the entropic force approach which is consistent with the result
obtained from different methods CaiKim ; Der . When $\beta_{i}=0$ ($i\geq 2$),
one recovers the standard Friedmann equation in Einstein gravity. The first
term in summation of the above equation is the Gauss-Bonnet leading correction
term derived in the previous section provided we define
$\hat{c}_{2}=\tilde{\alpha}$. In this case $\beta_{2}$ is exactly the
$\alpha^{\prime}$ of the previous section. Again, from Eqs. (60) and (61) we
see that the correction terms explicitly depend on the equation of state
parameter, $w$.
## V Conclusions and discussions
According to Verlinde’s argument, the total number of bits on the holographic
screen is proportional to the area, $A$, and can be specified as
$N=\frac{Ac^{3}}{G\hbar}$. Indeed, the derivation of Newton’s law of gravity
as well as Friedmann equations, in Verlide formalism, depend on the entropy-
area relationship $S=\frac{Ac^{3}}{4G\hbar}$, where $A=4\pi R^{2}$ represents
the area of the horizon Ver . However, it is well known that the area formula
of black hole entropy no longer holds in higher derivative gravities. So it
would be interesting to see whether one can derive Newton’s law of gravity as
well as the corresponding Friedmann equations in these gravities in the
framework of entropic force perspective developed by Verlinde Ver .
In this paper, starting from first principles and assuming the entropy
associated with the holographic screen/apparent horizon given by the
expression previously known via black hole thermodynamics, we were able to
derive the Newton’s law of gravity as well as the cosmological equations
(Friedmann equations) governing the evolution of the universe in any gravity
theory including Einstein, Gauss-Bonnet and more general Lovelock gravity in
arbitrary dimensions. We derived the Newton’s law of gravitation from entropic
force directly. Then, we derived the Friedmann equation by equating the mass
in Newtonian gravity with active gravitational mass. Therefore, the Friedmann
equation derived here from entropic force too, but indirectly. In our
derivation the assumption that the entropy of the apparent horizon of FRW
universe in Gauss-Bonnet and Lovelock gravity have the same form as the
spherically symmetric black hole entropy in these gravities, but replacing the
black hole horizon radius by the apparent horizon radius, plays a crucial
role. Interestingly enough, we found that the higher dimensional gravitational
coupling constant of Friedmann/Einstein equation can be derived naturally from
this approach which coincides with the result obtained by comparing the weak
field limit of Einstein equation with Poisson equation in higher dimension.
Our study shows that the approach here is powerful enough to derive the
gravitational field equations in any gravity theory. The results obtained here
in the framework of Gauss-Bonnet gravity and more general Lovelock gravity
further support the viability of Verlinde’s formalism.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, Iran.
## References
* (1) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973);
S. W. Hawking, Commun Math. Phys. 43, 199 (1975);
S. W. Hawking, Nature 248, 30 (1974).
* (2) J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).
* (3) P. C. W. Davies, J. Phys. A: Math. Gen. 8, 609 (1975);
W. G. Unruh, Phys. Rev. D 14, 870 (1976);
L. Susskind, J. Math. Phys. 36, 6377 (1995).
* (4) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).
* (5) T. Padmanabhan, Class. Quantum. Grav. 19 (2002) 5387\.
* (6) C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett. 96, 121301 (2006).
* (7) M. Akbar and R. G. Cai, Phys. Rev. D 75, 084003 (2007).
* (8) R. G. Cai and L. M. Cao, Phys.Rev. D 75, 064008 (2007).
* (9) R. G. Cai and S. P. Kim, JHEP 0502, 050 (2005).
* (10) B. Wang, E. Abdalla and R. K. Su, Phys.Lett. B 503, 394 (2001);
B. Wang, E. Abdalla and R. K. Su, Mod. Phys. Lett. A 17, 23 (2002).
* (11) R. G. Cai, L. M. Cao and Y. P. Hu, JHEP 0808 (2008) 090.
* (12) S. Nojiri and S. D. Odintsov, Gen. Relativ. Gravit. 38, 1285 (2006);
A. Sheykhi, Class. Quantum Grav. 27 (2010) 025007;
A. Sheykhi, Eur. Phys. J. C 69 (2010) 265.
* (13) A. Sheykhi, B. Wang and R. G. Cai, Nucl. Phys. B 779 (2007)1;
R. G. Cai and L. M. Cao, Nucl. Phys. B 785 (2007) 135
* (14) A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D 76 (2007) 023515;
A. Sheykhi, B. Wang, Phys. Lett. B 678 (2009) 434;
A. Sheykhi, JCAP 05 (2009) 019
* (15) T. Padmanabhan, Rept. Prog. Phys. 73 (2010) 046901\.
* (16) E. Verlinde, JHEP 1104, 029 (2011).
* (17) T. Padmanabhan, Mod. Phys. Lett. A 25 (2010) 1129.
* (18) R.G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, (2010) 061501(R);
R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81 (2010) 084012\.
* (19) L. Smolin, arXiv:1001.3668.
* (20) M. Li and Y. Wang, Phys. Lett. B 687, 243 (2010).
* (21) Y. Tian and X. Wu, Phys. Rev. D 81, 104013 (2010); Y. S. Myung, arXiv:1002.0871.
* (22) I. V. Vancea and M. A. Santos, arXiv:1002.2454.
* (23) L. Modesto and A. Randono, arXiv:1003.1998.
* (24) A. Sheykhi, Phys. Rev. D 81, 104011 (2010).
* (25) B. Liu, Y. C. Dai, X. R. Hu and J. B. Deng, Mod. Phys. Lett. A 26, 489 (2011).
* (26) S. H. Hendi and A. Sheykhi, Phys. Rev. D 83 (2011) 084012 ;
A. Sheykhi and S. H. Hendi, Phys. Rev. D 84 (2011) 044023;
S. H. Hendi and A. Sheykhi, Int. J Theor. Phys. 51 (2012) 1125;
A. Sheykhi, Z. Teimoori, Gen Relativ Gravit 44 (2012) 1129;
A. Sheykhi, Int. J Theor. Phys. 51 (2012) 185.
* (27) W. Gu, M. Li and R. X. Miao, arXiv:1011.3419;
R. X. Miao, J. Meng and M. Li, arXiv:1102.1166.
* (28) Y. X. Liu, Y. Q. Wang and S.W. Wei, Class. Quantum Grav. 27, 185002 (2010);
V. V. Kiselev and S. A. Timofeev, Mod. Phys. Lett. A 25, 2223 (2010);
R. A. Konoplya, Eur. Phys. J. C 69, 555 (2010);
R. Banerjee and B. R. Majhi. Phys. Rev. D 81, 124006 (2010);
P. Nicolini, Phys. Rev. D 82, 044030 (2010);
C. Gao, Phys. Rev. D 81, 087306 (2010);
Y. S. Myung and Y. W. Kim, Phys. Rev. D 81, 105012 (2010);
H. Wei, Phys. Lett. B 692, 167 (2010);
Y. Ling and J.P. Wu, JCAP 1008, 017 (2010);
D. A. Easson, P. H. Frampton and G. F. Smoot, Phys. Lett.B 696 (2011)273;
D. A. Easson, P. H. Frampton and G. F. Smoot, arXiv:1003.1528;
S. W. Wei, Y. X. Liu and Y. Q. Wang, Commun. Theor. Phys.56 (2011) 455\.
* (29) R. B. Mann and J. R. Mureika, Phys. Lett. B 703 (2011) 167\.
* (30) D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).
* (31) N. Deruelle and L. Farina-Busto, Phys. Rev. D 41, 3696 (1990).
* (32) B. Zwiebach, Phys. Lett. B 156, 315 (1985);
B. Zumino, Phys. Rep. 137, 109 (1986);
D. J. Gross and J. H. Sloan, Nucl. Phys. B291, 41 (1987).
* (33) J.M. Maldacena, Adv. Theor.Math. Phys. 2, 231 (1998);
Int. J. Theor. Phys. 38, 1113 (1999).
* (34) L. Susskind, J. Math. Phys. 36, 6377 (1995);
R. Bousso, Rev. Mod. Phys. 74, 825 (2002).
* (35) T. Padmanabhan, Phys. Rev. D 81, 124040 (2010).
* (36) T. Padmanabhan, Class. Quantum Grav., 21, 4485 (2004).
* (37) Reza Mansouri and Ali Nayeri, Gravitation Cosmology, Vol. 4, No. 2 (1998) 142.
* (38) D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985);
J. T. Wheeler, Nucl. Phys. B 268, 737 (1986);
Nucl. Phys. B 273, 732 (1986);
R. C. Myers and J. Z. Simon, Phys. Rev. D 38, 2434 (1988).
* (39) R. G. Cai, Phys. Rev. D65, 084014 (2002);
R. G. Cai and Q. Guo, Phys. Rev. D69, 104025 (2004);
R. G. Cai and K. S. Soh, Phys. Rev. D59, 044013 (1999).
* (40) R. G. Cai and Y. S. Myang, Phys. Lett. B 559 (2003) 60.
* (41) R. G. Cai, Phys. Lett. B 582, 237(2004).
|
arxiv-papers
| 2011-09-11T18:57:41 |
2024-09-04T02:49:22.239937
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, H. Moradpour and N. Riazi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1109.3631"
}
|
1109.3840
|
COMMISSIONS 27 AND 42 OF THE IAU
INFORMATION BULLETIN ON VARIABLE STARS
Number 0000
Konkoly Observatory Budapest 16 September 2011 HU ISSN 0374 – 0676
V1162 ORION: UPDATED AMPLITUDE AND PERIOD VARIATION
KHOKHUNTOD, PONGSAK1,2; ZHOU, AI-YING3∗; BOONYARAK, CHAYAN1; JIANG, SHI-
YANG1,3
1 Department of Physics, Faculty of Science, Naresuan University, Phitsanulok
65000, Thailand
2 Department of Astronomy, Beijing Normal University, China
3 National Astronomical Observatory of China, Chinese Academy of Sciences,
Datun Road 20A, Beijing, 100086, China; *Correspondent author, E-mail:
aiying@nao.cas.cn
V1162 Orion is an intermediate-amplitude $\delta$ Scuti type variable. Its
variability was first discovered by Lampens (1985), who derived a period of
0.078686(2) days with a full amplitude of about 0.18 mag in $V$ band. Hintz et
al. (1998) observed this star and found its amplitude in $V$ had dropped to
about 0.10 mag. Arentoft & Sterken (2000) and Arentoft et al. (2001a, 2001b)
got the same amplitude of $\sim$0.10 mag in $V$.
We had observed this star from 2007 January to 2010 February using several
telescopes at Naresuan University, Thailand and at Yunnan Observatory and the
Xinglong Station of National Astronomical Observatories, China (NAOC). We
determined a sum of 39 new times of maximum light (refers to Table 1),
following the method described by Zhou & Liu (2003). Errors involved in the
maximum determination are around 0.00045 d or less. In this note, we present a
two-night sample of the newly observed light curves along with an updated
$O-C$ diagram. Figure 1 shows the light curve of V1162 Ori on 2010 November 24
and 25 from the 60-cm telescope of NAOC. We can see clearly that the peak-to-
peak full amplitude in $V$ recovered to about 0.18 mag. This conforms with the
amplitude variability declared in the literature. The differential photometric
light curve data are available upon request from the authors.
Figure 1: The light curves of V1162 Ori obtained on 2010 Nov. 24 and 25.
We have noted that Arentoft et al. (2001a) presented the results of multisite
observations contributed by 18 telescopes from 15 observatories around the
global. They measured 253 light extrema (145 maxima and 108 minima) during the
observing season from 1999 October to 2000 May. They got an overall tendency
of period decreasing with a very high rate of $\frac{{\rm d}P}{{\rm
d}t}$=$-5\times 10^{-9}$ s s-1 (equivalent to $\frac{1}{P}\frac{{\rm d}P}{{\rm
d}t}=-2.4\times 10^{-5}$ yr-1). Moreover, the $(O-C)$ residuals seemed to
follow a cyclic variation with a period of 285$\pm$3 d, which is on a time-
scale similar to that of the amplitude variations (282$\pm$6 d). In addition,
we found 40 maxima in Wils et al. (2010, 2011), we draw here a $(O-C)$ diagram
based on a total of 385 times of maximum light.
Table 1: The new times of maximum light of V1162 Ori obtained in 2007–2011. No. | HJD(max) | No. | HJD(max) | No. | HJD(max)
---|---|---|---|---|---
1 | 2454130.0693 | 14 | 2454178.0626 | 27 | 2455525.13603
2 | 2454131.0901 | 15 | 2454181.0580 | 28 | 2455525.21233
3 | 2454131.1701 | 16 | 2454184.0422 | 29 | 2455525.29404
4 | 2454132.1171 | 17 | 2454847.0331 | 30 | 2455526.15567
5 | 2454132.1882 | 18 | 2454847.1111 | 31 | 2455526.23844
6 | 2454133.0578 | 19 | 2454847.1889 | 32 | 2455526.31468
7 | 2454133.1367 | 20 | 2454871.1100 | 33 | 2455565.10686
8 | 2454133.2158 | 21 | 2454871.1910 | 34 | 2455565.18583
9 | 2454137.0715 | 22 | 2455249.1147 | 35 | 2455565.26503
10 | 2454137.1467 | 23 | 2455250.0539 | 36 | 2455602.08888
11 | 2454161.0694 | 24 | 2455250.1342 | 37 | 2455602.16764
12 | 2454164.0582 | 25 | 2455524.19384 | 38 | 2455610.03529
13 | 2454176.0189 | 26 | 2455524.26824 | 39 | 2455610.11388
To calculate $(O-C)$ residuals and their corresponding cycles (denoted by E
below) elapsed since an initial maximum epoch, we have defined a new ephemeris
$T_{\rm max}={\rm HJD}\,2451890.3708+0.0786869\times E.$ (1)
according to Arentoft & Sterken (2002). The cycle counts are usually the
results rounded off by rounding a numerical value to the nearest integer. For
those cases when a value is exactly half-way between two integers, it is
always rounded up following the so-called ‘round half up’ tie-breaking rule.
For example, the values (66.51, 23.5, –23.5, 15.49, –18.38, –18.67), get
rounded to (67, 24, –23, 15, –18, –19), respectively. This rounding policy
results in $(O-C)$ residuals within half a period, i.e. $|O-C|\leq 0.0393$
days. The resultant $O-C$ diagram was drawn in Fig. 2, which cannot be fitted
by either a single linear line or a simple parabolic curve. Part of the 385
maxima and $(O-C)$ data are shown in Table 2. The full list is available upon
request from the authors.
Figure 2: The $O-C$ diagram of V1162 Ori: the forced parabolic fitting curve is not justified. The new maxima in Table 1 are drawn in squares. Table 2: List of 385 light maxima of V1162 Ori and their $(O-C)$ residuals computed by Eq.(1). The reference ’L’ refers to Lampens(1985), ’A’ refers to Arentoft et al.(2000,2001a,2001b,2002), ’H’ stands for Hintz et al.(1998), ’P’ for Poretti et al.(1990), ’W’ for Wils et al.(2010,2011), while ’J’ for this work. No. | Maximum | Fractional Cycles | Cycles | $(O-C)$ | Ref.
---|---|---|---|---|---
1 | 2445347.02750 | –83156.70 | –83157 | 0.023243 | L
2 | 2447110.78000 | –60741.88 | –60742 | 0.008880 | P
3 | 2447110.85550 | –60740.92 | –60741 | 0.005693 | P
$\cdots\cdots$ | | | | |
247 | 2451890.37080 | 0.00 | 0 | 0.000000 | A
248 | 2451890.45000 | 1.00 | 1 | 0.000513 | A
$\cdots\cdots$ | | | | |
346 | 2455293.33020 | 43246.83 | 43247 | –0.012964 | W
347 | 2454130.06930 | 28463.42 | 28463 | 0.033265 | J
$\cdots\cdots$ | | | | |
384 | 2455610.03529 | 47271.71 | 47272 | –0.022647 | J
385 | 2455610.11388 | 47272.71 | 47273 | –0.022744 | J
References:
Arentoft, T., Sterken, C., 2000, A&A, 354, 589
Arentoft, T., et al., 2001a, A&A, 374, 1056
Arentoft, T., et al., 2001b, A&A, 378, L33
Arentoft, T., Sterken, C., 2002, ASP Conf. Ser., 256, 79, in Observational
Aspects of Pulsating B- & A Stars.,
Hintz, E. G., Joner, M. D., Kim, C., 1998, PASP, 110, 689
Lampens, P., 1985, IBVS, 2794
Wils, P., Hambsch, F. J., Lampens, P., et al., 2010, IBVS, 5928
Wils, P., Hambsch, F. J., Robertson, C. W., et al., 2011, IBVS, 5977
Zhou, A.-Y., Liu, Z.-L., 2003, AJ, 126, 2462
|
arxiv-papers
| 2011-09-18T03:23:52 |
2024-09-04T02:49:22.251320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pongsak Khokhuntod, Ai-Ying Zhou, Chayan Boonyarak, Shi-Yang Jiang",
"submitter": "Ai-Ying Zhou",
"url": "https://arxiv.org/abs/1109.3840"
}
|
1109.3868
|
# Carrier-induced modulation of radiation by a gated graphene
M.V. Strikha F.T. Vasko ftvasko@yahoo.com Institute of Semiconductor
Physics, NAS of Ukraine, Pr. Nauky 41, Kyiv, 03028, Ukraine
###### Abstract
The modulation of the transmitted (reflected) radiation due to change of
interband transitions under variation of carriers concentration by the gate
voltage is studied theoretically. The calculations were performed for strongly
doped graphene on high-$\kappa$ (Al2O3, HfO2, AlN, and ZrO2) or SiO2
substrates under normal propagation of radiation. We have obtained the
modulation depth above 10% depending on wavelength, gate voltage (i.e.
carriers concentration), and parameters of substrate. The graphene -
dielectric substrate - doped Si (as gate) structures can be used as an
effective electrooptical modulator of near-IR and mid-IR radiation for the
cases of high-$\kappa$ and SiO2 substrates, respectively.
###### pacs:
78.67.Wj, 42.79.Hp
## I Introduction
The essential feature of graphene’s optical properties is its substantial
interaction with radiation in the wide spectral region, from far-$IR$ up to
$UV$, due to effective interband transitions (see reviews 1 ). The enhancement
of this responce due to interference permits one to make the graphene on
dielectric substrate visible 2 . The other example of the exceptional optical
properties is the graphene-based saturable absorber for ultrafast lasers in
the telecommunication spectral region 3 . Besides this, the carriers
contribution modifies essentially the graphene response due to the Pauli
blocking effect, when absorption is supressed at
$\hbar\omega/2<\varepsilon_{F}$, where $\omega$ is the frequency of radiation,
$\varepsilon_{F}$ is the Fermy energy, see experimental data and discussion in
Refs. 4 and 5. Recently, modulation 5a and polarization 5b of IR radiation
were observed in the graphene structure integrated with an optical waveguide.
The efficiency of modulation can also be enhanced for the case of normal
propagation of radiation by the interference effect under an appropriate
thickness of substrate, see Figs. 1a and 1b. As a result, an improvement of
modulation for transmission and reflection coefficients of the graphene -
substrate - gate structure by a gate voltage takes place in contrast to the
case of graphene placed on a semi-infinete insulator. 5c In this paper, we
perform the calculation of the optical characteristics of such a structure,
and we discuss the conditions for realisation of the graphene-based modulator
in the near- and mid-IR spectral region.
Figure 1: (Color online) (a) Schematic view of graphene - high-$\kappa$
substrate - heavily doped Si structure ($V_{g}$ is gate voltage). (b) Normal
propagation of incident ($in$), reflected ($r$), and transmitted ($t$)
radiation through the graphene placed over substrate of thickness $d$ on doped
Si ($\epsilon_{s}$ and $\epsilon$ are the correspondent dielectric
permittivities). (b) Threshold wavelenthes $\lambda_{th}$ versus transverse
field $E_{\bot}$ for structures with different substrates: SiO2 (1), AlN (2),
Al2O3 (3), and HfO2 (4) (the results for ZrO2 are similar to curve (4)).
Dashed line is correspondent to the telecommunication wavelength, 1.55 $\mu$m.
(c) Spectral dependences of Re$\sigma_{\omega}$ and Im$\sigma_{\omega}$ (solid
and dotted curves) at room temperature for concentrations: 1012 cm-2 (1),
$2\times 10^{12}$ cm-2 (2), $4\times 10^{12}$ cm-2 (3).
At low temperatures, or high doping levels, the threshold frequency for the
jump of absorption is determined by the condition
$\hbar\omega_{th}=2\varepsilon_{F}\propto\sqrt{n}$, where concentration $n$
depends linearly on gate voltage, $V_{g}$, and is inverse to substrate
thickness, $d$, so that $n\propto V_{g}/d\equiv E_{\bot}$. The dependences of
the threshold wavelength $\lambda_{th}$ on homogeneous field $E_{\bot}$ are
presented in Fig. 1c for a number of substrates. One can see that the
modulation of near-IR radiation (1.55 $\mu$m) is possible at $E_{\bot}<$3
MV/cm for the high-$\kappa$ substrates (Al2O3, HfO2, AlN, and ZrO2 are
examined below), and for SiO2 substrate the applied field should be twice
stonger. The modulation of mid-IR radiation (10.6 $\mu$m) can be realised in
lower fieids, $E_{\bot}<$ 1 MV/cm. For the modulation of radiation in the
visible spectral range, a field $E_{\bot}$ comparable with the breakdown field
for the substrate under consideration is needed. The modulation depth can be
estimated through the amplitude of the absorbtion jump on the threshold, equal
to 2.3%. For the 5 layer graphene the modulation efficiency can exceed 10%,
taking into account that the interference on substrate influences this value.
The absorption edge spreads with the increase of temperature (see the real
part of dynamic conductivity $\sigma_{\omega}$ presented in Fig. 1d), and the
modulation efficiency decreases. At room temperature the effective modulation
for near-IR radiation only is possible, while the modulation of mid-IR
radiation needs cooling to temperatures about 77 K. The efficiency of
modulation obtained is comparable to results both in bulk semiconductors and
in heterostructures, see Refs. 9 and 10, respectively.
The analysis performed below is organized as follows. In Sect. II we describe
the carrier-induced modulation of the response of graphene and evaluate the
transmission and reflection coefficients. The efficiency of modulation versus
applied field and thickness of substrate is analyzed in Sect. III. Discussion
and concluding remarks are given in the last section.
## II Basic equations
The response of a doped graphene sheet on probe radiation is described by the
two-dimensional dynamic conductivity $\sigma_{\omega}$. In the colisionless
approximation, when $\omega$ exceeds any relaxation rate, the real and
imaginary parts of $\sigma_{\omega}$ are given by the expressions:
$\displaystyle{\rm
Re}\sigma_{\omega}\simeq\frac{e^{2}}{4\hbar}(1-f_{ep_{\omega}}-f_{hp_{\omega}}),$
(1) $\displaystyle{\rm
Im}\sigma_{\omega}\simeq\bar{\sigma}_{\omega}-\frac{e^{2}}{4\hbar
p_{\omega}}{\cal
P}\int\limits_{0}^{\infty}\frac{dpp^{2}}{p_{\omega}^{2}-p^{2}}\left(f_{ep}+f_{hp}\right).$
In a doped graphene, when $E_{\bot}$ is strong enough, the Pauli blocking
factor in ${\rm Re}\sigma_{\omega}$ is written through electron and hole
distribution functions, $f_{e,h~{}p}$, taken at $p_{\omega}=\hbar\omega/(2v)$,
where $v=10^{8}$ cm/s is the velocity of quasiparticles. Mention, that in the
absence of carriers ${\rm Re}\sigma_{\omega}$ does not depend on any material
parameters. 6 In the imagionary part of $\sigma_{\omega}$ we eliminate the
contribution of the non-doped graphene $\bar{\sigma}_{\omega}$, see Refs. 12
and 13, and the carriers contribution is connected with ${\rm
Re}\sigma_{\omega}$ through the Kramers-Kronig relation, where $\cal P$ means
the principal value of integral. The response of $N$-layer epitaxial graphene
9 is described by the total conductivity $N\sigma_{\omega}$.
We restrict ourselvers to the geometry of normal propagation of the incident
($in$-), reflected ($r$-), and transient ($t$-) waves through the structure
”N-layer graphene - dielectric substrate - doped Si”, as it is shown in Fig.
1a. The in-plane electric field $E(z)\exp(-i\omega t)$ is governed by the wave
equation: 1 ; 8
$\frac{d^{2}E(z)}{dz^{2}}+\epsilon_{\omega}(z)\left(\frac{\omega}{c}\right)^{2}E(z)=0,$
(2)
where $z\neq 0$ and the permittivity $\varepsilon_{\omega}(z)$ is equal to the
constant $\varepsilon_{s}$ in the substrate layer with the thickness $d$ (at
$0<z<d$), while in the thick Si the dispersion $\varepsilon_{\omega}$ should
be taken into account, see Ref. 15. The boundary conditions at $N$-layer
graphene sheet, where $z\rightarrow 0$, takes the form
$\left.\frac{dE(z)}{dz}\right|_{-0}^{+0}+i\frac{4\pi\omega}{c^{2}}N\sigma_{\omega}E(z=0)=0,~{}~{}~{}~{}\left.E(z)\right|_{-0}^{+0}=0.$
(3)
These expressions contain the contribution of surface current, proportional to
$N\sigma_{\omega}$, that determines the jump of $[dE(z)/dz]$, while $E(z)$ is
continuous. At the substrate-Si interface we use the two conditions of
continuity:
$\left.\frac{dE(z)}{dz}\right|_{d-0}^{d+0}=0,~{}~{}~{}~{}E(z)|_{d-0}^{d+0}=0.$
(4)
Outside of the graphene sheet the solution of electrodynamic problem (2)-(4)
should be written in the form
$E(z)=\left\\{\begin{array}[]{*{20}c}E_{in}e^{ik_{\omega}z}+E_{r}e^{-ik_{\omega}z}&z<0\\\
E_{+}e^{i\tilde{k}_{\omega}z}+E_{-}e^{-i\tilde{k}_{\omega}z}&0<z<d\\\
E_{t}e^{i\bar{k}_{\omega}z}&d<z\end{array}\right..$ (5)
Here the amlitudes for $in$\- and $r$-waves ($z<0$), $t$-wave ($z>d$), and for
the field $E_{\pm}$ in the dielectric substrate (at $0<z<d$) are introduced.
In Eq.(5) the wave vectors $k_{\omega}=\omega/c$ (to the left),
$\tilde{k}_{\omega}=\sqrt{\epsilon_{s}}\omega/c$ (to the right), and
$\bar{k}_{\omega}=\sqrt{\epsilon_{\omega}}\omega/c$ (in the dielectric
substrate) are introduced as well. Using the boundary conditions (3) and (4)
we get the system of linear equations for the amlitudes above. The solution of
such a system determines the transition and reflection coefficients,
$T_{\omega}$ and $R_{\omega}$, according to
$T_{\omega}=\sqrt{\epsilon_{\omega}}\frac{|E_{t}|^{2}}{E_{in}^{2}},~{}~{}~{}~{}~{}R_{\omega}=\frac{|E_{r}|^{2}}{E_{in}^{2}}.$
(6)
According to energy conservation law, which connects $T_{\omega}$ and
$R_{\omega}$ with the relative absorption coefficient $\xi_{\omega}$, one
obtains:
$T_{\omega}+R_{\omega}+\xi_{\omega}=1.$ (7)
As a result, variations of $T_{\omega}$ and $R_{\omega}$ are correlated due to
the Pauli blocking effect which leads to a jump of $\xi_{\omega}$.
The direct expressions for the transmission and reflection coefficients take
form
$T_{\omega}=\frac{4\sqrt{\epsilon_{\omega}}}{|{A_{\omega}^{(+)}}|^{2}},~{}~{}~{}~{}~{}~{}R_{\omega}=\left|\frac{A_{\omega}^{(-)}}{A_{\omega}^{(+)}}\right|^{2}.$
(8)
where $A^{(\pm)}_{\omega}$ are expressed through the dynamic conductivity and
the structure parameters according to
$\displaystyle
A_{\omega}^{(\pm)}=\sqrt{\epsilon}\cos\tilde{k}_{\omega}d-i\sqrt{\epsilon_{s}}\sin\tilde{k}_{\omega}d$
(9) $\displaystyle+\left(\frac{4\pi\sigma_{\omega}}{c}\pm
1\right)\left({\cos\tilde{k}_{\omega}d-i\sqrt{\varepsilon/\varepsilon_{s}}\sin\tilde{k}_{\omega}d}\right).$
In the case $\epsilon_{\omega}=\epsilon_{s}$ the oscillating factors eliminate
from (9), and Eqs. (8) are in agreement with the previous results. 1 ; 8
Taking into consideration the interference on the substrate, when
$\epsilon_{\omega}\neq\epsilon_{s}$, the spectral dependences of $T_{\omega}$
and $R_{\omega}$ are determined by carriers concentration (through variations
of $E_{\bot}$ or $V_{g}$), the dielectric substrate thickness $d$, and the
permittivities $\varepsilon_{s}$ and $\varepsilon_{\omega}$. We have neglected
a weak absorption in Si and used $\varepsilon_{s}$ for SiO2 and high-$\kappa$
dielectrics from Refs. 15 and 16, respectively.
## III Results
Performing the numerical integration in Eq. (1) and using Eqs. (8, 9) one
obtains the transmission and reflection coefficients. Below we analyze the
dependences of $T_{\omega}$ and $R_{\omega}$ on the applied field $E_{\bot}$,
which determines the carriers concentration, for the substrates of various
thickness on the base of high-$\kappa$ dielectrics or SiO2. The computations
were performed for near-IR and mid-IR spectral regions ($\lambda=$1.55 $\mu$m
and 10.6 $\mu$m).
Figure 2: (Color online) (a) Contour plot of transmission coefficient of
graphene over Al2O3 substrate as a function of applied field $E_{\bot}$ and
thickness $d$. (b) The same for reflection coefficient.
### III.1 High-$\kappa$ substrates
We examine first the modulation of the telecommunication range radiation,
$\lambda=$1.55 $\mu$m, by the structures of 5-layer graphene on high-$\kappa$
substrate at room temperature. Figures 2(a) and 2(b) show the contour plots of
$T$ and $R$ versus $E_{\bot}$ and $d$ for the case of Al2O3 substrate. One can
see, that the change of $T$ versus $E_{\bot}$ is $\sim$10% near the
transmission maximum, and the modification of $T$ by interference can be as
large as 0.3 if $d$ is in the range 0.1 - 0.5 $\mu$m. Similarly, the change of
$R$ versus $E_{\bot}$ does not exceed several %, while the modification of $R$
versus $d$ can be of 0.3 order. The maximum of $T$ corresponds the minimum of
$R$ and vise versa.
Figure 3: (Color online) Transmissivity $T$ at wavelength 1.55 $\mu$m versus
$E_{\bot}$ for different substrates: (a) Al2O3, (b) HfO2, (c) AlN, and (d)
ZrO2. Curves 1 - 5 are correspondent to the thicknesses $d=$0.08, 0.12, 0.16,
0.2, and 0.24 $\mu$m. Figure 4: (Color online) Reflectivity $R$ at wavelength
1.55 $\mu$m versus $E_{\bot}$ for the same substrates as in Fig. 3(a-d).
Curves 1 - 5 are correspondent to the thicknesses $d=$0.28, 0.32, 0.36, 0.4
and 0.44 $\mu$m.
Gate-voltage-induced modification of transmission is presented in Fig. 3 for
different high-$\kappa$ substrates at several thicknesses near the maximum of
$T$. Similar dependences for $R$ are presented in Fig. 4 near the reflection
maximum, corresponding to the greater thicknesses. Besides the essential
dependence on thickness, $T$ and $R$ depend also on high-frequency and static
permittivity of the materials under consideration. Therefore, the effective
modulation for Al2O3 and AlN occurs in the range $E_{\bot}\sim$8 - 12 MV/cm,
and for HfO2 and ZrO2 the weaker fields $E_{\bot}\sim$4 - 6 MV/cm are needed.
The modulation depth for transmission exceeds in several times the modulation
depth for reflection. The effective modulation interval, corresponding the
region of the jump in absorption, becomes narrower with the decrease of
temperature.
### III.2 SiO2 substrate
Now we are going to examine the structures ”graphene - SiO2 \- Si”, where the
permittivities are smaller. Therefore the effective modulation for
transmission of near-IR radiation takes place at $E_{\bot}\sim$25 - 35 MV/cm,
see Fig. 5(a), i.e. it needs stronger fields than the threshold of single-
layer graphene on SiO2 substrate $\sim$6 MV/cm, see Fig. 1(b). The modulation
of $R$ in the same range of fields does not exceed several percents, see Fig.
5(b). It should be noted, that these fields $E_{\bot}$ are of the same order
of values, as a breakdown field, therefore a possibility of modulation in this
case needs a special verification.
Figure 5: (Color online) Transmissivity (a) and reflectivity (b), $T$ and $R$,
at wavelength 1.55 $\mu$m versus $E_{\bot}$ for SiO2 substrate. Curves 1 - 5
are correspondent to the thicknesses $d=$0.24, 0.28, 0.32, 0.36 and 0.4
$\mu$m.
The effective modulation of transmission (over 5%, see Fig. 6(a)) takes place
in mid-IR spectral region, for $\lambda=$10.6 $\mu$m. The applied field in
this case does not exceed 2 MV/cm, but the substrate thickness should be
greater because of the increase of $\lambda$. The corresponding modulation of
reflection does not exceed several percents as well, see Fig. 6(b).
Figure 6: (Color online) Transmissivity (a) and reflectivity (b) versus
$E_{\bot}$ for five-layer graphene over SiO2 substrate at wavelength 10.6
$\mu$m and room temperature. Curves 1 - 8 are correspondent to the thicknesses
$d=$0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1 and 2.4 $\mu$m.
For the single layer graphene the modulation depth obviously can not be
greater, than 2.3%. However, this modulation of transmission occurs for much
lower fields, than in the previous cases under examination, 200 kV/cm, and the
jump region becomes rather narrow with the decrease of temperature, see Figs.
7a, b.
Figure 7: (Color online) Transmissivity at wavelength 10.6 $\mu$m versus
$E_{\bot}$ for a sigle layer graphene over SiO2 substrate at temperatures 77 K
(a) and 20 K (b). Curves 1 - 8 are correspondent to the same thicknesses as in
Fig. 6.
## IV Concluding remarks
The results obtained clearly demonstrate the possibility for realization of
the modulator for telecommunication spectral range ($\sim$1.5 $\mu$m) on the
base of multilayer graphene, placed over high-$\kappa$ substrate. The
effective modulation can be realized in this case for the applied fields
$\sim$5 MV/cm, while for the case of SiO2 substrate the field should be
$\sim$20 MV/cm comparable to a breakdown value. The modulation depth for
multi-layer ($N=$5-10) graphene can be as large as 10-20% ($\sim$2% per one
layer). The same efficiency of modulation for the mid-IR radiation ($\sim$10.6
$\mu$m) can be realized for the applied fields not stronger than 2 MV/cm for
the low temperature region.
The consideration performed takes into account the contribution of interband
transitions, described by the complex dynamic conductivity, and the radiation
interference on the structure ”vacuum - graphene - substrate - doped Si” for
the case of normal propagation of radiation. The modulation is determined by
the Pauli blocking effect under the change of the carriers concentration by
the gate voltage, therefore the time of modulation is governed by the
recombination time of the excess concentration of carriers, or by the time of
injection from contacts.
Next, we discuss the assumptions used in our calculations, which are rather
standart ones. The dynamic conductivity of the carriers in the spectral region
under examination is described properly with the use of the linear dispersion
law of carriers in graphene. The phenomenological description of the
dispersion of $Im\sigma_{\omega}$ due to the transitions from the valence band
(see Refs. 12 and 13) does not change the results essentially due to the
smallness of its contribution for the spectral range under consideration. The
study of the declined propagation of radiation is more complicated, and the
modulation efficiency in this case decreases. Moreover, the modulation
efficiency can also be reduced in mid-IR range due to the absorption of
radiation in the doped Si. It should be noted, that modulation of electron
concentration in the gate gives a weak contribution to IR response and can be
neglected in comparison to the Pauli blocking effect in graphene.
In conclusion, the results obtained should stimulate the experimental study of
the electrooptical modulation of the near-IR radiation by the structure of
multilayer graphene over high-$\kappa$ dielectric substrate at room
temperature and high gate voltages (concentrations). For the mid-IR spectral
region the effective modulation can be realized at low temperatures.
###### Acknowledgements.
This work was supported by the State Fundamental Research Fund of Ukraine
(Grant 40.2/069).
## References
* (1) M. Orlita and M. Potemski, Semicond. Sci. Technol. 25 063001 (2010); F. Bonaccorso, Z. Sun, T. Hasan and A. C. Ferrari, Nature Photonics 4, 611 (2010); L.A. Falkovsky, Phys. Usp. 51, 887 (2008).
* (2) P. Blake, K. S. Novoselov, A. H. Castro Neto, D. Jiang, R. Yang, T. J. Booth, A. K. Geim, and E. W. Hill, Appl. Phys. Lett. 91, 063124 (2007); D. S. L. Abergel, A. Russell, and V. I. Falko, Appl. Phys. Lett. 91, 063125 (2007).
* (3) Z. Sun, D. Popa, T. Hasan, F. Torrisi, F. Wang, E. J. R. Kelleher, J. C. Travers, V. Nicolosi, and A. C. Ferrari, Nano Res. 3, 653 (2010); F.T. Vasko, Phys. Rev. B 82, 245422 (2010).
* (4) Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nature Physics 4, 532 (2008).
* (5) F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, Science 320, 206 (2008).
* (6) M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature, 474, 64 (2011).
* (7) Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, Nature Photonics 5, 411 (2011).
* (8) L. A. Falkovsky and S. S. Pershoguba, Phys. Rev. B 76, 153410 (2007); L. A. Falkovsky, Phys. Usp. 51, 887 (2008).
* (9) M. Cardona, Modulation Spectroscopy (Academic Press, New York, 1969).
* (10) F.T. Vasko and A.V. Kuznetsov, Electron States and Optical Transitions in Semiconductor Heterostructures (Springer, New York, 1998).
* (11) T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B78, 085432 (2008).
* (12) M. Bruna and S. Borini, Appl. Phys. Lett. 94 031901 (2009).
* (13) M. V. Strikha and F.T. Vasko, Phys. Rev. B 81, 115413 (2010).
* (14) W. A. de Heer, C. Berger, X. Wu, M. Sprinkle, Y. Hu, M. Ruan, J. A. Stroscio, P. N. First, R. Haddon, B. Piot, C. Faugeras, M. Potemski, J.-S. Moon, J. Phys. D: Appl. Phys. 43 374007 (2010).
* (15) C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson J. Appl. Phys. 83, 3323 (1998).
* (16) M. V. Fischetti, D. A. Neumayer, E. A. Cartier, J. Appl. Phys. 90, 4587 (2001).
|
arxiv-papers
| 2011-09-18T13:08:39 |
2024-09-04T02:49:22.256704
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.V. Strikha and F.T. Vasko",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/1109.3868"
}
|
1109.3881
|
]Author to whom the correspondence should be addressed
# Ejection dynamics of a ring polymer out of a nanochannel
Junfang Sheng CAS Key Laboratory of Soft Matter Chemistry, Department of
Polymer Science and Engineering, University of Science and Technology of
China, Hefei, Anhui Province 230026, P. R. China Kaifu Luo [
kluo@ustc.edu.cn CAS Key Laboratory of Soft Matter Chemistry, Department of
Polymer Science and Engineering, University of Science and Technology of
China, Hefei, Anhui Province 230026, P. R. China
###### Abstract
We investigate the ejection dynamics of a ring polymer out of a cylindrical
nanochannel using both theoretical analysis and three dimensional Langevin
dynamics simulations. The ejection dynamics for ring polymers shows two
regimes like for linear polymers, depending on the relative length of the
chain compared with the channel. For long chains with length $N$ larger than
the critical chain length $N_{c}$, at which the chain just fully occupies the
nanochannel, the ejection for ring polymers is faster compared with linear
chains of identical length due to a larger entropic pulling force; while for
short chains ($N<N_{c}$), it takes longer time for ring polymers to eject out
of the channel due to a longer distance to be diffused to reach the exit of
the channel before experiencing the entropic pulling force. These results can
help understand many biological processes, such as bacterial chromosome
segregation.
###### pacs:
87.15.-v, 82.35.Lr, 87.15.H-
## I Introduction
The properties of a polymer confined in a nanochannel have attracted broad
interest Brochard ; Kremer ; Sotta ; Cifra ; Gong ; Klushin ; Yang because
they are of fundamental relevance in polymer physics and are also related to
many biological processes, such as double-stranded DNA genomes packaging
inside the phage capsid Smith , polymers transport through nanopore Luo1 ;
Luo2 and viruses injecting their DNA into a host cell Miller .
The importance of cyclic structures in biological macromolecular science is
strikingly demonstrated by the existence of circular DNA, cyclic peptides and
cyclic oligosaccharides and polysaccharides Cyclic . Ring closure of a polymer
is one of the important factors influencing its statistical mechanical
properties. Understanding the static and dynamic properties of ring polymer is
a challenging problem due to the difficulties inherent to a systematic
theoretical analysis of such objects constrained to a unique topology. The
scaling behavior of isolated, highly diluted, ring polymers has been studied.
des Cloizeaux Cloizeaux , Deutsch Deutsch and Grosberg Grosberg discussed
the effect of topological constraints on the properties of ring polymers, and
found that the topological constraint and the excluded volume have similar
effects. The radius of gyration for large single ring polymers obey the same
scaling relationship as that of linear chains Deutsch ; Grosberg , although
this is not true for ring polymers in a melt or ring polymer brushes Halverson
; Sakaue ; Reith .
Ring closure acts as an important role in a wide range of biophysical contexts
where DNA is constrained: segregation of the compacted circular genome of some
bacteria Jun , formation of chromosomal territories in cell nuclei Dorier ,
compaction and ejection of the knotted DNA of a virus Marenduzzo ; Matthews ,
migration of a circular DNA in an electrophoresis gel Obukhov or in a
nanochannel Reisner .
After three decades of intensive research, the conformational properties of a
self-avoiding polymer chain confined in a slit or in a cylindrical nanochannel
are relatively well understood. Daoud ; Gennes ; Milchev10 ; Milchev11 ;
Arnold . However, a deeper understanding of the basic properties of ring
polymer in confined environments is a field in its infancy Persson ; Witz .
Only few studies have addressed semiflexible ring polymers. Ostermeir et al.
Frey investigated the internal structure of semiflexible ring polymers in
weak spherical confinement and found buckling and a conformational transition
to a figure eight form. Fritsche and Heermann Heermann examined the
conformational properties of a semiflexible ring polymer confined to different
geometrical constraints and found that the geometry of confinement plays a
important role in shaping the spatial organization of polymers. Most recently,
we have found the helix chain conformation of flexible ring polymers confined
to a cylindrical nanochannel, and demonstrated that the longitudinal size
along the channel for a ring polymer scales as $N\sigma(\sigma/D)^{2/3}$, the
same as that for a linear chain but with different prefactors. Here $D$ is the
radius of the channel, $N$ the chain length and $\sigma$ the Kuhn length of
the chain Sheng . We further gives the theoretical ratio value 0.561 of the
longitudinal size for a ring polymer and a linear chain of the same $N$.
As to the dynamics of the polymer under confinements, Milchev et al. Milchev10
have investigated the ejection of linear chain out of nanopore using Monte
carlo simulation and found that the ejection dynamics depends on the chain
length. Unlike its linear polymer counterpart, the dynamics of confined ring
polymers is still lacking, although many bimolecules are circular. To this
end, in this work we study the ejection dynamics of a ring polymer confined in
a nanochannel by means of analytical techniques and Langevin dynamics
simulations. The basic questions associated with this process are the
following: (a) what’s effect of the chain length and the channel length on the
ejection dynamics? (b) what’s the difference of the ejection dynamics for ring
polymers compared with the linear one? For a fixed channel, which one is
faster compared a ring polymer with a linear chain of the identical length?
We believe that this work is interesting and important for understanding
biological systems with more complexity, such as viruses injecting their DNA
into a host cell, the behavior of DNA inside phages or the spatial
organization of the bacterial nucleoid in E. coli.
## II Model and methods
In our numerical simulations, the polymer chains are modeled as bead-spring
chains of Lennard-Jones (LJ) particles with the Finite Extension Nonlinear
Elastic (FENE) potential. Excluded volume interaction between beads is modeled
by a short range repulsive LJ potential:
$U_{LJ}(r)=4\varepsilon[{(\frac{\sigma}{r})}^{12}-{(\frac{\sigma}{r})}^{6}]+\varepsilon$
for $r\leq 2^{1/6}\sigma$ and 0 for $r>2^{1/6}\sigma$. Here, $\sigma$ is the
diameter of a bead, and $\varepsilon$ is the depth of the potential. The
connectivity between neighboring beads is modeled as a FENE spring with
$U_{FENE}(r)=-\frac{1}{2}kR_{0}^{2}\ln(1-r^{2}/R_{0}^{2})$, where $r$ is the
distance between consecutive beads, $k$ is the spring constant and $R_{0}$ is
the maximum allowed separation between connected beads.
Figure 1: A schematic representation of the initial and the finial
conformation of a ring polymer ejection from a cylindrical channel in three
dimensions. Here, the channel diameter $D=5$, the channel height $h=20.5$ and
the polymer length $N=100$.
We consider a schematic representation as shown in Fig. 1, where a ring
polymer is confined in a cylindrical channel with one end sealed. The
nanochannel and the sealed surface are described by stationary particles
within distance $\sigma$ from one another which interact with the beads by the
repulsive Lennard-Jones potential. The particle positions of the nanochannel
and the sealed surface are not changed in the simulations.
In the Langevin dynamics simulation, each bead is subjected to conservative,
frictional, and random forces, respectively, with Allen
$m{\bf\ddot{r}}_{i}=-{\bf\nabla}({U}_{LJ}+{U}_{FENE})-\xi{\bf v}_{i}+{\bf
F}_{i}^{R}$. Here $m$ is the bead’s mass, $\xi$ is the friction coefficient,
${\bf v}_{i}$ is the bead’s velocity, and ${\bf F}_{i}^{R}$ is the random
force which satisfies the fluctuation-dissipation theorem. In the present
work, the LJ parameters $\varepsilon$, $\sigma$, and $m$ fix the system
energy, length and mass units respectively, leading to the corresponding time
scale $t_{LJ}=(m\sigma^{2}/\varepsilon)^{1/2}$ and force scale
$\varepsilon/\sigma$, which are of the order of ps and pN, respectively. The
dimensionless parameters in the model are then chosen to be $R_{0}=1.5$,
$k=15$, $\xi=0.7$.
In our model, each bead corresponds to a Kuhn length (twice of the persistence
length) of a polymer. For a single-stranded DNA (ssDNA), the persistence
length of the ssDNA is sequence and solvent dependent and varies in a wide
range, to our knowledge, usually from about 1 to 4 nm. We assume the value of
$\sigma\sim 2.8$ nm for a ssDNA containing approximately four nucleotide
bases. The average mass of a base in DNA is about 312 amu, so the bead mass
$m\approx 1248$ amu. We set $k_{B}T=1.2\varepsilon$, which means that the
interaction strength $\varepsilon$ is $3.39\times 10^{-21}$ J at actual
temperature 295 K. This leads to a time scale of 69.2 ps and a force scale of
1.2 pN. The Langevin equation is then integrated in time by a method described
by Ermak and Buckholz Ermak .
We initially fix the last monomer of the linear chain but anyone of the ring
polymer at the sealed bottom of the nanochannel, while the remaining monomers
are under thermal collisions described by the Langevin thermostat to obtain an
equilibrium configuration. In order to learn the mechanism of chain ejection
out of the nanochannel, the link of the monomer with the bottom of the channel
is removed, then the chain is released to diffuse along the channel. The
residence time $\tau$ is measured, once all monomers pass the opening at $x=h$
and leave the channel. Typically, we average our data over 700 independent
runs.
## III Results and discussion
### III.1 Scaling arguments
#### III.1.1 Longitudinal size of a polymer in infinitely long nanochannel
According to the blob picture, for a linear polymer confined in a infinitely
long three-dimensional nanochannel of diameter $D$, the chain will extend
along the channel axis forming a string of blobs of size $D$. The center of
the blob is on the axis of the nanochannel. For each blob, $D=Ag^{\nu}\sigma$
due to the dominant excluded volume effects, where $g$ is the number of
monomers in a blob, $\sigma$ is the Kuhn length of the chain, $\nu$ is the
Flory exponent in three dimensions, and $A$ is a constant. Thus, each blob
contains $g=(\frac{D}{A\sigma})^{\frac{1}{\nu}}$ monomers, and the number of
blobs is $n_{b}=N/g=N(\frac{A\sigma}{D})^{\frac{1}{\nu}}$. The free energy
cost for the chain confinement is proportional to the number of blobs, thus
the free energy in units of $k_{B}T$ is
$\mathcal{F}=B_{l}N(A\sigma/D)^{1/\nu}$, with $B_{l}$ being a constant. The
blob picture then predicts the longitudinal size of the linear chain to be
$R_{{\parallel},l}=n_{b}D=ND(\frac{A\sigma}{D})^{\frac{1}{\nu}}=(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}$.
Using $\nu=3/5$ in three dimensions, we obtain
$R_{\parallel,l}=(A\sigma)^{\frac{5}{3}}ND^{-\frac{2}{3}}.$ (1)
In order to model the the chain conformation for a ring polymer confined in a
nanochannel, we have extended the blob picture Sheng . For a ring polymer, the
chain will extend along the channel axis forming two strings of blobs of
$D/2$,the two strings of blobs show helix structure. For each blob of size
$D/2$, $D/2=Ag_{r}^{\nu}\sigma$ with $g_{r}$ being the the number of monomers
in a blob. Here, the same prefactor $A$ for ring polymers and linear chains is
due to the same solution environment. Thus, each blob contains
$g_{r}=(\frac{D}{2A\sigma})^{\frac{1}{\nu}}$ monomers, and the number of blobs
is $n_{b}=N/g_{r}=N(\frac{2A\sigma}{D})^{\frac{1}{\nu}}$. The free energy cost
in units of $k_{B}T$ is
$\mathcal{F}=B_{r}N(\frac{2A\sigma}{D})^{\frac{1}{\nu}}$, with $B_{r}$ being a
constant.
By geometrical analysis, the distance between two successive layers is
$\frac{\sqrt{2}}{4}D$, and so the total length occupied by blobs in the
channel is
$R_{\parallel,r}=\frac{\sqrt{2}}{4}D(\frac{N}{2g_{r}}-1)+\frac{1}{4}D+\frac{1}{4}D=D(\frac{\sqrt{2}}{8}\frac{N}{g_{r}}+\frac{1}{2}-\frac{\sqrt{2}}{4})$.
When $\frac{N}{g_{r}}$ is very large,
$R_{\parallel,r}\approx\frac{\sqrt{2}}{8}D\frac{N}{g_{r}}=\frac{\sqrt{2}}{8}DN(\frac{A\sigma}{D/2})^{\frac{1}{\nu}}=\frac{\sqrt{2}}{8}2^{\frac{1}{\nu}}(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}$.
Using $\nu=3/5$ in three dimensions, we obtain
$R_{\parallel,r}=\frac{\sqrt{2}}{8}2^{\frac{5}{3}}(A\sigma)^{\frac{5}{3}}ND^{-\frac{2}{3}}.$
(2)
Therefore, the longitudinal size along the channel for a ring polymer scales
as $R_{\parallel}\sim N\sigma(\sigma/D)^{2/3}$, the same as that for a linear
chain but with a different prefactor. The ratio of the longitudinal sizes
along the nanochannel (or the prefactors) for a ring polymer and a linear
chain is
$\frac{R_{\parallel,r}}{R_{\parallel,l}}=\frac{\sqrt{2}}{8}2^{\frac{5}{3}}=0.561.$
(3)
If using more accurate value of $\nu=0.588$, we have
$\frac{R_{\parallel,r}}{R_{\parallel,l}}=0.575$.
The simulation results Sheng confirm the above predictions and give
$(A\sigma)^{1/\nu}=1.367\pm 0.009$ for the parameters used in the model.
#### III.1.2 Ejection dynamics of a polymer confined in a nanochannel
Intuitively, for the ejection of a polymer out of a nanochannel, the dynamics
is controlled by the relative length of polymer compared with the channel
height $h$. There exists a critical polymer length $N_{c}$, where the polymer
just fully occupies the channel, namely $R_{\parallel}(N_{c})=h$. Thus, for
linear chains the critical length $N_{c,l}$ is
$N_{c,l}=\frac{h}{(A\sigma)^{\frac{5}{3}}D^{-\frac{2}{3}}},$ (4)
while for ring polymers the critical length $N_{c,r}$ reads
$N_{c,r}=\frac{4\sqrt{2}h}{2^{\frac{5}{3}}(A\sigma)^{\frac{5}{3}}D^{-\frac{2}{3}}}=\frac{1.782h}{(A\sigma)^{\frac{5}{3}}D^{-\frac{2}{3}}}.$
(5)
The ratio of the critical length for the ring polymers and the linear polymer
is
$\frac{N_{c,r}}{N_{c,l}}=\frac{4\sqrt{2}}{2^{\frac{5}{3}}}=1.782.$ (6)
Short chains with chain length of $N<N_{c}$ are initially fully confined in
the nanochannel while long polymers with chain length of $N>N_{c}$ initially
occupy the whole channel with several segments outside the channel exit. For
long chains with $N>N_{c}$, the ejection is a driven process where the pulling
force $f$ is from the entropy and is induced by already ejected monomers
Klushin . For short chains with $N<N_{c}$, polymer needs to move to the
channel exit by a diffusive process, and then experiences a pulling force as
for long chains.
We assume the ejection process to be quasi-equilibrium. For long chains, the
pulling force can be estimated from the free energy $\mathcal{F}$ of a chain
partially confined in the nanochannel with the innermost monomer being at
distance $x$ from the channel exit. For long linear chains,
$x=n(t)(A\sigma)^{1/\nu}D^{1-1/\nu}$ and the free energy
$\mathcal{F}=B_{l}n(t)(A\sigma/D)^{1/\nu}k_{B}T=B_{l}\frac{k_{B}T}{D}x$, with
$n(t)$ being the number of monomers inside the channel at time $t$. The
differential of the free energy allows an estimate of the pulling force
$f_{l}=B_{l}\frac{k_{B}T}{D}.$ (7)
It is worthy of noting that $f$ is independent of the tail length as well as
$h$, but inversely proportional to $D$. For long ring chains,
$x=n(t)\frac{\sqrt{2}}{8}2^{\frac{1}{\nu}}(A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}$
and the free energy
$\mathcal{F}=B_{r}n(t)(2A\sigma/D)^{1/\nu}k_{B}T=4\sqrt{2}B_{r}\frac{k_{B}T}{D}x$.
Thus, the pulling force
$f_{r}=4\sqrt{2}B_{r}\frac{k_{B}T}{D}.$ (8)
We further have the ratio of the pulling force for long ring polymers and
linear chains
$\frac{f_{r}}{f_{l}}=\frac{4\sqrt{2}B_{r}}{B_{l}},$ (9)
which is only determined by the universal prefactors for ring polymers and
linear chains.
During the ejection process, the pulling force induced by the tail is balanced
by the total friction. Namely, for long chains we have
$\xi n(t)\frac{dx}{dt}=-f,$ (10)
where $\xi$ is the friction coefficient per monomer. Taking into account the
relationship of $x(t)$ and $n(t)$, we obtain the ejection time
$\tau_{long,l}=\frac{\xi h^{2}}{2f_{l}(A\sigma)^{1/\nu}D^{1-1/\nu}}=\frac{\xi
h^{2}D^{1/\nu}}{2B_{l}(A\sigma)^{1/\nu}k_{B}T}$ (11)
for long linear chains, and
$\tau_{long,r}=\frac{2\sqrt{2}\xi
h^{2}}{f_{r}(2A\sigma)^{1/\nu}D^{1-1/\nu}}=\frac{\xi
h^{2}D^{1/\nu}}{2B_{r}(2A\sigma)^{1/\nu}k_{B}T}$ (12)
for long ring polymers. Therefore, the ratio of the ejection time for long
ring polymers and linear chains is
$\frac{\tau_{long,r}}{\tau_{long,l}}=\frac{B_{l}}{B_{r}2^{1/\nu}}=0.315\frac{B_{l}}{B_{r}},$
(13)
where $\nu=3/5$ is used.
As noted above, for short polymers ($N<N_{c}$), it undergoes a diffusive
process before the first segment exiting the channel, and subsequently the
ejection process driven by a pulling force. Accordingly, we divide the total
ejection time $\tau$ into two parts, $\tau_{1}$ for the diffusive process and
$\tau_{2}$ for the driven process.
For the the diffusive process of short linear chains, $\tau_{1}$ is
$\tau_{1,l}=\frac{(h-R_{\parallel,l})^{2}}{2D_{diff}}=\frac{N\xi[h-(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}]^{2}}{2k_{B}T},$
(14)
with $D_{diff}=\frac{k_{B}T}{N\xi}$ being the diffusion constant. In addition,
for the driven process $\tau_{2}$ can be written as
$\tau_{2,l}=\frac{\xi
R_{\parallel,l}^{2}D^{1/\nu}}{2B_{l}(A\sigma)^{1/\nu}k_{B}T}=\frac{\xi
N^{2}(A\sigma)^{\frac{1}{\nu}}D^{2-\frac{1}{\nu}}}{2B_{l}k_{B}T}.$ (15)
Here, $\tau_{2,l}$ is negligible compared to $\tau_{1,l}$ for quite short
chains, and then the ejection time $\tau_{l}\approx\tau_{1,l}$. Based on the
differential of the ejection time with $N$, $\frac{\partial\tau_{l}}{\partial
N}=[h-(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}][h-3(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}]$=0,
we obtain the critical chain length
$N_{c,l}=\frac{h}{(A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}}$ as shown in
Eq. (4) and another resolution
$N^{\ast}_{l}=N_{c,l}/3=\frac{h}{3(A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}},$
(16)
where the residence time $\tau_{l}$ reaches to its maximum value
$\tau_{max,l}=\frac{2\xi
h^{3}}{27(A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}k_{B}T}.$ (17)
For the the diffusive process of short ring chains, $\tau_{1}$ is
$\tau_{1,r}=\frac{(h-R_{\parallel,r})^{2}}{2D_{diff}}=\frac{N\xi[h-\frac{\sqrt{2}}{8}2^{\frac{1}{\nu}}(A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}]^{2}}{2k_{B}T},$
(18)
In addition, for the driven process $\tau_{2}$ can be written as
$\tau_{2,r}=\frac{\xi
R_{\parallel,r}^{2}D^{\frac{1}{\nu}}}{2B_{r}(2A\sigma)^{1/\nu}k_{B}T}=\frac{\xi
N^{2}(2A\sigma)^{\frac{1}{\nu}}D^{2-\frac{1}{\nu}}}{64B_{r}k_{B}T}.$ (19)
Again, $\tau_{2,r}$ is negligible compared to $\tau_{1,r}$ for quite short
chains, and then $\tau_{r}\approx\tau_{1,r}$. Based on the differential of the
ejection time with $N$, $\frac{\partial\tau_{r}}{\partial
N}=[h-\frac{\sqrt{2}}{8}(2A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}][h-\frac{3\sqrt{2}}{8}(2A\sigma)^{\frac{1}{\nu}}ND^{1-\frac{1}{\nu}}]=0$,
we obtain the critical chain length
$N_{c,r}=\frac{4\sqrt{2}h}{(2A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}}$ as
in Eq. (5) and another resolution
$N^{\ast}_{r}=N_{c,r}/3=\frac{4\sqrt{2}h}{3(2A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}},$
(20)
where the ejection time $\tau$ reaches to its maximum value
$\tau_{max,r}=\frac{8\sqrt{2}\xi
h^{3}}{27(2A\sigma)^{\frac{1}{\nu}}D^{1-\frac{1}{\nu}}k_{B}T}.$ (21)
Thus, we have
$\frac{N^{\ast}_{r}}{N^{\ast}_{l}}=\frac{\tau_{max,r}}{\tau_{max,l}}=\frac{4\sqrt{2}}{2^{\frac{1}{\nu}}}=1.782.$
(22)
### III.2 Simulation results
Figure 2: (a) Average ejection time $\tau$ as a function of the polymer length
$N$ for the channel height $h=20.5$ and different channel diameters. (b) The
same for channel diameter $D=7$ and different channel heights. The full lines
are plotted using Eqs. (18) and (19).
The average ejection time $\tau$ as a function of the ring polymer length $N$
for different channel diameters ($D=5,7$ and 9) at fixed channel height
$h=20.5$ and for different channel heights ($h=20.5,30.5$, and 40.5) at
channel diameter $D=7$ are shown in Fig. 2a and Fig. 2b, respectively. The two
pictures show that ejection time increases with the increase of channel
diameter and channel height. Moreover, we get a special polymer length
$N^{\ast}$ at which the ejection time meets its maximum. Fig. 3 shows the plot
of $N^{\ast}_{r}$ against $hD^{2/3}$ for different $D$ and $h$. All the data
points collapse on the same line, which is in agreement with the prediction in
Eq. (20). The line plotted in Fig. 4 proves the prediction in Eq. (21).
Figure 3: Scaling of $N_{c}$ at which the ejection time owns its maximum as a
function of $hD^{2/3}$ for different $h$ and $D$. Here, $h=20.5$, 30.5, and
40.5 and $D=5$, 7 and 9. Figure 4: Scaling of the maximum of ejection time of
different polymer sizes as a function of $h^{3}D^{2/3}$ for different $h$ and
$D$. Here, we use $D=5$, 7 and 9 for $h=30.5$ and 40.5, and $D=4$, 5, 7 and 9
for $h=20.5$. Figure 5: The ejection time $\tau_{long}$ as a function of
$h^{2}D^{5/3}$: (a) ring chain with $N=250,300$ ($D=5$, 7 and 9 for $h=20.5$,
30.5 and 40.5); (b) linear chain with $N=250$, 300 ($D=3$, 5 and 7 for
$h=20.5$ and 30.5), $N=350$, and 360 ($D=9$ for $h=40.5$).
As noted before, there exists a critical polymer length $N_{c}$ at which the
polymer just fully occupies the channel. Short chains ($N<N_{c}$) are
initially fully confined in the nanochannel while long polymers ($N>N_{c}$)
initially occupy the whole channel with several segments outside the channel
exit. From the platforms in Fig. 2, we obtain the ejection time $\tau_{long}$
for long polymers ($N>N_{c}$).
Fig. 5a and Fig. 5b show the scaling plot of $\tau_{long}$ with $h^{2}D^{5/3}$
for both ring polymers and linear chains, respectively. For different polymer
lengths, channel heights and channel diameters, all data points collapse on
the same line in Fig. 5a and Fig. 5b, respectively. These results confirm the
predictions in Eqs. (11) and (12). In addition, the slopes are 0.042 and 0.053
for ring polymer and linear chain, respectively. This indicates
$\frac{\tau_{long,r}}{\tau_{long,l}}=\frac{0.042}{0.053}=0.792$. Based on Eqs.
(11) and (12), we have $B_{r}=1.60$, $B_{l}=4.02$ and thus
$\frac{B_{r}}{B_{l}}=0.398$ using the parameters $\xi=0.7$, $T=1.2$ and
$(A\sigma)^{1/\nu}=1.367$. Moreover, we further obtain
$\frac{f_{r}}{f_{l}}=2.250$ through Eq. (9), which demonstrates that the
driving force induced by confinement for long ring polymers is larger than
that for linear chains. Using Eqs. (18) and (19) to fit curves in Fig. 2, we
find that the numerical results are qualitatively described by theoretical
findings.
Figure 6: Average ejection time as a function of the polymer length for the
linear chain and the ring polymer. Here, $D=7$, and $h=20.5$. The full line
plotted to fit the data points for linear chain is based on Eqs. (14) and
(15). Figure 7: The force $f$ exerted on the innermost monomer of the ring
polymer and the linear chain. Here, $N=300$, $D=3$, and $h=20.5$. Figure 8:
Squared distance $x^{2}(t)$ of the last monomer from the channel exit during
the ejection as a function of the time for the ring polymer and the linear
chain and different channel diameters. Here, $h=20.5$ and $N=250$.
To compare the ejection dynamics for ring polymers with that for linear
chains, we show the ejection time as a function of the chain length $N$ for
$D=7$ and $h=20.5$ in Fig. 6. One does see characteristic differences: for
short chains ($N<N_{c}$), it takes longer time for ring polymers to eject out
of the channel than that for linear chains; while for long chains ($N>N_{c}$),
linear chains need longer time. These findings are in agreement with the
predictions in Eqs. (13) and (22). Ring polymers has smaller $R_{\parallel}$
than that for linear chains of the same $N$ and thus ring polymers must
diffuse longer distance to reach the exit of the channel. When the chain
length is larger than the critical chain length ($N>N_{c}$), the force exerted
on the residual segments for ring polymer is larger than that for linear chain
due to the smaller blob size in the channel for ring polymers than that for
linear chains as predicted $\frac{f_{r}}{f_{l}}=2.250$. The platform of the
force at small time $t$ shown in Fig. 7 for both ring polymer and linear chain
confirms this prediction.
In Fig. 6, we find $\frac{N^{\ast}_{r}}{N^{\ast}_{l}}=\frac{34}{24}=1.417$,
$\frac{\tau_{max,r}}{\tau_{max,l}}=\frac{2383.031}{1637.134}=1.456$, which are
predicted as 1.782. The difference may be from the non-equilibrium process of
the ejection. In addition, $\frac{N_{c,r}}{N_{c,l}}=\frac{110}{64}=1.719$ and
it is predicted as 1.782 in Eq. (6).
The mean-squared distance $x^{2}(t)$ of the last monomer with respect to the
channel exit against elapsed time after the release of the last monomer is
shown in Fig. 8 for both ring polymers and linear chains. The lines plotted
according to the curves are based on the equation
$x^{2}(t)=x^{2}(0)-\frac{2f_{l}(A\sigma)^{1/\nu}D^{1-1/\nu}}{\xi}t$ for a
linear polymer and the equation
$x^{2}(t)=x^{2}(0)-\frac{f_{r}(2A\sigma)^{1/\nu}D^{1-1/\nu}}{2\sqrt{2}\xi}t$
for a ring polymer, which indicates that the Eq. (10) can correctly describe
the ejection dynamics.
Figure 9: Numbers of monomers $n(t)$ inside the channel normalized by its
value at $t=0$ as a function of the time for (a) ring polymers with different
lengths; (b) the ring polymer and the linear chain of length $N=100$. Here,
$D=3$ and $h=20.5$. Figure 10: The distribution of the ejection time: (a)
ring polymers of different chain lengths; (b) the ring polymer and the linear
chain of length $N=20$; (c) the ring polymer and the linear chain of length
$N=300$. Here, $D=3$ and $h=20.5$.
In order to know the details in the ejection process, we record the number of
residual monomers inside the channel in the total process, $n(t)/n(0)$
(normalized by its value at $t=0$). We see that the short ring polymer
experiences a diffusion process before it starts to eject out of the channel,
corresponding to the platform in the plot for $N=30$ as shown in Fig. 9a. When
the chain length $N>N_{c}$, the ejection process is faster for the ring
polymer than that for a linear one, which can also be inferred from the
portion of residual monomers at time $t$, as presented in Fig. 9b.
Fig. 10a shows the histograms of the ejection time for ring polymers with
different chain lengths. The ejection time distribution for polymer of length
$N=30$ has a long tail and is much wider than that for $N=50$. The ejection
time distributions for ring polymers and linear chains at both short and long
chain lengths are given in Fig. 10b and Fig. 10c, respectively. For short
chain $N=20$, it takes longer time for ring polymer to leave the channel than
that for linear chain, and the ejection time distribution for the ring polymer
is wider and has a long tail. For long chains $N=300$, however, the result is
opposite, reflecting the larger driving force for the ring polymer than that
for the linear chain.
Nature not only imposes geometrical constraints on biopolymers by confinement
through cell membrane, the cell nucleus or viral capsid, but also exploits the
advantages of certain underlying chain topologies, such as the ring structure.
In fact, E. coli has a rod-shaped geometry and its chromosome is not a linear
polymer but a circular one. Based on Monte Carlo simulations, Jun and Mulder
Jun addressed a basic physical issue associated with bacterial chromosome
segregation in rod-shaped cell-like geometry. By simulations of two ring
polymers, in the same setting as the linear ones and they found that two ring
polymers segregate more readily than linear ones in confinement. According to
our above theoretical analysis and simulation results, for ring polymers
confined in a cylindrical nanochannel the blob size for ring polymers is
smaller than that for linear polymers, which indicates that during the
chromosome segregation the driving force for ring polymers is larger than that
for linear one, leading to faster segregation.
## IV Conclusions
We investigate the ejection dynamics of a ring polymer out of a cylindrical
nanochannel using both theoretical analysis and three dimensional Langevin
dynamics simulations. The ejection dynamics for ring polymers shows two
regimes like for linear polymers, depending on the relative length of the
chain compared with the channel. For long chains with length $N$ larger than
the critical chain length $N_{c}$, at which the chain just fully occupies the
nanochannel, the ejection for ring polymers is faster compared with linear
chains of identical length due to a larger entropic pulling force; while for
short ($N<N_{c}$), it takes longer time for ring polymers to eject out of the
channel due to a longer distance to be diffused to reach the exit of the
channel before experiencing the entropic pulling force. These results can help
understand many biological processes.
Our results should enable a new understanding of the conformational statistics
and dynamics of confined ring biopolymers such as DNA. The concrete graph
about ring polymer confined in a nanochannel needs more deep study so as to
realize many complex problems in both biochemistry and theoretical study. Our
findings are of interest for (molecular) biological/biochemical, technology as
well as physics reasons.
###### Acknowledgements.
This work is supported by the National Natural Science Foundation of China
(Grant No. 21074126, 21174140), the Specialized Research Fund for the Doctoral
Program of Higher Education (Grant No. 20103402110032), and the “Hundred
Talents Program” of Chinese Academy of Science (CAS).
## References
* (1) F. Brochard-Wyart, P. G. de Gennes, J. Chem. Phys. 67, 52 (1977).
* (2) K. Kremer, K. Binder, J. Chem. Phys. 81, 6381 (1984).
* (3) P. Sotta, A. Lesne, J. M. Victor, J. Chem. Phys. 112, 1565 (2000).
* (4) P. Cifra, J. Chem. Phys. 131, 224903 (2009).
* (5) Y. Gong, Y. Wang, Macromolecules. 35, 7492 (2002).
* (6) L. I. Klushin, A. M. Skvortsov, H. P. Hsu, K. Binder, Macromolecules. 41, 5890 (2008).
* (7) Y. Yang, T. M. Burkhard, G. Gompper, Phys. Rev. E. 76, 011804 (2007).
* (8) D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, Nature 413, 748 (2001).
* (9) K. Luo, T. Ala-Nissila, S. C. Ying, J. Chem. Phys. 124, 034714 (2006).
* (10) K. Luo, T. Ala-Nissila, S. C. Ying, Phys. Rev. Lett. 100, 058101 (2008).
* (11) R. V. Miller, Sci. Am. 278, 66 (1998).
* (12) J. A. Semlyen, Cyclic Polymers, 2nd ed. (Springer, Dordrecht, 2000).
* (13) J. des Cloizeaux, J. Phys. Lett. 42, 433 (1981).
* (14) J. M. Deutsch, Phys. Rev. E. 59, R2539 (1999).
* (15) A. Y. Grosberg, Phys. Rev. Lett. 85, 3858 (2000).
* (16) J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204904 (2011).
* (17) T. Sakaue, Phys. Rev. Lett. 106, 167802 (2011).
* (18) D. Reith, A. Milchev, P. Virnau, and K. Binder, EPL 95, 28003 (2011).
* (19) S. Jun and B. Mulder, Proc. Natl. Acad. Sci. U.S.A. 103, 12388 (2006).
* (20) J. Dorier and A. Stasiak, Nucleic Acids Res. 37, 6316 (2009).
* (21) D. Marenduzzo and C. Micheletti, J. Mol. Biol. 330, 485 (2003).
* (22) R. Matthews, A. A. Louis, and J. M. Yeomans, Phys. Rev. Lett. 102, 088101 (2009).
* (23) S. P. Obukhov, M. Rubinstein, and T. Duke, Phys. Rev. Lett. 73, 1263 (1994).
* (24) W. Reisner, K. J. Morton, R. Riehn, Y. M. Wang, Z. Yu, M. Rosen, J. C. Sturm, S. Y. Chou, E. Frey, and R. H. Austin1, Phys. Rev. Lett. 94, 196101 (2005).
* (25) M. Daoud and P. G. de Gennes, J. Physique 38, 85 (1977).
* (26) P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1979).
* (27) A. Milchev, L. Klushin, A. Skvortsov and K. Binder, Macromolecules 43, 6877 (2010).
* (28) A. Milchev, J. Phys.: condens. Matter 23, 103101 (2011).
* (29) A. Arnold, B. Bozogui, D. Frenkel, B. Y. Ha, and S. Jun, J. Chem. Phys. 127, 164903 (2007).
* (30) F. Persson, P. Utko, W. Reisner, N. B. Larsen, and A. Kristensen, Nano Lett. 9, 1382 (2009).
* (31) G. Witz, K. Rechendorff, J. Adamcik, and G. Dietler, Phys. Rev. Lett. 106, 248301 (2011).
* (32) K. Ostermeir, K. Alim, and E. Frey, Phys. Rev. E 81, 061802 (2010); Soft Matter 6, 3467 (2010)
* (33) M. Fritsche, and D. Heermann, Soft Matter 7, xxx (2011), See Doi: 10.1039/c1sm05445g.
* (34) J. Sheng and K. Luo, to be published.
* (35) M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University, New York, 1987)
* (36) D. L. Ermak and H. Buckholz, J. Comput. Phys. 35, 169 (1980).
|
arxiv-papers
| 2011-09-18T15:44:17 |
2024-09-04T02:49:22.263157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junfang Sheng and Kaifu Luo",
"submitter": "Kaifu Luo",
"url": "https://arxiv.org/abs/1109.3881"
}
|
1109.4048
|
# Implementing Continuation based language in GCC
Shinji KONO, Kento YOGI
e-mail:kono@ie.u-ryukyu.ac.jp
Information Engineering, University of the Ryukyus
Nishihara-cyo 1, Okinawa, 903-01, Japan
(2008 Apr 13)
We have implemented C like Continuation based programming language.
Continuation based C, CbC was implemented using micro-C on various
architecture, and we have tried several CbC programming experiments. Here we
report new implementation of CbC compiler based on GCC 4.2.3. Since it
contains full C capability, we can use CbC and C in a mixture.
## 1 A Practical Continuation based Language
If CPS theory is successful, it should also be working well in practical area.
Our idea is simple. How about a programming language which has continuation
passing style only? How about it runs as fast as current GNU C compiler?
Instead of creating complete new programming language, we designed a lower
language of C, so called Continuation based C, here after CbC. Using CPS
transformation like method, we can compile C into CbC, that is, we have some
kind of backward compatibility.
We have implemented CbC using micro-C on various architecture, and we have
tried several CbC programming experiments. Here we report new partial
implementation of CbC compiler[5] based on GCC 4.2.3[1]. Since it contains
full C capability, we can use CbC and C in a mixture, so when call the mixture
C with C, here after CwC.
First we show CbC language overview.
## 2 Continuation based C
CbC’s basic programming unit is a code segment. It is not a subroutine, but it
looks like a function, because it has input and output. We can use C struct as
input and output interfaces.
struct interface1 { int i; };
struct interface2 { int o; };
__code f(struct interface1 a) {
struct interface2 b; b.o=a.i;
goto g(b);
}
In this example, a code segment `f` has `input a` and sends `output b` to a
code segment `g`. There is no return from code segment `b`, `b` should call
another continuation using `goto`. Any control structure in C is allowed in
CwC language, but in case of CbC, we restrict ourselves to use `if` statement
only, because it is sufficient to implement C to CbC translation. In this
case, code segment has one input interface and several output interfaces
(fig.1).
Figure 1: code
`__code` and parameterized global goto statement is an extension of
Continuation based C. Unlike `C--` [4]’s parameterized goto, we cannot goto
into normal C function.
### 2.1 Intermix with C
In CwC, we can go to a code segment from a C function and we can call C
functions in a code segment. So we don’t have to shift completely from C to
CbC. The later one is straight forward, but the former one needs further
extensions.
void *env;
__code (*exit)(int);
__code h(char *s) {
printf(s);
goto (*exit)(0),env;
}
int main() {
env = __environment;
exit = __return;
goto h("hello World\n");
}
In this hello world example, the environment of `main()` and its continuation
is kept in global variables. The environment and the continuation can be get
using `__environment`, and `__return`. Arbitrary mixture of code segments and
functions are allowed (in CwC). The continuation of `goto` statement never
returns to original function, but it goes to caller of original function. In
this case, it returns result 0 to the operating system.
## 3 What’s good?
CbC is a kind of high level assembler language. It can do several original C
language cannot do. For examples,
Thread Scheduler
Context Switch
Synchronization Primitives
I/O wait semantics
are impossible to write in C. Usually it requires some help of assembler
language such as `__asm` statement extension which is of course not portable.
### 3.1 Scheduler example
We can easily write these things in CbC, because CbC has no hidden information
behind the stack frame of C. A thread simply go to the scheduler,
goto scheduler(self, task_list);
and the scheduler simply pass the control to the next thread in the task
queue.
code scheduler(Thread self,TaskPtr list)
{
TaskPtr t = list;
TaskPtr e;
list = list->next;
goto list->thread->next(list->thread,list);
}
Of course it is a simulator, but it is an implementation also. If we have a
CPU resource API, we can write real multi CPU scheduler in CbC.
This is impossible in C, because we cannot access the hidden stack which is
necessary to switch in the scheduler. In CbC, everything is visible, so we can
switch threads very easily.
This means we can use CbC as an executable specification language of OS API.
### 3.2 Self Verification
Since we can write a scheduler in CbC, we can also enumerate all possible
interleaving of a concurrent program. We have implement a model checker in
CwC. CbC can be a self verifiable language[7].
SPIN[3] is a very reliable model checker, but it have to use special
specification language PROMELA. We cannot directly use PROMELA as an
implementation language, and it is slightly difficult to study its concurrent
execution semantics including communication ports.
There are another kind of model checker for real programming language, such as
Java PathFinder[2]. Java PathFinder use Java Virtual Machine (JVM) for state
space enumeration which is very expensive some time.
In CbC, state enumerator itself is written in CbC, and its concurrency
semantics is written in CbC itself. Besides it is very close to the
implementation. Actually we can use CbC as an implementation language. Since
enumerator is written in the application itself, we can perform abstraction or
approximation in the application specific way, which is a little difficult in
Java PathFinder. It is possible to handle JVM API for the purpose, although.
We can use CPS transformed CbC source code for verification, but we don’t have
to transform all of the source code, because CwC supports all C constructs.
(But not in C++… Theoretically it is possible with using cfront converter, it
should be difficult).
### 3.3 As a target language
Now we have GCC implementation of CbC, it runs very fast. Many popular
languages are implemented on top of C. Some of them uses very large switch
statement for the byte code interpreter. We don’t have to use these hacks,
when we use CbC as an implementation language.
CbC is naturally similar to the state charts. It means it is very close to UML
diagrams. Although CbC does not have Object Oriented feature such as message
passing nor inheritance, which is not crucial in UML.
## 4 Transformation (C2CbC)
Conversion from C to CbC is straight forward, but it generates a lot of code
segments. Since CbC does not have heap management itself, the stack area have
to be allocated explicitly.
We find GCC can perform better optimization in translated code segment. We
will discuss it later.
We have an easy implementation of C to CbC compilation, but it is not a
practical level, but we need good converter for backward compatibility.
We can also consider possible conversion from C++ to CbC. In this case, all
hidden operation in C++ should become explicit, for examples, object
allocations and deallocations in the stack, handling of auto pointer and so
on.
## 5 GNU CC implementation
So how to implement CwC in GCC. The idea itself is simple[6], forcing C tail
call elimination for all code segment.
But before GCC version 4.x, tail call elimination (here after TCE) is not so
cleanly implemented , it is very difficult to implement it. But in GCC 4.x,
basically TCE can be applied for all possible functions.
`__code` is implemented as a new type keyword in GCC. You may think `__code`
is an attribute of a function, which means that the function can call in tail
call elimination only.
Because of this implementation, we can actually call code segment as a normal
function call.
### 5.1 How to force tail call elimination
There many enable conditions for tail call elimination, for example, there
should be no statement after tail call, return value type have to be the same,
arguments size should be compatible, and so on. We find almost half of lines
in `calls.c` spends to check TCE possibilities.
Our conclusion is this. It is not practical to make sure to pass all the TCE
tests, instead, we write TCE only version of `expand_call()` separately in 783
lines.
4463 18527 145469 calls.c
expand_call() for function
783 2935 23651 cbc-goto.h
expand_cbc_goto() for code segment
All code segment has the same virtual argument size and void return type, that
is argument register or argument value in the memory is shared among all code
segments. This leads a problem.
### 5.2 Parallel Assignment
Consider the next code,
__code carg4(struct arg args0,struct arg args1,
int i, int j,int k,int l)
{
goto carg5(args1,args0,j,k,l,i);
}
In this case, simple sequential assignments does not work. It override `args1`
or `args0`. In normal function case, GCC simply give up TCE, and pushes all
arguments in new register or stack area. We are not allowed that. That is we
have to implement parallel assignment in the code segment goto.
This is done by simple copy overlapped arguments in a stack. We hope to
eliminate unnecessary copy during GCC optimization.
### 5.3 Not yet done
Currently we have not yet implemented goto with environment and `__return`,
`__environment`.
In some GCC 4.x supported architecture, TCE itself is not supported in special
case. Our method does not work for the architecture.
Since we made modifications on GCC compiler itself, our method is GCC version
sensitive. We have to do necessary modifications for coming new version of
GCC.
## 6 Result
Here is our bench mark program.
f0(int i) {
int k,j;
k = 3+i;
j = g0(i+3);
return k+4+j;
}
g0(int i) {
return h0(i+4)+i;
}
h0(int i) {
return i+4;
}
It is written in C, we perform CPS transformation in several steps by hands.
There are several optimization is possible.
/* straight conversion case (1) */
typedef char *stack;
struct cont_interface {
// General Return Continuation
__code (*ret)();
};
__code f(int i,stack sp) {
int k,j;
k = 3+i;
goto f_g0(i,k,sp);
}
struct f_g0_interface {
// Specialized Return Continuation
__code (*ret)();
int i_,k_,j_;
};
__code f_g1(int j,stack sp);
__code f_g0(int i,int k,stack sp) { // Caller
struct f_g0_interface *c =
(struct f_g0_interface *)(
sp -= sizeof(struct f_g0_interface));
c->ret = f_g1;
c->k_ = k;
c->i_ = i;
goto g(i+3,sp);
}
__code f_g1(int j,stack sp) { // Continuation
struct f_g0_interface *c =
(struct f_g0_interface *)sp;
int k = c->k_;
sp+=sizeof(struct f_g0_interface);
c = (struct f_g0_interface *)sp;
goto (c->ret)(k+4+j,sp);
}
__code g_h1(int j,stack sp);
__code g(int i,stack sp) { // Caller
struct f_g0_interface *c =
(struct f_g0_interface *)(
sp -= sizeof(struct f_g0_interface));
c->ret = g_h1;
c->i_ = i;
goto h(i+3,sp);
}
__code g_h1(int j,stack sp) {
// Continuation
struct f_g0_interface *c =
(struct f_g0_interface *)sp;
int i = c->i_;
sp+=sizeof(struct f_g0_interface);
c = (struct f_g0_interface *)sp;
goto (c->ret)(j+i,sp);
}
__code h(int i,stack sp) {
struct f_g0_interface *c =
(struct f_g0_interface *)sp;
goto (c->ret)(i+4,sp);
}
struct main_continuation {
// General Return Continuation
__code (*ret)();
__code (*main_ret)();
void *env;
};
__code main_return(int i,stack sp) {
if (loop-->0)
goto f(233,sp);
printf("#0103:%d\n",i);
goto (( (struct main_continuation *)sp)->main_ret)(0),
((struct main_continuation *)sp)->env;
}
This is awfully long, but it is straight forward. Several forward prototyping
is necessary, and we find strict prototyping is painful in CbC, because we
have to use many code segments to perform simple thing. CbC is not a language
for human, but for automatic generation, verification or IDE directed
programming.
We can shorten the result in this way.
/* little optimized case (3) */
__code f2_1(int i,char *sp) {
int k,j;
k = 3+i;
goto g2_1(k,i+3,sp);
}
__code g2_1(int k,int i,char *sp) {
goto h2_11(k,i+4,sp);
}
__code f2_0_1(int k,int j,char *sp);
__code h2_1_1(int i,int k,int j,char *sp) {
goto f2_0_1(k,i+j,sp);
}
__code h2_11(int i,int k,char *sp) {
goto h2_1_1(i,k,i+4,sp);
}
__code f2_0_1(int k,int j,char *sp) {
goto (( (struct cont_interface *)
sp)->ret)(k+4+j,sp);
}
__code main_return2_1(int i,stack sp) {
if (loop-->0)
goto f2_1(233,sp);
printf("#0165:%d\n",i);
goto (( (struct main_continuation *)sp)->main_ret)(0),
((struct main_continuation *)sp)->env;
}
In this example, CPS transformed source is faster than original function call
form. There are not so much area for the optimization in function call form,
because function call API have to be strict. CbC does not need standard call
API other than interface which is simply a struct and there are no need for
register save. (This bench mark is designed to require the register save).
Here is the result in IA32 architecture (Table.1). Micro-C is our previous
implementation in tiny C. `conv1 1` is function call. `conv1 2`, `conv1 3` is
optimized CPS transformed source.
| ./conv1 1 | ./conv1 2 | ./conv1 3
---|---|---|---
Micro-C | 8.97 | 2.19 | 2.73
GCC | 4.87 | 3.08 | 3.65
GCC (+omit) | 4.20 | 2.25 | 2.76
GCC (+fast) | 3.44 | 1.76 | 2.34
Table 1: Micro-C, GCC bench mark (in sec)
There are two optimization flag for GCC. `-fomit-frame-pointer` eliminates
frame pointer (%ebp). The frame pointer itself is useful in code segment, but
it generates unnecessary push and pop or `leave` instruction. Using `fastcall`
option, GCC ignore the standard call convention such as all argument have be
on stack in IA32. In Micro-C implementation, these optimization is naturally
implemented in code segment, so it is faster than GCC without these options.
But with these options, GCC is faster than Micro-C. Of course, in more complex
source, GCC’s complex optimization should work well.
## 7 Conclusion
We have designed and implemented Continuation based language for practical
use. We have partial implementation of CwC using GCC 4.2.3. Using suitable
optimized options CPS transformed source sometimes runs faster than original
function call version.
This gcc implementation should be portable on all architectures supporting
tail call elimination, but we have tested only on i386 now.
## References
* [1] Free Software Foundation, Inc. GCC, the GNU Compiler Collection, March 2008.
* [2] K. Havelund and T. Pressburger. Model checking java programs using java pathfinder, 1998.
* [3] Gerard J. Holzmann. The model checker SPIN. Software Engineering, Vol. 23, No. 5, pp. 279–295, 1997.
* [4] Norman Ramsey and Simon Peyton Jones. A single intermediate language that supports multiple implementations of exceptions. In ACM SIGPLAN 2000 Conference on Programming Language Design and Implementation, June 2000.
* [5] Shinji KONO. CbC, March 2008.
* [6] 河野 真治. 継続を基本とした言語CbCのgcc上の実装. 日本ソフトウェア科学会第19回大会論文集, Sep 2002.
* [7] 河野 真治 . 検証を自身で表現できるハードウェア、ソフトウェア記述言語 Continuation based C と¿、そのCell への応用. 電子通信学会VLD研究会, March 2008.
|
arxiv-papers
| 2011-09-19T14:51:27 |
2024-09-04T02:49:22.276750
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shinji Kono and Kento Yogi",
"submitter": "Shinji Kono Dr.",
"url": "https://arxiv.org/abs/1109.4048"
}
|
1109.4085
|
# Tropical geometries and dynamics of biochemical networks. Application to
hybrid cell cycle models.
V. Noel1, D. Grigoriev2, S. Vakulenko3 and O. Radulescu4
1IRMAR UMR 6625, University of Rennes 1, Rennes, France,
2 CNRS, Mathématiques, Université de Lille, 59655, Villeneuve d’Ascq, France,
3 Saint Petersburg State University of Technology and Design, St.Petersburg,
Russia,
4 DIMNP UMR CNRS 5235, University of Montpellier 2, Montpellier, France.
Abstract
We use the Litvinov-Maslov correspondence principle to reduce and hybridize
networks of biochemical reactions. We apply this method to a cell cycle
oscillator model. The reduced and hybridized model can be used as a hybrid
model for the cell cycle. We also propose a practical recipe for detecting
quasi-equilibrium QE reactions and quasi-steady state QSS species in
biochemical models with rational rate functions and use this recipe for model
reduction. Interestingly, the QE/QSS invariant manifold of the smooth model
and the reduced dynamics along this manifold can be put into correspondence to
the tropical variety of the hybridization and to sliding modes along this
variety, respectively
Keywords: systems biology, model reduction, hybrid models, tropical geometry
## 1 Introduction.
Systems biology develops biochemical dynamic models of various cellular
processes such as signalling, metabolism, gene regulation. These models can
reproduce complex spatial and temporal dynamic behavior observed in molecular
biology experiments. In spite of their complex behavior, currently available
dynamical models are relatively small size abstractions, containing only tens
of variables. This modest size results from the lack of precise information on
kinetic parameters of the biochemical reactions on one hand, and of the
limitations of parameter identification methods on the other hand. Further
limitations can result from the combinatorial explosion of interactions among
molecules with multiple modifications and interaction sites [DFF+07]. In
middle out modeling strategies small models can be justified by saying that
one looks for an optimal level of complexity that captures the salient
features of the phenomenon under study. The ability to choose the relevant
details and to omit the less important ones is part of the art of the modeler.
Beyond modeler’s art, the success of simple models relies on an important
property of large dynamical systems. The dynamics of multiscale, dissipative,
large biochemical models, can be reduced to that of simpler models, that were
called dominant subsystems [RGZL08, GRZ10, GR08]. Simplified, dominant
subsystems contain less parameters and are more easy to analyze. The choice of
the dominant subsystem depends on the comparison among the time scales of the
large model. Among the conditions leading to dominance and allowing to
generate reduced models, the most important are quasi-equilibrium (QE) and the
quasi-steady state (QSS) approximations [GRZ10]. In nonlinear systems,
timescales and together with them dominant subsystems can change during the
dynamics and undergo more or less sharp transitions. The existence of these
transitions suggests that a hybrid, discrete/continous framework is well
adapted for the description of the dynamics of large nonlinear systems with
multiple time scales [CDR09, NVR10, NVR11].
The notion of dominance can be exploited to obtain simpler models from larger
models with multiple separated timescales and to assemble these simpler models
into hybrid models. This notion is asymptotic and a natural mathematical
framework to capture multiple asymptotic relations is the tropical geometry.
Motivated by applications in mathematical physics [LM96], systems of
polynomial equations [Stu02], etc., tropical geometry uses a change of scale
to transform nonlinear systems into discontinuous piecewise linear systems.
The tropicalization is a robust property of the system, remaining constant for
large domains of parameter values; it can reveal qualitative stable features
of the system’s dynamics, such as various types of attractors. Thus, the use
of tropicalization to model large systems in molecular biology could be a
promising solution to the problem of incomplete or imprecise information on
the kinetic parameters.
In this paper we propose a method for reduction and hybridization of
biochemical networks. This method, based on tropical geometry, could be used
to automatically produce the simple models that are needed in middle-out
approaches of systems biology.
## 2 Biochemical networks with rational rate functions.
Systems biology models use the formalism of chemical kinetics to model
dynamics of cellular processes. We consider here that the molecules of various
species are present in sufficient large numbers and that stochastic
fluctuations are negligible as a consequence of the law of large numbers
and/or of the averaging theorem [CDR09]. We also consider that space transport
phenomena are sufficiently rapid such that the well stirred reactor hypothesis
is valid. In these conditions, the dynamics of the biochemical system can be
described by systems of differential equations. In chemical kinetics,
enzymatic reactions are often presented as indivisible entities characterized
by stoichiometry vectors and rate functions. However, each enzymatic reaction
can be decomposed into several steps that define the reaction mechanism. The
resulting stoichiometry and global rate depend on the mechanism. Several
methods were designed for calculating effective rates of arbitrarily complex
mechanisms. For linear mechanisms King and Altman [KA56] proposed a graphical
method to compute global rates; these are rational functions of the
concentrations (an example is the Michaelis-Menten equation). Yablovsky and
Lazman [LY08b] studied the same problem for non-linear mechanisms and found
that in this case the reaction rates are solutions of polynomial equations;
these can be solved by radicals in a few number of cases and can be calculated
by multi-variate hypergeometric series in general [LY08b]. Truncation of these
series to finite order leads to rational approximations of the reaction rates.
In chemical kinetics with rational reaction rates the concentration $x_{i}$ of
the $i$-th component follows the ordinary differential equation:
$\frac{dx_{i}}{dt}=P_{i}(\mathbold x)/Q_{i}(\mathbold x),$ (1)
where $P_{i}(\mathbold x)=\sum_{\alpha\in A_{i}}a_{i,\alpha}\mathbold
x^{\alpha}$, $Q_{i}(\mathbold x)=\sum_{\beta\in B_{i}}b_{i,\beta}\mathbold
x^{\beta}$, are polynomials and we have $1\leq i\leq n$. Here $\mathbold
x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{n}^{\alpha_{n}}$,
$\mathbold x^{\beta}=x_{1}^{\beta_{1}}x_{2}^{\beta_{2}}\ldots
x_{n}^{\beta_{n}}$, $a_{i,\alpha},b_{i,\beta}$, are nonzero real numbers, and
$A_{i},B_{i}$ are finite subsets of ${\mathbb{N}}^{n}$ called supports of
$P_{i}$ and $Q_{i}$.
A simple example of model with rational reaction rates is the minimal cell
cycle oscillator model proposed by Tyson [Tys91]. This example will be studied
throughout the paper. The dynamics of this nonlinear model that contains 5
species and 7 reactions is described by a system of 5 polynomial differential
equations:
$\displaystyle y_{1}^{\prime}$
$\displaystyle=k_{9}y_{2}-k_{8}y_{1}+k_{6}y_{3},$ $\displaystyle
y_{2}^{\prime}$ $\displaystyle=k_{8}y_{1}-k_{9}y_{2}-k_{3}y_{2}y_{5},$
$\displaystyle y_{3}^{\prime}$
$\displaystyle=k_{4}^{\prime}y_{4}+k_{4}y_{4}y_{3}^{2}/C^{2}-k_{6}y_{3},$
$\displaystyle y_{4}^{\prime}$
$\displaystyle=-k_{4}^{\prime}y_{4}-k_{4}y_{4}y_{3}^{2}/C^{2}+k_{3}y_{2}y_{5},$
$\displaystyle y_{5}^{\prime}$ $\displaystyle=k_{1}-k_{3}y_{2}y_{5},$ (2)
where $\displaystyle y_{1}+y_{2}+y_{3}+y_{4}=C.$
## 3 Hybridization and tropical geometry.
Tropical geometry is a new branch of algebraic geometry that studies the
asymptotic properties of varieties. While algebraic geometry deals with
polynomial functions, tropical geometry deals with piecewise linear functions
with integer directing slopes. Tropical geometry has a growing number of
applications in enumerative problems in nonlinear equation solving [Roj03],
statistics [PS04], traffic optimization [Aub10].
The logarithmic transformation $u_{i}=logx_{i},\,1\leq i\leq n$, well known
for drawing graphs on logarithmic paper, plays a central role in tropical
geometry [Vir08]. By abus de langage, here we call logarithmic paper the image
of ${\mathbb{R}}^{n}_{+}$ by the logarithmic transformation, even if $n>2$.
Monomials $M(\mathbold x)=a_{\alpha}\mathbold x^{\alpha}$ with positive
coefficients $a_{\alpha}>0$, become linear functions,
$logM=loga_{\alpha}+<\alpha,log(\mathbold x)>$, by this transformation.
Furthermore, the euclidian distance on the logarithmic paper is a good measure
of separation (see next section).
Litvinov and Maslov [LMS01, LM96] proposed a heuristic (correspondence
principle) allowing to transform mathematical objects (integrals, polynomials)
into their quantified (tropical) versions. According to this heuristic, to a
polynomial with positive real coefficients $\sum_{\alpha\in
A}a_{\alpha}\mathbold x^{\alpha}$, one associates the max-plus polynomial
$max_{\alpha\in A}\\{log(a_{\alpha})+<log(\mathbold x),\alpha>\\}$.
We adapt this heuristic to associate a piecewise-smooth hybrid model to the
systems of rational ODEs (1).
###### Definition 1
We call tropicalization of the smooth ODE system (1) the following piecewise-
smooth system:
$\frac{dx_{i}}{dt}=s_{i}exp[max_{\alpha\in
A_{i}}\\{log(|a_{i,\alpha}|)+<\mathbold u,\alpha>\\}-max_{\beta\in
B_{i}}\\{log(|b_{i,\beta}|)+<\mathbold u,\beta>\\}],$ (3)
where $\mathbold u=(logx_{1},\ldots,logx_{n})$,
$s_{i}=sign(a_{i,\alpha_{max}})sign(b_{i,\beta_{max}})$ and
$a_{i,\alpha_{max}},\,\alpha_{max}\in A_{i}$ (respectively,
$b_{i,\beta_{max}},\,\beta_{max}\in B_{i}$) denotes the coefficient of a
monomial of the numerator (respectively, of the denominator) for which the
maximum occurring in (3) is attained.
In a different notation this reads:
$\frac{dx_{i}}{dt}=Dom\\{a_{i,\alpha}\mathbold x^{\alpha}\\}_{\alpha\in
A_{i}}/Dom\\{b_{i,\beta}\mathbold x^{\beta}\\}_{\alpha\in B_{i}},$ (4)
where $Dom\\{a_{i,\alpha}\mathbold x^{\alpha}\\}_{\alpha\in
A_{i}}=sign(a_{i,\alpha_{max}})exp[max_{\alpha\in
A_{i}}\\{log(|a_{i,\alpha}|)+<\mathbold u,\alpha>\\}]$.
Finally, the tropicalization can be written with Heaviside functions:
$\frac{dx_{i}}{dt}=\frac{\sum_{\alpha\in A_{i}}a_{i,\alpha}\mathbold
x^{\alpha}\prod_{\alpha^{\prime}\neq\alpha}\theta(<\alpha-\alpha^{\prime},log(\mathbold
x)>+log(|a_{i,\alpha}|)-log(|a_{i,\alpha^{\prime}}|))}{\sum_{\beta\in
B_{i}}b_{i,\beta}\mathbold
x^{\beta}\prod_{\beta^{\prime}\neq\beta}\theta(<\beta-\beta^{\prime},log(\mathbold
x)>+log(|b_{i,\beta}|)-log(|b_{i,\beta^{\prime}}|))},$ (5)
where $\theta(x)=1$ if $x>0$, $0$ if not.
The following definitions are standard and will be used throughout the paper:
###### Definition 2
The Newton polytope of a polynomial $P(\mathbold x)=\sum_{\alpha\in
A}a_{\alpha}\mathbold x^{\alpha}$ is defined as the convex hull of the support
of $P$, $New(P)=conv(A)$.
###### Definition 3
The max-plus polynomial $P^{\tau}(\mathbold
x)=max\\{log|a_{\alpha}|+<\alpha,log(\mathbold x)>\\}$ is called the
tropicalization of $P(\mathbold x)$. The logarithmic function is defined as
$log(\mathbold x):{\mathbb{R}}_{+}^{n}\to{\mathbb{R}}^{n}$, $log(\mathbold
x)_{i}=log(x_{i})$.
###### Definition 4
The set of points $\mathbold x\in{\mathbb{R}}^{n}$ where $P^{\tau}(\mathbold
x)$ is not smooth is called tropical variety. Alternative names are used such
as logarithmic limit sets, Bergman fans, Bieri-Groves sets, or non Archimedean
amoebas [PT05].
In two dimensions, a tropical variety is a tropical curve made of several
half-lines (tentacles) and finite intervals [Mik]. A tropical line corresponds
to only three monomials and is made of three half lines sharing a common
point. The tentacles and the intervals of the tropical variety are orthogonal
to the edges and point to the interior of the Newton polygon [PT05] (see
Fig.1).
## 4 Dominance and separation.
The above heuristic is related to the notion of dominance. Actually we have
replaced each polynomial in the rational function by the dominant monomial.
Dominance of monomials has an asymptotic meaning inside cones of the
logarithmic paper. For instance ${\mathbold x}^{\alpha}$ dominates ${\mathbold
x}^{\beta}$ on the half plane $<log(\mathbold x),\alpha-\beta>>0$ of the
logarithmic paper. We have ${\mathbold x}^{\beta}/{\mathbold x}^{\alpha}\to 0$
when the limit is taken along lines in this half plane.
For practical applications, we would also need a finite scale notion of
dominance.
Let $M_{1}(\mathbold x)=a_{\alpha_{1}}\mathbold x^{\alpha_{1}}$ and
$M_{2}(\mathbold x)=a_{\alpha_{2}}\mathbold x^{\alpha_{2}}$ be two monomials.
We define the following binary relations:
###### Definition 5 (Separation)
$M_{1}$ and $M_{2}$ are separated on a domain $D\subset R^{n}_{+}$ at a level
$\rho>0$ if $|log(|a_{\alpha_{1}}|\mathbold
x^{\alpha_{1}})-log(|a_{\alpha_{2}}|\mathbold x^{\alpha_{2}})|>\rho$ for all
$\mathbold x\in D$.
On logarithmic paper, two monomials are separated on the domain $D$, if $D$ is
separated by the euclidian distance $\rho$ from the hyperplane $<log(\mathbold
x),\alpha_{1}-\alpha_{2}>=log|a_{\alpha_{2}}|-log|a_{\alpha_{1}}|$.
###### Definition 6 (Dominance)
The monomial $M_{1}$ dominates the monomial $M_{2}$ at the level $\rho>0$,
$M_{1}\succ_{\rho}M_{2}$, if $log(|a_{\alpha_{1}}|\mathbold
x^{\alpha_{1}})>log(|a_{\alpha_{2}}|\mathbold x^{\alpha_{2}})+\rho$ for all
$\mathbold x\in D\subset{\mathbb{R}}_{+}^{n}$.
Dominance is a partial order relation on the set of multivariate monomials
defined on subsets of ${\mathbb{R}}_{+}^{n}$.
## 5 Dominance and global reduction of large models.
There are two simple methods for model reduction of nonlinear models with
multiple timescales: the quasi-equilibrium (QE) and the quasi-steady state
(QSS) approximations. As discussed in [GRZ10], these two approximations are
physically and dynamically distinct. Here we present a method allowing to
detect QE reactions and QSS species.
Like in [RGZL08], the first step of the method is to detect the ”slaved”
species, i.e. the species that obey quasi-steady state equations. These can be
formally defined by introducing the notion of imposed trace. Given the traces
$\mathbold x(t)$ of all the species, the imposed trace of the $i$-th species
is a real solution $x_{i}^{*}(t)$ of the polynomial equation
$P_{i}(x_{1}(t),\ldots,x_{i-1}(t),x_{i}^{*}(t),x_{i+1}(t),\ldots,x_{n}(t))=0$.
Eventually, there may be several imposed traced, because a polynomial equation
can have several real solutions.
###### Definition 7
We say that a species is slaved if the distance between the traces $x_{i}(t)$
and some imposed trace $x_{i}^{*}(t)$ is small on some interval, $sup_{t\in
I}|log(x_{i}(t))-log(x_{i}^{*}(t))|<\delta$, for some $\delta>0$ sufficiently
small. A species is globally slaved if $I=(T,\infty)$ for some $T\geq 0$.
Slaved species are good candidates for QSS species and this criterion was used
to identify QSS species in [RGZL08]. More generally, slaved species are
involved in rapid processes, but are not always QSS. Actually, two distinct
cases lead to slaved species.
Quasi-equilibrium. A system with fast, quasi-equilibrium reactions has the
following structure [GRZ10]:
$\frac{d\mathbold x}{dt}=\sum_{s,slow}R_{s}(\mathbold
x)\mathbold\gamma^{s}+\frac{1}{\epsilon}\sum_{f,fast}R_{f}(\mathbold
x)\mathbold\gamma^{f},$ (6)
where $\epsilon>0$ is a small parameter
$\mathbold\gamma^{s},\mathbold\gamma^{f}\in{\mathbb{Z}}^{n}$ are
stoichiometric vectors. The reaction rates $R_{s}(\mathbold x)$,
$R_{f}(\mathbold x)$ are considered rational functions of $\mathbold x$.
To separate slow/fast variables, we have to study the spaces of linear
conservation law of the initial system (6) and of the following fast
subsystem:
$\frac{d\mathbold x}{dt}=\frac{1}{\epsilon}\sum_{f,fast}R_{f}(\mathbold
x)\mathbold\gamma^{f}.$ (7)
In general, the system (6) can have several conservation laws. These are
linear functions $b^{1}(\mathbold x),\ldots,b^{m}(\mathbold x)$ of the
concentrations that are constant in time. The conservation laws of the system
(7) provide variables that are constant on the fast timescale. If they are
also conserved by the full dynamics, the system has no slow variables
(variables are either fast or constant). In this case, the dynamics of the
fast variables is simply given by Eq.(7). Suppose now that the system (7) has
some more conservation laws $b^{m+1}(\mathbold x),\ldots,b^{m+l}(\mathbold
x),$ that are not conserved by the full system (6). Then, these provide the
slow variables of the system. The fast variables are those $x_{i}$ such that
$(\mathbold\gamma^{f})_{i}\neq 0$, for some fast reaction $f$.
Let us suppose that the fast system (7) has a stable steady state that is a
solution of the QE equations (augmented by the conservation laws of the fast
system):
$\displaystyle\sum_{f,fast}R_{f}(\mathbold x)\mathbold\gamma^{f}=0,$ (8)
$\displaystyle b^{i}(\mathbold x)=C_{i},\quad 1\leq i\leq m+l.$ (9)
By classical singular perturbation methods [TVS85, Was65] one can show that
the fast variables can be decomposed as $x_{i}=\tilde{x}_{i}+\eta_{i}$ where
$\tilde{x}_{i}$ satisfy the QE equations (8) and $\eta_{i}={\cal
O}(\epsilon)$, meaning that the fast variables $x_{i}$ are slaved [GRZ10].
Let $P_{i}$, $\tilde{P}_{i}$ be the numerators of the rational functions
$\sum_{s,slow}R_{s}(\mathbold
x)\mathbold\gamma^{s}_{i}+\frac{1}{\epsilon}\sum_{f,fast}R_{f}(\mathbold
x)\mathbold\gamma^{f}_{i}$ and $\sum_{f,fast}R_{f}(\mathbold
x)\mathbold\gamma^{f}_{i}$, respectively. We call $\tilde{P}_{i}$ the pruned
version of $P_{i}$. When $\epsilon$ is small enough, the monomials of the
pruned version $\tilde{P}_{i}$ dominate the monomials of $P_{i}$. This
suggests a practical recipe for identifying QE reactions:
Algorithm 1
Step 1: Detect slaved species.
Step 2: For each $P_{i}$ corresponding to slaved species, compute the pruned
version $\tilde{P}_{i}$ by eliminating all monomials that are dominated by
other monomials of $P_{i}$.
Step 3: Identify, in the structure of $\tilde{P}_{i}$ the forward and reverse
rates of QE reactions. This step could be performed by recipes presented in
[SH10].
Quasi-steady state. In the most usual version of QSS approximation [SS89], the
species are split in two groups with concentration vectors $\mathbold x^{s}$
(“slow” or basic components) and $\mathbold x^{f}$ (“fast” or QSS species).
Quasi-steady species (also called radicals or fast intermediates) are low-
concentration, slaved species. Typically, QSS species are consumed (rather
than produced) by fast reactions. The small parameter $\epsilon$ used in
singular perturbation theory is now the ratio of small concentrations of fast
intermediates to the concentration of other species. After rescaling
$\mathbold x^{s}$ and $\mathbold x^{f}$ to order one, the set of kinetic
equations reads:
$\displaystyle\frac{d\mathbold x^{s}}{dt}$ $\displaystyle=\mathbold
W^{s}(\mathbold x^{s},\mathbold x^{f}),$ (10) $\displaystyle\frac{d\mathbold
x^{f}}{dt}$ $\displaystyle=(1/\epsilon)\mathbold W^{f}(\mathbold
x^{s},\mathbold x^{f}),$ (11)
where the functions $\mathbold W^{s}$, $\mathbold W^{f}$ and their derivatives
are of order one ($0<\epsilon<<1$).
Let us suppose that the fast dynamics (11) has a stable steady state. The
standard singular perturbation theory[TVS85, Was65] provides the QSS algebraic
condition $\mathbold W^{f}(\mathbold x^{s},\mathbold x^{f})=0$ which means
that fast species $\mathbold x^{f}$ are slaved. These equations, together with
additional balances for $x^{f}$ (conservation laws) are enough to deduce the
fast variables $x^{f}$ as functions of the slow variables $x^{s}$ and to
eliminate them [YBGE91, LY08a, RGZL08]. The slow dynamics is given by Eq.(10).
In networks with rational reaction rates the components of $\mathbold
W^{f}(\mathbold x^{s},\mathbold x^{f})$ are rational functions. Like for QE we
can define $P_{i}$ as numerators of $\mathbold W^{f}_{i}$. The difference
between QSS conditions with respect to QE situation is that in the pruned
polynomial $\tilde{P}_{i}$ one can no longer find forward and backward rates
of QE reactions, ie the step 3 of Algorithm 1 will not identify reversible
reactions. Alternatively, one can realize that slaved species can have
relatively large concentrations, in which case they are not QSS species.
However, it is difficult to say which concentration value separates QSS from
non QSS species among slaved species, hence the former, qualitative criterion
is better.
## 6 Sliding modes of the tropicalization.
A notable phenomenon resulting from tropicalization is the occurrence of
sliding modes. Sliding modes are well known for ordinary differential
equations with discontinuous vector fields [FA88]. In such systems, the
dynamics can follow discontinuity hypersurfaces where the vector field is not
defined.
The conditions for the existence of sliding modes are generally intricate.
However, when the discontinuity hypersurfaces are smooth and $n-1$ dimensional
($n$ is the dimension of the vector field) then the conditions for sliding
modes read:
$<n_{+}(x),f_{+}(x)><0,\quad<n_{-}(x),f_{-}(x)><0,\quad x\in\Sigma,$ (12)
where $f_{+},f_{-}$ are the vector fields on the two sides of $\Sigma$ and
$n_{+}=-n_{-}$ are the interior normals.
Let us consider that the smooth system (1) has quasi-steady state species or
quasi-equilibrium reactions. In this case, the fast dynamics reads:
$\frac{dx_{i}}{dt}=\frac{1}{\epsilon}\tilde{P}_{i}(\mathbold
x)/\tilde{Q}_{i}(\mathbold x),\quad i\quad\text{fast},$ (13)
where $\tilde{P}_{i}(\mathbold x)$, $\tilde{Q}_{i}(\mathbold x)$ are pruned
versions of $P_{i}$, $Q_{i}$, and $\epsilon$ is the small, singular
perturbation parameter.
For sufficiently large times, the fast variables satisfy (to ${\cal
O}(\epsilon)$):
$\tilde{P}_{i}(\mathbold x)=0,\quad i\quad\text{fast}.$ (14)
The pruned polynomial is usually a fewnomial (contains a small number of
monomials). In particular, let us consider the case when only two monomials
remain after pruning, $\tilde{P}_{i}(\mathbold x)=a_{1}\mathbold
x^{\alpha_{1}}+a_{2}\mathbold x^{\alpha_{2}}$. Then, the equation (14) defines
a hyperplane $S=\\{<log(\mathbold
x),\alpha_{1}-\alpha_{2}>=log(|a_{1}|/|a_{2}|)\\}$. This hyperplane belongs to
the tropical variety of $\tilde{P}_{i}$, because it is the place where the
monomial $\mathbold x^{\alpha_{1}}$ switches to $\mathbold x^{\alpha_{2}}$ in
the max-plus polynomial defined by $\tilde{P}_{i}$. For $\epsilon$ small, the
QE of QSS conditions guarantee the existence of an invariant manifold
${\mathcal{M}}_{\epsilon}$, whose distance to $S$ is ${\cal O}(\epsilon)$.
Let $n_{+},n_{-}$ defined as above and let
$(f_{+})_{i}=\frac{1}{\tilde{Q}_{i}(\mathbold x)}a_{1}\mathbold
x^{\alpha_{1}}$, $(f_{-})_{i}=\frac{1}{\tilde{Q}_{i}(\mathbold
x)}a_{2}\mathbold x^{\alpha_{2}}$,
$f_{i}=\frac{1}{\epsilon}[(f_{+})_{i}+(f_{-})_{i}]$ for $i$ fast,
$(f_{+})_{j}=(f_{-})_{j}=f_{j}=\frac{\tilde{P}_{j}}{\tilde{Q}_{j}}$, for $j$
not fast. Then, the stability conditions for the invariant manifold read
$<n_{+}(x_{+}),f(x_{+})><0$, $<n_{-}(x_{-}),f(x_{-}))><0$, where $x_{+},x_{-}$
are close to ${\mathcal{M}}_{\epsilon}$ on the side towards which points
$n_{+}$ and $n_{-}$, respectively. We note that
$|(f_{+})_{i}(x_{+})|>|(f_{-})_{i}(x_{+})|$. Thus,
$<n_{+},f>=\frac{1}{\epsilon}(n_{+})_{i}[(f_{+})_{i}+(f_{-})_{i}]+\sum_{j,notfast}(n_{+})_{j}(f_{+})_{j}$
and
$<n_{+},f_{+}>=\frac{1}{\epsilon}(n_{+})_{i}(f_{+})_{i}+\sum_{j,notfast}(n_{+})_{j}(f_{+})_{j}$.
Thus, if $<n_{+},f><0$, then for $\epsilon$ small enough
$(n_{+})_{i}(f_{+})_{i}<0$ and $<n_{+},f_{+}><0$ because
$<n_{+},f>><n_{+},f_{+}>$. Similarly, we show that $<n_{-},f><0$ implies
$<n_{-},f_{-}><0$. This proves the following
###### Theorem 1
If the smooth dynamics obeys QE or QSS conditions and if the pruned polynomial
$\tilde{P}$ defining the fast dynamics is a 2-nomial, then the QE or QSS
equations define a hyperplane of the tropical variety of $\tilde{P}$. The
stability of the QE of QSS manifold implies the existence of a sliding mode of
the tropicalization along this hyperplane.
The converse result, i.e. deducing the stability of the QE/QSS manifold from
the existence of a sliding mode on the tropical variety may be wrong. Indeed,
it is possible for a trajectory of the smooth system to be close to a
hyperplane of the tropical variety carrying a sliding mode and where the
QE/QSS equations are satisfied identically. However, as we will see in the
next section, this trajectory can leave the hyperplane sooner than the sliding
mode.
Figure 1: (top left) Detection of slaved species by comparing traces to
imposed traces: the species $y_{1},y_{2},y_{5}$ are slaved globally, the
species $y_{3},y_{4}$ are slaved on intervals $Q_{3}$,$Q_{4}$, respectively.
(top right) Comparison of monomials of the polynomial systems of quasi-steady
state equations. (bottom left) Newton polygons and inner normals of the
reduced two dimensional polynomial model. (bottom right) Phase portrait on
logarithmic paper of the reduced two dimensional model. We represent the two
tropical curves (the tripods graphs, a red and a blue one), the modes (smooth
vector fields within domains bordered by tropical curves tentacles), the
smooth and tropicalized limit cycles. The tropicalized cycle contains two
sliding modes $S_{3}$,$S_{4}$ corresponding to the intervals $Q_{3}$, $Q_{4}$
on which $y_{3}$, $y_{4}$ are quasi-stationary, respectively.
## 7 From smooth to hybrid models via reduction.
Starting with the system (2) we first reduce it to a simpler model. The
analysis of the model is performed for the values of parameters from [Tys91],
namely
$k_{1}=0.015,k_{3}=200,k_{4}=180,k_{4}^{\prime}=0.018,k_{6}=1,k_{7}=0.6,k_{8}=1000000,k_{9}=1000$;
In order to do that we generate one or several traces (trajectories)
$y_{i}(t)$. The smooth system has a stable periodic trace which is a limit
cycle attractor. We also compute the imposed traces $y_{i}^{*}(t)$ that are
solutions of the equations:
$\displaystyle k_{9}y_{2}(t)-k_{8}y_{1}^{*}(t)+k_{6}y_{3}(t)=0,$
$\displaystyle k_{8}y_{1}(t)-k_{9}y_{2}^{*}(t)-k_{3}y_{2}(t)y_{5}(t)=0,$
$\displaystyle
k_{4}^{\prime}y_{4}(t)+k_{4}y_{4}(t)y_{3}^{*2}(t)/C^{2}-k_{6}y_{3}^{*}(t)=0,$
$\displaystyle-
k_{4}^{\prime}y_{4}^{*}(t)-k_{4}y_{4}^{*}(t)y_{3}^{2}(t)/C^{2}+k_{3}y_{2}(t)y_{5}(t)=0,$
$\displaystyle k_{1}-k_{3}y_{2}(t)y_{5}^{*}(t)=0.$ (15)
We find that, for three species $y_{1}$,$y_{2}$, and $y_{5}$, the distance
between the traces $y_{i}^{*}(t)$ and $y_{i}(t)$ is small for all times which
means that these species are slaved on the whole limit cycle (Figure 1 top
left). Also, we have a global conservation law $y_{1}+y_{2}+y_{3}+y_{4}=C$,
that can be obtained by summing the first four differential equations in (2).
The three quasi-steady state equations for the three slaved species have to be
solved jointly with the global conservation law:
$\displaystyle k_{9}y_{2}-k_{8}y_{1}+k_{6}y_{3}=0,$ $\displaystyle
k_{8}y_{1}-k_{9}y_{2}-k_{3}y_{2}y_{5}=0,$ $\displaystyle
k_{1}-k_{3}y_{2}y_{5}=0,$ $\displaystyle y_{1}+y_{2}+y_{3}+y_{4}=C.$ (16)
Comparison of the monomials (for values of parameters as above) in this system
shows that $max(k_{8}y_{1},k_{9}y_{2})\succ k_{6}y_{3}$, and
$max(k_{8}y_{1},k_{9}y_{2})\succ k_{3}y_{2}y_{5}$ (Fig.1 top right) which
leads to the pruned system:
$\displaystyle k_{8}y_{1}-k_{9}y_{2}=0,$ $\displaystyle
k_{8}y_{1}-k_{9}y_{2}=0,$ $\displaystyle k_{1}-k_{3}y_{2}y_{5}=0,$
$\displaystyle y_{1}+y_{2}+y_{3}+y_{4}=C.$ (17)
The first two equations are identical and correspond to quasi-equilibrium of
the reaction between $y_{1}$ and $y_{2}$. The third equation means that
$y_{5}$ is a quasi-steady state species. The pruned system allows the
elimination of the variables $y_{1},y_{2},y_{5}$. The slow variable
$y_{12}=y_{1}+y_{2}$ demanded by the quasi-equilibrium condition (this is a
conservation law of the fast system) can be eliminated by using the global
conservation law.
We note that the dominance relations leading to the pruned equations were
found numerically in a neighborhood of the periodic trace. This means that QE
and QSS approximations are valid at least on the limit cycle. More global
testing of these relations will be presented elsewhere. Note that the system
(16) can be solved also without pruning. However, (16) has four independent
equations allowing to eliminate four of the five dynamic variables leading to
a one dimensional dynamical system. It turns out that the correct application
of the QE and QSS approximations has to use (17) and not (16).
After elimination, we obtain the following reduced differential-algebraic
dynamical system:
$\displaystyle y_{3}^{\prime}$
$\displaystyle=k_{4}^{\prime}y_{4}+k_{4}y_{4}y_{3}^{2}/C^{2}-k_{6}y_{3},$
$\displaystyle y_{4}^{\prime}$
$\displaystyle=-k_{4}^{\prime}y_{4}-k_{4}y_{4}y_{3}^{2}/C^{2}+k_{1},$
$\displaystyle y_{1}$ $\displaystyle=(C-y_{3}-y_{4})k_{9}/(k_{8}+k_{9}),$
$\displaystyle y_{2}$ $\displaystyle=(C-y_{3}-y_{4})k_{8}/(k_{8}+k_{9}),$
$\displaystyle y_{5}$
$\displaystyle=k_{1}(k_{8}+k_{9})/(k_{3}k_{8}(C-y_{3}-y_{4}).$ (18)
Now we tropicalize this reduced system. The tropicalization could have been
done on the initial system in which case the pruned equations (17) would
indicate that the reduced dynamics is a sliding mode of the tropicalized
system on the two dimensional hypersurface
$k_{8}y_{1}=k_{9}y_{2},k_{1}=k_{3}y_{2}y_{5},y_{1}+y_{2}+y_{3}+y_{4}=C$.
However, although the result (concerning the dynamics on the QE/QSS manifold)
should be the same, it is much handier to tropicalize the reduced system (18).
Indeed, the tropicalization of the full 5D system is difficult to visualize
and would also produce complex modes that can not be reduced to 2D (these
modes describe the fast relaxation to the QE/QSS manifold).
The resulting hybrid model reads:
$\displaystyle y_{3}^{\prime}$
$\displaystyle=Dom\\{k_{4}^{\prime}y_{4},k_{4}y_{4}y_{3}^{2}/C^{2},-k_{6}y_{3}\\},$
$\displaystyle y_{4}^{\prime}$
$\displaystyle=Dom\\{-k_{4}^{\prime}y_{4},-k_{4}y_{4}y_{3}^{2}/C^{2},k_{1}\\},$
or equivalently using Heaviside functions:
$\displaystyle y_{3}^{\prime}$
$\displaystyle=k_{4}^{\prime}y_{4}\theta(-h_{1}-2u_{3})\theta(h_{2}+u_{4}-u_{3})+\frac{k_{4}}{C^{2}}y_{4}y_{3}^{2}\theta(h_{1}+2u_{3})\theta(h_{1}+h_{2}+u_{4}+u_{3})$
$\displaystyle-
k_{6}y_{3}\theta(-h_{2}-u_{4}+u_{3})\theta(-h_{1}-h_{2}-u_{4}-u_{3}),$
$\displaystyle y_{4}^{\prime}$
$\displaystyle=-k_{4}^{\prime}y_{4}\theta(-h_{3}-2u_{3})\theta(-h_{4}+u_{4})-\frac{k_{4}}{C^{2}}y_{4}y_{3}^{2}\theta(h_{3}+2u_{3})\theta(h_{3}-h_{4}+2u_{3}+u_{4})$
(20) $\displaystyle
k_{1}\theta(h_{4}-u_{4})\theta(-h_{3}+h_{4}-2u_{3}-u_{4}),$
where $h_{1}=h_{3}=log(k_{4}/(k_{4}^{\prime}C^{2}))$,
$h_{2}=log(k_{4}^{\prime}/k_{6})$, $h_{4}=log(k_{1}/k_{4}^{\prime})$.
The Newton polygons of the polynomials
$k_{4}^{\prime}y_{4}+k_{4}y_{4}y_{3}^{2}/C^{2}-k_{6}y_{3}$ and
$-k_{4}^{\prime}y_{4}-k_{4}y_{4}y_{3}^{2}/C^{2}-k_{6}y_{3}$ are triangles
(Fig.1 bottom left). The two triangles share a common edge which is a
consequence of the fact that the reduced model have two reactions each one
acting on the two species. The tentacles of the two tropical curves (in red
and blue in Fig.1 bottom right) point in the same directions as the inner
normals to the edges of the Newton polygons (the corresponding equations are
$h_{1}+2u_{3}=0$, $h_{2}+u_{4}-u_{3}=0$, $h_{1}+h_{2}+u_{4}+u_{3}=0$ for one
and $h_{3}+2u_{3}=0$, $h_{4}+u_{4}=0$, $h_{3}-h_{4}+2u_{3}+u_{4}=0$ for the
other). These tentacles (half lines) decompose the positive quarter plane into
6 sectors corresponding to the 6 modes of the hybrid model.
In Fig.1 bottom right we have also represented the phase portrait of the
reduced model on logarithmic paper. The dynamical variables are
$u_{3}=log(y_{3})$ and $u_{4}=log(y_{4})$. The vector field corresponding to
$u_{3}^{\prime}=y_{3}^{\prime}/y_{3}$ and
$u_{4}^{\prime}=y_{4}^{\prime}/y_{4}$ was computed with the dominant monomials
in each plane sector as follows:
$\displaystyle u_{4}^{\prime}$
$\displaystyle=-k_{4}y_{3}^{2},u_{3}^{\prime}=-k_{6}\text{ for the mode 1},$
$\displaystyle u_{4}^{\prime}$
$\displaystyle=-k_{4}y_{3}^{2},u_{3}^{\prime}=k_{4}y_{3}y_{4}\text{ for the
mode 2},$ $\displaystyle u_{4}^{\prime}$
$\displaystyle=k_{1}y_{4}^{-1},u_{3}^{\prime}=k_{4}y_{3}y_{4}\text{ for the
mode 3},$ $\displaystyle u_{4}^{\prime}$
$\displaystyle=k_{1}y_{4}^{-1},u_{3}^{\prime}=k_{4}^{\prime}y_{4}y_{3}^{-1}\text{
for the mode 4},$ $\displaystyle u_{4}^{\prime}$
$\displaystyle=k_{1}y_{4}^{-1},u_{3}^{\prime}=-k_{6}\text{ for the mode 5},$
$\displaystyle u_{4}^{\prime}$
$\displaystyle=-k_{4}^{\prime},u_{3}^{\prime}=k_{4}^{\prime}y_{4}y_{3}^{-1}\text{
for the mode 6}.$ (21)
Like the smooth system, the tropicalization has a stable periodic trajectory
(limit cycle). This is represented together with the limit cycle trajectory of
the smooth system in Fig.1 bottom right. The period of the tropicalized limit
cycle is slightly changed with respect to the period of the smooth cycle.
However, we can modulate the period of the tropicalized cycle and make it fit
the period of the smooth cycle by acting on the moments of the mode change.
This stands to displacing the tentacles of the tropical varieties parallel to
the initial positions or equivalently, to changing the parameters
$h_{1},h_{2},h_{3},h_{4}$ while keeping $h_{1}=h_{3}$ which is a symmetry of
the problem.
The tropicalized system has piecewise smooth hybrid dynamics. Typically, it
passes from one type of smooth dynamics (mode) described by one set of
differential equations to another smooth dynamics (mode) described by another
set of differential equations (the possible modes are listed in Eq.(21)). The
command to change the mode is intrinsic and happens when the trajectory
attains the tropical curve. However, if the sliding mode condition (12) is
fulfilled the trajectory continues along some tropical curve tentacle instead
of changing plane sector and evolve according to one of the interior modes
(21). The tropicalized limit cycle has two sliding modes ($S_{4}$ and $S_{3}$
in Fig.1). The first one is along the half-line $h_{3}-h_{4}+2u_{3}+u_{4}=0$
on the logarithmic paper (tentacle $S_{4}$ on the red tropical curve in
Fig.1). In order to check (12) we note that $f^{+}=(k_{1}y_{4}^{-1},-k_{6})$,
$f^{-}=(-k_{4}y_{3}^{2},-k_{6})$, $n^{+}=-n^{-}=(-1,-2)$. We have a sliding
mode if $-k_{1}y_{4}^{-1}+2k_{6}<0$, meaning that the exit from the sliding
mode occurs when $u_{4}>log(k_{1}/(2k_{6}))$. The second sliding mode is along
the tentacle $h_{2}+u_{4}-u_{3}=0$ ($S_{3}$ on the blue tropical curve in
Fig.1). We have $f^{+}=(k_{1}y_{4}^{-1},-k_{6})$,
$f^{-}=(k_{1}y_{4}^{-1},k_{4}^{\prime}y_{4}y_{3}^{-1})$,
$n^{+}=-n^{-}=(-1,1)$. The conditions (12) are fulfilled when
$k_{1}y_{4}^{-1}-k_{4}^{\prime}y_{4}y_{3}^{-1}<0$ which is satisfied on the
entire tentacle. The exit from this second mode occurs at the end of the blue
tropical curve tentacle. Interestingly, the sliding modes of the
tropicalization can be put into correspondence with places on the smooth limit
cycle where the smooth limit cycle acquires new QSS species. This can be seen
in Fig.1 top left. The species $y_{3}$ becomes quasi-stationary on time
intervals $Q_{3}$ that satisfy (with good approximation) the relation
$h_{2}+u_{4}-u_{3}=0$ and correspond to the sliding mode on the blue tropical
curve. Also, the species $y_{4}$ becomes quasi-stationary on very short time
intervals $Q_{4}$ that satisfy $h_{3}-h_{4}+2u_{3}+u_{4}=0$ and correspond to
the sliding mode on the red tropical curve. As pointed out in the preceding
section, the trajectories of the smooth dynamics can evolve close to the
tentacles, but leave them sooner than the sliding modes.
We end this section with a study of the bifurcations of the ODE model and of
its tropicalization. It is easy to check that there is only one degree of
freedom describing the relative position of the two tropical curves. This is
the distance between the origins of the tropical curves, that is given by the
combination $k_{1}k_{4}^{\prime-1/2}k_{4}^{1/2}k_{6}^{-1}$. Thus, by changing
any one of the parameters $k_{1},k_{4}^{\prime},k_{4},k_{6}$ we can invert the
relative position of the tropical curves and change the partition of the
logarithmic paper into domains. This leads to two Hopf bifurcations of the ODE
model and also two Hopf bifurcations of the tropicalization. The bifurcation
of the tropicalization is discontinuous and can also be delayed with respect
to the continuous bifurcation of the ODE model (Fig.2).
Figure 2: Hopf bifurcations of the smooth and tropicalized system. (left) The
relative positions of the tropical curves can be changed by changing the
combination $k_{1}k_{4}^{\prime-1/2}k_{4}^{1/2}k_{6}^{-1}$. The first Hopf
bifurcation corresponds to $k_{1}k_{4}^{\prime-1/2}k_{4}^{1/2}k_{6}^{-1}=1$,
i.e. $log(k_{1})=-4.61$, when the tropical curves intersect in a single point.
For the second Hopf bifurcation the relative position of the two tropical
curves is no longer exceptional; the position of the bifurcation results from
sliding modes stability analysis. (right) Amplitudes of oscillation are shown
for the tropicalization (red) and for the smooth system (blue);
## 8 Solving ordinary differential equations in triangular form
We give a digest of a general algorithm for solving systems of the type (1)
and more generally, an arbitrary system of ordinary differential equations:
$G_{j}(x_{1},x_{1}^{(1)},\dots,x_{1}^{r},x_{2},x_{2}^{(1)},\dots,x_{2}^{(r)},\dots,x_{n},x_{n}^{(1)},\dots,x_{n}^{(r)},t)=0,\,1\leq
j\leq N,$ (22)
where $G_{j}$ are differential polynomials of the order at most $r$ in the
derivatives $x_{i}^{(s)}=\partial^{s}x_{i}/\partial t^{s},\,s\leq r$. Let the
degrees of the differential polynomials $G_{j}$ do not exceed $d$. Finally,
for algorithmic complexity purposes we assume that the coefficients of $G_{j}$
are integers with absolute values less than $2^{l}$, the latter means that the
bit-size of the coefficients $l(G_{j})\leq l$.
In [Sei56] an algorithm was designed which works not only for ordinary
differential systems like (22), but even for systems of partial differential
equations. For ordinary systems (22) the algorithm was improved in [Gri89],
although still its complexity is rather big (see below). We describe the
ingredients of the output (which has a triangular form) of the latter improved
algorithm and provide for it the complexity bounds.
The algorithm executes the consecutive elimination of the indeterminates
$x_{n},\dots,x_{1}$. The algorithm yields a partition $P=\\{P_{i}\\}_{1\leq
i\leq M}$ of the space of the possible functions $x_{1}$. Each $P_{i}$ is
given by a system of an equation $f_{i,1}(x_{1},t)=0$ and an inequality
$g_{i,1}(x_{1},t)\neq 0$ for suitable differential polynomials
$f_{i,1},\,g_{i,1}$. Then the algorithm yields an equation
$f_{i,2}(x_{1},x_{2},t)=0$ and an inequality $g_{i,2}(x_{1},x_{2},t)\neq 0$
for $x_{2}$ for suitable differential polynomials $f_{i,2},\,g_{i,2}$. We
underline that the latter equation and inequality hold on $P_{i}$. One can
treat the system $f_{i,2}=0,\,g_{i,2}\neq 0$ as the conditions on $x_{2}$ with
the coefficients being some differential polynomials in $x_{1}$ (satisfying
$P_{i}$).
Continuing in a similar way, the algorithm produces a triangular system of
differential polynomials $f_{i,3}(x_{1},x_{2},x_{3},t)$,
$g_{i,3}(x_{1},x_{2},x_{3},t),\dots$,$f_{i,n}(x_{1},\dots,x_{n},t)$,
$g_{i,n}(x_{1},\dots,x_{n},t)$. Thus, at the end $x_{n}$ satisfies (on
$P_{i}$) the equation $f_{i,n}(x_{1},\dots,x_{n},t)=0$ and the inequality
$g_{i,n}(x_{1},\dots,x_{n},t)\neq 0$ treated as a system with the coefficients
being differential polynomials in $x_{1},\dots,x_{n-1}$.
In other words, suppose that one has a device being able to solve an ordinary
differential system $f(x)=0,\,g(x)\neq 0$ in a single indeterminate $x$. Then
the algorithm would allow one to solve the system (22) consecutively: first
producing $x_{1}$ satisfying $f_{i,1}(x_{1},t)=0,\,g_{i,1}(x_{1},t)\neq 0$,
after that producing $x_{2}$ satisfying
$f_{i,2}(x_{1},x_{2},t)=0,\,g_{i,2}(x_{1},x_{2},t)\neq 0$ and so on.
This completes the description of the output of the algorithm. Now we turn to
the issue of its complexity. One can bound the orders of the differential
polynomials $ord(f_{i,s}),\,ord(g_{i,s})\leq r\cdot 2^{n}\,:=R,\,1\leq i\leq
M,\,1\leq s\leq n$, the number of the elements in the partition and the
degrees $M,\,deg(f_{i,s}),\,deg(g_{i,s})\leq(Nd)^{2^{R}}\,:=Q$. Finally, the
bit-size of the integer coefficients of $f_{i,s},\,g_{i,s}$ and the complexity
of the algorithm can be bounded by a certain polynomial in $l,\,Q$.
Thus, the number $n$ of the indeterminates brings the main contribution into
the complexity bound, which is triple exponential in $n$. Of course, the above
bounds have an a priori nature: they take into the account all the conceivable
possibilities in the worst case, whereas in practical computations
considerable simplifications are usually expected.
This illustrates the gain that one can obtain from using tropical geometry to
guide model reduction and obtain systems with smaller numbers of
indeterminates.
## 9 Conclusion.
Tropical geometry offers a natural framework to study biochemical networks
with multiple timescales and rational reaction rate functions. First, and
probably most importantly, tropicalization can guide model reduction of ODE
systems. We have shown that the existence of quasi-equilibrium reactions and
of quasi-stationary species implies the existence of sliding modes along the
tropical variety. Conversely, when the tropicalization has sliding modes along
hyperplanes defined by the equality of two monomials, we propose an algorithm
to decide whether the system has quasi-equilibrium reactions or quasi-
equilibrium species. This distinction allows correct model reduction, and
represents an improvement of methods proposed in [RGZL08].
The tropicalization represents an abstraction of the ODE model. This
abstraction may be not sound for some dynamic properties, but may conserve
others. If the trajectories of the ODE model are either very far or very close
to the tropical varieties, they tend to remain close to the trajectories of
the tropicalization for a while. However, the quality of the approximation is
not guaranteed at finite distance from the tropical variety. For instance, the
exit of tropicalized trajectories from a sliding mode tends to be delayed, and
smooth trajectories leave earlier neighborhoods of tropical varieties. The
example studied in this paper also illustrates some properties of bifurcations
of the tropicalization, that we have tested numerically. The tropicalization
qualitatively preserves the type and stability of attractors, but can also
introduce delays of a Hopf bifurcation. Thus, the tropicalization can only
roughly indicate the position of the bifurcation of the ODE model.
Furthermore, for Hopf bifurcations, the amplitude of the oscillations behaves
differently for the ODE model and for the tropicalization. In fact, Hopf
bifurcations are continuous for the ODE model and discontinuous for the
tropicalization.
The tropicalization provides in the same time a reduced model and a ”skeleton”
for the hybrid dynamics of the reduced model. This skeleton, specified by the
tropical varieties, is robust. As a matter of fact, monomials of parameters
are generically well separated [GR08]. This implies that tropicalized and
smooth trajectories are not that far one from another. Furthermore, because
the tropicalized dynamics is robust, it follows that the system can tolerate
large relative changes of the parameters without strong modifications of its
dynamics.
The dynamics of the model studied in this paper is relatively simple: it has a
limit cycle embedded in a two dimensional invariant manifold. As future work
we intend to extend the approach to more complex attractors, such as cycles in
dimension larger than two and chaotic attractors. Methods to compute tropical
varieties in any dimension are well developed in tropical algebraic geometry
[BJS+07]. Given the tropical variety, the existence of sliding modes can be
easily checked and the pruned polynomials defining the fast dynamics
calculated. This should lead directly to identification of quasi-equilibrium
reactions and quasi-stationary species, without the need of simulation
(replaces Step 1 in the Algorithm 1). Proposing simplified descriptions of the
dynamics of large and imprecise systems, tropical geometry techniques could
find a wide range of applications from synthetic biology design to
understanding emerging properties of complex biochemical networks.
## Aknowlegments
VN was supported by University of Rennes 1. SV was supported by the Russian
Foundation for Basic Research (Grant Nos. 10-01-00627 s and 10-01-00814 a) and
the CDRF NIH (Grant No. RR07801) and by a visiting professorship grant from
the University of Montpellier 2.
## References
* [Aub10] J.P. Aubin. Macroscopic traffic models: Shifting from densities to ”celerities”. Applied Mathematics and Computation, 217:963–971, 2010.
* [BJS+07] T. Bogart, A.N. Jensen, D. Speyer, B. Sturmfels, and R.R. Thomas. Computing tropical varieties. Journal of Symbolic Computation, 42(1-2):54–73, 2007.
* [CDR09] A. Crudu, A. Debussche, and O. Radulescu. Hybrid stochastic simplifications for multiscale gene networks. BMC Systems Biology, 3(1):89, 2009.
* [DFF+07] V. Danos, J. Feret, W. Fontana, R. Harmer, and J. Krivine. Rule-based modelling of cellular signalling. CONCUR 2007–Concurrency Theory, pages 17–41, 2007.
* [FA88] A.F. Filippov and FM Arscott. Differential equations with discontinuous righthand sides, volume 18. Springer, 1988.
* [GR08] AN Gorban and O. Radulescu. Dynamic and static limitation in reaction networks, revisited . In David West Guy B. Marin and Gregory S. Yablonsky, editors, Advances in Chemical Engineering - Mathematics in Chemical Kinetics and Engineering, volume 34 of Advances in Chemical Engineering, pages 103–173. Elsevier, 2008.
* [Gri89] D. Grigoriev. Complexity of quantifier elimination in the theory of ordinary differential equations. Lect. Notes Comput. Sci., 378:11–25, 1989.
* [GRZ10] AN Gorban, O. Radulescu, and AY Zinovyev. Asymptotology of chemical reaction networks. Chemical Engineering Science, 65:2310–2324, 2010.
* [KA56] E.L. King and C. Altman. A schematic method of deriving the rate laws for enzyme-catalyzed reactions. The Journal of Physical Chemistry, 60(10):1375–1378, 1956.
* [LM96] G.L. Litvinov and V.P. Maslov. Idempotent mathematics: a correspondence principle and its applications to computing. Russian Mathematical Surveys, 51(6):1210–1211, 1996.
* [LMS01] G.L. Litvinov, V.P. Maslov, and G.B. Shpiz. Idempotent functional analysis: an algebraic approach. Mathematical Notes, 69(5):696–729, 2001.
* [LY08a] Mark Z. Lazman and Gregory S. Yablonsky. Chapter 2 overall reaction rate equation of single-route complex catalytic reaction in terms of hypergeometric series. In David West Guy B. Marin and Gregory S. Yablonsky, editors, Advances in Chemical Engineering - Mathematics in Chemical Kinetics and Engineering, volume 34 of Advances in Chemical Engineering, pages 47 – 102. Elsevier, 2008.
* [LY08b] M.Z. Lazman and G.S. Yablonsky. Overall reaction rate equation of single-route complex catalytic reaction in terms of hypergeometric series. Advances in Chemical Engineering, 34:47–102, 2008.
* [Mik] G. Mikhalkin. Enumerative tropical algebraic geometry in r2. J. AMS, 18(2):313–377.
* [NVR10] V. Noel, S. Vakulenko, and O. Radulescu. Piecewise smooth hybrid systems as models for networks in molecular biology. In Proceedings of JOBIM 2010. Jobim, 2010.
* [NVR11] V. Noel, S. Vakulenko, and O. Radulescu. Algorithm for identification of piecewise smooth hybrid systems; application to eukaryotic cell cycle regulation. In Proceedings of WABI 2011, to appear, 2011.
* [PS04] L. Pachter and B. Sturmfels. Tropical geometry of statistical models. Proceedings of the National Academy of Sciences of the United States of America, 101(46):16132, 2004.
* [PT05] M. Passare and A. Tsikh. Amoebas: their spines and their contours. In Idempotent mathematics and mathematical physics: international workshop, February 3-10, 2003, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria, volume 377, page 275. Amer Mathematical Society, 2005.
* [RGZL08] O. Radulescu, A.N. Gorban, A. Zinovyev, and A. Lilienbaum. Robust simplifications of multiscale biochemical networks. BMC systems biology, 2(1):86, 2008.
* [Roj03] J.M. Rojas. Why polyhedra matter in non-linear equation solving. In Topics in algebraic geometry and geometric modeling: Workshop on Algebraic Geometry and Geometric Modeling, July 29-August 2, 2002, Vilnius University, Lithuania, volume 334, page 293. Amer Mathematical Society, 2003\.
* [Sei56] A. Seidenberg. An elimination theory for differential algebra. Univ. of California Press, 3(2):31–66, 1956.
* [SH10] S. Soliman and M. Heiner. A unique transformation from ordinary differential equations to reaction networks. PloS one, 5(12):e14284, 2010.
* [SS89] L.A. Segel and M. Slemrod. The quasi-steady-state assumption: a case study in perturbation. SIAM Review, 31(3):446–477, 1989.
* [Stu02] B. Sturmfels. Solving systems of polynomial equations. American mathematical society, 2002.
* [TVS85] A. N. Tikhonov, A. B. Vasileva, and A. G. Sveshnikov. Differential equations. Springer, Berlin, 1985.
* [Tys91] J.J. Tyson. Modeling the cell division cycle: cdc2 and cyclin interactions. Proceedings of the National Academy of Sciences of the United States of America, 88(16):7328, 1991.
* [Vir08] O. Viro. From the sixteenth hilbert problem to tropical geometry. Japanese Journal of Mathematics, 3(2):185–214, 2008.
* [Was65] W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Wiley, New York, 1965.
* [YBGE91] GS Yablonskii, VI Bykov, AN Gorban, and VI Elokhin. Kinetic models of catalytic reactions. Comprehensive Chemical Kinetics, volume 32. Elsevier, Amsterdam, 1991.
|
arxiv-papers
| 2011-09-19T16:41:36 |
2024-09-04T02:49:22.283741
|
{
"license": "Public Domain",
"authors": "Vincent Noel, Dima Grigoriev, Sergei Vakulenko and Ovidiu Radulescu",
"submitter": "Ovidiu Radulescu",
"url": "https://arxiv.org/abs/1109.4085"
}
|
1109.4096
|
# Main Belt Asteroids with WISE/NEOWISE I:
Preliminary Albedos and Diameters
Joseph R. Masiero11affiliation: Jet Propulsion Laboratory/California Institute
of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
Joseph.Masiero@jpl.nasa.gov , A. K. Mainzer11affiliation: Jet Propulsion
Laboratory/California Institute of Technology, 4800 Oak Grove Dr., MS 321-520,
Pasadena, CA 91109, USA, Joseph.Masiero@jpl.nasa.gov , T. Grav22affiliation:
Johns Hopkins University, Baltimore, MD 21218 USA , J. M. Bauer11affiliation:
Jet Propulsion Laboratory/California Institute of Technology, 4800 Oak Grove
Dr., MS 321-520, Pasadena, CA 91109, USA, Joseph.Masiero@jpl.nasa.gov
33affiliation: Infrared Processing and Analysis Center, California Institute
of Technology, Pasadena, CA 91125 USA , R. M. Cutri33affiliation: Infrared
Processing and Analysis Center, California Institute of Technology, Pasadena,
CA 91125 USA , J. Dailey33affiliation: Infrared Processing and Analysis
Center, California Institute of Technology, Pasadena, CA 91125 USA , P. R. M.
Eisenhardt11affiliation: Jet Propulsion Laboratory/California Institute of
Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
Joseph.Masiero@jpl.nasa.gov , R. S. McMillan44affiliation: Lunar and Planetary
Laboratory, University of Arizona, 1629 East University Blvd, Kuiper Space
Science Bldg. #92, Tucson, AZ 85721-0092 USA , T. B. Spahr55affiliation: Minor
Planet Center, Harvard-Smithsonian for Astrophysics, 60 Garden Street,
Cambridge, MA 02138 USA , M. F. Skrutskie66affiliation: Department of
Astronomy, P.O. Box 3818, University of Virginia, Charlottesville, VA
22903-0818 USA , D. Tholen77affiliation: Institute for Astronomy, University
of Hawaii, Honolulu, HI 96822 USA , R. G. Walker88affiliation: Monterey
Institute for Research in Astronomy, Monterey, CA USA , E. L.
Wright99affiliation: UCLA Astronomy, PO Box 91547, Los Angeles, CA 90095-1547
USA , E. DeBaun11affiliation: Jet Propulsion Laboratory/California Institute
of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
Joseph.Masiero@jpl.nasa.gov 1010affiliation: Dartmouth College, Hanover, NH
03755 USA , D. Elsbury11affiliation: Jet Propulsion Laboratory/California
Institute of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109,
USA, Joseph.Masiero@jpl.nasa.gov 1111affiliation: Notre Dame High School,
13645 Riverside Dr, Sherman Oaks, CA 91423 USA , T. Gautier IV11affiliation:
Jet Propulsion Laboratory/California Institute of Technology, 4800 Oak Grove
Dr., MS 321-520, Pasadena, CA 91109, USA, Joseph.Masiero@jpl.nasa.gov
1212affiliation: Flintridge Preparatory School, 4543 Crown Ave, La Canada, CA
91101 USA , S. Gomillion11affiliation: Jet Propulsion Laboratory/California
Institute of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109,
USA, Joseph.Masiero@jpl.nasa.gov 1313affiliation: Embry-Riddle Aeronautical
University, 600 S. Clyde Morris Blvd, Daytona Beach, FL 32114, USA , A.
Wilkins11affiliation: Jet Propulsion Laboratory/California Institute of
Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
Joseph.Masiero@jpl.nasa.gov 1414affiliation: University of Maryland, College
Park, MD 20742, USA
###### Abstract
We present initial results from the Wide-field Infrared Survey Explorer
(WISE), a four-band all-sky thermal infrared survey that produces data well
suited to measuring the physical properties of asteroids, and the NEOWISE
enhancement to the WISE mission allowing for detailed study of Solar system
objects. Using a NEATM thermal model fitting routine we compute diameters for
over 100,000 Main Belt asteroids from their IR thermal flux, with errors
better than $10\%$. We then incorporate literature values of visible
measurements (in the form of the H absolute magnitude) to determine albedos.
Using these data we investigate the albedo and diameter distributions of the
Main Belt. As observed previously, we find a change in the average albedo when
comparing the inner, middle, and outer portions of the Main Belt. We also
confirm that the albedo distribution of each region is strongly bimodal. We
observe groupings of objects with similar albedos in regions of the Main Belt
associated with dynamical breakup families. Asteroid families typically show a
characteristic albedo for all members, but there are notable exceptions to
this. This paper is the first look at the Main Belt asteroids in the WISE
data, and only represents the preliminary, observed raw size and albedo
distributions for the populations considered. These distributions are subject
to survey biases inherent to the NEOWISE dataset and cannot yet be interpreted
as describing the true populations; the debiased size and albedo distributions
will be the subject of the next paper in this series.
## 1 Introduction
Since the discovery of (1) Ceres in 1801 (Piazzi, 1801) the majority of
observations of asteroids and other minor planets have been conducted in
visible wavelengths. While visible light can provide very accurate positions,
the interpretation of photometry is rendered ambiguous by the dependence of
the observed flux on both the size and albedo of the asteroid. This
relationship is described by the equation:
$D=\frac{1329}{\sqrt{p_{V}}}10^{-H/5}$
(see Harris & Lagerros, 2002, for an overview and references therein for its
derivation), where $D$ is the diameter in kilometers, $p_{V}$ is the visible
geometric albedo, and $H$ is the absolute magnitude which is defined as the
apparent magnitude the body would have $1~{}$AU from the Sun, $1~{}$AU from
the observer, and at $0^{\circ}$ phase angle. Albedos of Solar system objects
are observed to vary from only a few percent up to nearly $100\%$ percent for
icy surfaces, causing diameters inferred from visible data alone to have
nearly an order of magnitude uncertainty. Multiwavelength visible surveys have
found statistical correlations between the albedos and the visible colors of
asteroids (e.g. Ivezić et al., 2001), however the accuracy of the mapping
between these two properties for individual objects is currently being
examined (Mainzer et al., 2011c). Using an independent method of measurement
for either the albedo or the diameter allows for the unique solution of the
other, given a visible $H$ value.
Accurate diameters and albedos for a large number of asteroids enable a number
of important areas of research into the history and formation of the Solar
system. Diameter measurements of asteroids based on infrared flux allow us to
quantify the size-frequency distribution (SFD) of the bodies in a way
independent of assumptions about the translation from H magnitude to diameter
that are typically required. Both the initial accretion and formation process
of asteroids as well as the subsequent collisional and orbital evolution
affect the current Main Belt SFD, and a well-measured SFD will allow us to put
constraints on these processes. The albedo of an asteroid, meanwhile, is a
strong function of its composition. Compositional gradients for the Main Belt
have been shown in the past from both infrared and spectral surveys (e.g.
Zellner, 1979; Gradie & Tedesco, 1982; Tedesco et al., 2002; Bus & Binzel,
2002), but only for a limited number of objects. A large survey conducted in
mid-infrared wavelengths will allow Main Belt albedos and diameters to be
produced with good accuracy; this in turn will allow us to study the
compositional gradient of the Solar System and may ultimately allow us to set
constraints on any major planetary migration that may have occurred.
Recent work has attempted to understand the SFD of asteroids as it relates to
the impact physics dominating the evolution of the Main Belt with the aid of
numerical simulations. O’Brien & Greenberg (2003) analytically calculated the
behavior of a population in a steady-state collisionally dominated regime to
determine how the slope of a population’s SFD behaved as a function of the
mechanical strength of the material composing the body. They find that while a
strength-less regime yields a nominal power-law slope of $a=-3.5$, including
the strength of the body can vary the result over a range of slopes depending
on the specific circumstances. Expanding on this, Durda et al. (2007)
simulated a variety of impacts with numerical hydrocodes to look for the
effect of internal structure, impact angle, impact velocity, and impactor-
target size ratio on the resultant SFD of the shattered products. These
authors have also attempted to match their numerical results to measured SFDs
for asteroid families to determine the initial size of the parent body,
however the data used have depended on assumptions for the albedos of the
asteroids. For the larger Main Belt population, Bottke et al. (2005a) have
modeled the evolution of various initial size distributions to determine what
the current MBA SFD can indicate about the SFD present at the formation of the
asteroids. Their results show a clear peak in formation SFD, with few objects
smaller than $D\sim 100~{}$km forming directly from the protosolar nebula,
however these results are also based on diameters determined from assumed
albedos. Similarly, Bottke et al. (2005b) have investigated the sources and
sinks of excited MBAs and the effect orbital evolution has the SFD of the Main
Belt.
Studies of asteroid albedo distributions have in the past been limited by
small data sets. Early measurements were made using thermal infrared detectors
on ground-based telescopes to determine radiometric diameters and thus albedos
for tens of objects (e.g. Morrison, 1974). The InfraRed Astronomical Satellite
(IRAS) revolutionized studies of physical properties of the Main Belt by
observing over $2000$ asteroids in the thermal infrared, determining albedos
and diameters for these bodies in a uniform way (Tedesco et al., 2002, 2004).
To date, IRAS represents the largest and most complete survey of asteroid
albedos in the literature.
If the sizes and albedos of the members of an asteroid family are known, it is
possible to use their orbital evolution to study their age. The Yarkovsky
effect (see Bottke et al., 2006, for a review of the subject) occurs when
incident optical light is absorbed and re-emitted as thermal infrared photons
in a different direction, usually due to the rotation of the body. This
difference in momentum causes changes in the orbit of the body over long time
intervals. Nesvorný & Bottke (2004) use this effect to refine predictions for
the age of the Karin family based on backwards integration of orbital
parameters. As Karin is one of the youngest families (Nesvorný et al., 2002)
this effect becomes increasingly important for accurate age-dating of families
the older they are. The accuracy of this technique, however, depends on
knowing sizes and albedos of the objects to a high degree of certainty.
The distribution of albedos in the Main Belt as a whole and broken down by
region can provide us a window into the changing chemical and mineralogical
processes active in different regions of the early Solar system. Although the
current understanding of the history of the Main Belt has the asteroids
forming in or near their current locations, new theories are being proposed
that the Main Belt may in fact be the result of a mixing of two distinct
populations from different regions of the Solar system. Migrations of the
giant planets may have both cleared many of the objects that initially formed
in the Main Belt region and repopulated this area with objects from beyond the
“snow line” (Morbidelli et al., 2010; Walsh & Morbidelli, 2011). We then might
expect the Main Belt to be composed of two overlapping populations, one having
formed in a volatile-poor region of the protosolar disk and one forming in a
volatile rich area, though the latter population would lose any surface
volatiles over the age of the Solar system. Under this scenario albedo may be
able to trace dynamical evolution as well as chemical processing.
There are very few ways to measure the albedo of an asteroid directly.
Observations taken in situ by spacecraft can be used to measure both the
absolute albedo and its variation across a body’s surface (e.g. Howett et al.,
2010), as well as its diameter, though only a handful of objects have been
visited by spacecraft. Imaging polarimetry can also be used to determine the
integrated surface albedo for a number of objects (Cellino et al., 1999), but
appropriately calibrated instrumentation is uncommon. Similarly, direct
measurements of asteroid diameters can come from a variety of techniques.
Resolved imaging of an asteroid provides the simplest method of size
determination, however the use of e.g. HST or Keck adaptive optics (AO) only
allows the largest few asteroids to be resolved (e.g. Schmidt et al., 2009; Li
et al., 2010). Asteroid occultations of background stars also provide a robust
method of diameter measurement (Shevchenko & Tedesco, 2006), however the
logistical constraints of occultation events make obtaining a large sample
difficult, and shape measurements are only instantaneous projections. Radar
measurements can provide precise distances, rotation rates, projected
profiles, and with sufficient data 3D shape models (Ostro et al., 2002), but
returned fluxes fall off quickly with distance which limits the number of
objects observable with this technique.
Indirect measurements of asteroid diameters and albedos can provide a wealth
of information both for individual objects and the population as a whole.
These techniques typically can observe a large number of objects in a
relatively short period of time, provide uniform data for an entire
population, and have understood biases that allow for determination of the
true, underlying distributions. Careful calibration of indirect measurements
is required, but once established these techniques can provide highly accurate
measurements of asteroid physical properties for a large number of bodies.
For objects that have known orbits, measurement of the infrared flux emitted
from the surface can be used to constrain the diameter of the body (see §3 for
a discussion of the method used here). Infrared imaging can be accomplished
rapidly when integrated in an all-sky survey. Thermal infrared measurements of
a large sample of asteroids represent the best way to determine robust
diameters rapidly for many thousands of objects. In this paper, we present
preliminary results from the Wide-field Infrared Survey Explorer (WISE) space
telescope, the next-generation all-sky infrared survey, focusing here on the
cryogenic observations of Main Belt asteroids.
## 2 Observations
### 2.1 WISE and NEOWISE
Launched on 2009 December 14, WISE is a thermal infrared space telescope that
performed an all-sky survey from 2010 January 14 until it exhausted the
telescope-cooling hydrogen ice on 2010 August 5. The survey continued during
the warmup of this secondary tank with limited sensitivity in the longer
wavelengths until the primary coolant, responsible for maintaining detector
temperature, exhausted 2010 October 1. WISE subsequently entered a Post-
Cryogenic Mission to complete the survey of the largest MBAs, continue
discovering new NEOs, and complete a second-pass survey of the inertial sky in
the two shortest wavelengths. During its cryogenic mission, WISE imaged the
sky in four infrared wavelengths simultaneously using dichroic beamsplitters
to produce co-boresight images with band centers at $3.4~{}\mu$m,
$4.6~{}\mu$m, $12~{}\mu$m and $22~{}\mu$m (W1, W2, W3, and W4, respectively).
The latter two bands are particularly important for Solar system studies as
the dominant amount of the flux received from asteroids is from thermal
emission peaking at these wavelengths. First-pass calibration of the WISE data
was tuned to the fully cryogenic mission and while the final calibration
currently being undertaken will finalize measurements obtained during each
warm-up stage, we restrict our current analysis to objects observed during
this fully cryogenic stage. Pre-launch descriptions of WISE were given by
Mainzer et al. (2006) and Liu et al. (2008), while post-launch overviews,
including initial calibrations and color corrections, are presented by Wright
et al. (2010) and Mainzer et al. (2011b).
The WISE survey follows a continuous scan along lines of ecliptic longitude at
a solar elongation of $\sim 90^{\circ}$ as the spacecraft orbits above the
terminator of the Earth. An oscillating scan mirror compensates for this
motion, providing stable images with effective integration times of
$8.8~{}$seconds. The positions of all known minor planets are propagated to
the time of each observation and checked against all transient sources in the
field of view, recording the appropriate calibrated magnitudes if observed. In
this way, thermal measurements of each minor planet cataloged by the Minor
Planet Center (MPC)111see the MPCORB.DAT file available here:
http://www.minorplanetcenter.net/iau/MPCORB.html can be found. In addition to
previously known objects, the WISE processing pipeline includes the capability
to discover new objects via the NEOWISE enhancement (Mainzer et al., 2011a).
While comprehensive followup of new potential near-Earth objects (NEOs) has
been one of the priorities of the WISE team (Mainzer et al., 2011d), the large
number of Main Belt discoveries ($>34,000$) has prohibited sufficient
immediate ground-based followup. Large-scale asteroid surveys such as
Spacewatch and the Catalina Sky Survey have already provided serendipitous
followup of many of the new WISE MBAs and will continue to do so, while future
surveys are expected to recover the remainder. As described in Wright et al.
(2010) and Mainzer et al. (2011a), the WISE cadence resulted in an average of
10-12 observations of each minor planet detected over $\sim 36~{}$hours. For a
subset of the Main Belt, predominantly in the outer regions, observations were
obtained at two or more epochs depending on the relative motion of the object
and Earth around the Sun.
### 2.2 Data and calibration
In this paper, we consider only those Main Belt asteroids detected by
NEOWISE/WMOPS during the cryogenic portion of the mission, shown in a top-down
view of the inner Solar system in Fig 1. This consists of $129,750$ unique
objects. We obtained our data used for fitting in a method identical to the
one described in Mainzer et al. (2011b) and Mainzer et al. (2011d), though
tuned for MBAs. Specifically, we queried the Minor Planet Center observation
file222http://www.minorplanetcenter.net/iau/ECS/MPCAT-OBS/MPCAT-OBS.html for
all observations submitted from WISE (observatory code C51). We then used the
resultant RA-Dec-time values as input for a query of the WISE individual
exposure archive, the “Level 1b” data, through the Gator tool provided by the
InfraRed Science Archive
(IRSA333http://irsa.ipac.caltech.edu/applications/Gator/). To ensure that only
the observations of the moving objects were returned, we restricted our search
radius to 0.3 arcsecs from the position and 2 seconds from the time obtained
for the detection from the MPC. Additionally, we set a constraint of
JD$<2455413.5$ to ensure that only data during the fully cryogenic mission was
used for this initial survey. This method of data acquisition has the benefit
of ensuring all observations used have been vetted both internally by the WISE
data pipeline and again by the MPC.
Figure 1: Top-down view of the inner Solar system showing the location of all
objects observed during the fully-cryogenic mission. Positions were propagated
to 2010 August 5, the date of the exhaustion of coolant from the secondary
tank. Black points indicate MBAs while grey points are all other Solar system
objects. Axes’ units are AU.
All data were processed using the first pass version of the WISE pipeline,
which computed dark current/sky offset levels, flagged instrumental artifacts
such as latent images and diffraction spikes and performed linearity
compensation. Only observations with an artifact flag cc_flags$=0$ or $p$ in a
band were accepted: a value of $0$ indicates no evidence of artifact was found
by the pipeline, while $p$ indicates the possibility of contamination by a
latent image. As discussed in Mainzer et al. (2011b) we find that the pipeline
was overly conservative in artifact flagging and cc_flags$=p$ values have
similar fluxes to cc_flags$=0$ detections while increasing the number of
usable observations by $\sim 20\%$. The ph_qual flag was required to have a
value of A, B or C to again ensure only valid detections were used. Non-
linearity and saturation are a particular concern for the brightest MBAs,
especially in bands W3 and W4. The WISE data reduction pipeline applies a non-
linearity and saturation correction for all observations brighter than the
threshold of $W1=7.8$ mag, $W2=6.5$ mag, $W3=3.6$ mag, $W4=-0.6$ mag. Objects
with magnitudes brighter than $W3=4$ and $W4=3$ were assigned errors of
$0.2~{}$mag to account for the change in the point-spread function for very
bright objects, and a linear correction to the magnitudes of sources with
$-2<W3<4$ was applied (Mainzer et al., 2011b; Cutri et al., 2011). Following
those authors, we did not use objects brighter than $W3=-2$ and $W4=-6$ for
thermal modeling.
Each object was required to have been observed at least 3 times in one WISE
band with magnitude error $\sigma_{mag}\leq 0.25$ to undergo thermal modeling,
as a precaution against contamination by spurious sources (e.g. background
noise, cosmic rays, stars, etc.). For multiple-band thermal models we required
other bands to have at least $40\%$ of the detection rate of the band with the
largest number of detections, usually W3 for MBAs. To reduce the possibility
of confusion with inertially fixed sources such as stars and galaxies, we
searched each position retrieved from the Level 1b catalog in the Daily and
Atlas Coadded Catalogs (also served by IRSA) within 6.5 arcsecs, equivalent to
the W1, W2 and W3 beam sizes. These searches looked for sources that appeared
at least twice and in at least $30\%$ of the images covering that location.
Any sources returned from these searches were considered to be inertial, which
could contaminate the observation of the asteroid at that position. Thus that
asteroid detection was discarded from the thermal modeling routine.
In Fig 2 we show the mean colors of inner Solar system objects as observed
during the cryogenic phase of the WISE mission with the MBAs highlighted in
black (see Mainzer et al., 2011d; Grav et al., 2011, for discussions on the
other populations shown in this figure). The bifurcation in the W1-W2 color
observed for the MBAs traces the two dominant albedo groupings in the Main
Belt (see below for further discussion). The MBAs span a wide range of colors
and are bounded by the NEO and Jupiter Trojan populations. Note that MBAs
occupying color-space typically associated with one of the other populations
are candidates for objects that may have been misidentified as MBAs during
preliminary orbit fitting. We show in Fig 3 the color for each object as a
function of heliocentric distance at the time of observation. Color,
especially in bands dominated by thermal emission, is a strong function of
temperature of the body and thus distance from the Sun. Figure 4 shows the
sky-plane velocity for each object compared with its W3-W4 color. The mean
sky-plane velocity for MBAs is $0.2~{}$deg/day; objects with velocities
significantly larger than this are candidates for NEOs misclassified as MBAs,
requiring further followup. Future work will combine color and sky-plane
velocity cuts to distinguish MBAs with poorly known orbits from NEOs and
Trojans.
Figure 2: (a)W1-W2 vs. W3-W4 colors for all MBAs (black), Trojans (green), and
NEOs (magenta); (b) W2-W3 vs. W3-W4 colors for these same populations. Figure
3: (a)W1-W2 vs. heliocentric distance for all MBAs (black), Trojans (green),
and NEOs (magenta); (b) as in (a) but for W2-W3 color; (c) as in (a) but for
W3-W4 color. Figure 4: (a)W3-W4 color vs. sky-plane velocity for all MBAs
(black), Trojans (green), and NEOs (magenta); (b) differential distribution of
sky-plane velocities for all MBAs.
## 3 Diameter and albedo determination through thermal modeling
In contrast to the visible flux received from an asteroid, which is reflected
sunlight, the mid-IR flux beyond $\sim 6~{}\mu$m from an object in the Main
Belt is almost completely thermal emission from that body. For an object with
an established orbit, the phase angle and distances to the asteroid from the
Earth and the Sun are well known, thus the observed thermal flux can be
converted into a total emitted flux at the surface. By making assumptions for
some physical surface properties, a diameter can be computed from a single
band detection. When simultaneous thermal measurements at multiple wavelengths
are available the beaming parameter of the surface material may also be fit to
the data. The beaming parameter ($\eta$) represents the deviation of the
thermal emission from that of a smooth non-rotating sphere due to rotation and
surface roughness, and is used to consolidate the uncertainty in the values of
the surface thermal properties, including emissivity. When visible-light data
are additionally available, the visible albedo ($p_{V}$) becomes a free
parameter that can now also be fit using the two data sets in conjunction.
Further data such as W1 or W2 measurements that are dominated by reflected
sunlight allow us to derive an independent measure of the ratio of visible to
NIR reflectance as well.
In order to analyze the thermal infrared asteroid measurements from the IRAS
satellite, Lebofsky et al. (1986) developed the “Standard Thermal Model” (STM)
for asteroids, calibrated against measurements from (1) Ceres and (2) Pallas.
In this model, the beaming parameter was held constant to $\eta=0.756$ based
on the ground-truth occultation observations of the calibrator asteroids.
Subsequent work indicated that this model might not be appropriate for smaller
asteroids, thus Harris (1998) modified STM to a form appropriate for a “Near-
Earth Asteroid Thermal Model” (NEATM), where $\eta$ is allowed to vary. While
designed specifically to account for the breakdown of STM when considering
NEOs, NEATM can be applied quite readily to a wide range of bodies in the
Solar system (e.g. Mueller et al., 2010; Ryan & Woodward, 2010). Wright (2007)
compared NEATM with a full thermophysical model of a cratered surface and
found that for low phases both models produce consistent results. Lebofsky &
Spencer (1989) present an investigation of objects with a beaming parameter at
the theoretical maximum of $\eta=\pi$ which occurs for a body rotating so
quickly it is latitudinally isothermal.
We have performed preliminary thermal modeling of MBAs based on the WISE
First-Pass Data Processing Pipeline described above, covering observations
taken during the cryogenic phase of the mission. We modeled each object as a
non-rotating sphere with triangular facets and variable diameter, beaming
parameter, visible albedo, and NIR reflectance ratio as appropriate for the
data. Relative distances and phase angles were computed for each measurement
to ensure changing distances do not bias the resultant fits. Although we do
not expect all, or even most, asteroids to have a spherical shape, our
observations covering $\sim 36~{}$hours smooth out rotation effects and allow
us to determine the effective diameter of a spherical body with the same
physical properties. Long period rotators ($P\sim$days) with large amplitudes,
e.g. binary asteroids with mass fraction $\sim 1$, will have poor fits
resulting in a moderate mis-measurement of albedo and diameter. Future work
will address the light curve component of our data set to determine the
minimum and maximum sizes of our targets in order to estimate first-order
shape models as well as estimates of the fraction of binaries in the Main
Belt.
At each instance, the temperature on every facet was computed and color
corrected based on the values in Wright et al. (2010). The emitted thermal
flux for each facet was computed with the NEATM model and night-side facets
were assumed to contribute zero flux. A reflected light model was used to
determine the reflected component in each band for all illuminated facets
visible to WISE at the time of observation; this was most important for W1 and
W2. The model reflected and emitted light was summed for all facets and
converted to a model magnitude using the Jansky flux of a zero magnitude
source provided in Wright et al. (2010) and modified according to the text for
red sources: $306.681~{}$Jy for W1, $170.663~{}$Jy for W2, $31.3684~{}$Jy for
W3 and $7.9525~{}$Jy for W4. Note that the modifications in the W3 and W4
zeropoints are the result of adjusting the central wavelengths of these bands
to $\lambda_{0,W3}=11.0984~{}\mu$m and $\lambda_{0,W4}=22.6405~{}\mu$m to
correct the discrepancy observed in the calibration tests between blue-
spectrum and red-spectrum objects. These model magnitudes were then compared
with the measured magnitudes, and the model was iterated through a least-
squares fitting routine until the best fit was found.
For objects with two thermally dominated bands, the beaming parameter was
allowed to vary, while for those with only one thermal band we used a fixed
value of $\eta=1.0$, based on the peak of the $\eta$ distribution of MBAs that
were fit with a variable beaming parameter (see below). For objects where no
detected band was dominated ($>75\%$) by reflected light we assumed a NIR
reflectance ratio of $1.5$ as was found for MBAs with fitted NIR reflectances
(see below). These objects typically were not detected in W1 and either were
not detected or had both thermal emission and reflected light in W2.
Monte Carlo simulations were performed in each case to determine the errors on
all variable parameters. For high S/N cases where the quoted error on the
measured magnitude only represented the statistical error, we set a floor of
$\sigma_{mag}=0.03~{}$, representing the absolute error on the photometry
(Wright et al., 2010). For all objects we assumed that the emissivity
$\epsilon=0.9$ and we assumed that the magnitude-phase slope parameter (c.f.
Bowell et al., 1989) was $G=0.15$ unless otherwise given by the MPC or in the
Lightcurve Database
(LCDB444http://www.minorplanet.info/lightcurvedatabase.html, Warner et al.,
2009a). The quoted errors on the modeled parameters are equal to the weighted
standard deviation of all Monte Carlo trial values. For objects with fixed
beaming parameter, an error of $\sigma_{\eta}=0.2$ was assumed to allow for
proper error determination of derived parameters based on the mean and
standard deviation of all best-fitting beaming parameters for objects with
fitted values (see §5). Similarly, for objects with fixed IR reflectance
ratios, we assume an error bar of $\sigma_{ratio}=0.5$ based on the mean and
standard deviation of objects with fitted IR reflectance ratios (see §8). We
note that as the flux calibrations presented by Mainzer et al. (2011b) set a
limit on the computed diameter accuracy for sources in the WISE data of
$\sigma_{D}=10\%$. This error implies a minimum fractional error for albedo of
$\sigma_{p_{V}}=20\%\times p_{V}$ assuming a perfect H magnitude. These values
are in addition to any Poissonian error inherent to the observations, though
for most of the objects presented here the calibration errors dominate our
solutions.
In total $129,750$ Main Belt asteroids, selected from the cryogenic phase of
the survey, had sufficient number and quality of detections to allow us to
perform thermal modeling and determine their effective diameter. Of these,
$17,482$ objects had orbital arcs shorter than $30$ days; as such their orbits
have a larger uncertainty than the rest of the population, which corresponds
to uncertainty in their geocentric and heliocentric distances, which will
naturally increase the error on their calculated diameters. Additionally,
$112,265$ objects also had available optical data allowing us to calculate
albedo as well as diameter. Both of these latter two populations are changing
continuously, as ground-based surveys submit serendipidous visible
observations of NEOWISE-discovered asteroids (allowing us to fit albedos and
allowing the MPC to fit better orbits), and as the MPC links WISE observations
with previous one-night stands and lost asteroids.
We provide a table of our best fits for Main Belt asteroids from the Pass 1
processed cryogenic survey data online at:
http://wise2.ipac.caltech.edu/staff/bauer/NEOWISE_pass1/. This table contains:
the MPC-packed format name of the object; the $H$ and $G$ values used; the
diameter, albedo, beaming parameter, and infrared albedo as well as associated
error bars; the number of observations in each WISE band that were used for
fitting; and the mean modified Julian date of the observations. For objects
observed at multiple epochs with fits consistent across all observations, each
epoch is presented as a separate row in the table. As discussed in §10 objects
where multiple epochs were forced to fit to a single model because the
separate fits were divergent are presented as a single row. Objects without
optical data at the time of publication have “nan” (“not a number”) values for
absolute magnitude and albedo. Similarly, objects for which an infrared albedo
could not be fit nor had literature optical data that could be used with an
assumed reflectance ratio to estimate an infrared albedo (see §8) have “nan”
values in this entry.
## 4 Preliminary raw size-frequency distribution of MBAs
Using fluxes from the WISE data, and a faceted NEATM model, we are able to
determine diameters for our observed objects. In Fig 5 we show the cumulative
preliminary raw size-frequency distribution (PRSFD) for the three major
regions of the Main Belt: the inner-Main Belt (IMB, those objects with
$1.8~{}$AU $<a<2.5~{}$AU), the middle-Main Belt (MMB, objects with $2.5~{}$AU
$<a<2.82~{}$AU), and the outer-Main Belt (OMB, objects with $2.82~{}$AU
$<a<3.6~{}$AU). In all cases, perihelion distance was required to be beyond
the orbit of Mars, $q>1.666~{}$AU. Also plotted on the distribution histograms
are 100-trial Monte Carlo (MC) simulations of the diameter distribution
including the appropriate measured errors. This mean MC distribution and
associated error are shown as points; the error bar sizes are smaller than the
point size. We see no significiant change between the MC distribution from the
distribution of best-fit diameters. We find the slope of the PRSFD for smaller
objects in all three subpopulations to be consistent with the $a=-2.5$ value
found by Gladman et al. (2009), however debiasing will be critical to
determining the true value of this slope. We observe a significant change in
slope for the PRSFD between $15-25~{}$km, consistent with the location of the
“kink” seen in the debiased $H$ distribution by Jedicke & Metcalfe (1998). A
debiasing campaign of the observed population currently underway will allow us
to explore the true SFDs of the populations and will be discussed in future
work.
Figure 5: Cumulative raw size-frequency distribution of MBAs in the inner-
(red), middle- (blue), and outer-Main Belt (black). Plotted under the
distributions are grey points showing the Monte Carlo simulation for each data
set; the error bars are the size of the points.
## 5 Variable beaming parameters
For objects with detections in at least two thermal bands, we allow the
beaming parameter to vary during the thermal model fitting. We are able to fit
beaming parameters for $66,406$ MBAs. We find a wide range of best-fit beaming
parameters between the theoretical limits of $0.3$ and $\pi$, with a peak
value of $\eta=1.0$ and standard deviation of $\sigma_{\eta}=0.2$. In Fig 6 we
show the beaming parameter found for all objects with fitted values as a
function of a variety of orbital and physical parameters. Beaming parameter
has a weak dependence on semi-major axis (Fig 6a), an effect that is more
pronounced for the relationship with phase angle (Fig 6g). Note that due to
the constraints of the pointing of WISE over the course of the survey to solar
elongations $\approx 90^{\circ}$, phase angle, heliocentric distance and
geocentric distance are strongly correlated, though ecliptic latitude of the
observations as well small changes in the exact pointing over the survey
weaken this relation. Wright (2007) has shown that for the phase angles we
typically observe MBAs at
($14^{\circ}\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}\alpha\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}27^{\circ}$)
the differences in calculated diameter and beaming parameter between NEATM and
more realistic thermophysical models is minimal over a large range of
observing geometries. From the raw distribution, the beaming parameter shows
no dependence on size (Fig 6b), eccentricity (Fig 6c), inclination (Fig 6d) or
absolute magnitude (Fig 6e).
We find a best-fit linear relation to the running average of beaming parameter
as a function of phase of:
$\eta=(0.79\pm 0.01)+\alpha(0.011+0.001)$
where $\eta$ is the beaming parameter and $\alpha$ is the phase angle in
degrees. This is consistent with the results found in Mainzer et al. (2011d)
for the NEOs from WISE, but differs significantly from the results of Wolters
& Green (2009) who found a best-fitting line of $\eta=1.08+0.007\alpha$. For
phase angles within the Main Belt ($14^{\circ}-32^{\circ}$) the average
beaming ranges from $0.94<\eta<1.14$ although the spread around this value is
large. Thus $\eta_{assumed}=1.0$ for objects with only a single thermal band
is a reasonable assumption for objects in the Main Belt. Debiasing will allow
us to account for any detection-limit effects that may bias the selection of
objects that have sufficient data for fitting of the beaming parameter.
Note that while the running average over the beaming parameter shows a
dependence on albedo (Fig 6f) and subsolar temperature (Fig 6h), albedo
determinations are very sensitive to survey biases, both from WISE and optical
followup, while subsolar temperature is a function of beaming parameter as per
the equation from the definition of NEATM in Harris (1998):
$T_{SS}=[(1-A)S/(\eta\epsilon\sigma)]^{0.25}$ (1)
(where $T_{SS}$ is the subsolar temperature, $A$ is the Bond albedo, $S$ the
incident Solar flux, $\eta$ the beaming parameter, $\epsilon$ the emissivity,
and $\sigma$ the Stefan-Boltzmann constant), and so this cannot be used as an
independent constraint.
Figure 6: Beaming parameter for all objects with fitted values, compared to
(a) semi-major axis, (b) diameter, (c) eccentricity, (d) inclination, (e)
absolute magnitude, (f) albedo, (g) phase and (h) subsolar temperature. The
thick red line shows the running average for $1000$ object-wide bins stepped
by $100$ objects. The picket-fence effect in the absolute magnitude is an
artifact of the $0.1~{}$mag reported precision of $H$ for most objects.
We show in Fig 7 histograms of the beaming parameter distribution for the
inner-, middle-, and outer-Main Belt populations. Also shown as points are the
mean distribution and error from a 100-trial Monte Carlo simulation of the
distribution using the error bar on each fitted beaming parameter. All
populations show longer tails toward higher values of beaming parameter than
toward lower, and have consistent shapes. The shift in peak beaming parameter
with phase can be seen as the change in distribution between populations with
different average phase angles.
Figure 7: Histograms show the beaming parameter distribution for the IMB (red
solid), MMB (blue dashed) and OMB (black dotted) populations. The points with
error bars show the mean Monte Carlo distribution and associated error.
## 6 Preliminary raw albedo distribution of MBAs
With the inclusion of visible data in our modeling, we determine albedos as
well as diameters for the asteroids discussed here. We use the published $H$
and $G$ values for all asteroids, available from the MPC. During the
confirmation of the calibration of WISE for asteroids, Mainzer et al. (2011b)
investigated the need for an offset in H to account for systematic errors in H
values, but found that no offset was required (c.f. Jurić et al., 2002, who
found a $0.2~{}$mag shift). The $H$ magnitudes were assigned a random error of
$\sim 0.2~{}$mag. We perform Monte Carlo simulations of our visible light
measurements as well as of the thermal measurements to quantify the error on
albedo, however in all cases the minimum error on albedo will be $20\%$
(Mainzer et al., 2011b) for objects with optical data and one good thermal
band. We have sufficient optical data to determine albedos for $112,265$ MBAs.
Though our individual albedos have large error bars, the population statistics
can still provide us with a window into the state of the surface composition
of the Main Belt.
Figure 8: Preliminary raw differential albedo distributions for all inner-,
middle-, and outer-Main Belt asteroids, shown as red solid, blue dashed, and
black dotted histograms respectively. The points show Monte Carlo simulations
of the albedos and their error bars, and the smooth curves the best fitting
double-Gaussian distributions.
In Fig 8 we show the differential preliminary raw albedo distribution (PRAD)
of all the inner-, middle-, and outer-Main Belt asteroids in our survey. We
then take the fitted albedos and their respective error bars and perform a
100-trial MC simulation of these values to find a mean distribution with
errors, shown as points. In all cases, the peaks of the distributions broaden
slightly in the MC simulation, which is expected. In log-albedo-space the
differential distribution is well described by a bimodal Gaussian
distribution. We show our best-fitting double-Gaussian (fitted to the mean
distribution found through the MC simulations) as the smooth dotted curve
under each set of points.
The bimodality in albedo likely traces the difference between the two major
branches of asteroid composition: the S-type asteroids with high albedos and
the C-type asteroids with low albedos (Chapman et al., 1975; Tedesco et al.,
1989). Mainzer et al. (2011c) investigate the specific link between albedo and
a variety of taxonomic classification systems. As discussed above, however,
the PRAD will naturally include the observational biases of the ground-based
telescopes used to determine the optical magnitudes needed to find the visible
albedo, favoring higher albedo asteroids and over-representing their
contribution to the total population, particularly in the Inner Main Belt.
Debiasing, currently being undertaken, will allow us to quantify and remove
this effect.
The mean value and width of the Gaussian that best describes the dark peak of
the PRAD for each population is consistent across populations, with mean
albedo $\mu=0.06$ and a dispersion of $\sigma=^{+.03}_{-.02}$. Note that the
Gaussian error bars on the $\mu$ value are in log space, and thus asymmetric
in native units. Unlike the dark asteroids, the bright complex shows a
distinct change in the mean value in the PRAD as one moves out in the Main
Belt. The mean albedo of the bright peak for the Gaussian describing each
population is: $\mu_{IMB}=0.28$, $\mu_{MMB}=0.25$, and $\mu_{OMB}=0.17$, with
widths of $\sigma_{IMB}=^{+.13}_{-.09}$, $\sigma_{MMB}=^{+.11}_{-.08}$, and
$\sigma_{OMB}=^{+.08}_{-.05}$.
The objects discovered by NEOWISE that have not had optical followup will add
a significant bias to the PRAD as these objects are most likely to be ones
missed by optical surveys, i.e. lower albedo asteroids. During the cryogenic
portion of the survey, NEOWISE observed $23,616$ previously unknown Main Belt
asteroids with data of sufficient quality for thermal modeling and with
orbital arcs longer than $1$ day (and thus not considered “one night stand”
observations by the MPC). Some $\sim 10,000$ additional asteroids were given
temporary designations by the MPC but did not have sufficient arc length to
calculate an orbit. While they do not have known orbits and thus can’t be
classified as members of the Main Belt or not, we can use this to set an upper
limit on the number of lost MBAs. As these objects are linked to older
precovery data or are serendipitously followed up our count of discovered
objects with computed orbits will increase. Out of these discovered objects
$19,178$ have optical photometric data as well as thermal infrared, allowing
for albedo determination.
It should be noted that while many of the discovered objects have optical
observations, there is a strong bias in favor of recovery of the highest
albedo discoveries by subsequent ground based observations. The visible light
received from an object is directly proportional to the albedo of that object,
while the thermal infrared flux is driven by the temperature of the surface,
which is only weakly dependent on albedo as shown in Eq 1. As such, optical
surveys show a significant albedo bias towards brighter objects for both
discovery and recovery observations, while the NEOWISE infrared survey is more
sensitive to detection of low albedo objects. Any survey will have inherent
biases in the data set and it is necessary to account for them before the true
albedo distribution can be determined for a population.
Figure 9: The same as Fig 8, but for only the NEOWISE-discovered MBAs that
also received optical photometric followup.
We show in Fig 9 the albedo distribution of the NEOWISE-discovered MBAs with
optical photometry for the inner-, middle-, and outer-belt subpopulations.
Even though the optical followup will be biased toward favoring higher albedo
objects, the NEOWISE discoveries are dominated by low albedos, as these are
the objects that were initially missed by the ground-based optical surveys.
All three distributions can be described by a single Gaussian function with
mean albedo between $0.05<\mu<0.06$ and $0.02<\sigma<0.03$, consistent with
the values found for the dark complex in the whole population above. We can
use the albedos of the discovered objects and the roughly equal number of
objects in each subgroup as an initial attempt to constrain the effect of
these lost objects on the greater albedo distribution. We show in Fig 10 a
revised albedo distribution including this toy model for the albedos of the
$\sim 15,000$ Main Belt asteroids without followup photometry, based on the
albedo distribution of the NEOWISE-discovered objects. As expected the primary
effect is to increase the relative abundance of dark objects in each region of
the Belt.
Figure 10: The same as Fig 8, but now including a simple model for the objects
discovered by NEOWISE that have no optical followup photometry.
We note that the more detailed way to properly account for the uncertainties
introduced by objects without follow-up and objects for which reliable orbits
could not be determined is through careful modeling of survey biases. A
debiased study of the Main Belt asteroids will be subject of future work, and
will allow us to determine the true size and albedo distributions of these
objects.
## 7 Dynamically grouped albedos in the Main Belt
We also investigate the distribution of albedo as a function of orbital
parameters, in particular semimajor axis ($a$), eccentricity ($e$) and
inclination ($i$). Figure 11 shows the distribution of albedos as a function
of semimajor axis. We have color-coded the points by albedo using a “weather-
map” palette divided evenly in $\log(p_{V})$ space, and use this same color
code for all subsequent plots. Dark colors (black, grey, dark blue, blue)
indicate objects in the low albedo complex, while brighter colors (magenta,
red, orange, yellow) indicate members of the high albedo complex. Objects
colored yellow have very high albedos and are concentrated in the Hungaria
region and inner-Main Belt near the orbit of (4) Vesta. These largest albedos
found are likely artifacts of using $G=0.15$ to calculate the $H$ value: while
the thermal models for the diameters of these objects show no errors, using
the literature $H$ value forces anomalously high albedos. We investigate the
use of different $H$ and $G$ values for these highest albedo objects in a
future publication.
Figure 11: Asteroid albedo vs. semimajor axis. Colors denoted here are split
evenly in $\log(p_{V})$ space and are used to denote albedo in subsequent
plots.
We show in Figures 12 and 13 the plot of semimajor axis vs. inclination and
eccentricity, respectively, using the same colors denoted above. Asteroid
albedos are not homogeneously distributed throughout the Main Belt, but rather
are clumped in a-e-i space, correlating with the positions of known asteroid
families (Nesvorný et al., 2006a). MBAs have been previously shown to cluster
in color-space (Ivezić et al., 2002; Parker et al., 2008) and show similar
reflectance spectra (e.g. Binzel & Xu, 1993; Cellino et al., 2001, etc.)
indicative of the common origin of members of families, which are the result
of a catastrophic breakup of a single parent body (Hirayama, 1918). The
clustering of albedos is a further confirmation of this origin. A future paper
in this series will investigate the use of albedo as an added criterion to the
orbital parameters typically used to determine family membership.
Figure 12: Orbital inclination vs. semimajor axis. Colors are the same as Fig
11. Figure 13: Orbital eccentricity vs. semimajor axis. Colors are the same as
Fig 11.
## 8 IR reflectance
The reflectance spectrum for most asteroids (but not all, e.g. B-types) show
an increasing value as the wavelength moves from the visible to near-infrared
(NIR) region of the spectrum (e.g. DeMeo et al., 2009, etc.). If these trends
continue into the W1 and W2 bandpasses we would expect the reflectance
observed there to be higher than observed in the optical. We assume in our
thermal model that the reflectance in W1 and W2 is the same; depending on the
location and depth of absorption bands this may not be universally true, but
this assumption provides a generic constraint from which we can identify
interesting objects that do not follow this assumption. Reflectance depends on
both the albedo and the $G$ slope parameter, both of which cannot be assumed
to be wavelength independent. We do not have sufficient phase coverage to fit
$G_{NIR}$ and thus disentangle its effect from that of $p_{NIR}$, so we
present only the NIR reflectance ratio for objects in the Main Belt with
sufficient signal in W1 and/or W2 to be able to fit this value, a total of
$4194$ objects.
We show in Fig 14 plots of the NIR reflectance ratio compared to a range of
physical and orbital parameters, as well as running averages for those
distributions. All objects were detected in thermal emission, however only
objects with sufficient reflected light were able to provide fitted NIR
reflectance ratios. As such objects with higher IR albedos will be more likely
to have a fitted reflectance ratio (this is comparable to the biases inherent
in optical surveys). Evidence of this is apparent in Fig 14e, where the
running average of the ratio climbs for objects with fainter $H$ values. We
note that as the visible albedo $p_{V}$ is intricately linked with the NIR/Vis
ratio it cannot be considered an independent variable. The structure seen in
the running average in Fig 14f is expected to be heavily influenced by the
debiasing of the Main Belt population currently being undertaken. Future work
(Mainzer et al., 2011c) will explore the connection between taxonomic types
derived from spectroscopy and photometry and the IR reflectance ratio found in
the WISE data.
Figure 14: NIR reflectance ratio for all objects with fitted values, compared
to (a) semi-major axis, (b) diameter, (c) eccentricity, (d) inclination, (e)
absolute magnitude, (f) albedo, (g) beaming parameter and (h) phase. The thick
red line shows the running average for $400$ object-wide bins stepped by $40$
objects. The picket-fence effect in the absolute magnitude is an artifact of
the $0.1~{}$mag accuracy of $H$ for most objects.
In Fig 15 we show the raw differential distribution of NIR reflectance ratios
for the IMB, MMB, and OMB asteroids with fitted ratios. Also shown are the
mean distribution and associated errors derived from a 100-trial Monte Carlo
simulation of all measured reflectance ratios. All three populations show a
peak between $1.2<$ratio$<1.4$ while the running average vs. semi-major axis
varies across the Main Belt from $1.3<$ratio$<1.6$. This is consistent with
the values found for the NEOs by Mainzer et al. (2011d). For objects without
fitted NIR reflectance ratios, we use the mean of all fitted values of $1.5$
and an error bar based on the associated standard deviation of
$\sigma_{ratio}=0.5$ for modeling purposes.
Figure 15: Distribution of NIR reflectance ratio for the IMB (red solid), MMB
(blue dashed) and OMB (black dotted) populations. Shown as points are the mean
distribution and associated errors from Monte Carlo simulations of each ratio.
## 9 Asteroid families
Asteroid families were first identified as groups of objects that clustered
tightly in orbital element-space by Hirayama (1918) nearly a century ago.
Subsequent work has confirmed that families originate from the catastrophic
breakup of a single parent asteroid after an impact (see Cellino, et al.,
2009, for a recent review of the current state of the field). This single
mineralogical origin causes families to cluster tightly not only when
comparing orbital elements but also when investigating colors (Ivezić et al.,
2002; Parker et al., 2008) and reflectance spectra (e.g. Binzel & Xu, 1993;
Cellino et al., 2001, etc.). The SFD of asteroid family members can also act
as a tracer of the physical properties of the original parent body and can
even be used to constrain the impact velocity and angle (Durda et al., 2007).
However, a major deficiency in models to date has been the lack of measured
diameters for the family members, forcing these values to be assumed based on
the apparent magnitude of the object. Albedo measurements of the largest
bodies in a family are often available from the IRAS data set (Tedesco et al.,
2002) and can be used to assume an albedo for all family members, but this can
add a significant and systematic error to the diameters used, especially in
the cases where families may be mixed or where the largest body in a family
may not be associated with the other members (e.g. Cellino et al., 2001).
There are a number of methods that can be used to determine which asteroids
are members of a given family. The Hierarchical Clustering Method (HCM,
Zappalà, et al., 1990), a commonly used technique, takes the differences in
velocities between the proper orbital elements of objects to reveal dynamical
associations. Nesvorný (2010) used this method to identify $55$ families out
of $293,368$ MBAs with low inclinations. We use these $55$ families as the
baseline for our analysis, selecting those objects that appear both in that
list and in the WISE observations. As all these objects were discovered by
optical surveys, there will be an inherent bias in the albedos favoring
brighter family members. Future work will address this bias and explore the
use of albedo in conjunction with dynamical orbital properties to identify
members of asteroid families and to reject interloper objects.
Of the $55$ families identified by Nesvorný (2010) we find that $46$ have more
than $20$ members observed during the cryogenic WISE mission. Due to the
limitations of proper orbital element calculations, high inclination objects
are not included in the AstDys
list555http://hamilton.dm.unipi.it/astdys/index.php of proper orbital elements
(Milani & Knez̀ević, 1994). As such, high inclination families are likewise
not represented in the list of family members. We have, however, included in
our analysis the Pallas family, identified by Nesvorny (private communication)
through the same methods as the $55$ published families. We also include the
asteroids located in the Hungaria region; while not canonically included in
the list of dynamical families, recent work by Warner, et al. (2009b) and
Milani, et al. (2010) support the classification of objects near Hungaria in
orbital space as a true dynamical family. These two added groups bring our
total considered population up to $48$ families.
We show in Fig 16 the proper inclination against the proper semimajor axis for
all objects identified as a member of one of the families considered here.
Note that for the Hungaria and Pallas families proper orbital elements were
not available and so osculating elements were used. Similarly, we show in Fig
17 the proper eccentricity against the proper semimajor axis. In both plots,
we use the same color scheme as shown in Fig 11. It is quite apparent in these
plots that families characteristically have uniform albedos, however there are
notable exceptions.
Figure 16: Proper orbital inclination vs. proper semimajor axis for asteroid
families. Colors are the same as Fig 11. Figure 17: Proper orbital
eccentricity vs. proper semimajor axis for asteroid families. Colors are the
same as Fig 11.
We show in Fig 18 the cumulative PRSFD for each of the families observed
during the WISE survey, as well as Monte Carlo simulations of all
distributions with appropriate error bars. Without debiasing, the PRSFD cannot
be assumed to represent the true size distribution of the entire family
population, as neither the NEOWISE-inherent biases nor the biases in family
selection have been accounted for. Approximately $25\%$ of these families show
kinks at the large end of the distribution inconsistent with a simple power-
law. As small number statistics dominate families especially at the largest
sizes, and because WISE did not survey the entire Main Belt before the
exhaustion of cryogen, precise debiasing is required to confidently measure
the shape of the true SFD, especially at the largest sizes for each family.
Figure 18: Cumulative raw size-frequency distribution for each asteroid family
considered here. The solid black line shows the family indicated by the name
in each plot. The dotted red line shows the PRSFD for the Vesta family is all
plots, for ease of comparison.
In Fig 19 we show the normalized PRAD for each family, along with the Monte
Carlo results for each distribution. As was evident in Fig 16 and Fig 17 most
families show a single-peaked albedo distribution, however as mentioned above
debiasing will be critical to proper interpretation of these distributions.
About $15\%$ of families include a small population of objects with non-
characteristic albedos; these may be the result of an improper association of
background objects into the family. However, another $10\%$ of families show
significant mixing between two albedo types that cannot be solely due to the
intrusion of a few background objects. In particular the Nysa-Polana and
Tirela families show nearly parity between the low and high albedo objects in
the preliminary raw distributions.
Among the families with sufficient data for study were the Karin and Koronis
families. The Karin family is believed to be a very young family, with an age
of $\sim 5.8~{}$Myr, that formed from the breakup of a member of the much
older ($2-3~{}$Gyr) Koronis family (Nesvorný et al., 2006b). We fit a Gaussian
to the observed family albedo distributions. We find that the Karin family has
a lower mean albedo from this fit ($p_{V-Karin}=0.18\pm 0.05$) than the
Koronis family ($p_{V-Koronis}=0.24\pm 0.05$) where the error bars represent
the width of the best-fit Gaussian. Chapman (2004) gives an overview of space
weathering effects, a process that is generally thought to darken and redden
surfaces of atmosphereless bodies in the Solar system. Our result is in
apparent contradiction with this hypothesis under the assumption that the
compositions of both families are identical. It is possible that variations in
the composition, the presence of interlopers, or differentiation of the proto-
Koronis parent body could result in this observation. This analysis will be
strengthened by the identification of more NEOWISE-observed family members
(there were $31$ identified members of the Karin family and $1079$ members of
the Koronis family in this data set) and the acquisition of additional
compositional information.
We also observed $984$ asteroids that were identified as members of the
Baptistina family. Bottke et al. (2007) postulate that a fragment from the
breakup of the Baptistina family was the impactor responsible for the K/T mass
extinction event. However, these authors assumed an albedo for the family
members of $0.05$. We find that the best fitting Gaussian to the Baptistina
family member albedos has a mean of $p_{V-Baptistina}=0.21^{+0.13}_{-0.08}$
where the error bar indicates the width of the Gaussian distribution. The
method of age determination used by Bottke et al. (2007) depends on the albedo
assumed, and the calculated age $T$ is proportional to albedo following
$T\propto p_{V}^{-0.5}$. Thus our measured albedo results in an age for the
breakup approximately $\sim 2$ times younger than that found with the lower
assumed albedo (from $160~{}$Myr to $\sim 80~{}$Myr), reducing the likelihood
that the Baptistina breakup generated the K/T impactor (c.f. Reddy et al.,
2009).
Also of interest are the families that show characteristic albedos distinct
from the distribution observed in the Main Belt in Fig 8. For example, the Eos
and Aeolia families have characteristic albedos that fall in between the two
peaks of the MBA albedos. Standing out as an outlier is the Hungaria family:
while showing a fairly strong coherence in albedo within the family, that
characteristic albedo is incredibly high (see below). Future work will
investigate whether variations in the $G$ slope parameter used to compute the
$H$ absolute magnitude (and thus the reflected-light albedo) from the assumed
value of $G=0.15$ typically used could account for the very large albedos
calculated in these preliminary results.
Figure 19: The same as Fig 18 but for the PRAD. Monte Carlo simulations of the
distributions are plotted as grey points with error bars. The distributions
have been normalized to unit area for easier comparison.
## 10 Unusual objects
A small fraction of our objects had fits that did not conform to the general
trends seen for the population as a whole. These highly unusual fits may be
caused by strange physical parameters, incorrect associated properties (e.g.
$H$ mag, orbital elements, etc), or a break-down of the NEATM model. In any
case, these objects warrant further inspection.
The most obvious candidates for this category of unusual fits are the
asteroids with very high visible geometric albedos ($p_{V}\geq 0.70$). We find
$193$ objects in our survey have computed albedos that fall into this range,
mostly contained within the region of orbital element space occupied by the
Hungaria and Vesta families (out of $343$ objects found in the Hungaria region
and $1938$ in the Vesta family), implying a possible mineralogical origin.
The asteroid (434) Hungaria, the lowest numbered member of its namesake
region, has a relatively large albedo of $p_{V}=0.46$ which can be explained
by a composition dominated by the iron-poor mineral enstatite however some of
the observed spectral features may require contamination from a darker,
external source (Kelley & Gaffey, 2002). This may indicate that albedos can
range larger than $p_{V}\sim 0.5$, but albedos significantly higher than this
are likely suspect.
Visual inspection of the thermal model fits of these objects shows that these
high albedos are not due to a failure of the thermal model. For those objects
where IR reflectance factor could also be fitted, we find values typically
with reflectance ratio$\leq 1$, while the distribution of beaming parameters
for these objects is similar to that of the general Main Belt population. If
the $H$ or $G$ values for these objects were very far from the true values
this could result in the unusual calculated albedos.
By increasing the error bar we assume for the $H$ value and setting the IR
reflectance ratio to a constant value of ratio$=1.5$ (the average for MBAs) we
are able to find good fits of objects with reflected light in W1 and/or W2.
Under this assumption, these objects all return fits with albedos comparable
to that of (434) Hungaria. However all these cases require the $H$ magnitudes
to be $0.4-1.0~{}$mag fainter than the values given by the Minor Planet
Center.
A mis-identification of $H$ could be a symptom of an improper assumption for
the $G$ slope parameter. In particular if $G$ should be much larger (e.g.
equal to or greater than the value for (44) Nysa of $G=0.46$666as given by the
Small Body Database: http://ssd.jpl.nasa.gov/sbdb.cgi this would result in an
$H$ value that was too bright by $\sim 0.4~{}$mag or more, which would thus
give an albedo that was a factor of $\sim 1.4$ too large. This can account for
some, but not all, of our improbably high albedos. Large light curve
amplitudes may also contribute to an absolute magnitude that is too bright.
Future work will address these objects in detail.
In addition to the high albedo objects, we find a very small number of objects
with beaming parameters at or close to the theoretical limit of $\eta=\pi$.
While some of these fits can be rejected upon visual inspection e.g. because
of a single bad point in W4 dragging the fit to higher beaming parameter, at
least $6$ objects appear to be legitimate fits with beaming parameters
indicative of latitudinally-isothermal surfaces, though none of the fits show
any significant lightcurve variations. Further investigation will be critical
to determine if they show rapid rotation or very high thermal inertia needed
to explain this beaming parameter.
Approximately $10,000$ MBAs were observed by WISE at two different epochs. We
initially treated each epoch separately for fitting purposes, and in the
majority of cases the fits were within the expected error of each other. For
those that weren’t, we recomputed the best fitting model using both epochs
together and forcing the physical parameters to be identical. We find $36$
objects for which the two-epoch fits could not produce a good fit at one or
both epochs. While some of these objects may have very long rotation
periods($P>10~{}$days) and thus different projected diameters between epochs,
others do not appear to show any light curve variation during our observations
and may be cases of objects showing a significant difference between the
temperatures of the morning and afternoon hemispheres of the body.
We also observe $151$ objects with peak-to-trough light curve variations
larger than $1.5~{}$mag in W3 and average magnitude measurement error smaller
than $\sigma_{W3}<0.2$ after removing spurious measurements. As these are not
fitted amplitudes they represent a minimum for the light curve amplitude for
the body observed. An example of one of these objects, (61469), is shown in
Fig 20. There are also many objects with amplitudes smaller than this cutoff
with readily apparent rotational effects, too. A future work will investigate
specific light curves to determine the period and amplitude of the objects.
Figure 20: Light curve for asteroid (61469), with magnitudes for $W4$, $W3$
and $W2$ shown in red, green and cyan (top to bottom) respectively. The dashed
line shows the modeled magnitudes for the best-fitting sphere. The period of
this asteroid is approximately $\sim 40~{}$hr assuming a double-peaked
lightcurve.
## 11 Conclusions
We have presented an initial analysis of Main Belt asteroids detected by
NEOWISE during the cryogenic portion of the WISE mission. With infrared fluxes
of sufficient quality to determine diameters and albedos for $129,750$ MBAs,
we show the power and great potential contained in this dataset. These data
allow us to probe the composition, structure, and history of the Main Belt in
ways that were previously impossible.
For objects with thermal emission detected in two or more bands we allowed the
beaming parameter to vary. We find a mean beaming parameter of $\eta\sim 1.0$
however we do see evidence of a phase dependence for the beaming parameter,
ranging from $\eta\sim 0.94$ for low phases to $\eta\sim 1.14$ for higher
phases (within the Main Belt). The best fit linear relation between beaming
and phase is $\eta=0.79+0.011\alpha$ which is a much shallower relation than
seen previously in Wolters & Green (2009), but consistent with Mainzer et al.
(2011d) which included the near Earth objects with fitted beaming parameters
as well.
As was observed in the IRAS data (e.g. Tedesco et al., 1989, 2002, 2005) the
albedos of Main Belt asteroids are strongly bimodal: a bright complex
($p_{V}\sim 0.25$) and the dark complex ($p_{V}\sim 0.06$). We find both peaks
to be well described by Gaussian distributions in log-albedo space.
We find that the reflectance of asteroids in the W1 and W2 bandpasses is
typically larger than the albedo found in visible light. The best fit ratio of
reflectance ranges from $\sim 1.6$ in the IMB to $\sim 1.3$ in the OMB however
the spread of values is large, and the final distribution will depend strongly
on the debiasing.
We identify albedo clusters in $a-e-i$ space corresponding to the locations of
asteroid families. Albedo is another coherent property of dynamical families
in addition to orbit (Hirayama, 1918), color (Ivezić et al., 2002) and
reflectance spectrum (e.g. Binzel & Xu, 1993; Cellino et al., 2001, etc.).
Albedo can also be used to trace the halos of similar objects that surround
some families (e.g. Vesta, Eos, etc., Parker et al., 2008) that may be
evidence of a collisional breakup very early in the age of the Solar System.
Using asteroids previously identified though HCM techniques to be members of
collisional families we show that most, but not all, families have a
characteristic albedo.
Critical to any interpretation of the observations presented here is a careful
accounting of the biases in both the WISE survey data as well as the optical
data used to derive albedos. We are currently undertaking an extensive
debiasing campaign with the goal of producing unbiased size and albedo
distributions for the Main Belt. This will be the subject of the next paper in
this series.
## Acknowledgments
J.R.M. was supported by an appointment to the NASA Postdoctoral Program at
JPL, administered by Oak Ridge Associated Universities through a contract with
NASA. J.R.M. thanks M. Delbó and M. Mueller for providing access to their
thermal modeling code which was helpful in early test cases. This publication
makes use of data products from the Wide-field Infrared Survey Explorer, which
is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration. This publication also makes use
of data products from NEOWISE, which is a project of the Jet Propulsion
Laboratory/California Institute of Technology, funded by the Planetary Science
Division of the National Aeronautics and Space Administration. This research
has made use of the NASA/IPAC Infrared Science Archive, which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration. We thank the
worldwide community of dedicated amateur and professional astronomers devoted
to minor planet follow-up observations. We are deeply grateful for the
outstanding contributions of all members of the WISE and NEOWISE teams.
## References
* Binzel & Xu (1993) Binzel, R.P. & Xu, S., 1993, Science, 260, 186.
* Bottke et al. (2005a) Bottke, W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, A., Vokrouhlický & Levison, H.F., 2005a, Icarus, 175, 111.
* Bottke et al. (2005b) Bottke, W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, A., Vokrouhlický & Levison, H.F., 2005b, Icarus, 179, 63.
* Bottke et al. (2006) Bottke, W.F., Vokrouhlický, D., Rubincam, D.P. & Nesvorný, D., 2006, AREPS, 34, 157.
* Bottke et al. (2007) Bottke, W.F., Vokrouhlický, D. & Nesvorný, D., 2007, Nature, 449, 48.
* Bowell et al. (1989) Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J. & Harris, A.W., 1989, Asteroids II, University of Arizona Press, 524.
* Bus & Binzel (2002) Bus, S.J. & Binzel, R.P., 2002, Icarus, 158, 146.
* Cellino et al. (1999) Cellino, A., Gil Hutton, R., Tedesco, E.F., di Martino, M. & Brunini, A., 1999, Icarus, 138, 129.
* Cellino et al. (2001) Cellino, A., Zappalà, V., Doressoundiram, A., di Martino, M., Bendjoya, Ph., Dotto, E. & Migliorini, F., 2001, Icarus, 152, 225.
* Cellino, et al. (2009) Cellino, A., Del’Orro, A. & Tedesco, E.F., 2009, P&SS, 57, 173.
* Chapman et al. (1975) Chapman, C.R., Morrison, D. & Zellner, B., 1975, Icarus, 25, 104.
* Chapman (2004) Chapman, C.R., 2004, AREPS, 32, 539.
* Cutri et al. (2011) Cutri, R.M., Wright, E.L., Conrow, T., Bauer, J., Benford, D., Brandenburg, H, Dailey, J., Eisenhardt, P.R.M., et al., 2011, http://wise2.ipac.caltech.edu/docs/release/prelim/expsup/wise_prelrel_toc.html
* DeMeo et al. (2009) DeMeo, F.E., Binzel, R.P., Slivan, S.M. & Bus, S.J., 2009, Icarus, 202, 160.
* Durda et al. (2007) Durda, D.D., Bottke, W.F., Nesvorný, D., Enke, B.L., Merline, W.J., Asphaug, E. & Richardson, D.C., 2007, Icarus, 186, 498.
* Gladman et al. (2009) Gladman, B.J., David, D.R., Neese, C., Jedicke, R., Williams, G., Kavelaars, J.J., Petit, J.-M., Scholl, H., Holman, M., Warrington, B., Esquerdo, G., & Tricarico, P., 2009, Icarus, 202, 104.
* Gradie & Tedesco (1982) Gradie, J. & Tedesco, E., 1982, Science, 216, 1405.
* Grav et al. (2011) Grav, T., Mainzer, A.K., Bauer, J.M., Masiero, J., Spahr, T., McMillan, R.S., Walker, R., Cutri, R., Wright, E., 2011, submitted to ApJ.
* Harris (1998) Harris, A.W., 1998, Icarus, 131, 291.
* Harris & Lagerros (2002) Harris, A.W. & Lagerros, J.S.V., 2002, Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, 205.
* Hirayama (1918) Hirayama, K., 1918, Icarus, 31, 185.
* Howett et al. (2010) Howett, C.J.A., Spencer, J.R., Pearl, J. & Segura, M., 2010, Icarus, 206, 573.
* Ivezić et al. (2001) Ivezić, Z̆., Tabachnik, S., Rafikov, R., Lupton, R.H., Quinn, T., Hammergren, M., Eyer, L., Chu, J., et al., 2001, AJ, 122, 2749.
* Ivezić et al. (2002) Ivezić, Z̆., Lupton, R.H., Jurić, M., Tabachnik, S., Quinn, T., Gunn, J.E., Knapp, G.R., Rockosi, C.M. & Brinkmann, J., 2002, AJ, 124, 2943.
* Jedicke & Metcalfe (1998) Jedicke, R. & Metcalfe, T.S., 1998, Icarus, 131, 245.
* Jurić et al. (2002) Jurić, M., Ivezić, Z̆., Lupton, R.H., Quinn, T., Tabachnik, S., Fan, X., Gunn, J.E., Hennessy, G.S., Knapp, G.R., Munn, J.A., Pier, J.R., Rockosi, C.M., Schneider, D.P., Brinkmann, J., Csabai, I. & Fukugita, M., 2002, AJ, 124, 1776.
* Kelley & Gaffey (2002) Kelley, M.S. & Gaffey, M.J., 2002, M&PS, 37, 1815.
* Lebofsky et al. (1986) Lebofsky, L.A., Sykes, M.V., Tedesco, E.F., Veeder, G.J., Matson, D.L., Brown, R.H., Gradie, J.C., Feierberg, M.A. & Rudy, R.J., 1986, Icarus, 68, 239.
* Lebofsky & Spencer (1989) Lebofsky, L.A. & Spencer, J.R., 1989, Asteroids II, University of Arizona Press, 128.
* Li et al. (2010) Li, J.-Y., McFadden, L.A., Thomas, P.C., Mutchler, M.J., Parker, J.W., Young, E.F., Russell, C.T., Sykes, M.V., Schmidt, B.E., 2010, Icarus, 208, 238.
* Liu et al. (2008) Liu, F., Cutri, R., Greanias, G., Duval, V., Eisenhardt, P., Elwell, J., Heinrichsen, I., Howard, J., et al., 2008, SPIE, 7017E, 16.
* Mainzer et al. (2006) Mainzer, A.K., Eisenhardt, P., Wright, E.L., Liu, F.-C., Irace, W., Heinrichsen, I., Cutri, R. & Duval, V., 2006, SPIE, 6265, 21.
* Mainzer et al. (2011a) Mainzer, A.K., Bauer, J.M., Grav, T., Masiero, J., Cutri, R.M., Dailey, J., Eisenhardt, P., McMillan, R.M. et al., 2011a, ApJ, in press.
* Mainzer et al. (2011b) Mainzer, A.K., Grav, T., Masiero, J., Bauer, J.M., Wright, E., Cutri, R.M., McMillan, R.S., Cohen, M., Ressler, M., Eisenhardt, P., 2011b, submitted to ApJ.
* Mainzer et al. (2011c) Mainzer, A.K., Grav, T., Masiero, J., Hand, E., Bauer, J.M., Tholen, D., McMillan, R.S., Spahr, T. et al., 2011c, submitted to ApJ.
* Mainzer et al. (2011d) Mainzer, A.K., et al., 2011d, in preparation.
* Milani & Knez̀ević (1994) Milani, A. & Knez̀ević, Z., 1994, Icarus, 107, 219.
* Milani, et al. (2010) Milani, A., Knez̀ević, Z., Novaković, B. & Cellino, A., 2010, Icarus, 207, 769.
* Morbidelli et al. (2010) Morbidelli, A., Brasser, R., Gomes, R., Levison, H.F. & Tsiganis, K., 2010, AJ, 140, 1391.
* Morrison (1974) Morrison, D., 1974, ApJ, 194, 203.
* Mueller et al. (2010) Mueller, T.G., Lellouch, E., Stansberry, J., Kiss, C., Santos-Sanz, P., Vilenius, E., Protopapa, S., Moreno, R., et al., 2010, A&A, 518L, 146.
* Nesvorný et al. (2002) Nesvorný, D., Bottke, W.F., Dones, L. & Levison, H.F., 2002, Nature, 417, 720.
* Nesvorný & Bottke (2004) Nesvorný, D. & Bottke, W.F., 2004, Icarus, 170, 324.
* Nesvorný et al. (2006a) Nesvorný, D., Bottke, W.F., Vokrouhlický, D., Morbidelli, A. & Jedicke, R., 2006a, Asteroids, Comets, Meteors, Proc. of the 229th Symp. of the IAU, Daniela, L., Sylvio Ferraz, M., and Angel, F.J. (eds), Cambridge University Press, 289.
* Nesvorný et al. (2006b) Nesvorný, D., Enke, B.L., Bottke, W.F., Durda, D.D., Asphaug, E. & Richardson, D.C., 2006b, Icarus, 183, 296.
* Nesvorný (2010) Nesvorny, D., 2010, EAR-A-VARGBDET-5-NESVORNYFAM-V1.0, NASA Planetary Data System.
* O’Brien & Greenberg (2003) O’Brien, D.P. & Greenberg, R., 2003, Icarus, 164, 334.
* Ostro et al. (2002) Ostro, S.J., Hudson, R.S., Benner, L.A.M., Giorgini, J.D., Magri, C., Margot, J.L. & Nolan, M.C., 2002, Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, 151.
* Parker et al. (2008) Parker, A., Ivezić, Z̆., Jurić, M., Lupton, R.H., Sekora, M.D. & Kowalski, A., 2008, Icarus, 198, 138.
* Piazzi (1801) Piazzi, G., 1801, Nella Reale Stamperia, Palermo, 25.
* Reddy et al. (2009) Reddy, V., Emery, J.P., Gaffey, M.J., Bottke, W.F., Cramer, A. & Kelley, M.S., 2009, M&PS, 44, 1917.
* Ryan & Woodward (2010) Ryan, E.L. & Woodward, C.E., 2010, AJ, 140, 933.
* Schmidt et al. (2009) Schmidt, B.E., Thomas, P.C., Bauer, J.M., Li, J.-Y., McFadden, L.A., Mutchler, M.J., Radcliffe, S.C., Rivkin, A.S., et al., 2009, Science, 326, 275.
* Shevchenko & Tedesco (2006) Shevchenko, V.G. & Tedesco, E.F., 2006, Icarus, 184, 211.
* Tedesco et al. (1989) Tedesco, E.F., Williams, J.G., Matson, D.L., Veeder, G.J., Gradie, J.C. & Lebofsky, L.A., 1989, AJ, 97, 580.
* Tedesco et al. (2002) Tedesco, E.F., Noah, P.V., Noah, M. & Price, S.D., 2002, AJ, 123, 1056.
* Tedesco et al. (2004) Tedesco, E.F., Noah, P.V., Noah, M. & Price, S.D., 2004, IRAS-A-FPA-3-RDR-IMPS-V6.0, NASA Planetary Data System.
* Tedesco et al. (2005) Tedesco, E.F., Cellino, A. & Zappalá, V., 2005, AJ, 129, 2869.
* Walsh & Morbidelli (2011) Walsh, K.J. & Morbidelli, A., 2011, A&A, 526, 126.
* Warner et al. (2009a) Warner, B.D., Harris, A.W. & Pravec, P., 2009a, Icarus, 202, 134.
* Warner, et al. (2009b) Warner, B.D., Harris, A.W., Vokrouhlický, D., Nesvorný, D. & Bottke, W.F., 2009b, Icarus, 204, 172.
* Wolters & Green (2009) Wolters, S.D. & Green, S.F., 2009, MNRAS, 400, 204.
* Wright (2007) Wright, E.L., 2007, arXiv:astro-ph/0703085.
* Wright et al. (2010) Wright, E.L., Eisenhardt, P., Mainzer, A.K., Ressler, M.E., Cutri, R.M., Jarrett, T., Kirkpatrick, J.D., Padgett, D., et al., 2010, AJ, 140, 1868.
* Zappalà, et al. (1990) Zappalà, V., Cellino, A., Farinella, P. & Knezevic, Z., 1990, AJ, 100, 2030.
* Zellner (1979) Zellner, B., 1979, Asteroids (University of Arizona Press), 783.
|
arxiv-papers
| 2011-09-19T17:09:36 |
2024-09-04T02:49:22.293903
|
{
"license": "Public Domain",
"authors": "Joseph R. Masiero, A. K. Mainzer, T. Grav, J. M. Bauer, R. M. Cutri,\n J. Dailey, P. R. M. Eisenhardt, R. S. McMillan, T. B. Spahr, M. F. Skrutskie,\n D. Tholen, R. G. Walker, E. L. Wright, E. DeBaun, D. Elsbury, T. Gautier IV,\n S. Gomillion, and A. Wilkins",
"submitter": "Joseph Masiero",
"url": "https://arxiv.org/abs/1109.4096"
}
|
1109.4097
|
# Intermittent Josephson effect with feedback voltage and temperature
oscillations in graphite-coated nanocapsules with superconducting TaC core
Dianyu Geng Zhenhua Wang zhwang@imr.ac.cn Da Li Zhidong Zhang Shenyang
National Laboratory for Materials Science, Institute of Metal Research, and
International Center for Materials Physics, Chinese Academy of Sciences,
Shenyang 110016, People’s Republic of China Xiaolin Wang Institute for
Superconducting and Electronic Materials, Faculty of Engineering, Australian
Institute for Innovative Materials, University of Wollongong, Wollongong,
Australia
###### Abstract
An intermittent Josephson effect in the form of voltage and temperature
oscillations in the voltage - current curves near 2 K is observed in pellets
consisting of superconducting TaC nanocapsules coated with graphite. This
phenomenon is attributed to non-equilibrium conditions, when Cooper pairs
across a junction, which stimulate the emission of photons and the feedback
temperature change of the junction. It occurs in a three-dimensional granular
framework composed of TaC/carbon/TaC tunneling junctions with a Mott metal-
insulator transition, below the critical temperature Tc of non-ideal type-II
superconductor TaC.
###### pacs:
75.20.-z,75.75.Lf,61.46.Hk
The Josephson tunneling effect in superconductors is one of the most important
quantum effects 01jose ; 02jose ; 03jean . Nanocapsules (or core-shell
nanostructures) have already revealed abundant new physics and potential
applications 04sait ; 05zhang . Superconducting nanocapsules consisting of
superconductors (of any type) as core and normal (i.e. non-superconducting)
materials as shells are particularly interesting in terms of the fabrication
of junctions. The behavior of tunneling supercurrents in Josephson junctions
may be very unlike ordinary experience. For instance, Josephson predicted non-
equilibrium properties of coupled superconductors02jose , such as the
occurrence of dc supercurrents (i.e. the transfer of Cooper pairs across the
barrier), provided the potential difference is such that energy can be
conserved by absorption or stimulated emission of a photon (or multiphoton).
Intermittent-type chaos occurring in rf- and dc-current-driven Josephson
junctions was also investigated 06ben ; 07kautz ; 08goldhirsch ; 09octavio ;
zumofen10 .
It is particularly important to detect the transformation between Cooper pairs
and electrons during non-equilibrium processes in Josephson junctions. These
inspire us to fabricate Josephson junctions using carbon-coated TaC
nanocapsules and to investigate their superconductive characteristics related
to the junction tunneling effect. It is expected that a collective variant of
the single-electron tunneling and/or supercurrent effects may take place in a
three-dimensional framework of granular superconductors with tunneling
junctions li11 ; wang12 ; wang13 . Although there have been many studies on
intermittent-type Josephson effect 06ben ; 07kautz ; 08goldhirsch ; 09octavio
; zumofen10 , up to our knowledge, there has been no report on intermittent
Josephson effects with feed-back temperature change on materials. In this
Letter, we report the structure and electric, magnetic, and superconducting
properties of TaC(C) nanocapsules. Near 2 K, voltage and temperature
oscillations are observed in the current-voltage curves, which are attributed
to that electron pairs intermittently tunneling through the Josephson non-zero
voltage junctions emit photons 02jose . At low temperature, the
superconductive currents first take place in the junctions. emitting photons,
arising temperature and resistance, and then the superconductive currents
decrease (or stop), decreasing the Cooper pairs and the temperature until the
next cycle (the supercurrents take place again at temperature low enough).
There is a feedback of the temperature change and the voltage - current
function of the junctions. The combination effect of the superconductor TaC
and the Mott metal-insulator transition of graphite are likely responsible for
the oscillations.
The carbon-coated TaC nanocapsules studied in this work have been fabricated
via an arc discharge process in ethanol vapor and argon atmosphere. The
details of this method have been described elsewhere li11 ; wang12 ; wang13 .
A piece of pure Ta metal and a graphite rod were used as anode and cathode,
respectively. The phase and structure of the TaC(C) nanocapsules were
investigated by X-ray diffraction (XRD) using Cu-Kα ($\lambda$ = 0.15405 nm)
radiation. The particle size was analyzed by high-resolution transmission
electron microscopy (TEM). The transport and magnetic properties were measured
in a Quantum Design superconducting quantum interference device (SQUID)
magnetometer, with maximum temperature tolerance of less than 5 mK below 5 K.
The relative error of the temperature and voltage data is less than 0.1%. For
the resistivity measurements, as-prepared TaC(C) nanocapsules were pressed
into pellets under 1.2 GPa. I-V curves were measured with a Keithley 2400
Source Meter and a Keithley 2182 Nanovoltmeter. The temperature dependence of
the resistivity and the I-V curves were measured using the SQUID in the sweep
mode.
The TEM image in Fig. 1 shows the core/shell structure of the TaC(C)
nanocapsules. The crystal plane spacings of the cores for some of the
nanocapsules in Fig. 1 are only in one direction, and they usually consist of
only one TaC grain coated with a carbon shell. The left inset of Fig. 1 shows
a single nanocapsule which contains a single grain of TaC as core and carbon
as shell. The characteristic spacing of the strong-reflection (111) plane of
TaC is 0.258 nm. The size of the nanocapsules ranges between 10 and 20 nm. The
XRD pattern shown in the right inset of Fig. 1 indicates that the phases in
the nanocapsules are indeed TaC and C. The thickness of the spacing in the
carbon shell is 0.34 nm, which indicates that the shell is graphite with (002)
orientation. The TaC grains are separated by the graphite, which acts as a
conductance barrier between two TaC grains. The electrons are trapped inside
isolated TaC grains. The grain connections in the system of nanocapsules may
be considered as zero-dimensional, although the grain sizes in three
dimensions are in nanometers. The tunneling framework consists of
superconductor-insulator-superconductor ($\ldots$ TaC/(C)/TaC$\ldots$)
tunneling junctions, which vary among themselves because of the different
thicknesses of the graphite shells and the different sizes of the TaC core.
Figure 1: TEM morphology and core/shell structure of TaC(C) nanocapsules. The
left inset shows a crystal grain of TaC(C), and the right inset contains the
XRD pattern.
The temperature dependence of the zero-field-cooled (ZFC) and field-cooled
(FC) magnetization of TaC(C) nanocapsules, measured in a magnetic field of 50
Oe, which is lower than the first critical field, is presented in Fig. 2. The
ZFC curve shows a clear superconductive transition temperature Tc of 10.2 K
from paramagnetism to diamagnetism, at the point where the TaC cores become
superconducting. The FC curve initially exhibits Pauli paramagnetism and
indicates strong magnetic vortex pinning features. The inset in Fig. 2 shows
the magnetic hysteresis loop at 2 K, which is asymmetric because TaC(C) is a
non-ideal type-II superconductor with Tc = 10.2 K shig14 ; toth15 ; fink16 ;
yosi17 , and there is in the magnetic fields from 0 Oe to 1000 Oe, and then to
-1000 Oe to 1000 Oe. On the initial magnetization curve, at the initial
critical field Hic = 250 Oe, the negative magnetization begins to increase. On
the loop, at H = 100 Oe, the magnetization begins to shift. Hc1 = 100 Oe is
the lower critical field, and Hc2$\approx$920 Oe is the upper critical field,
showing that the superconductivity of the TaC(C) nanocapsules is typical of a
non-ideal type-II superconductor, for which the magnetization process is
irreversible. For ideal type-II superconductors, the remanent magnetization at
Hc2 is zero, and the magnetic flux lattice is symmetric yosi17 . For the
TaC(C) nanocapsule conductor in the first quadrant for H = 920 Oe, MHc2 is not
zero, but approximately 0.01 emu/g. (H is from 1000 Oe to 920 Oe.) The
magnetic hysteresis loop changes with the magnetic field, and the magnetic
flux lattice is not symmetrical in the TaC(C) nanocapsules. For the upper
critical field, the higher the magnetic-field range, the higher the MHc2 is.
It trends toward a limiting value which is smaller than the remanent
magnetization. This property is useful for obtaining a high critical current
density in superconductors bean18 .
Figure 2: Zero-field-cooled (ZFC) and field-cooled (FC) magnetization of
TaC(C) nanocapsules as a function of temperature, measured in a magnetic field
of 50 Oe. The inset shows a hysteresis loop at 2 K, which is typical for a
non-ideal type-II superconductor.
Graphite is a semiconductor with zero activation energy at zero temperature
wall19 and a Mott insulator with a metal-insulator transition case20 ; mott21
. Figure 3 presents the temperature dependence of the resistivity ($\rho$)
measured at zero field at a current of 100 nA by means of the four-probe dc
method. The $\rho$-T curve shows a metal-insulator transition, which can be
ascribed to the graphite shells of the TaC nanocapsules. Above 2.8 K, the
resistivity is semiconductor-like, and below 2.8 K, it is metallic and
precipitous decreased. At 2 K the resistivity of the nanocapsules has reached
zero, which means that the TaC nanocrystals are in the superconducting state.
The right inset shows a magnification of the behavior of $\rho$ near 2 K. The
left inset of Fig. 3 shows a plot of $\rho$ on a logarithmic scale against
T-1/4. It illustrates that the resistivity is that of a Mott insulator, which
is similar to what has been reported for NbC(C)-C li11 , carbon-coated Sn
nanoparticles wang12 , and Mo carbide nanoparticles embedded in a carbon
matrix wang13 . This explains the electrical-conduction mechanism of the
TaC(C) nanocapsule composite, which contains an intrinsic framework of
nanocrystalline TaC and graphite. The main process in the resistivity is a
Mott metal-insulator transition case20 ; mott21 .
Figure 3: Temperature dependence of the resistivity ($\rho$) of a pellet of
TaC(C) nanocapsules measured at a current of 100 nA. The right inset presents
a magnification of at the lowest temperatures (point S). In the left inset, is
plotted on a logarithmic scale against T-1/4.
Figure 4 shows V-I curves that were subsequently measured at 11, 9, 7, 5, 3,
and 2 K by means of the two-probe dc method (foil: length 3.34 mm; width, 1.88
mm; thickness, 0.06 mm) in the current range from 300 $\mu$A to 0 $\mu$A, with
decreasing steps of 2.5 $\mu$A. During the measurement process the time from
300 $\mu$A to 0 $\mu$A was 1800 s. The top scale of time(s) corresponds to the
time for every curve measurement. At 11 K, which is above Tc, the V-I curve
follows Ohmic behavior, with V/I = R = constant. Below Tc, single-electron
tunneling takes place with conductance close to 4e2/h. Comparing the curves at
9 K, 7 K, 5 K, and 3 K, the voltage increases with decreasing temperature, and
the curves gradually become curved, indicating that the conduction mechanism
starts deviating from Ohm’s law.
Figure 4: Current (I) versus Voltage (V) and temperature fluctuations
($\triangle$ T= T-2). V-I curves of a pellet of TaC(C) nanocapsules
successively measured at 11, 9, 7, 5, 3, and 2 K. The lowest curve presents
the temperature fluctuations $\triangle$ T(= T-2) of V-I curve measured at 2
K. The top scale of time(s) corresponds to the time for every curve. The
voltage oscillation, 90 mV = $V_{s}=\Sigma^{i=n}_{i=1}V_{si}$, as described
below in the text, results from the supercurrent at that moment; combined
voltage from the Ohmic part and the single-electron tunneling part is 164 mV.
The maximum temperature difference is 36 mK for the I-$\triangle$T curve.
In Fig. 3, on the $\rho$-T curve, from 10 K to 3 K, the higher the
temperature, the higher $\rho$ is, which corresponds to what is shown in Fig.
4 that the lower the temperature, the higher the voltage is. Very
interestingly, at 2 K, the voltage exhibits oscillatory behavior as a function
of current, which can not arise from Ohmic current and single-electron
tunneling.
During measurement in the sweep mode at 2 K, the temperature of the sample was
observed to oscillate despite the temperature control, because when H = 0
there are still some remaining magnetic fluxes inside. To indicate how the
temperature oscillates with the current, an I-$\Delta$T ($\Delta$T = T - 2 K)
curve is presented at the bottom of Fig. 4. In the oscillations, the voltage
and temperature increase over a short time ($<$ 0.05 s) and then decrease
slowly ($>$ 120 s). The oscillations in temperature and voltage are
synchronous.
From Figs.2 and 3, we know that at 2 K, the TaC nanograins are
superconductors. The Cooper pairs tunneling at non-zero voltage through the
TaC/C/TaC junctions give rise to the emission of photons, which results in an
increase in the temperature. In fact, the voltage exhibits oscillatory
behavior accompanied by temperature change ($\Delta$T = T-2). Figure 5
represents that the voltage (V) and $\Delta$T versus time (t) curves, which
were successively measured at 2 K, with a current I = 50 nA at the absence of
a magnetic field (H = 0). The measurement method is shown as Fig. 4. The V-t
and $\Delta$T-t curves oscillate like a music score, and the big oscillation
amplitude is 0.16 mV. The resistance, V/I, does not follow Ohmic and single
tunneling electron behavior and the voltage jumps with T, which indeed
exhibits a quantum effect.
Figure 5: Voltage (V) versus time (t) and temperature fluctuations ( T
=$\Delta$T-2) curves of a pellet of TaC(C) nanocapsules successively measured
at 2 K, with H=0 and I=50 nA. The V-t and $\Delta$T-t curves oscillate like a
music score, and the big oscillation amplitude is 0.16 mV. The changing range
for $\Delta$T is 68 mK. V and $\Delta$T also exhibit as functions of time and
the characteristic quantum indeterminacy.
In Fig. 2, Tc is observed to be 10.2 K, and in Fig. 3, the maximum resistance
temperature is 2.8 K. In superconductors, the free energy, which corresponds
to the condensation energy, equals E $\propto$ (Tc-T)2 anne22 . For the
supercurrent through TaC/C/TaC tunneling junctions, the temperature difference
needs to be at least over Tc-T $\approx$ 10.2K - 2.8 K = 7.4K to overcome the
barriers between TaC and graphite.
The supercurrent, the electric Ohmic current, the single-electron tunneling
current, and their corresponding voltages coexist. To explain the
oscillations, all TaC nanocapsules are assumed to be isolated from each other.
The TaC(C) nanocapsules are pressed into a specimen, which forms a three-
dimensional granular framework composed of TaC/C/TaC tunneling junctions. We
assume the number of tunnel junctions in the pellet to be equal to n and use
the equivalent lumped circuit model mccu23 ; hans24 for the experiments.
For the ith tunneling junction, the total current Ii is composed of the
supercurrent, the electric Ohmic current, and the single-electron tunneling
current. The total current measured in the I-V curves is
$\sum^{n}_{i=1}I_{i}=I.$
In the system for the equivalent model, the ith tunneling junction may be
written as follows 02jose :
$\hbar\omega_{i}=2eV_{si},$
where $\omega_{i}$ is the frequency of the emitted photon when a Cooper pair
is tunneling through the ith TaC/C/TaC tunneling junction; Vsi is the non-zero
potential difference between the two sides of the ith junction (see Fig. 6).
Then:
$V_{s}=\sum^{n}_{i=1}V_{si}=\frac{\hbar}{2e}\left(\sum^{n}_{i=1}\omega_{i}\right)(i=1,2,3,\ldots,n),$
(1)
where Vs is the voltage barrier that must be surmounted for Cooper pairs to
cross the junctions and emit photons. There are involved the special rules for
the interfering amplitudes for the Cooper pairs and electrons that occur in
the processes. The amplitude for the Bose particles is the sum of the two
interfering amplitudes, and for the Fermi ones it is minus.
Figure 6: (a) A Cooper pair tunnels through the $i^{th}$ Josephson junction,
emitting a photon, ${\hbar\omega_{i}=2eV_{si}}$ , as it crosses the interface
between the two TaC(C) nanocapsules. E is the free energy in the TaC
nanocapsules. (b) TEM morphology of one Josephson junction in the TaC (C)
nanocapsules, and the thickness of graphite layers between the two TaC grains
is 1 nm.
When measured the I-V curves some Cooper pairs are interference. For the
states with Bose particles, the probability in the Bose case is larger than
that of the particles acted independently anne22 ; Feyn25 . When the tunneling
effects take place, the probability amplitudes of Cooper pairs in graphite
abruptly increase and give rise to the emission of photons and consequently,
the temperature jumps up; increasing the resistance of graphite. Then the
supercurrent stops and the number of Copper pairs is quickly decreased, which
are transformed to single electrons. Subsequently, the temperature begins to
drop down. With decreasing the temperature of the system, the number of Cooper
pairs is gradually increased, and the resistance of graphite decreases until
the next cycle. The probability amplitudes of Copper pairs also are a function
of the time (See Fig. 5) and exhibited quantum indeterminacy. In fact, there
is a competitive mechanism between the junction barrier and the free energy of
the superconducting carriers in the TaC nanograins. A phase transition occurs
during the supercurrent goes through the junctions, where a boson (Cooper
pair) transforms into two fermions (electrons), and this process stimulates
the emission of a photon. After photons are emitted, the temperature of the
TaC/C/TaC nanocapsules rises abruptly; the voltage then jumps, and the
supercurrent stops because the resistance has been quickly risen (See Fig. 3
from 2 K to 2.8 K). The probability amplitudes of Cooper pairs are driven by
the particle statistics of Cooper pairs, in which the thermodynamic observable
has an abrupt change, leading to a jump of voltage, a quantum behavior anne22
; Feyn25 . The graphite barrier is a Mott insulator with a metal-insulator
transition. As the temperature gradually decreases, the barrier is reduced.
However, in nano-TaC grain superconductors the free energy is increased as the
temperature falls. The barrier is decreased and the free energy is increased,
until the free energy overcomes the barrier, and the another cycle begins
again. The temperature and the voltage are gradually reduced, which is driven
by electron interactions. The Mott metal-insulator transition of the graphite
shells promotes oscillations, as is shown in Fig. 6.
Therefore, the tunneling mechanisms of the Cooper pairs in the non-ideal type-
superconductor TaC and the Mott metal-insulator transition in graphite would
be responsible for the voltage and temperature oscillations in the
nanocapsules. As shown in Fig. 4, while this occurs, $V_{s}$ =
$\sum_{i=1}^{n}$$V_{si}$ = 90 mV. The combined voltage of the Ohmic part and
the single-electron tunneling part is 164 mV.
In summary, oscillations in voltage and temperature have been observed in
TaC(C) nanocapsules. The Josephson intermittent effect occurs in a non-
equilibrium system of ($\ldots$TaC/(C)/TaC$\ldots$) tunneling junctions, with
a function of the time. The mechanism is that the supercurrents go through the
Josephson junctions and emit photons with the feedback resistance,temperature
change of the TaC(C) junctions.
The authors acknowledge the support of the National Natural Science Foundation
of China under grant number 51102244 and the National Basic Research Program
(No. 2012CB933103), the Ministry of Science and Technology of China. This work
is also partly supported by the Australian Research Council through a
Discovery project.
## References
* (1) B. D. Josephson, Phys. Lett. 1, 251 (1962).
* (2) B. D. Josephson, Rev. Mod. Phys. 36, 216 (1964).
* (3) B. Jeanneret and S. P. Benz, Eur. Phys. J. Special Topics 172, 181 (2009).
* (4) Y. Saito, Carbon 33, 979 (1995).
* (5) Z. D. Zhang, in: H.S. Nalwa (Ed.), Encyclopedia of Nanoscience and Nanotechnology,Vol. 6, American Scientific, California, (2004) 77-160.
* (6) E. Ben-Jacob, I. Goldhirsch, Y. Imry and S. Fishman, Phys. Rev. Lett. 49, 1599 (1982).
* (7) R. L. Kautz, Rep. Prog. Phys. 59, 935-992 (1996).
* (8) I. Goldhirsch Y. Imry, G. Wasserman and E. Ben-Jacob Phys. Rev. B. 29, 1218 (1984).
* (9) M. Octavio, Phys. Rev. B 29, 1231 (1984).
* (10) G. Zumofen and J. Klafter, Phys. Rev. E 47, 851 (1993).
* (11) D. Li, W.F. Li, S. Ma, and Z.D. Zhang, Phys. Rev. B 73, 193402 (2006).
* (12) Z. H. Wang, D. Y. Geng, Z. Han, and Z. D. Zhang, J. Appl. Phys. 108, 013903 (2010).
* (13) Z. H. Wang, D. Li, D. Y. Geng, S. Ma, W. Liu, and Z. D. Zhang, J. Mater. Res. 24, 2229 (2009).
* (14) T. Shigaki, S. M. Oh, J.G. Li, and D. W. Park, Sci. Technol. Adv. Mater. 6 (2005) 111-118.
* (15) L. E. Toth, Transition Metal Carbides and Nitrides, Academic Press: New York, (1971).
* (16) H. J. Fink, A. C. Thoresen, E. Parker, V. F. Aackay, and L. Toth, Phys. Rev. 138, A1170 (1965).
* (17) Y. Yosida and I. Oguro, Physica C 434, 173 (2006).
* (18) C.P. Bean, Phys. Rev. Lett. 8, 250 (1962).
* (19) P. R. Wallace, Phys. Rev. 71, 622 (1947).
* (20) A. Casey, H. Patel, J. Nyéki, B. P. Cowan, and J. Saunders, Phys. Rev. Lett. 93, 115301 (2003).
* (21) N. F. Mott, Rev. Mod. Phys. 40, 677 (1968).
* (22) J. F. Annett, Superconductivity, Superfluids and Condensates, Oxford University Press, (2004) 6, 73.
* (23) D.E. McCumber, J. Appl. Phys. 39, 2503 (1968).
* (24) P.K. Hansma, G.I. Rochlin, and J.N. Sweet, Phys. Rev. B. 4, 3003 (1971).
* (25) R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics 3, Printed in the United States of America, (1964).
|
arxiv-papers
| 2011-09-19T17:12:51 |
2024-09-04T02:49:22.308776
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dianyu Geng, Zhenhua Wang, Da Li, Zhidong Zhang, and Xiaolin Wang",
"submitter": "Zhenhua Wang",
"url": "https://arxiv.org/abs/1109.4097"
}
|
1109.4288
|
# Adding Logical Operators to Tree Pattern Queries on Graph-Structured Data
Qiang Zeng1,2 Xiaorui Jiang1,2 Hai Zhuge1
1Key Lab of Intelligent Information Processing Institute of Computing
Technology Chinese Academy of Sciences
2Graduate University of Chinese Academy of Sciences
{zengqiang xiaoruijiang}@kg.ict.ac.cn zhuge@ict.ac.cn
###### Abstract
As data are increasingly modeled as graphs for expressing complex
relationships, the tree pattern query on graph-structured data becomes an
important type of queries in real-world applications. Most practical query
languages, such as XQuery and SPARQL, support logical expressions using
logical-AND/OR/NOT operators to define structural constraints of tree
patterns. In this paper, (1) we propose generalized tree pattern queries
(GTPQs) over graph-structured data, which fully support propositional logic of
structural constraints. (2) We make a thorough study of fundamental problems
including satisfiability, containment and minimization, and analyze the
computational complexity and the decision procedures of these problems. (3) We
propose a compact graph representation of intermediate results and a pruning
approach to reduce the size of intermediate results and the number of join
operations – two factors that often impair the efficiency of traditional
algorithms for evaluating tree pattern queries. (4) We present an efficient
algorithm for evaluating GTPQs using 3-hop as the underlying reachability
index. (5) Experiments on both real-life and synthetic data sets demonstrate
the effectiveness and efficiency of our algorithm, from several times to
orders of magnitude faster than state-of-the-art algorithms in terms of
evaluation time, even for traditional tree pattern queries with only
conjunctive operations.
## 1 Introduction
Graphs are among the most ubiquitous data models for many areas, such as
social networks, semantic web and biological networks. As the most common tool
for data transmissions, XML documents are desirably modeled as graphs rather
than trees to represent flexible data structures by incorporating the concept
of ID/IDREFs. Semantic Web data are also modeled as graphs, e.g. in RDF/RDFS.
On graph data, tree pattern queries (TPQs) are one of important queries of
practical interest. In query languages such as XQuery and SPARQL, many queries
can be regarded as TPQs over graphs. As most of them support logical
operations including conjunction ($\wedge$), disjunction ($\vee$) and negation
($\neg$) in the query conditions, it is necessary to study TPQs over graphs
with multiple logical predicates, as illustrated in the following example.
###### Example 1.
A DBLP XML document separately stores inproceeding records for papers and
proceeding records for volumes, linked by crossref elements indicating where a
paper is published [24]. The underlying data structure is clearly a graph.
Consider the following three queries which ask for information of publications
for which a certain tree pattern of data holds.
1. $Q_{1}$:
Retrieve the information about Alice’s conference papers that are published
from 2000 to 2010 and co-authored with Bob.
2. $Q_{2}$:
Retrieve the information about the conference papers of either Alice or Bob
published from 2000 to 2010.
3. $Q_{3}$:
Retrieve the information about Alice’s conference papers that are not co-
authored with Bob and published from 2000 to 2010.
Figure 1: The tree representation of $Q_{1}$, $Q_{2}$, and $Q_{3}$ in Example
1. Document elements matching the starred query nodes are required to be
returned and the single-/double-lined edges denote the parent-child/ancestor-
descendant relationships between elements.
They can be expressed in XQuery and are essentially TPQs on graph-structured
data (see the Appendix), but $Q_{2}$ and $Q_{3}$ cannot be expressed in
traditional TPQs, which only contain conjunctive predicates. Indeed, they
share the same tree representation as depicted in Fig. 1, but different
structural predicates should be imposed on the inproceedings element $u_{1}$.
For example, in $Q_{1}$, each embedding of the pattern should satisfy all
paths specified in the query; but for $Q_{2}$, the two path conditions
“$u_{1}$–$u_{2}$” and “$u_{1}$–$u_{3}$” are not required to be satisfied
simultaneously. A predicate that specifies those edge constraints and
incorporates disjunction and negation needs to be attached to each query node
in order to express $Q_{2}$ and $Q_{3}$. In general, (1) it is common in
practice that logical expressions on query nodes needs to be imposed to
specify complex relationships for not only attribute predicates (e.g.
$2000\leq\textrm{year}\leq 2010$) but also structural constraints
$\big{(}$e.g. ($u_{1}$–$u_{2}$ or $u_{1}$–$u_{3}$) in $Q_{2}$ and
$not(u_{1}$–$u_{3})$ in $Q_{3}$$\big{)}$; (2) some of the nodes $\big{(}$e.g.
$u_{i}(i\in\\{1,2,3,6,8\\})$$\big{)}$ in the query pattern only serve as
filters for pruning unexpected results, which means that the results of a TPQ
should consist of matches for a portion of the query nodes only. ∎
Although TPQs have been widely studied for many years, few of the proposed
processing algorithms can be used to efficiently evaluate such queries over
general graphs. They can neither support disjunction and negation on
structural constraints nor be optimized for the situation where output nodes
take only a portion of query nodes (see Related work for details).
###### Contributions & Roadmap.
This work makes the first effort to deal with TPQ over general graph-
structured data with Boolean logic support. The contributions are summarized
as follows.
(1) We introduce a new class of tree pattern queries over graph-structured
data, called generalized tree pattern queries (GTPQs) (Section 2). In a GTPQ,
a node is not only associated with an attribute predicate, which specifies the
property conditions, but also a structural predicate in terms of propositional
logic with logic connectives including conjunction, negation and disjunction
to specify structural conditions with respect to its descendants. The query
allows a portion of the query nodes to be output nodes. We also show that our
formalization of query is advantageous over those in the literature on queries
against tree-structured data.
(2) We investigate fundamental problems for GTPQs, including satisfiability,
containment, equivalence and minimization (Section 3). We show that the
satisfiability of a special GTPQ with only conjunction and disjunction is
solvable in linear time, but the satisfiability and the other three problems
become computationally intractable when disjunction is incorporated. We
propose an exact algorithm to minimize GTPQs, which is supposed to be
sufficiently efficient, since the query sizes are typically small in practice.
(3) We propose a graph representation of intermediate results and a pruning
approach to address notable problems in evaluating query patterns over graphs,
develop an algorithm for GTPQs with ancestor-descendant edges and its
extension to deal with parent-child edges (Section 4). The algorithm can
largely filter nodes that cannot contribute to the final results, wisely avoid
generating redundant intermediate results, and compactly represent the
matches.
(4) We implement our algorithm and conduct an experimental study using
synthetic and real-life data (Section 5). We find that our evaluation
algorithm performs significantly better than state-of-art algorithms even for
conjunctive TPQs. It also has better scalability and is robust for different
queries on different graphs. The experiments also demonstrate the
effectiveness of the graph representation of results and the efficiency of the
pruning method.
###### Related work.
There is a large body of research work on TPQs over tree-structured data (see
[14] for a survey). However, all studies heavily relied on the relatively
simple structure of trees and employed the node encoding schemes (including
the interval [3], Dewey [21] and sequence [28] encodings) that are not
applicable to graphs for determining structural relationships. Techniques
critical for their efficiency, such as stack encoding and nodes skipping, can
be only applied to tree-structured data. For some sparse graph data whose
structures can be modeled by disjoint trees connected by edges, such as many
XML documents with ID/IDREFs, although one can apply those existing algorithms
for tree-structured data to evaluate a query over such graphs by first
decomposing it to several TPQs over different trees and then merging the
results of distinct queries to form the final results, it is inefficient due
to large redundant intermediate results and costly merging processes.
Some studies extended the traditional TPQs by incorporating additional
functions and restrictions. Chen et al. [9] included optional nodes to
patterns and investigated efficient evaluation plans upon native XML database
systems. The generalized tree pattern is still against tree-structured data,
which differs from this work that studies TPQs over graph-structured data with
logical predicates. Jiang et al. [16] proposed new holistic algorithms based
on a concept of OR-blocks to process AND/OR-twigs, TPQs with OR-predicates. In
the end of Section 2, we shall show that (1) our query size can be always no
larger than the size of element nodes of AND/OR-twig for expressing a
semantically identical query; (2) constructing OR-blocks involves converting a
propositional formula to conjunctive normal form, thus taking exponential time
in the worst case; (3) the proposed algorithms only support tree-struct-ured
data as input. [17] studied path queries with negation, while [29] and [20]
added negation to TPQs. They cannot be applied to GTPQs either, since they are
based on the classical holistic twig join algorithm [3] that only works on
tree-structured data.
There has been work on pattern queries for graph-structured data. TwigStackD
[6] generalized the holistic algorithms, but it takes considerable time and
space without a pre-filtering process [30]. HGJoin [27] can evaluate general
graph pattern queries using OPT-tree-cover [1] as the underlying reachability
indexing approach. It decomposes a pattern into a set of complete bipartite
graphs and generates matches for them in order according to a plan. The time
cost of plan generation is always exponential since it has to produce a state
graph with exponential nodes no matter for obtaining an optimal or suboptimal
plan. Cheng et al. [11] proposed R-join/R-semijoin processing for the graph
pattern matching problem. It relies on a cluster-based R-join index whose size
is typically prohibitively large, as the index stores matches for every two
labels derived from 2-hop indexing [12]. Unlike the plan generation of HGJoin,
it adopts left-join to reduce the cost, but in the worst case the time
complexity is still exponential. Since both HGJoin and R-join/R-semijoin use
structural joins similar to the earlier work on tree-structured data, they
typically have large intermediate results and need to perform large amounts of
expensive join operations. All these three algorithms also do not directly
support queries with negative/disjunctive predicates. A straightforward
approach to apply them to the GTPQ processing is to decompose the query into
multiple conjunctive TPQs and perform the difference and merge operations on
results of the decomposed queries. However, the number of the resultant
conjunctive TPQs may be exponential and large intermediate results may need to
be generated and merged.
A number of studies investigated various graph pattern matching problems [15,
13, 31]. [15] proposed a graph query language GraphQL and studied graph-
specific optimization techniques for graph pattern matching that combines
subgraph isomorphism and predicate evaluation. While the language is able to
express queries with ancestor-descendant edges and disjunctive predicates, the
work focused on processing non-recursive and conjunctive graph pattern
queries, where all edges of a query pattern correspond to the parent-child
edges of GTPQs, specifying the adjacent relationship between desired matching
nodes. [13] defined matching in terms of bounded simulation to reduce its
computation complexity. [31] studied distance pattern matching, in which query
edges are mapped to paths with a bounded length. Queries of [13] and [31] do
not support negative/disjunctive predicates on edges and have quite different
semantics with ours.
Most existing algorithms are to find all instances of patterns containing
matches of all query nodes. In real-world applications, however, the answer to
the query often only require matches of several but not all query nodes.
Indeed, many query nodes only serve as filters for imposing structural
constraints on output nodes. Our framework can avoid generating redundant
matches at run time.
Satisfiability, containment, equivalence and minimization are fundamental
problems for any query languages. The minimization of TPQs over tree-
structured data has been investigated in several papers. Amer-Yahia et al. [2]
proposed algorithms for the minimization with and without integrity
constraints. Ramanan [23] studied this problem for TPQs defined by graph
simulation. Chen et al. [5] used a richer class of integrity constraints for
query minimization of TPQs with an unique output node. However, we are not
aware of previous work on minimization as well as the other three problems for
TPQs with logical predicates either over tree-structured data or over graph-
structured data.
## 2 Data model and generalized
tree pattern queries
###### Data graphs.
A data graph is a directed graph $G=(V,E,f)$, where (1) $V$ is a finite set of
nodes; (2) $E\subseteq V\times V$ is finite set of edges, in which each pair
($v,v^{\prime}$) denotes an edge from $v$ to $v^{\prime}$; (3) $f$ is a
function on $V$ defining attribute values associated with nodes. For each node
$v\in V$, $f(v)$ is a tuple ($A_{1}=a_{1},\ldots,A_{n}=a_{n}$), where the
expression $A_{i}=a_{i}(i\in[1,n])$ represents that $v$ has a attribute
denoted by $A_{i}$ and its value is a constant $a_{i}$. For example, in a data
graph $G=(V,E,f)$ of a DBLP document, the node properties in $f$ may include
tags, string values, typed values, and attributes specified in the elements.
Abusing notions for trees and traditional tree pattern queries, we refer to a
node $v_{2}$ as a _child_ of a node $v_{1}$ (or $v_{1}$ as a parent of
$v_{2}$) and say they have a _parent-child_ (PC) relationship if there is an
edge $(v_{1},v_{2})$ in $E$, and refer to $v_{2}$ as a descendant of $v_{1}$
(or $v_{1}$ as an _ancestor_ of $v_{2}$) and say they have an _ancestor-
descendant_ (AD) relationship if there is a nonempty path from $v_{1}$ to
$v_{2}$ in $G$.
###### Generalized tree pattern queries.
A generalized tree pattern query (GTPQ)
$Q=(V_{b},V_{p},V_{o},E_{q},f_{a},f_{e},f_{s})$, where:
(1) $V_{b}$ and $V_{p}$ are both a finite set of nodes, called _backbone
nodes_ and _predicate nodes_ , respectively. The complete set of query nodes
is denoted as $V_{q}$, i.e., $V_{q}=V_{b}\cup V_{p}$.
(2) $V_{o}\subseteq V_{b}$. The nodes in $V_{o}$ are called _output nodes_.
(3) $E_{q}\subseteq\\{(u_{1},u_{2})|u_{1},u_{2}\in
V_{b}\\}\cup\\{(u_{1},u_{2})|u_{1}\in V_{b}\cup V_{p},u_{2}\in V_{p}\\}$, is a
finite set of edges. Here, $(V_{q},E_{q})$ is restricted to a directed tree .
(4) $f_{a}$ is a function defined on $V_{q}$ such that for each node $u\in
V_{q}$, $f_{a}(u)$ is an _attribute predicate_ that is a conjunction of atomic
formulas of the form of “$A$ op $a$”, in which $A$ is an attribute name, $a$
is a constant and op is a comparison operator in $\\{<,\leq,=,\neq,>,\geq\\}$.
(5) $f_{e}$ is a function on $E_{q}$ to specify the type of the edge. Each
edge $(u_{1},u_{2})$ represents either PC relationship or AD relationship.
(6) $f_{s}$ is a function defined on internal nodes. For each internal node
$u\in V_{q}$ with $k$ children being predicate nodes, $f_{s}(u)$, called a
_structural predicate_ , is a propositional formula in $k$ variables
$p_{u^{\prime}_{1}},\ldots,p_{u^{\prime}_{k}}$, each corresponding to a tree
edge directing to a predicate child of $u$. In particular, if $u$ has no
predicate children, $f_{s}(u)=1$. Each node $u$ is associated with a distinct
propositional variable denoted by $p_{u}$.
We call a GTPQ a union-conjunctive GTPQ if the structural predicates on all
query nodes are negation-free, and call it a conjunctive GTPQ if the
structural predicates on all the query nodes only have conjunction
connectives.
Before giving the semantics of GTPQs, we add variables for non-root backbone
nodes to extend the structural predicate. For an internal node $u$ with
$k^{\prime}$ backbone children, denoted by $u_{1},\ldots,u_{k^{\prime}}$, the
_extended structural predicate_ $f_{ext}(u)=p_{u_{1}}\wedge\ldots\wedge
p_{u_{k^{\prime}}}\wedge f_{s}(u)$.
###### Example 2.
In Example 1, $Q_{1}=(V_{b},V_{p},V_{o},E_{q},f_{s},f_{e},f_{s})$ is a
conjunctive GTPQ, in which (1) $V_{b}=\\{u_{1},u_{4},u_{5},u_{6},u_{7}\\}$,
$V_{p}=\\{u_{2},u_{3},u_{8}\\}$, $V_{o}=\\{u_{4},u_{5},u_{7}\\}$; (2) the
attribute predicate $f_{a}$ for a query node is a conjunction of comparisons
among tags and typed values $\big{(}$e.g. $f_{a}(u_{2})=({\rm tag}=$ “author”
$\wedge$ value $=$ “Bob”)$\big{)}$; (3) $f_{s}(u_{1})=p_{u_{2}}\wedge
p_{u_{3}}$, and $f_{s}(u_{6})=p_{u_{8}}$. The only difference between $Q_{2}$
and $Q_{1}$ is that in $Q_{2}$, $f_{s}(u_{1})=p_{u_{2}}\vee p_{u_{3}}$. In
$Q_{3}$, $f_{s}(u_{1})=p_{u_{2}}\wedge\neg p_{u_{3}}$. As an example of
extended structural predicates, for $Q_{2}$, $f_{ext}(u_{1})=(p_{u_{2}}\vee
p_{u_{3}})\wedge p_{u_{4}}\wedge p_{u_{5}}\wedge p_{u_{6}}$. ∎
###### Semantics.
Consider a data graph $G=(V,E,f)$ and a GTPQ
$Q=(V_{b},V_{p},V_{o},E_{q},f_{a},f_{e},f_{s})$. We say that a data node $v$
in $G$ _downwardly matches_ a query node $u$ in $Q$, denoted by $v\models u$,
if the following conditions are satisfied:
(1) $v$ satisfies the attribute predicate of $u$, denoted by $v\sim u$. That
is, for each formula “$A$ op $a$” in $f_{a}(u)$, there is an element
($A=a^{\prime}$) in $f(v)$ such that $a^{\prime}$ op $a$. $v$ is called a
_candidate matching node_ of $u$. $mat(u)$ denotes the set of candidate
matching nodes of $u$, i.e., $mat(u)=\\{v|v\in V,v\sim u\\}$.
(2) If $u$ is an internal node, the data node $v$ determines a truth
assignment to the variables of $f_{ext}(u)$ such that $f^{v}_{ext}(u)=1$,
where $f^{v}_{ext}(u)$ denotes the truth-value of $f_{ext}$ under the
assignment. For each variable $p_{u^{\prime}}$, the truth-value
$p^{v}_{u^{\prime}}$ is assigned as follows: for each PC (resp. AD) child
$u^{\prime}$ of $u$, $p^{v}_{u^{\prime}}=1$ if there exists a child (resp.
descendant) $v^{\prime}$ of $v$ such that $v^{\prime}\models u^{\prime}$;
otherwise, $p^{v}_{u^{\prime}}=0$.
Let $V_{b}=\\{u_{1},\ldots,u_{m}\\}$. A $m$-ary tuple ($v_{1},\ldots,v_{m}$)
of nodes in $G$ is said to be a match of $Q$ on $G$, if the following
conditions hold: (1) for each $v_{i}(i\in[1,m])$, $v_{i}\models u_{i}$; (2)
for each edge $(u_{i},u_{j})\in E_{q}(i,j\in[1,m])$, if $u_{j}$ is a PC child
of $u_{i}$, $v_{j}$ is a child of $v_{i}$; otherwise, $v_{j}$ is a descendant
of $v_{i}$.
The answer $Q(G)$ to $Q$ is a set of results in the form of tuples, where each
tuple consists of the images of output nodes $V_{o}$ in a match of $Q$. For
each match, there is at least an assignment for all variables that makes the
extended structural predicates of all internal backbone nodes and some of
internal predicate nodes evaluate to true, which we call a certificate of the
match. For a match and an assignment as a certificate of the match, an
instance of $Q$ on $G$ is a tuple consisting of such nodes that each of them
matches a distinct query node whose corresponding propositional variable is
true under the assignment. In particular, an instance of conjunctive GTPQ is
exactly a match of the query.
(a) Data graph $G$
(b) GTPQ $Q$ on $G$
Figure 2: Example of a data graph and a GTPQ. We use a rectangle to represent
a predicate node and a circle to represent a backbone node.
(a) B-twig query
(b) GTPQ
Figure 3: Comparison between a B-twig query and a GTPQ
###### Example 3.
For simplicity of presentation, a lower-case letter $x_{i}$ in all figures
throughout this paper denotes $f(v)$ for a data node $v$ and a capital letter
$Y_{j}$ denotes $f_{a}(u)$ for a query node $u$ such that $v\sim u$ if $j\leq
i$ and $X=Y$.
Consider the data graph and the query shown in Fig. 2. $v_{13}\sim
u_{5},v_{15}\not\sim u_{5}$. Accordingly,
$mat(u_{5})=\\{v_{13}\\},mat(u_{10})=\\{v_{9},v_{10},v_{13},v_{15}\\}$. The
answer $Q(G)=\\{(v_{3},v_{11}),(v_{3},v_{12}),(v_{3},\\\
v_{14}),($$v_{8}$,$v_{12}),(v_{8}$,$v_{14})\\}$. One of the query matches
leading to $(v_{3},v_{11})$ is $(v_{1},v_{3},v_{3},v_{11})$, where elements
are sorted in the ascending order of the subscripts of corresponding query
nodes. An instance of this match is
$\\{u_{1}:v_{1},u_{2}:v_{3},u_{3}:v_{3},u_{4}:v_{11},u_{7}:v_{6},u_{8}:v_{11},u_{9}:v_{15}\\}$,
where ‘$u:v$’ means $v$ is a match of $u$. Indeed, $v_{3}\models u_{3}$,
because (1) $v_{3}\sim u_{3}$, and (2) $f^{v_{3}}_{ext}(u_{3})=1$ since
$v_{6}\models u_{7}$ and $v_{11}\models u_{8}$. Also, $v_{5}\models u_{3}$,
because $v_{5}$ cannot reach a node matching $u_{6}$ and hence
$p^{v_{5}}_{u_{3}}=0$, thereby $f^{v_{5}}_{ext}(u_{3})=1$. ∎
For simplicity of semantics, we require a query to explicitly specify backbone
nodes and predicate nodes and restrict output nodes to backbone ones. The
distinction between the two types of nodes is that propositional variables
associated with backbone nodes are disallowed to be operands of negation and
disjunction as those associated with predicate nodes, which guarantees that
each backbone node has an image in a match of the query. Permitting negation
and disjunction on any query nodes leads to issues that are not
computationally desirable. If each query result is still required to have an
image for each output node, the expressive power does not change; but to
determine whether a query is valid is effectively to check whether the
variables associated with output nodes are always true for all certificates of
matches, which is a co-NP-complete problem. Otherwise, the output structures
become not fixed. They can either be specifically defined in the query, or
consist of exponential combinations of output nodes by default. Our algorithm
described in Section 4 can be straightforwardly extended to process queries
with multiple output structures (see the Appendix).
We now compare GTPQ with the works in [29] and [4]. [29] deals with _AND/OR-
twig_ against tree-structured data. [4] further extends [29] to handle _B-
twig_ , which additionally introduces the logical-NOT operation into the
query. Both represent a query by defining special types of nodes for
operators, namely logical-AND nodes, logical-OR nodes and logical-NOT nodes.
For each occurrence of a variable in a structural predicate of a GTPQ, the
corresponding AND/OR-twig or B-twig needs to use a distinct subtree to express
the structural constraints with respect to descendants as specified by the
variable, since in AND/OR-twigs and B-twigs, the query nodes connected to
different operator nodes are considered as distinct. The query size of AND/OR-
twigs or B-twigs hence may be much larger than the size of a GTPQ for
expressing complex tree patterns. In Fig. 3, the B-twig query has to use two
paths $u_{2}$–$u_{4}$ and $u_{5}$–$u_{6}$ to represent the constraints that
can be imposed by a single path $u_{2}$–$u_{5}$ in the semantically equivalent
GTPQ. Moreover, before evaluating the query, [29] and [4] have to construct
OR-blocks to normalize the twig. The normalization process is essentially a
CNF conversion of propositional formulas. Since a CNF conversion can lead to
an exponential explosion of the formula, the time cost of a conversion is
exponential in the size of original query, and the resulting query size also
becomes exponential in the worst case. Therefore, our query representation is
more powerful and compact than the tree representation of [29] and [4].
## 3 Fundamental Problems for Generalized Tree Pattern Queries
In this section, we study the problems of satisfiability, containment,
equivalence, and minimization of GTPQs, which are important for query analysis
and optimization.
### 3.1 Satisfiability
A GTPQ $Q$ is _satisfiable_ if there is a data graph $G$ on which the answer
$Q(G)$ to $Q$ is nonempty. We first introduce some definitions before showing
how to determine the satisfiability and establishing the property of the
problem.
We say $u$ is an _independently constraint_ node if (1) the formula
$\big{(}f_{s}(u^{\prime})[p_{u}/1]\oplus
f_{s}(u^{\prime})[p_{u}/0]\big{)}\wedge f_{s}(u)$ is satisfiable, in which
$u^{\prime}$ is the parent of $u$, $f_{s}(u^{\prime})[p_{u}/x]$ is the formula
produced by assigning $x$ to the variable $p_{u}$ $(x\in\\{0,1\\})$, and
$\oplus$ is the exclusive-or logical operator; (2) all ancestors of $u$ are
independently constraint nodes. Intuitively, the variables of independently
constraint nodes can independently affect the resulting truth-value of the
structural predicates of their parents and ancestors. Backbone nodes are
clearly independently constraint nodes, if their structural predicates are
satisfiable.
A _transitive structural predicate_ $f_{tr}(u)$ for a node $u$ is constructed
from $f_{ext}(u)$ in a bottom-up sweep as follows. (1) For each leaf node and
each non-independently constraint node $u$ , the transitive structural
predicate is the same as the extended structural predicate, i.e.
$f_{tr}(u)=f_{ext}(u)$. (2) For an internal node $u$ such that the transitive
structural predicates of all children have been defined, $f_{tr}(u)$ is
produced by substituting $\big{(}p_{u^{\prime}}\wedge
f_{tr}(u^{\prime})\big{)}$ for each variable $p_{u^{\prime}}$ of independently
constraint node $u^{\prime}$ in $f_{s}(u)$.
For two non-root nodes $u_{1},u_{2}$ in $Q$, we say that $u_{2}$ is _similar_
to $u_{1}$, denoted by $u_{1}\triangleleft u_{2}$, if the following conditions
hold. (1) For each formula “$A$ op $a_{1}$” in $f_{a}(u_{1})$, there is a
formula “$A$ op $a_{2}$” in $f_{a}(u_{2})$ such that (a) if $\text{op
}\in\\{\leq,<\\}$, $a_{2}\leq a_{1}$, (b) if $\text{op }\in\\{\geq,>\\}$,
$a_{2}\geq a_{1}$, (c) if $\text{op }\in\\{=,\neq\\}$, $a_{1}=a_{2}$. We use
$u_{2}\vdash u_{1}$ to denote that $u_{1}$ and $u_{2}$ satisfy this condition.
(2) For each PC (resp. AD) child $u^{\prime}_{1}$ of $u_{1}$ such that
$u^{\prime}_{1}$ is an independently constraint node, there is a PC child
(resp. a descendant) $u^{\prime}_{2}$ of $u_{2}$ such that
$u^{\prime}_{1}\triangleleft u^{\prime}_{2}$. (3) The formula
$f_{tr}(u_{2})\to f_{tr}(u_{1})[u_{1}\mapsto u_{2}]$ is a tautology, where
$f_{tr}(u_{1})[u_{1}\mapsto u_{2}]$ is a formula transformed from
$f_{tr}(u_{1})$ by replacing $p_{u^{\prime}}$ with $p_{u^{\prime\prime}}$ for
each pair ($u^{\prime},u^{\prime\prime}$) such that (a) $u^{\prime}$ is a
descendant of $u_{1}$, (b) $u^{\prime\prime}$ is a descendant of $u_{2}$ and
(c) $u^{\prime}\trianglelefteq u^{\prime\prime}$. We say that $u_{1}$ is
_subsumed_ by $u_{2}$, denoted by $u_{1}\trianglelefteq u_{2}$, if (1)
$u_{1}\triangleleft u_{2}$, and (2) the parent of $u_{1}$ is the lowest common
ancestor $u_{lca}$ of $u_{1}$ and $u_{2}$, and (a) if $u_{1}$ is a PC child of
$u_{lca}$, $u_{2}$ is also a PC child of $u_{lca}$; (b) otherwise $u_{2}$ is a
descendant of $u_{lca}$.
We finally define _complete structural predicates_ to characterize the whole
structural constraints of a GTPQ. For a node $u$, the complete structural
predicate $f_{cs}(u)$ is created from the corresponding transitive structural
predicate $f_{tr}(u)$ by performing the following operations: (1) for each
descendant $u^{\prime}$ of $u$, if its attribute predicate is unsatisfiable,
$f^{new}_{cs}(u)=f^{old}_{cs}(u)[p_{u^{\prime}}/0]$, where $f^{old}_{cs}(u)$
is the old formula before this transformation and $f^{new}_{cs}(u)$ is the
newly generated formula; (2) for every two nodes $u_{1}$ and $u_{2}$ in two
distinct subtrees of $u$ such that $u_{2}\trianglelefteq u_{1}$,
$f^{new}_{cs}(u)=f^{old}_{cs}(u)\wedge\big{(}\neg
p_{u_{1}}\vee(p_{u_{2}}\wedge f_{ext}(p_{u_{2}})\big{)}$, where
$f^{old}_{cs}(u)$ and $f^{new}_{cs}(u)$ have the same meaning as above in (1).
Theorem 1 shows that the satisfiability of a GTPQ is equivalent to the
satisfiability of the complete structural predicate of the root, if given that
the attribute predicate of the root is satisfiable. If the query is a
conjunctive or union-conjunctive GTPQ, the problem of satisfiability can be
solved in linear time. When negation is added into the query, the
satisfiability becomes NP-complete.
###### Theorem 1.
A GTPQ $Q$ is satisfiable if and only if for the root node $u$ of $Q$,
$f_{a}(u)$ and $f_{cs}(u)$ are both satisfiable. ∎
###### Theorem 2.
1. 1.
The satisfiability of a union-conjunctive GTPQ can be determined in linear
time.
2. 2.
The satisfiability of a GTPQ is NP-complete.∎
(a) $Q_{1}$
(b) $Q_{2}$
(c) $Q_{3}$
Figure 4: Examples for four fundamental problems of GTPQs
###### Example 4.
Consider the query in Fig. 2(b). All query nodes are independently constraint
nodes. Replacing $p_{u_{7}}$ with $p_{u_{7}}\wedge(p_{u_{9}}\vee p_{u_{10}})$
in $f_{ext}(u_{3})$, we have $f_{tr}(u_{3})=\neg
p_{u_{6}}\vee(p_{u_{7}}\wedge(p_{u_{9}}\vee p_{u_{10}})\wedge p_{u_{8}})$.
Since there are no two nodes $u$ and $u^{\prime}$ such that $u\trianglelefteq
u^{\prime}$, $f_{cs}(u_{1})=f_{tr}(u_{1})=p_{u_{5}}\wedge p_{u_{4}}\wedge
p_{p_{5}}\wedge p_{u_{3}}\wedge\big{(}\neg
p_{u_{6}}\vee(p_{u_{7}}\wedge(p_{u_{9}}\vee p_{u_{10}})\wedge
p_{u_{8}})\big{)}$. Due to the satisfiability of $f_{cs}(u_{1})$, we see that
the query is satisfiable. Indeed, we can get a nonempty answer by posing $Q$
on $G$ in Fig. 2(b) as shown in Example 3.
Let us turn to $Q_{1}$ and $Q_{2}$ depicted in Fig. 4. The following table
presents structural predicates of internal nodes for $Q_{1}$ and $Q_{2}$.
$f_{s}(u_{1})=\neg p_{u_{2}}$ | $f_{s}(u_{2})=p_{u_{4}}$ | $f_{s}(u_{5})=p_{u_{8}}$
---|---|---
$f_{s}(u_{3})=(p_{u_{5}}\wedge p_{u_{6}})\vee(\neg p_{u_{5}}\wedge p_{u_{6}})$ | $f_{s}(u_{6})=p_{u_{7}}$
For both queries, $u_{5}$ and $u_{8}$ are two non-independently constraint
nodes. In $Q_{1}$, we have $u_{2}\trianglelefteq u_{6}$, because (1)
$u_{6}\vdash u_{2}$, (2) $u_{4}\trianglelefteq u_{7}$, (3)
$f_{tr}(u_{6})\rightarrow f_{tr}(u_{2})[u_{2}\mapsto
u_{6}]=p_{u_{7}}\rightarrow p_{u_{7}}$, which is a tautology, (4) $u_{2}$ is
an AD child of $u_{1}$ which is an ancestor of $u_{6}$. In contrast, for
$Q_{2}$, $u_{2}\not\trianglelefteq u_{6}$, since now $u_{2}$ is a PC child of
$u_{1}$ but $u_{6}$ is not. Suppose attribute predicates of all nodes are
satisfiable. Then for $Q_{2}$, $f^{2}_{cs}(u_{1})=\neg(p_{u_{2}}\wedge
p_{u_{4}})\wedge p_{u_{3}}\wedge\big{(}(p_{u_{5}}\wedge p_{u_{6}}\wedge
p_{u_{7}})\vee(\neg p_{u_{5}}\wedge p_{u_{6}}\wedge p_{u_{7}})\big{)}$, which
is satisfiable; but for $Q_{1}$,
$f^{1}_{cs}(u_{1})=f^{2}_{cs}(u_{1})\wedge\big{(}p_{u_{6}}\rightarrow(p_{u_{2}}\wedge
p_{u_{4}})\big{)}$, which is unsatisfiable. Therefore, we know that $Q_{2}$ is
satisfiable and $Q_{1}$ not. ∎
### 3.2 Containment and Equivalence
For two GTPQs $Q_{1}$ and $Q_{2}$, $Q_{1}$ is _contained_ in $Q_{2}$, denoted
by $Q_{1}\sqsubseteq Q_{2}$, if for any data graph $G$, $Q_{1}(G)\subseteq
Q_{2}(G)$. $Q_{1}$ and $Q_{2}$ is _equivalent_ , denoted by $Q_{1}\equiv
Q_{2}$, if $Q_{1}(G)\subseteq Q_{2}(G)$ and $Q_{2}(G)\subseteq Q_{1}(G)$.
###### Homomorphism.
Given two GTPQs $Q_{1}$ with query nodes $V^{1}_{q}$ and $Q_{2}$ with query
nodes $V^{2}_{q}$, a homomorphism from $Q_{1}$ to $Q_{2}$ is a mapping
$\lambda$ from $V^{1}_{q}$ to $V^{2}_{q}\cup\\{\perp\\}$ such that (1) the two
sets of output nodes of $Q_{1}$ and $Q_{2}$ are bijective; (2) for any non-
independently constraint node $u\in V^{1}_{q}$, $\lambda(u)=\perp$; (3) for
any independently constraint node $u_{1}$ in $V^{1}_{q}$, (a) for any PC
(resp, AD) child node $u^{\prime}_{1}$ of $u_{1}$ such that $u^{\prime}_{1}$
is also an independently constraint node, $\lambda(u^{\prime}_{1})$ is a PC
child (resp, a descendant) of $\lambda(u_{1})$, and (b) $\lambda(u_{1})\vdash
u_{1}$; (4) the formula $f_{cs}(u^{2}_{root})\to
f_{cs}(u^{1}_{root})[u^{1}_{root}\mapsto\lambda(u^{1}_{root})]$ is a
tautology, where $u^{1}_{root}$ is the root node of $Q_{1}$ and
$f_{cs}(u^{1}_{root})[u^{1}_{root}\mapsto\lambda(u^{1}_{root})]$ is a formula
transformed from $f_{cs}(u^{1}_{root})$ by replacing $p_{u^{\prime}}$ with
$p_{\lambda(u^{\prime})}$ for each independently constraint node
$u^{\prime}\in V^{1}_{q}$.
Theorem 3 yields a decision procedure for containment and equivalence between
two GTPQs. Theorem 4 states the intractability of the two problems of
containment and equivalence.
###### Theorem 3.
For two GTPQs $Q_{1}$ and $Q_{2}$, $Q_{1}\sqsubseteq Q_{2}$ iff there exists a
homomorphism from $Q_{2}$ to $Q_{1}$. ∎
###### Theorem 4.
The containment checking for GTPQs is co-NP-hard. ∎
###### Example 5.
Recall the queries in Fig. 4. We now assume $f_{s}(u_{1})=p_{u_{2}}$ and
others the same as in Example 4. Let $Q_{3}$ be a conjunctive GTPQ, and
$u^{j}_{i}$ denote $u_{i}$ in $Q_{j}$ to distinguish nodes in different
queries. We have that $Q_{2}\sqsubseteq Q_{3}$, $Q_{2}\sqsubseteq Q_{1}$ and
$Q_{1}\equiv Q_{3}$. Indeed, there is a homomorphism $\lambda_{3,2}$ from
$Q_{3}$ to $Q_{2}$, where
$\lambda_{3,2}(u^{3}_{1})=u^{2}_{1},\lambda_{3,2}(u^{3}_{2})=u^{2}_{3},\lambda_{3,2}(u^{3}_{3})=u^{2}_{6},\lambda_{3,2}(u^{3}_{4})=u^{2}_{7}$.
There is also $\lambda_{1,3}$ from $Q_{1}$ to $Q_{3}$, in which
$\lambda_{1,3}(u^{1}_{i})=\perp(i=5,8),\lambda_{1,3}(u^{1}_{j})=u^{3}_{3}(j=2,6),\lambda_{1,3}(u^{1}_{k})=u^{3}_{4}(k=4,7),\lambda_{1,3}(u^{1}_{1})\\\
=u^{3}_{1},\lambda_{1,3}(u^{1}_{3})=u^{3}_{2}$. We can also derive
$\lambda_{3,1}$ and $\lambda_{1,2}$. ∎
Input: GTPQ $Q=(V_{b},V_{p},V_{o},E_{q},f_{a},f_{e},f_{s})$ with the root
$u_{r}$.
Output: A minimum equivalent GTPQ $Q_{m}$ of $Q$.
1 construct an equivalent query $Q_{m}$ from $Q$ by removing subtrees rooted
at a node whose attribute predicate is unsatisfiable and assigning the
variables of the removed nodes to 0 for respective structural predicates
2
3check each structural predicate to determine for each node whether it is an
independently constraint node and remove all non-independently constraint
nodes followed by assigning the variables of them to 0 for respective
structural predicates
4
5compute the complete structural predicate $f_{cs}(u)$ for each node $u$ in
$Q_{m}$ in bottom-up order
6 for each $u\in V^{m}_{q}$ in bottom-up order do do
7 if $f_{cs}(u)$ is unsatisfiable then
8 $f_{s}\big{(}parent(u)\big{)}:=f_{s}\big{(}parent(u)\big{)}[p_{u}/0]$
9 remove the whole subtree rooted at $u$ from $Q_{m}$
10
11
12
13for each node $u\in V^{m}_{q}$ do
14 if the formula $f_{cs}(u_{r})\rightarrow p_{u}$ is a tautology then
15 for each $u^{\prime}$ such that $u^{\prime}\trianglelefteq u$ do
16
$f_{s}\big{(}parent(u^{\prime})\big{)}:=f_{s}\big{(}parent(u^{\prime})\big{)}[p_{u^{\prime}}/1]$
17 for each output node $u_{o}$ in the subtree rooted at $u^{\prime}$ do
18 if there exists $u^{\prime\prime}$ such that $u_{o}\triangleleft
u^{\prime\prime}$ and the subtree query pattern rooted at $u^{\prime\prime}$
and that rooted at $u_{o}$ are isomorphic then
19 remove $u_{o}$ from the set of output nodes and add $u^{\prime\prime}$ into
it
20
21
22 remove nodes in the subtree rooted at $u^{\prime}$ from $Q_{m}$ that are
not ancestors of any output nodes and corresponding edges they connect
23
24
25 else if the formula $f_{cs}(u_{r})\rightarrow\neg p_{u}$ is a tautology
then
26 for each pair $(u,u^{\prime})\in S$ do
27
$f_{s}\big{(}parent(u^{\prime})\big{)}:=f_{s}\big{(}parent(u^{\prime})\big{)}[p_{u^{\prime}}/0]$
28 remove the whole subtree rooted at $u^{\prime}$ from $Q_{m}$
29
30
31
32
return $Q_{m}$
Algorithm 1 minGTPQ
### 3.3 Minimization
Since the efficiency of processing a query depends on the size of it, it is
necessary to identify and eliminate redundant nodes. For a GTPQ with query
nodes $V_{q}$, we define its size as $|Q|=|V_{q}|$.
###### Minimization.
Given a GTPQ $Q$, the minimization problem is to find another GTPQ $Q_{m}$
such that (1) $Q\equiv Q_{m}$, (2) $|Q_{m}|\leq|Q|$, and (3) there exists no
other such $Q^{\prime}$ with $|Q^{\prime}|<|Q_{m}|$.
From Theorem 3, we have that for a GTPQ $Q$, there is a minimal equivalent
GTPQ of $Q$ whose query nodes are a subset of query nodes of $Q$. We say two
GTPQs $Q_{1}$ and $Q_{2}$ are isomorphic, if there is a homomorphism between
them that is a one-to-one mapping. The following proposition shows that the
minimal equivalent query of a GTPQ is unique up to isomorphism.
###### Proposition 5.
Let GTPQs $Q_{1}$ and $Q_{2}$ be minimal and equivalent. Then $Q_{1}$ and
$Q_{2}$ are isomorphic.∎
Algorithm 1 shows how to minimize a GTPQ. We give an example to illustrate it.
###### Example 6.
In Fig. 4, the query $Q_{3}$ is a minimum equivalent query of $Q_{1}$ with
structural predicates given in Example 5. (1) Since we suppose all attribute
predicates are satisfiable, there are no nodes to be removed in this step, and
$Q_{m}=Q_{1}$ (line 1). (2) All nodes except $u_{5}$ and $u_{8}$ are
independently constraint nodes, hence we remove $u_{5}$ and $u_{8}$ and assign
0 to $p_{u_{5}}$ in $f_{s}(u_{3})$, thereby having that
$f_{s}(u_{3})=p_{u_{6}}$ (line 2). In this step, all propositional formulas of
structural predicates are _simplified_ to equivalent formulas with minimum
variables. (3) There are no nodes whose complete structural predicates are
unsatisfiable, and so none is removed (line 4–7). (4) The formula
$f_{cs}(u_{1})\rightarrow p_{u_{6}}$ is a tautology and $u_{2}\trianglelefteq
u_{6}$, so $u_{2}$ and its child $u_{4}$ is removed, and we have
$f_{s}(u_{1})=1$, thereby generating the query $Q_{3}$ (line 8–19). This step
is to remove subtrees which can be _semantically subsumed_ by others. ∎
The correctness can be proved based on Theorem 3. Since the algorithm involves
solving SAT problems, the worst-case time complexity is exponential in the
query size. In fact, Theorem 6 shows that the minimization problem is NP-hard
and hence it is difficult to find a polynomial-time algorithm. Nevertheless,
because there are many high-performance algorithms for SAT and the query size
is not much large in practice, it is still worth minimizing a GTPQ considering
the benefits of efficiency of evaluation.
###### Theorem 6.
The minimization problem for GTPQs is NP-hard.∎
## 4 Evaluating Generalized Tree
Pattern queries
### 4.1 Framework
Recall that two major problems that impair the efficiency of algorithms for
processing TPQs over graphs are large intermediate results and expensive join
operations on them. In the following, we propose two new techniques to address
them.
###### Graph representation of intermediate results.
To reduce the cost of storing intermediate results and avoid merge-join
operations, we represent intermediate results as a graph rather than sets of
tuples. Each match for a path or a substructure of the query pattern can be
embedded into the tree pattern and hence naturally can be represented as a
tree. By grouping all the candidate matches by the corresponding matched query
nodes and adding an edge to connect a pair of data nodes whenever there’s an
edge between the corresponding pair of query nodes in the query pattern, we
can represent the intermediate and final results as graphs. In such a graph
representation, each data node exists at most once, in contrast to the tuple
representation in which a data node may be in multiple tuples. Also, the AD or
PC relationship between two nodes is exactly represented by only one edge,
while in the tuple form the corresponding two nodes may be put as an element
in more than one tuple to repeatedly and explicitly represent their
relationship. Since the size of the intermediate matches may be huge, even
exponential in both the query size and the data size in the worst case, the
graph representation is much more compact with at most quadratic space cost.
Moreover, to enumerate all resulting matches of a pattern query, we only need
to perform one single graph traversal on a presumably small graph instead of
multiple merge-join operations over large intermediate results.
It is worth noting that such a way of representing intermediate results can be
also applied to algorithms for other graph pattern queries to boost their
evaluation. For TPQs, it is particularly optimal because we can enumerate
matches directly from the graph. However, for graph pattern queries,
additional matching operations including joins may be unavoidable because it
is difficult to locally determine which nodes should be traversed to form a
match. The additional matching operations are in essence an easier evaluation
of a pattern matching on a smaller graph, such a technique can thus still be
expected to speed up the whole processing.
###### Reachability index enhanced effective pruning.
Since the number of data nodes to be processed significantly affects the
efficiency of pattern query evaluation, it is desirable to perform effective
pruning to reduce the number of candidate matching nodes. In the literature,
[6] and [11] have developed two pruning approaches for reachability query
pattern matching. TwigStackD [6] proposed a pre-filtering approach that can
select nodes guaranteed to be in final matches. Since it has to perform two
graph traversals on the data graph, it is likely unfeasible for large-scale
real-world graphs. The work [11] on pattern queries over labeled graphs
proposed another pruning process, namely R-semijoin, using a special index
called cluster-based R-join index. It can filter nodes that cannot possibly
contribute to partial matches for an AD edge between two labeled query nodes.
However, (1) the selected nodes may be still redundant since the nodes only
satisfy the reachability condition imposed by one edge and the global
structural satisfaction is not checked. (2) It is highly costly to construct
and store the R-join index for a large data graph since the index essentially
precomputes and stores all matches for pairwise labels and the index size is
quadratic in the graph size. (3) It cannot be used to perform pruning for
queries that have expressive attribute predicates rather than a fixed set of
labels associated with nodes. Since predicates of query nodes are often not
fixed and predictable, the index actually cannot be precomputed and this
approach cannot be used.
We explore the potentials of existing reachability index for effective
pruning. It is interesting to note that most reachability indexing schemes
follow a paradigm. They first utilize a relatively simple reachability index
which often assigns two or three labels to each node in order to cover the
reachability of a substructure, called a cover, such as tree-cover in [1, 26],
path-tree in [18], and chain-cover in [8, 19]. To cover the remaining
reachability information, each node keeps one or two lists where complete or
just a portion of ancestors and descendants are stored. When answering whether
a node can reach another, the algorithms typically use nodes stored in the
lists as the intermediate to determine the reachability.
When it comes to answer a number of reachability queries between two sets of
nodes, the following two observations are helpful: (1) the lists of different
nodes often share a number of nodes, (2) the nodes in different lists have
rich reachability information. If we merge the lists of a set of nodes by
eliminating the duplicates and those whose reachability information can be
derived from others, the merged list “subsumes” all the reachability
information in the original lists of the node set but the size will not be
much larger, and possibly even much smaller, than the list size of any
individual node. Using the merged list, reachability patterns are likely to be
evaluated more efficiently.
For example, considering a reachability pattern $u_{A}$—$u_{B}$, we want to
filter data nodes in $mat(u_{A})$ that cannot reach any nodes in $mat(u_{B})$.
Instead of performing $|mat(u_{A})|\times|mat(u_{B})|$ pairwise reachability
queries to check for each node $v\in mat(u_{A})$ whether it can reach a node
$v^{\prime}\in mat(u_{B})$, (1) we merge all index lists of $mat(u_{B})$ to a
single list of the minimum size that preserves all the reachability
information saved in the original lists; and (2) for each $v\in mat(u_{A})$,
use the list of $v$ and the merged list rather than individual lists for
$mat(u_{B})$ to holistically determine whether $v$ reaches some node in
$mat(u_{B})$. Intuitively, we can regard the set $mat(u_{B})$ as a single
dummy node which is reachable from all nodes that are ancestors of nodes in
$mat(u_{B})$.
In this paper, we use 3-hop [19] as the underlying reachability index scheme,
as 3-hop has both a very compact index size and reasonable query processing
time. As different labeling schemes are often preferable to different graph
structures, it is also very flexible for our framework to use other labeling
schemes to efficiently process different types of graphs.
We restrict our attention to in-memory processing and do not address the
issues relating to disk-based access methods and physical representation of
graph data.
###### Algorithm outline.
Our GTPQ evaluation algorithm (referred to as GTEA) is outlined as follows.
First, it prunes candidate matching nodes that do not satisfy downward
structural constraints (i.e. not satisfy the subtree pattern query rooted at
the corresponding query node). Second, it performs the second round pruning
process on a carefully selected subtree pattern, called prime subtree, to
remove nodes not satisfying upward structural constraints (i.e. not reachable
from any candidate nodes of the root). Third, the prime subtree is further
shrunk if possible, and GTEA generates the matches of the shrunk prime subtree
while representing the intermediate results as a graph, from which the final
results can be efficiently obtained. We begin with focusing on evaluating
GTPQs with AD edges only and show how to extend the algorithm to process PC
edges in Section 4.4.
### 4.2 Pruning Candidate Matching Nodes
We use a two-round pruning process to filter unqualified data nodes. The first
round selects data nodes that satisfy downward structural constraints of the
query pattern for each query node. At the second round, we then obtain a
minimum subtree that contains all output nodes having more than one candidate
matching node, and select necessary edges from this subtree to find nodes
satisfying upward structural constraints.
Figure 5: Chain decomposition and 3-hop index
#### 4.2.1 Preliminary: Merging 3-hop index
3-hop is a recent graph reachability indexing scheme well-known for its
compact index size and reasonable query time. It follows the indexing paradigm
mentioned in Section 4.1. It uses the chain-cover which consists of a set of
disjoint chains covering all nodes in the graph. Each node in the graph is
assigned a chain ID $cid$ and its sequence number $sid$ on its chain. For two
nodes $v$ and $v^{\prime}$ on the same chain (i.e., $v.cid=v^{\prime}.cid$),
$v\leq_{c}v^{\prime}$, if $v.sid\leq v^{\prime}.sid$. In particular, if
$v.sid<v^{\prime}.sid$, we say $v$ is _smaller_ than $v^{\prime}$. Obviously,
reachability on the chain-cover can be answered using chain IDs and sequence
numbers. To encode the remaining reachability information outside chain-cover,
3-hop records a successor list $L_{out}(v)$ $\big{(}$resp. predecessor list
$L_{in}(v)$$\big{)}$ of “entry” (resp. “exit”) nodes to (resp. from) other
chains for each node $v$. The entry (resp. exit) node to (resp. from) a chain
is the smallest (resp. largest) one on that chain that $v$ reaches (resp.
reaches $v$). See [19] for details of 3-hop index construction. For answering
the reachability between two nodes $v_{1}$ and $v_{2}$ on different chains,
3-hop takes the following steps. (1) Collect the smallest nodes on any other
chain that $v_{1}$ can reach through exit nodes of chain $v_{1}.cid$. That is,
we get a set of nodes
$X_{v_{1}}=\\{x|x\in\bigcup_{v_{1}\leq_{c}v^{\prime}}L_{out}(v^{\prime})$$\textrm{
and }\forall
v^{\prime}$$\geq_{c}$$v_{1},x\leq_{c}L^{x.cid}_{out}(v^{\prime})\\}$
$\cup\\{v_{1}\\}$, where $L^{x.cid}_{out}(v^{\prime})$ is the entry node of
$v^{\prime}$ on chain $x.cid$. We call $X_{v_{1}}$ the _complete successor
list_ of $v_{1}$. (2) Collect the largest nodes on any chain that can reach
$v_{2}$ through entry nodes of chain $v_{2}.cid$. In this step, we get a set
of nodes
$Y_{v_{2}}=\\{y|y\in\bigcup_{v^{\prime}\leq_{c}v_{2}}L_{in}(v^{\prime})\textrm{
and }\forall v^{\prime}$$\leq_{c}$$v_{2}$,
$L^{y.cid}_{in}(v^{\prime})\leq_{c}y\\}\cup\\{v_{2}\\}$, where
$L^{y.cid}_{in}(v^{\prime})$ is the exit node of $v^{\prime}$ on chain
$y.cid$. We call $Y_{v_{2}}$ the _complete predecessor list_ of $v_{2}$. (3)
If there is a pair $(x,y)(x\in X_{v_{1}},y\in Y_{v_{2}})$ such that
$x\leq_{c}y$, then we can conclude that $v_{1}$ can reach $v_{2}$.
###### Example 7.
Fig. 5 gives a chain decomposition of $G$ of Fig. 2(a) and the corresponding
3-hop index. Chain IDs and sequence numbers are omitted. As an example,
$v_{3}.cid=v_{11}.cid=1$, $v_{11}.sid=4$ and $v_{3}.sid=2$. Because
$v_{3}.sid<v_{11}.sid$, $v_{3}\leq_{c}v_{11}$ and $v_{11}$ is reachable from
$v_{3}$. To answer whether $v_{3}$ can reach $v_{9}$, we collect the entry
nodes in $L_{out}(v_{i})(i=3,7,11,16)$ into $X_{v_{3}}=\\{v_{3},v_{4}\\}$.
Then we look up the exit nodes in $L_{in}(v_{j})(j=9,5)$ and get
$Y_{v_{9}}=\\{v_{9},v_{12}\\}$. Since there is a pair $(v_{4},v_{12})$ such
that $v_{4}\in X_{v_{3}},v_{12}\in Y_{v_{9}}$, and $v_{4}\leq_{c}v_{12}$, we
say $v_{3}$ can reach $v_{9}$. ∎
Note that to obtain the complete predecessor (resp. successor) lists, the
original 3-hop needs to visit all larger (resp. smaller) nodes. We can assign
a forward (and backward) tracing pointer to each node which points to the
smallest larger (resp. largest smaller) node whose $L_{out}$ (resp. $L_{in}$)
list is nonempty so that nodes with empty lists can be skipped. We define two
operations _next_($v$) and _pre_($v$) on each node $v$, which return the node
that the forward and the backward tracing pointer points to respectively. For
example, since $v_{6}$ is the largest smaller node that has a non-empty
$L_{in}$ w.r.t. $v_{15}$, prev$(v_{15})=v_{6}$.
A basic operation of the pruning process is merging the complete
predecessor/successor lists for a given set of data nodes (denoted by $S$).
For the 3-hop case, it picks the largest (resp. smallest) nodes on each chain
from the complete predecessor (resp. successor) list and we call the resultant
list _predecessor contour_ $C^{p}$ (resp. _successor contour_ $C^{s}$). A node
$v$ is said to reach (resp. be reachable from) $S$ if $v$ reaches (resp. is
reachable from) at least one node in $S$. We have the following proposition.
###### Proposition 7.
A data node $v$ reaches $mat(u)$ iff there is a pair
$\textup{(}x,y\textup{)}\in X_{v}\times C^{p}$ such that $x\leq_{c}y$, while
$mat(u)$ reaches $v$ iff there exists a pair $\textup{(}x,y\textup{)}\in
C^{s}\times Y_{v}$ such that $x\leq_{c}y$. ∎
Input: A set of nodes $S$.
Output: The predecessor contour $C^{p}$ of $S$.
1 for each node $v\in S$ do
2 if $C^{p}[v.cid]<v.sid$ then $C^{p}[v.cid]:=v.sid$
3
4 $v^{\prime}:=v$
5 repeat
6 for each index node $v^{\prime\prime}\in L_{in}(v^{\prime})$ do
7 if $C^{p}[v^{\prime\prime}.cid]<v^{\prime\prime}.sid$ then
8 $C^{p}[v^{\prime\prime}.cid]:=v^{\prime\prime}.sid$
9
10
11 $v^{\prime}:=\textrm{prev}(v^{\prime})$
12 until $v^{\prime}=null$ or $visited_{v^{\prime}.cid}\geq v^{\prime}.sid$
13 if $visited_{v.cid}<v.sid$ then $visited_{v.cid}:=v.sid$
14
15
return $C^{p}$
Procedure 2 MergePredLists
Procedure 2 sketches the process of calculating the predecessor contour
$C^{p}$, where $visited_{i}$ records the largest node on chain $i$ whose
predecessor list has been looked up. For each node $v\in S$, MergePredLists
processes $v$ and those smaller nodes whose predecessor lists have not been
looked up as follows. For each node $v^{\prime}$ to be processed and each exit
node $v^{\prime\prime}$ in $L_{in}(v^{\prime})$, it compares
$v^{\prime\prime}$ with the nodes in $C^{p}$ on the same chain of
$v^{\prime\prime}$, and update $C^{p}$ if $v^{\prime\prime}$ is larger (line
4–9). To retrieve nodes from $C^{p}$ efficiently, $C^{p}$ can be implemented
as a map that uses chain IDs as keys and the sequence numbers as values.
###### Example 8.
We show how to compute the predecessor contour of $mat(u_{10})$ for the query
$Q$ of Fig. 2. Example 3 have given that
$mat(u_{10})=\\{v_{9},v_{10},v_{13},v_{15}\\}$. The procedure collects the
complete predecessor lists for each of $mat(u_{10})$ one by one, but no
predecessor list is repeatedly visited. For example, assume that $v_{10}$ is
read before $v_{15}$. When collecting $Y_{v_{15}}$, although prev($v_{15}$)
points to $v_{6}$, MergePredLists needs not look up $L_{in}(v_{6})$, because
the list has been looked up when collecting $Y_{v_{10}}$. The predecessor
contour of $mat(u_{10})$ is $\\{v_{3},v_{9},v_{13},v_{15}\\}$. It can be
easily verified that the size of this predecessor contour is a half of the
total size of the four individual complete lists of $v_{9},v_{10},v_{13}$ and
$v_{15}$. Note that the size of a predecessor contour is bounded by the number
of chains. This example actually gives the worst case but still has a high
compression rate (50%). ∎
Input: 3-hop index $L_{out}$, a GTPQ $Q$.
Output: Updated candidate matching nodes satisfying downward structural
constraints.
1 for each node $u\in V_{q}$ do $mat(u):=\\{x|x\in V,x\sim u\\}$
2
3 for each leaf node $u^{\prime}$ in $V_{q}$ do
$C^{p}_{u^{\prime}}:=\textrm{MergePredLists}\big{(}mat(u^{\prime})\big{)}$
4
5 $V^{\prime}_{q}=V_{q}\backslash\\{u^{\prime}|u^{\prime}\textrm{ is a leaf
node}\\}$
6 for each $u\in V^{\prime}_{q}$ in bottom-up order do
7 for each $v\in mat(u)$ do $chain_{v.cid}:=chain_{v.cid}\cup\\{v\\}$
8
9 for each $chain_{i}$ that is not empty do
10 for each child $u^{\prime}$ of $u$ do $val[p_{u^{\prime}}]:=0$
11
12
13 for each node $v_{i}\in chain_{i}$ do
14 for each child $u^{\prime}$ of $u$ _s.t._ $val[p_{u^{\prime}}]=0$ do
15 if $v_{i}$ reaches $mat(u^{\prime})$ then // using Proposition 7
16 $val[p_{u^{\prime}}]:=1$
17
18 if $f_{s}(u)$ evaluates to false with the valuation $val$ then
19 $mat(u):=mat(u)\backslash\\{v_{i}\\}$
20
21
22
23 $C^{p}_{u}:=\textrm{MergePredLists}\big{(}mat(u)\big{)}$
24
Procedure 3 PruneDownward
###### Time complexity.
The time complexity of the procedure is $O(|S|+|L_{in}|)$, where $|L_{in}|$ is
the total size of all predecessor lists in 3-hop index. It can be observed
from the fact that no index node in a predecessor list has been ever
repeatedly visited.
Following the same line of MergePredLists, we develop MergeSuccLists that
calculates the successor contour of a node set with time complexity of
$O(|S|+|L_{out}|)$, where $|L_{out}|$ is the total size of all successor lists
in 3-hop index.
#### 4.2.2 Pruning process for downward structural constraints
Procedure 6 describes the first round of the pruning process. In the
procedure, $val$ refers to a valuation for variables associated with query
nodes. PruneDownward first collects $mat(\cdot)$ sorted in the descending
order of sequence numbers for each query node and calculates the predecessor
contours for leaf nodes (line 1–2). Then, it processes each non-leaf query
node $u$ following a bottom-up fashion (line 4–14). For each node $u$, it
first groups nodes $mat(u)$ by chain ID (line 5). Then for each candidate
matching node $v_{i}$ of $u$ on each chain $i$, PruneDownward checks whether
$v_{i}$ satisfies downward structural constraints (line 8–13). To do this, (1)
it first assigns a valuation to $p_{u^{\prime}}$ for each child node
$u^{\prime}$ of $u$ according to the reachability from $v_{i}$ to
$mat(u^{\prime})$ (line 9–11) , (2) and then remove $v_{i}$ from $mat(u)$ if
the structural predicate $f_{s}(u)$ of $u$ evaluates to false under the
valuation (line 12–13). Note that when processing the next node on the same
chain, the valuation for the previous node is inherited due to the transitive
property of transitive closure in a chain. Therefore, no predecessor list is
repeatedly looked up. After all candidate matching nodes for $u$ have been
processed, the remaining data nodes in $mat(u)$ must satisfy the downward
structural constraints. Then the predecessor contour for $u$ is computed (line
14), and used in the pruning process of the parent node of $u$. The procedure
terminates after the root is processed.
###### Example 9.
We first show how procedure PruneDownward prunes
$mat(u_{3})$ of Fig. 2. In a bottom-up fashion, before pruning $mat(u_{3})$,
PruneDownward first processes its non-leaf child $u_{7}$. No nodes in
$mat(u_{7})($i.e. $\\{v_{6},v_{7}\\})$ are removed, because $v_{6}$ can reach
both $mat(u_{9})$ and $mat(u_{10})$ while $v_{7}$ can reach $mat(u_{10})$. The
predecessor contour for $mat(u_{7})$ is then computed and
$C^{p}_{u_{7}}=\\{v_{6},v_{7}\\}$. For determining whether $v_{5}$ should be
removed from $mat(u_{3})$, PruneDownward checks the reachability between
$v_{5}$ and $mat(u_{6})$, $mat(u_{7})$, $mat(u_{8})$ respectively by using the
predecessor contours. One can verify that $v_{5}$ cannot reach $mat(u_{6})$,
which means $val[p_{u_{6}}]\\\ =0$ and the structural predicate
$f^{v_{5}}_{s}(u_{3})$ evaluates to true. Thus, $v_{5}$ remains in
$mat(u_{3})$. Because the other two nodes $v_{3}$ and $v_{8}$ are in different
chains, they do not inherit the valuation determined by $v_{5}$ and
PruneDownward needs to check pairwise reachability between $\\{v_{3},v_{8}\\}$
and $\\{mat(u_{6})$, $mat(u_{7})$, $mat(u_{8})\\}$. Only $v_{8}$ is
subsequently removed, because $p_{u_{8}}=1,p_{u_{6}}=p_{u_{7}}=0$ and
$f^{v_{8}}_{ext}(u_{3})$ evaluates to false. Finally, after this pruning
round, $mat(u_{3})=\\{v_{3},v_{5}\\}$.
When PruneDownward refines $mat(u_{1})$ and reads $v_{2}$, the assignments of
$p_{u_{2}}$ and $p_{u_{3}}$ are directly inherited from the result computed in
the previous step of processing $v_{4}$ and $f^{v_{2}}_{ext}(u_{1})$
immediately evaluates to true without any index lookups.
PruneDownward gets the following refined candidate matching nodes which
satisfy the downward structural constraints: $mat(u_{2})\\\
=\\{v_{3},v_{8}\\},mat(u_{3})=\\{v_{3},v_{5}\\}.$∎
###### Time complexity.
Since no successor list is repeatedly checked, the 3-hop index is looked up
for at most $|E_{q}||L_{out}|$ times, where $|E_{q}|$ is the number of edges
in the tree pattern. MergePredLists is invoked ($|V_{q}|-1$) times to compute
predecessor contours for each non-root query node, and the total time cost is
$O(|V_{mat}|+|V_{q}||L_{in}|)$, where $|V_{q}|$ is the number of query nodes
and $|V_{mat}|$ is the total size of initial candidate matching nodes (i.e.
$|V_{mat}|=\Sigma_{i}|mat(u_{i})|$). Therefore, PruneDownward is in
$O(|V_{q}|(|L_{in}|+|L_{out}|)+|V_{mat}|)$ time.
Input: 3-hop index $L_{in}$, the prime subtree $(V_{t},E_{t})$.
Output: Updated candidate matching nodes satisfying upward structural
constraints.
1 $C^{s}_{u_{root}}:=\textrm{MergeSuccLists}\big{(}mat(u_{root})\big{)}$
2 $V_{t}:=V_{t}\backslash\\{u_{root}\\}$
3 for each node $u\in V_{t}$ in top-down order such that $|mat(u)|>1$ do
4 for each child $u^{\prime}$ of $u$ such that $|mat(u^{\prime})|>1$ do
5 for each node $v\in mat(u^{\prime})$ do
6 $chain_{v.cid}:=chain_{v.cid}\cup\\{v\\}$
7 $Group_{v}:=Group_{v}\cup\\{u^{\prime}\\}$
8
9
10 for each node $v_{i}$ in a nonempty $chain_{i}$ do
11
12 if $mat(u^{\prime})$ do not reach $v_{i}$ then // using Proposition 7
13 for each $u^{\prime}\in Group_{v_{i}}$ do
14 $mat(u^{\prime}):=mat(u^{\prime})\backslash\\{v_{i}\\}$
15
16
17 else break
18
19 for each non-leaf child $u^{\prime}$ of $u$ do
20 $C^{s}_{u^{\prime}}:=\textrm{MergeSuccLists}\big{(}mat(u^{\prime})\big{)}$
21
22
Procedure 4 PruneUpward
#### 4.2.3 Pruning process for upward structural constraints
After the fist-round pruning process, for each backbone node $u$, the
remaining nodes in $mat(u)$ satisfy all the structural constraints imposed by
predicates. Because the results of the query should consist of matches of
output nodes only, the matches for predicate nodes are no longer useful and do
not need to be considered. Moreover, some backbone nodes may not contribute to
determining which candidate matching output nodes are in the same instance and
hence can be also discarded. With these two observations, the structural
constraints of a backbone subtree are enough to derive the relationships among
candidate matching nodes for the output query nodes. Such a subtree, we call
the prime subtree, can be induced by the paths from the query root to all such
output nodes that $|mat(\cdot)|>1$. The next pruning step only needs to
consider this subtree pattern which in essence is reduced to a conjunctive
GTPQ.
In the opposite direction to PruneDownward, procedure PruneUpward (Procedure
7) traverses down the prime subtree. For each query node $u$, it filters the
candidate matching nodes of each child $u^{\prime}$ of $u$ (line 3–14). All
the candidate nodes to be processed are first clustered and merged into
duplicate-free sets according to their chain IDs, where the order of nodes is
reversed (line 4–7). As a data node can match multiple query nodes, the
algorithm uses $Group_{v}$ to record the corresponding query nodes that $v$
matches (line 7) in order to update $mat(\cdot)$ when a reachability condition
is determined (line 10–11). Then, for each node $v_{i}\in mat(u^{\prime})$ on
a nonempty $chain_{i}$, $v_{i}$ should be removed if $mat(u)$ cannot reach
$v_{i}$ according to Proposition 7. Observe that once a node is confirmed to
satisfy the condition of the incoming edge, all other larger nodes do not need
to be checked since they must also satisfy the condition.
###### Example 10.
In this example, assume that $u_{2}$ and $u_{3}$ are output nodes of $Q$ of
Fig. 2. The prime subtree is induced by $u_{1}$, $u_{2}$ and $u_{3}$.
PruneUpward starts from $u_{1}$ to refine $mat(u_{2})$ and $mat(u_{3})$. After
grouping distinct data nodes into $chain$, it gets $chain_{1}=\\{v_{3}\\}$,
$chain_{3}$$=$$\\{v_{8}\\}$, and $chain_{4}=\\{v_{5}\\}$. $v_{3}$ is in both
$mat(u_{2})$ and $mat(u_{3})$, but the procedure only stores one copy in
$chain$ to avoid processing it repeatedly when checking reachability with
$mat(u_{1})$. After the two query nodes whose matching candidate nodes have
the identical $v_{3}$ are inserted to $Group_{v_{3}}$,
$Group_{v_{3}}=\\{u_{2},u_{3}\\}$. Because $mat(u_{1})$ reaches $v_{3}$,
$v_{3}$ is not removed from either $mat(u_{2})$ or $mat(u_{3})$. Similarly, it
can be verified that $mat(u_{1})$ can reach $v_{8}$ and $v_{5}$. In the end,
none is removed from $mat(u_{2})$ and $mat(u_{3})$ after this pruning round.∎
###### Time complexity.
The time complexity is
$O(|V^{\prime}_{mat}|+(|L_{in}|+|L_{out}|)|V^{\prime}_{t}|)$, where
$|V^{\prime}_{t}|$ is the number of internal nodes in the prime subtree and
$|V^{\prime}_{mat}|$ is the total size of the remaining candidate matching
nodes after the first pruning round.
### 4.3 Computing Final Results
Figure 6: Example of the maximal matching graph for $Q$ over $G$ depicted in
Fig. 2
###### Shrunk prime subtree.
As a result of the pruning process, the matching output nodes are guaranteed
to be in the answer. The left to do is to identify how they form the final
results by computing the matches of edges in the prime subtree. Given a prime
subtree, assume that $u$ is the lowest common ancestor of all output nodes. We
can further shrink the subtree by (1) removing the ancestors of $u$ if $u$ is
not the root, and (2) removing all such nodes $u^{\prime}$ that
$|mat(u^{\prime})|=1$. If the removing process leads to disjoint subtrees, we
just compute results for each subtree, do a Cartesian product of them and add
the candidate matching nodes of removed output nodes to assemble the whole
final results. From now on, we only need to compute edge matches for the
shrunk prime subtree(s).
###### Example 11.
The shrunk prime subtree of $Q$ of Fig. 2 is induced by $u_{2}$ and $u_{4}$.
Even if we change the query to mark $u_{5}$ also as an output node, the shrunk
prime subtree is _still_ the same since $|mat(u_{5})|=|\\{v_{13}\\}|=1$ and
$v_{13}$ must be in every answer.∎
###### Maximal matching graph.
The full matches of the shrunk prime subtree can be represented by a _maximal
matching graph_ $Q_{g}(G)=(V_{r},E_{r})$, where (1) $V_{r}\subseteq V$ such
that $v\in V_{r}$, if there is a query node $u\in V_{q}$ such that $v\models
u$; (2) $E_{r}\subseteq V_{r}\times V_{r}$ such that $(v_{1},v_{2})\in E_{r}$,
if $(v_{1},v_{2})$ is a match of an edge $(u_{1},u_{2})\in E_{q}$.
We group the nodes and edges in the graph according to what query nodes and
edges they match. Specifically, in an implementation, each node $v$ has
several branch lists, each of which corresponds to the child of the query node
that $v$ matches and includes pointers pointing to nodes matching the child.
###### Example 12.
Recall the GTPQ $Q$ and data graph $G$ in Fig. 2. Let $u_{2}$, $u_{3}$ and
$u_{4}$ be output nodes. Fig. 6 shows the corresponding maximal matching
graph. As an example, $v_{1}$ has two branch lists corresponding to the two
incident query edges, denoted by $bch_{1}$ and $bch_{2}$ respectively.
$bch_{1}=\\{ptr_{v_{3}},ptr_{v_{8}}\\}$, and
$bch_{2}=\\{ptr_{v_{3}},ptr_{v_{5}}\\}$, where $ptr_{v_{i}}(i=3,5,8)$ is
pointer to $v_{i}$. ∎
Input: The maximal matching graph $MaximalGraph$, a query node $u$ and one of
its candidate matching node $v$.
Output: the answer to the subGTPQ rooted at $u$ and dominated by $v$.
1 if $v$ is a leaf node then return $\\{u:v\\}$
2
3else
4 $results:=\emptyset$
5 for each branch list $bch$ of $v$ do
6 $branchResults:=\emptyset$
7 for each node $v^{\prime}$ that a pointer in $bch$ points to do
8 $branchResults:=branchResults\cup\textrm{
CollectResults}(MaximalGraph,v^{\prime}$)
9
10 $results:=results\times branchResults$
11
12 if $u$ is an output node then $results:=\\{u:v\\}\times results$
13
14 return $results$
15
Procedure 5 CollectResults
###### Computing the maximal matching graph.
Since the nodes of the maximal matching graph have been obtained after the
pruning process, we only need to compute matches for each query edge whose
head and tail both have more than one matching node. Given a query edge
$(u_{1},u_{2})$, a straightforward way is to check the reachability between
nodes in $mat(u_{1})$ and $mat(u_{2})$ using 3-hop index. The time complexity
is $O((|L_{in}+L_{out}|)|E_{q}||V_{mat}|^{2}_{max})$, with $|V_{mat}|_{max}$
being the maximal size of the candidate matching nodes after the pruning
process. Since in practice many queries are highly selective and
$|V_{mat}|_{max}$ is presumably pretty small, the straightforward way is
expected to be fast and practical.
A more sophisticated approach that we choose is to utilize the similar
technique used in procedure PruneUpward. Observe that the loop from line 9 to
12 in PruneUpward is to determine whether a data node matching some child of
$u$ is reachable from $mat(u)$. By replacing $C^{s}_{u}$ with the successor
list of a node $v$, we can simultaneously get all edges from $v$ in the
maximal matching graph in $O(|L_{in}|+|L_{out}|+|E_{v}|)$, where $|E_{v}|$ is
the out-degree of $v$ in the resulting graph. The total time complexity then
is $O((|L_{in}|+L_{out})|V^{inter}_{mat}|+|E_{mg}|)$, where
$|V^{inter}_{mat}|$ is the number of candidate matching nodes for internal
query nodes and $|E_{mg}|$ is the number of edges in the resulting maximal
matching graph.
###### Enumerating results.
We next present procedure 5, referred to as CollectResults, which derives
final results from the maximal matching graph. Each result is in a tuple
format. To avoid ambiguity in presentation, we explicitly specify in the tuple
which query node a data node matches. Specifically, each element in a tuple is
of the form $u:v$, which means $v$ is an image of $u$ in a match.
Procedure CollectResults traverses down the maximal graph. For a leaf node,
since its corresponding query node must be an output node, the procedure
returns a tuple with only an element of it (line 1). For an internal node, it
collects results from each child for every branch list, and then does a
Cartesian product of them (line 4-8). If the query node it matches is an
output node, it is inserted into each result (line 9). The final answer to the
query is the union of the results of those nodes matching the query root. When
query nodes in the shrunk prime subtree are all output nodes, no redundant
intermediate results would be produced. Note that no existing algorithms for
pattern queries on graphs can achieve this. When there are non-output query
nodes in the shrunk prime subtree, our algorithm is not duplicate free. Recall
Example 12. The results obtained from $v_{1}$ are the same as those obtained
from $v_{3}$, since $u_{1}$ is not an output node and $v_{1}$ can reach
$v_{3}$. However, the duplicate intermediate tuples are a subset of the
counterpart of other works, because (1) the prime subtree we pick is a minimum
subtree of the original query pattern that contains all output nodes, (2) for
non-output nodes, the algorithm merges the intermediate partial results in
advance (line 5).
###### Remark.
In practical languages, there is also group operation that require grouping
the results. We can also easily adapt our algorithm to support the group
operator. In CollectResults, when $u$ is a group node, the result returned for
$v$ is a tuple containing $v$ and a special group element which is the set of
matches of the subtree dominated by $v$. That is, in line 9,
$result:=\\{u:v,(result)\\}$.
### 4.4 Evaluating Queries with PC Edges
In the context of graph database, the research on pattern queries often
focuses on reachability patterns. Indeed, the reachability pattern query is an
important building block for other queries. Adding PC edges to a pattern
significantly increases the complexity of evaluation. Even for tree-structured
data, [25] has theoretically demonstrated the difficulty of handling TPQs with
arbitrary combination of PC and AD edges. [25] has proved that no holistic
algorithms can achieve optimality as for queries with AD edges only. For
graph-structured data, the evaluation of conjunctive pattern queries whose
edges all represent PC relationship is essentially a computationally-hard
labeled graph isomorphism problem. Nevertheless, we can use the similar idea
of our framework to support GTPQs with PC edges.
When processing a node $u$ in PruneDownward: (1) if $u$ has only PC outgoing
edges, we merge the set of parents of $mat(u^{\prime})$ for each child
$u^{\prime}$ of $u$ into $P_{u^{\prime}}$, instead of computing the
predecessor contours. Then we sort $mat(u)$ and each $P_{u^{\prime}}$, and
check for each node $v$ in $mat(u)$ whether it is in some $P_{u^{\prime}}$ in
a multiway merge-sort style. If yes, then $val[p_{u^{\prime}}]:=1$, otherwise
$val[p_{u^{\prime}}]:=0$. (2) If $u$ has both AD and PC edges, we process
these two type of edges separately to refine $mat(u)$. Similarly, when
performing PruneUpward, we collect sets of children of $mat(u)$ instead of
computing the successor contour.
After the pruning stage, all candidate matching nodes are guaranteed to be in
final results. To compute the maximal matching graph, we can either do nested
joins to check the adjacent relationships, or perform multiway merge-join to
derive the adjacent edges in the resulting graph. Other operations including
determining the prime subtree and enumerating final results are the same.
Alternatively, we can also use another strategy to deal with PC edges.
Regarding PC edge as a special type of AD edge, we can first process PC edges
in the same way with AD edges in the process of pruning, except those whose
tail’s structural variable is the operand of a negation operator and which
need to be processed as stated before. The prime subtree becomes a minimum
subtree that contains all output nodes and those PC edges that are regarded as
AD edges when pruning. After computing the maximal matching graph, we check
whether the two incident nodes of the corresponding edge in the maximal
matching graph are adjacent in the data graph and remove them if not. Next,
the unsatisfied nodes are removed in a top-down fashion, followed by
enumerating final results. We use this strategy in our implementation.
## 5 Experimental Evaluation
In this section, we present an experimental study using both real-life and
synthetic data to evaluate (1) the efficiency and scalability of our
algorithm, (2) the effectiveness of representing intermediate results as
graphs, and (3) the efficiency of the pruning process.
We only give the experimental results for conjunctive TPQs with all query
nodes being output nodes (i.e. the traditional TPQs). We found that our
algorithm has better performance than other algorithms even for them. Since
there has been no other algorithms designed for GTPQs and the decomposition-
based approach that may be applied on top of them to process GTPQs incurs high
overhead as analyzed in Related work and empirically demonstrated in prior
studies [16] and [29], our algorithm can do even far better for general GTPQs
than those algorithms, compared to the results reported here. Additional
experimental results concerning I/O cost and the results on GTPQs with
disjunctive and negative predicates can be found in the Appendix.
###### Implementation.
We have implemented the algorithm proposed in Section 4 (GTEA), TwigStack [3],
Twig2Stack [7], TwigStackD [6] and HGJoin [27]. TwigStack is the classical
holistic twig join algorithm. Twig2Stack is the latest algorithm for
evaluating TPQs on tree-structured data which has a distinct feature of
representing results in hierarchical stacks. Other algorithms for tree-
structured data that can support disjunction and/or negation, such as
BTwigMerge [4] and TwigStackList$\neg$ [29], are in essence the same as
TwigStack with respect to the conjunctive TPQs and hence are not included in
our experiments. TwigStackD can evaluate conjunctive TPQs over graph-
structured data. In our implementation, we fixed the problems in the original
paper [30]. HGJoin is a hash-based structural join algorithm for processing
graph pattern queries. We did not implement the query plan generation in the
original algorithm which relies on selective estimation techniques [22] and
takes exponential time in the query size; instead, for each query, we
generated all valid plans and took evaluation on each. The minimum query
processing time on the best plan is reported; thus, the time presented in this
paper is always smaller than the real time of the original HGJoin. This
version is denoted by HGJoin+. By representing intermediate results as graphs,
we have also implemented another version denoted by HGJoin*. All experiments
are performed on a 2.4GHz Intel-Core-i3 CPU with 3.7 GB RAM.
### 5.1 On XMark Data
In this set of experiments, we use large synthetic XMark data [24] to evaluate
the efficiency and scalability of various algorithms. As mentioned in Section
1, many graph-structured XML database can be modeled by a special form of
graphs consisting of trees connected by cross edges (ID/IDREF links). In this
case, we can use existing twig join algorithms to process conjunctive TPQs by
decomposing them into a set of subqueries on separative trees. We use
TwigStack and Twig2Stack to investigate the efficiency of applying this
approach.
###### Datasets.
We generated five XMark datasets with the scaling factors from 0.5 to 4. For
each dataset, we generate a graph, where nodes correspond to XML elements and
edges represent the internal links (parent-child) and ID/IDREF links. The
attribute for graph nodes is the tag of elements except for nodes
corresponding to person, item elements, for each type of which we randomly
classify them into ten groups to represent different properties. A label is
assigned to each node according to the tag or the group it belongs to.
Distinct labels indicate different attribute values. The details of the
generated documents and graphs are presented in Table 1.
Table 1: Statistics of XMark datasets Scaling factor | 0.5 | 1 | 1.5 | 2 | 4
---|---|---|---|---|---
Dataset size (MB) | 55 | 111 | 167 | 223 | 447
Nodes (Million) | 0.64 | 1.29 | 1.94 | 2.52 | 5.17
Edges (Million) | 0.77 | 1.54 | 2.32 | 3.09 | 6.20
Table 2: The average size of query results on XMark Queries | 55M | 111M | 167M | 223M | 447M
---|---|---|---|---|---
$Q_{1}$ | 368 | 762.8 | 1115.8 | 1496.8 | 2986.8
$Q_{2}$ | 34.6 | 75.8 | 117.8 | 150.3 | 297.2
$Q_{3}$ | 1.9 | 4.1 | 5.8 | 6.1 | 17.1
(a) $Q_{1}$
(b) $Q_{2}$
(c) $Q_{3}$
Figure 7: Queries for XMark data
(a) Varying data size
(b) Varying query size
Figure 8: Performance results on XMark data
###### Queries.
Three types of queries we used for experiments are depicted in Fig. 7, where
dotted edges refer to ID/IDREF links in the original data. For each query
type, we generated ten queries by randomly choosing a label for each of person
and item nodes representing a different attribute predicate. The average is
reported.
###### Experimental results.
Fig. 8(a) shows the query evaluation time for $Q_{1}$ on datasets varying the
data size. The results for $Q_{2}$ and $Q_{3}$ are quite similar. The results
reveal the following. (1) GTEA constantly outperforms all other algorithms.
Specifically, GTEA is three times to more than one order of magnitude faster
than TwigStack and Twig2Stack, five times to more than two orders of magnitude
faster than HGJoin, and in the best cases three times faster than TwigStackD.
When data size becomes larger, the performance gain by GTEA becomes more
significant. (2) TwigStackD also has very good performance in this set of
experiments with the following reasons. (a) It utilizes SSPI, a reachability
index with pretty small size and good querying time for tree-like graphs. (b)
Its basic idea is extended from the holistic twig join algorithms, and so
TwigStackD also has the advantages taken by the stack encoding and the
blocking method for path results [3]. (c) Although TwigStackD has to buffer
every nodes in pools (a special structure used to store nodes popped from
stacks) and large amounts of the operations of checking edge conditions with
all nodes in pools have to be done (indicated as reasons of inefficiency in
[27] and [11]), the pre-filtering process it uses can filter redundant nodes
and relieve the cost of the above operations. Indeed, without the pre-
filtering process, TwigStackD is slower by orders of magnitude [30]. (3) It is
sort of surprising that TwigStack has slightly better performance than
Twig2Stack. The reason is that although Twig2Stack can avoid generating path
matches (as a primary reason for the efficiency in [7]), the overhead brought
by merging stack trees and maintaining the hierarchical structures overrides
the benefits in the experiments. The fact that the depth of XMark graphs is
small (with an average of 5), also make the hierarchical stack encoding have
not a strong advantage. Besides, the enumeration of path matches (as a reason
for inefficiency for TwigStack in [7]) can be done fast using the blocking
technique. (4) HGJoin has the worst performance, mainly because (a) the
structural-join way has to generate a large number of (largely redundant)
intermediate results for small substructures and (b) non-trivial merge-join
operations on them have to be done even with the best plan. The query
processing time increases significantly when the size of data graphs
increases.
Fig. 8(b) shows the results on the XMark dataset of scale 0.5 for different
queries. (1) The query processing time of GTEA nearly maintains the same as
the query size increases. In particular, the time cost for evaluating $Q_{2}$
is smaller than that for $Q_{1}$. It is because the size of the results of
$Q_{2}$ is much smaller than that for $Q_{1}$ as presented in Table 2,
resulting in smaller cost for enumerating the final results. (2) The
processing time of TwigStack and Twig2Stack does not increase significantly
over $Q_{1}$, $Q_{2}$ and $Q_{3}$, although they have to evaluate a increasing
number of subqueries and perform a growing number of merge operations. Indeed,
as shown in Table 2, the sizes of the results of $Q_{1}$ and $Q_{2}$, which
are a subquery of $Q_{2}$ and $Q_{3}$ respectively, are small and thus the
extra cost for evaluating $Q_{2}$ and $Q_{3}$ is very limited. (3) However,
HGJoin is much more sensitive to the increase of the query size, which is due
to the impact of the redundant intermediate results and expensive sort
operations involved in performing multi-structural joins. The results for
HGJoin highlight the crucial importance of using a pruning process to reduce
the size of intermediate results not contributing to the answer.
Figure 9: Performance results on arXiv data. (a) Distribution of the result
sizes. (b) Query processing time on the queries with small sizes of results.
(c) Query processing time on the queries with small sizes of results. (d)
Comparison with the pre-filtering process.
### 5.2 On arXiv Data
In this set of experiments, we used a real-life graph to evaluate the
performance of GTEA, TwigStackD and HGJoin for general graph data, verify the
effectiveness of graph representation of intermediate results and the
efficiency of the pruning process.
###### Dataset.
We generated a graph from the HEP-Th
database111http://kdl.cs.umass.edu/data/hepth/hepth-info.html, originally
derived from the arXiv222http://arxiv.org/. There are paper nodes and author
nodes, each associated with multiple properties. For simplicity, we assigned a
label to each author node according to the email domain, and a label to each
paper node based on its area and journal it is published in, to represent the
attributes. The edges of the graph represent author or citation relationships.
The graph has 9562 nodes, 28120 edges, and 1132 distinct labels.
###### Query generator.
We designed a query generator to randomly produce meaningful queries. Each
query node is associated with a label randomly chosen from the data graph to
represent attribute predicates. Two groups of queries are generated: one has a
small size of results between 2 and 50, the other has a large size between 200
and 1200. For each group, five sets of queries were generated with query size
varying from 5 to 13. We generated fifteen different queries for each size
scale and report the average. The average time can reflect the average case
performance of each algorithm, since the queries are generated in a random
way. The results for queries of distinct sizes in the same group are
comparable, because the differences of the result sizes of the queries have
little impact on the query processing time and the number of query results for
each size scale follow a close distribution as illustrated in Fig. 9(a).
###### Experimental results.
Fig. 9(b) and (c) report the results for the two groups of queries. They tell
us the following. (1) GTEA has the best query processing time, significantly
smaller than the processing time of other algorithms (more than one order of
magnitude in most cases). It also has the best scalability in both two groups
of experiments. (2) TwigStackD no longer has good performance as on XMark
data. In fact, it has the longest querying time for queries with size of 5 to
9. The arXiv graph is much denser and deeper than XMark data, causing the
inefficiency of the pool structure as well as SSPI. The problem of TwigStackD
is highlighted by Fig. 9(c) where it fluctuates sharply for queries with large
results. The results reflect that TwigStackD has rather poor performance for
particular queries. In contrast, GTEA is most robust since it always maintains
good performance for all experiments. (3) HGJoin+ is not quite scalable
similar to the performance on the XMark data. Yet it now has better
performance than TwigStackD when the query size is smaller than 11. (4) The
revised HGJoin (i.e. HGJoin*) has better scalability than HGJoin+. For the
group of queries with large results, the query processing time of HGJoin* is
smaller than that of HGJoin+ when the query size is larger than 7, compared to
11 for the group of queries with small results. This observation demonstrates
that graph representation of intermediate results can improve the performance
and achieve better scalability especially when there are many
intermediate/final results and when the query size is large. The reason why
the revised one takes more time than the original one for processing the
queries of small sizes is that HGJoin* incurs costs for dynamically and
recursively deleting unqualified nodes (not exist in our algorithm though),
which offset the benefits taken by avoiding merge-join operations on tuples.
Fig. 9(d) evaluates the efficiency of our pruning process and the pre-
filtering algorithm in TwigStackD, which clearly shows that our pruning method
greatly outperforms the counterpart and also has better scalability with the
query size. It is because the pre-fltering algorithm in TwigStackD requires
two traversals of the data graph.
## 6 Conclusions
We have proposed the GTPQ, a new class of tree pattern queries on graph-
structured data, which incorporates structural predicates defined in terms of
propositional logic to specify structural conditions. We studied several
fundamental problems, and established a general framework for evaluating GTPQs
using a graph representation of graphs and a pruning approach. An algorithm
has been developed for evaluating GTPQs, which can achieve a small size of
intermediate results due to the effective pruning process and largely avoid
generating redundant matches by dynamically shrinking the tree pattern during
pruning and enumerating processes.
Acknowledgement. This work is supported by the National Science Foundation of
China (61075074).
## References
* [1] R. Agrawal, A. Borgida, and H. V. Jagadish. Efficient management of transitive relationships in large data and knowledge bases. In SIGMOD, 1989.
* [2] S. Amer-yahia, S. Cho, L. V. Lakshmanan, and D. Srivastava. Minimization of tree pattern queries. In SIGMOD, 2001.
* [3] N. Bruno, N. Koudas, and D. Srivastava. Holistic twig joins: optimal XML pattern matching. In SIGMOD, 2002.
* [4] D. Che, T. Ling, and W. Hou. Holistic boolean-twig pattern matching for efficient XML query processing. TKDE, PP(99):1, 2011.
* [5] D. Chen and C.-Y. Chan. Minimization of tree pattern queries with constraints. In SIGMOD, 2008.
* [6] L. Chen, A. Gupta, and M. E. Kurul. Stack-based algorithms for pattern matching on DAGs. In VLDB, 2005.
* [7] S. Chen, H.-G. Li, J. Tatemura, W.-P. Hsiung, D. Agrawal, and K. S. Candan. Twig2stack: bottom-up processing of generalized tree pattern queries over XML documents. In VLDB, 2006.
* [8] Y. Chen and Y. Chen. An efficient algorithm for answering graph reachability queries. In ICDE, 2008.
* [9] Z. Chen, H. V. Jagadish, L. V. S. Lakshmanan, and S. Paparizos. From tree patterns to generalized tree patterns: on efficient evaluation of XQuery. In VLDB, 2003.
* [10] Z. Chen, B. Xu, and D. Ding. The complexity of variable minimal formulas. Chinese Science Bulletin, 55(18):1957–1960, 2010.
* [11] J. Cheng, J. X. Yu, and P. S. Yu. Graph pattern matching: A join/semijoin approach. TKDE, 23:1006–1021, 2011.
* [12] E. Cohen, E. Halperin, H. Kaplan, and U. Zwick. Reachability and distance queries via 2-hop labels. In SODA, 2002.
* [13] W. Fan, J. Li, S. Ma, N. Tang, Y. Wu, and Y. Wu. Graph pattern matching: from intractable to polynomial time. PVLDB, 3(1):264–275, 2010.
* [14] G. Gou and R. Chirkova. Efficiently querying large XML data repositories: A survey. TKDE, 19(10):1381–1403, October 2007.
* [15] H. He and A. K. Singh. Graphs-at-a-time: query language and access methods for graph databases. In SIGMOD, 2008.
* [16] H. Jiang, H. Lu, and W. Wang. Efficient processing of XML twig queries with or-predicates. In SIGMOD, 2004.
* [17] E. Jiao, T. W. Ling, and C. yong Chan. Pathstack$\neg$: A holistic path join algorithm for path query with not-predicates on XML data. In DASFAA, 2005.
* [18] R. Jin, N. Ruan, Y. Xiang, and H. Wang. Path-tree: An efficient reachability indexing scheme for large directed graphs. TODS, 36(1), 2011.
* [19] R. Jin, Y. Xiang, N. Ruan, and D. Fuhry. 3-hop: a high-compression indexing scheme for reachability query. In SIGMOD, 2009.
* [20] J. Lu, T. W. Ling, Z. Bao, and C. Wang. Extended XML tree pattern matching: Theories and algorithms. TKDE, 23(3):402 –416, 2011.
* [21] J. Lu, T. W. Ling, C.-Y. Chan, and T. Chen. From region encoding to extended dewey: on efficient processing of XML twig pattern matching. In VLDB, 2005.
* [22] N. Polyzotis, M. Garofalakis, and Y. Ioannidis. Selectivity estimation for XML twigs. In ICDE, 2004.
* [23] P. Ramanan. Efficient algorithms for minimizing tree pattern queries. In SIGMOD, 2002.
* [24] A. Schmidt, F. Waas, M. Kersten, M. J. Carey, I. Manolescu, and R. Busse. Xmark: a benchmark for XML data management. In VLDB, 2002.
* [25] M. Shalem and Z. Bar-Yossef. The space complexity of processing XML twig queries over indexed documents. In ICDE, 2008.
* [26] H. Wang, H. He, J. Yang, P. Yu, and J. Yu. Dual labeling: Answering graph reachability queries in constant time. In ICDE, 2006.
* [27] H. Wang, J. Li, J. Luo, and H. Gao. Hash-base subgraph query processing method for graph-structured XML documents. PVLDB, 1(1):478–489, 2008.
* [28] H. Wang, S. Park, W. Fan, and P. S. Yu. Vist: a dynamic index method for querying XML data by tree structures. In SIGMOD, 2003.
* [29] T. Yu, T. Ling, and J. Lu. Twigstacklist$\neg$: A holistic twig join algorithm for twig query with not-predicates on XML data. In DASFAA, 2006.
* [30] Q. Zeng and H. Zhuge. Comments on "stack-based algorithms for pattern matching on DAGs". PVLDB, 5(7):668–679, 2012.
* [31] L. Zou, L. Chen, and M. T. Özsu. Distance-join: pattern match query in a large graph database. PVLDB, 2(1):886–897, 2009.
## Appendix A XQuery Example
$Q_{1}$ in Example 1 can be expressed in XQuery:
let | $\textdollar$dblp $:=$ doc(dblp.xml)
---|---
for | $\textdollar$paper in $\textdollar$dblp//inproceedings,
| $\textdollar$conf in $\textdollar$dblp//proceedings
where | $\textdollar$paper/author $=$ “Alice” and $\textdollar$paper/author $=$ “Bob” and
| $\textdollar$paper/crossref $=$ $\textdollar$conf/@key and
data($\textdollar$conf/year) $\geq$ 2000 and
| data($\textdollar$conf/year) $\leq$ 2010
return
| if (exists($\textdollar$paper/year) and exists($\textdollar$conf/title))
| then | $<$paper$>$
| | $<$title$>\\{\textdollar$paper/title$\\}$$<$/title$>$
| | $<$year$>\\{\textdollar$paper/year$\\}$$<$/year$>$
| | $<$conf$>\\{\textdollar$conf/title$\\}$$<$/conf$>$
| | $<$/paper$>$
## Appendix B Proofs
###### Proof Sketch of Theorem 1.
Given a GTPQ $Q$, we can safely remove two kinds of nodes as well as their
descendants without changing the satisfiability: the nodes whose attribute
predicates are unsatisfiable and those non-independently constraint nodes. We
next only consider the case where there does not exist such two kinds of
nodes. We prove that $Q$ is satisfiable, iff for the root node $u_{r}$,
$f_{cs}(u_{r})$ is satisfiable.
(1) $\rightarrow:$ Suppose $G$ is such a data graph that $Q(G)$ is non-empty.
Let $C$ be a certificate and $T$ be the corresponding truth assignment $T$ on
variables in structural predicates: For a query node $u$, if there exists a
data node $v$ such that $\in C$ and $v\models u$, $p^{T}_{v}:=1$, otherwise
$p^{T}_{v}:=0$.
By the definition of semantics, if $p^{T}_{v}=1$, $f^{T}_{ext}=1$; thus,
$f^{T}_{tr}=1$. For each clause $(\neg p_{u_{1}}\vee(p_{u_{2}}\wedge
f_{ext}(p_{u_{2}}))$ in $f_{cs}(u_{r})$, because $u_{2}\trianglelefteq u_{1}$,
$p_{u_{1}}\rightarrow p_{u_{2}}$ and $f_{ext}(u_{1})\rightarrow
f_{ext}(u_{2})$ hold; thus, $(\neg p_{u_{1}}\vee(p_{u_{2}}\wedge
f_{ext}(p_{u_{2}}))$ is true. Therefore, $f^{T}_{cs}(u_{r})=1$.
(2)$\leftarrow:$ Suppose $T$ is a satisfying truth assignment of
$f_{cs}(u_{r})$. We initialize a data graph $G=(V,E,f)$ as follows.
1. (a)
For each variable $p_{u_{i}}$ in $f_{cs}(u_{r})$ such that $(p_{u_{i}}\wedge
f_{ext}(u_{i}))^{T}=1$, add a node $v_{i}$ to $G$.
2. (b)
Add an edge $(v_{i},v_{j})$ to $G$, iff $(u_{i},u_{j})$ is an edge in $Q$.
3. (c)
For each node $v_{i}$, choose $f(v_{i})$ such that $f(v_{i})$ satisfies
$f_{a}(u_{i})$.
We simulate the process of evaluating $Q$ on $G$ and denote the truth
assignment in the evaluation by $T^{\prime}$. We assign a truth value to each
node variable in a bottom-up process according to the semantics of GTPQ and at
the same time modify $G$ if necessary to make $V$ as a certificate.
For any query node $u_{i}$, if $p^{T}_{u_{i}}=0$ and
$p^{T^{\prime}}_{u_{i}}=1$, it can be inferred that there exists $v_{j}\in G$
such that $f(v_{j})$ satisfies $f_{a}(u_{i})$. If $f_{a}(u_{j})\not\rightarrow
f_{a}(u_{i})$, we change $f(v_{j})$ so that $f(v_{j})$ satisfies
$f_{a}(u_{j})$, but does not satisfy $f_{a}(u_{i})$, leading to
$p^{T^{\prime}}_{u_{i}}=0$.
We next prove by contradiction that after the above processing, if
$p^{T}_{u}=1$, $p^{T^{\prime}}_{u}=1$. Assume one node at the largest depth,
for which $p^{T}_{u}=1$ and $p^{T^{\prime}}_{u}=0$, is $u$. By assumption, for
any descendant $u_{d}$ of $u$, if $p^{T}_{u_{d}}=1$,
$p^{T^{\prime}}_{u_{d}}=1$. So there must be a child $u^{\prime}$ of $u$ for
which $p^{T}_{u^{\prime}}=0$ and $p^{T^{\prime}}_{u^{\prime}}=1$. From the way
$G$ is constructed, there is a mapping from $u^{\prime}$ and its descendants
to another descendant $u^{\prime\prime}$ of $u$ and its descendant such that
$u^{\prime}\trianglelefteq u^{\prime\prime}$ and $p^{T}(u^{\prime\prime})=1$.
However, since $p^{T}(f_{cs}(u_{r}))=1$, if $p^{T}(u^{\prime\prime})=1$,
$p^{T}(u^{\prime})=1$, which is contradictory to our assumption.
For each backbone node $u$, $(p_{u}\wedge
f_{ext}(u))^{T^{\prime}}=(p_{u}\wedge f_{ext}(u))^{T}=1$. So each output node
has a non-empty image in $V$ and those images constitute an answer to $Q$. ∎
###### Proof Sketch of Theorem 2.
Since attribute predicates are conjunctive, the satisfiability of them can be
determined in linear time. We assume in the following that all attribute
predicates are satisfiable.
(1) A union-conjunctive GTPQ where all attribute predicates are satisfiable is
always satisfiable.
(2) We prove that the satisfiability problem of a general GTPQ is NP-Complete
by a reduction from SAT.
Given any instance $\phi$ of SAT, we suppose $\phi$ has $n$ variables and
construct a GTPQ $Q$ with $n+1$ nodes as follows. (a) First, choose the first
$n$ nodes, each $v_{i}$ corresponding to a distinct variable $x_{i}$ in
$\phi$. Then, construct an edge from the $(n+1)$-th node to each of them. (b)
Each node is associated with a satisfiable attribute predicate with a distinct
attribute variable. The structural predicate of the root is $\phi$, with
$p_{v_{i}}$ replacing $x_{i}$ for each non-leaf node $v_{i}$. (c) The root
node, denoted by $u_{r}$, is the only output node.
Since $f_{cs}(u_{r})=f_{s}(u_{r})=\phi$, $f_{cs}(u_{r})$ is satisfiable iff
$\phi$ is satisfiable. By Theorem 1, the conclusion that $Q$ is satisfiable
iff $\phi$ is satisfiable immediately follows.
It is easy to check that the reduction takes linear time and the
satisfiability is in NP. ∎
###### Proof Sketch of Theorem 3.
(1) $\rightarrow.$ According to the truth table of the complete structural
predicate of the root node of $Q_{1}$, we can enumerate all (potentially
exponential) combinations of query nodes of $Q_{1}$ such that for each
combination, there exists a bijection $\lambda$ from a certificate to it.
Informally, for each combination as a GTPQ, we can construct a data graph $G$
from a satisfying truth assignment in the way we use in the proof of Theorem
1, so that the data nodes constitute a certificate. By assumption, $G$ is also
a certificate with respect to $Q_{2}$, and there is a mapping
$\lambda^{\prime}$ from $Q_{2}$ to the certificate. Further,
$\lambda^{\prime}\circ\lambda^{-1}$ is a mapping from $Q_{2}$ to $Q_{1}$
satisfying the first three conditions in the definition of homomorphism.
Finally, a homomorphism can be derived from all such mappings with respect to
the combinations.
(2) $\leftarrow.$ For the opposite direction, suppose there is a homomorphism
$\lambda$ from $Q_{2}$ to $Q_{1}$. Let $G$ be a data graph, on which the
answer of $Q_{1}$ is not empty. Suppose $res\in Q_{1}(G)$ and $C$ is a
corresponding certificate with the truth assignment denoted by $T$. It is
clear that $C$ is a certificate of $Q_{2}$ with a truth assignment
$T^{\prime}$ such that (a) $p^{T^{\prime}}_{u}=1$, iff $p^{T}_{\lambda(u)}=1$;
(2) $f^{T^{\prime}}_{cs}(u_{r})=1$ for the root $u_{r}$. ∎
###### Proof Sketch of Theorem 4.
The proof is based on a reduction from the tautology checking problem (TCP) of
propositional formulas to the containment problem of GTPQs by constructing a
GTPQ from an instance of TCP using the same technique in the proof of Theorem
2. ∎
###### Proof Sketch of Theorem 6.
The proof is based on a reduction from the variable minimal equivalence
problem (VME) [10] in propositional logic of propositional formulas to the
decision version of the minimization problem of GTPQs by constructing a GTPQ
from an instance of VME using the same technique in the proof of Theorem 2. ∎
## Appendix C Additional Experimental
Results
### C.1 Measuring I/O cost
We measure the I/O cost of each algorithm in terms of three metrics, namely
the number of data nodes accessed (#input), the number of index elements
looked up (#index), and the size of intermediate results
(#intermediate_results).
Regarding the number of index lookups, the value for GTEA is the total number
of elements retrieved from successor and predecessor lists in 3-hop index; the
value for HGJoin is the total number of ids and interval lables in tag lists
(called Alist and Dlist in [27]); the value for TwigStackD is the total number
of surrogate and surplus predecessors visited in SSPI. Since TwigStack and
Twig2Stack do not use a graph reachability index, they have no such cost.
The cost of intermediate results for each algorithm is computed as follows.
(1) The value for GTEA is twice the total number of the nodes and edges of the
maximal matching graph. (2) The values for HGJoin, TwigStack and Twig2Stack
include the cost of intermediate results for subqueries in the form of tuples.
(3) In addition, TwigStack and Twig2Stack also involves the space cost of
stack encoding. (4) Apart from the cost of stack encoding, TwigStackD
introduces the space cost of pool encoding. It is necessary to clarify that in
our experiments, all intermediate results are maintained in main memory and
not stored on disk. This metric is to evaluate the worst-case I/O cost caused
by the intermediate results. When measuring this cost, we assume that any
intermediate result is written to disk and read back to main memory when
needed.
Figure 10: I/O cost
Fig. 10 depicts the experimental results for processing $Q_{3}$ on the XMark
dataset with scale factor 1.5. The detailed costs are reported above columns.
Note that TwigStack and Twig2Stack involve exactly the same I/O cost.
From the results, TwigStack and Twig2Stack read the smallest number of data
nodes. They only need to scan those data nodes corresponding to all query
nodes for once. In comparison, GTEA accesses more, because it needs to perform
a two-round pruning process (bottom-up and top-down). The value, however, is
bounded by two times of that of TwigStack. As HGJoin splits a query to subtree
queries and the different subqueries have identical query nodes, HGJoin also
accesses some data nodes for more than once, with a bound of the maximum
number of children of any node in the tree pattern. TwigStackD reads far more
data nodes than others in the experiments, resulting from the two traversals
of the data graph in the pre-filtering process.
The results clearly show that GTEA creates much fewer intermediate results
than all other four algorithms. TwigStack and Twig2S-tack have more
intermediate results than GTEA by four orders of magnitude. The huge gap
results from the fact that TwigStack and Twig2Stack need to output a large
number of intermediate path and twig solutions to each subtree query which is
far less selective than the whole query. The structural joins adopted by
HGJoin also introduce many partial solutions and lead to a large size of
intermediate results as shown in the figure. For TwigStackD, its pre-filtering
process selects nodes potentially in the final answers and considerably saves
the space cost of stacks and pools. GTEA shows the best performance, as it can
prune non-answer nodes as TwigStackD and represent the intermediate results as
a maximal matching graph.
Fig. 10 shows that GTEA again outperforms HGJoin, due to the compact 3-hop
index and the effectiveness of the merging operations in the pruning process.
Yet GTEA incurs more cost for looking up indexes than TwigStackD. GTEA uses
the 3-hop index in the two-round pruning process and when constructing the
maximal matching graph, while TwigStackD looks up the reachability index only
when expanding the partial solutions in pools. However, the small cost
achieved by TwigStackD is at the expense of the large I/O cost for scanning
data nodes in the pre-filtering process which significantly reducing the
number of nodes to be processed in the stacks and pools. Moreover, since
indexes are often kept mostly in main memory, the difference in the number of
disk I/O’s needed for GTEA and TwigStackD to support the index lookup is
supposed to be actually small.
Overall, GTEA achieves good performance gain over other competitors in terms
of I/O cost. The results indicate that the pruning process does not incur high
I/O cost as TwigStackD and the graph representation can keep the space cost of
intermediate results pretty small.
Figure 11: The tree structure of tested queries
### C.2 GTPQ Processing
Table 3: The output nodes of the queries in Exp-1 $Q_{4}$ | open_auction
---|---
$Q_{5}$ | open_auction, bidder, seller
$Q_{6}$ | open_auction, bidder, seller, city, profile
$Q_{7}$ | open_auction, item, location
$Q_{8}$ | all query nodes
Table 4: The structural predicates of the queries in Exp-2 DIS1 | $f_{s}(\textrm{open\\_auction})=\textrm{bidder}\vee\textrm{seller}$
---|---
DIS2 | $f_{s}(\textrm{open\\_auction})=\textrm{bidder}\vee\textrm{seller}$
$f_{s}(\textrm{item})=\textrm{mailbox}\vee\textrm{location}$
DIS3 | $f_{s}(\textrm{open\\_auction})=\textrm{bidder}\vee\textrm{seller}\vee\textrm{item}$
NEG1 | $f_{s}(\textrm{person})=\neg\textrm{education}$
NEG2 | $f_{s}(\textrm{open\\_auction})=\neg\textrm{bidder},f_{s}(\textrm{person})=\neg\textrm{education}$
NEG3 | $f_{s}(\textrm{open\\_auction})=\neg\textrm{bidder}\wedge\neg\textrm{seller}$
$f_{s}(\textrm{person})=\neg\textrm{education}$
DIS_NEG1 | $f_{s}(\textrm{open\\_auction})=\neg\textrm{bidder}\vee\textrm{seller}$
$f_{s}(\textrm{person})=\neg\textrm{education}$
DIS_NEG2 | $f_{s}(\textrm{open\\_auction})=(\neg\textrm{bidder}\wedge\textrm{seller})\vee(\textrm{bidder}\wedge\neg\textrm{seller})$
DIS_NEG3 | $f_{s}(\textrm{open\\_auction})=(\neg\textrm{bidder}\wedge\textrm{seller})\vee(\textrm{bidder}\wedge\neg\textrm{seller})$
$f_{s}(\textrm{person})=\neg\textrm{education}$
DIS_NEG4 | $f_{s}(\textrm{open\\_auction})=$
$(\neg\textrm{bidder}\wedge\textrm{seller}\wedge\textrm{item})\vee(\textrm{bidder}\wedge\neg\textrm{seller}\wedge\neg\textrm{item}),$
$f_{s}(\textrm{person})=\neg\textrm{education}$
Table 5: Numbers of query results $Q_{4}$ | $Q_{5}$ | $Q_{6}$ | $Q_{7}$ | $Q_{8}$
---|---|---|---|---
88 | 98 | 98 | 88 | 151
DIS1 | DIS2 | DIS3 | NEG1 | NEG1
1236 | 26352 | 2052 | 456 | 1938
NEG3 | DIS_NEG1 | DIS_NEG2 | DIS_NEG3 | DIS_NEG4
1240 | 4156 | 2328 | 2300 | 5643
(a) Varying the number of non-output nodes
(b) Union-conjunctive queries
(c) Queries with negation
(d) Queries with disjunction and negation
Figure 12: GTPQ Processing
In this section, we present the experimental results for GTPQs with the same
structure (Fig. 11) on the XMark data set with scale factor 4. Since HGJoin
and TwigStackD need to do the same deco-mpose-and-merge operations to process
GTPQs and our experiments for conjunctive queries have shown that TwigStackD
significantly outperforms HGJoin, we did not include HGJoin in this set of
experiments. Twig2Stack was also not included as it has comparable performance
to TwigStack and the post-process on top of the two algorithms for processing
GTPQs is also the same.
###### Exp-1 Optimization for non-output nodes.
We first compare GT-EA, TwigStack and TwigStackD for processing conjunctive
queries with varying the size of output nodes. The output nodes for each
tested query are given in Table 3. The result sizes of those queries are
presented in Table 5. Because TwigStack and TwigStackD are not optimized for
queries with non-output nodes and the differences in the result sizes of the
tested queries are small, the processing time on different queries is close to
each other for both algorithms. Fig. 12(a) depicts the results of GTEA only.
Recall that GTEA uses a prime subtree obtained based on the output nodes and
the specific matching nodes in procedure PruneDownward and constructing the
maximal matching graph for avoiding creating useless matches to non-output
nodes. Hence, the processing time of GTEA depends on the structure of the
prime subtree and the size of the final answers. The results show that the
less the number of output nodes is, the less processing time the evaluation
generally takes.
###### Exp-2 GTPQ processing.
We next show the experimental results for queries that may contain negation
and disjunction. Three classes of tested queries, namely the queries with
disjunction only (DIS), those with negation only (NEG) and with both
disjunction and negation (DIS_NEG), are shown in Table 4. All potentially
valid backbone nodes are set as output nodes for all queries. Fig. 12(b), (c)
and (d) depict the results for the tested GTPQs. All of them consistently
verify the significant performance gain of GTEA (from several times to three
orders of magnitude). Indeed, as mentioned in the related work, TwigStack and
TwigStackD need to process a number of small subqueries and do expensive post
merge-join operations for processing GTPQs. It is non-trivial to fine tune the
two algorithms for GTPQs. It may be possible to derive an efficient mechanism
that makes the intermediate results output by TwigStack and TwigStackD in
sorted order so that the merge-join operations take less cost. However, it is
difficult to reduce the large size of intermediate results which considerably
impairs the efficiency of TwigStack and TwigStackD, so they are unlikely to
outperform our algorithm anyway.
## Appendix D Processing queries with multiple output structures
GTEA can be straightforwardly extended to process queries not restricted to
backbone nodes. The only modification is in procedure CollectResults. For an
internal node in the maximal matching graph, instead of doing one Cartesian
product of the results of branches, the procedure may perform several
Cartesian products of the results of different branches depending on the
specified result structures. Take the query DIS1 (the query structure is shown
in Fig. 11 and the predicates are defined in Table 4) for example, and suppose
that the results of the query should be of the form (open_auction, bidder,
item) or (open_auction, seller, item). The (shrunk) prime subtree is
constructed by considering bidder and seller as the originally defined
backbone nodes. In the maximal matching graph, for each matching nodes of
open_auction, the CollectResults procedure performs two Cartesian products to
derive the answers: one product of the two branch results corresponding to
bidder and item, and the other product of the two branch results corresponding
to seller and item.
## Appendix E Expanded Algorithms
We show procedure PruneDownward and PruneUpward in more details in Procedure 6
and Procedure 7.
Input: 3-Hop index $L_{out}$, a GTPQ $Q$
Output: Candidate matching nodes satisfying downward structural constraints.
1 for each node $u\in V_{q}$ do $mat(u):=\\{x|x\in V,x\sim u\\}$
2
3 for each leaf node $u^{\prime}$ in $V_{q}$ do
$C^{p}_{u^{\prime}}:=\textrm{MergePredLists}(mat(u^{\prime}))$
4
5 $V^{\prime}_{q}=V_{q}\backslash\\{u^{\prime}|u^{\prime}\textrm{ is a leaf
node}\\}$
6 for each $u\in V^{\prime}_{q}$ in bottom-up order do
7 for each $v\in mat(u)$ do $chain_{v.cid}:=chain_{v.cid}\cup\\{v\\}$
8
9 for each $chain_{i}$ that is not empty do
10 for each child $u^{\prime}$ of $u$ do $val[p_{u^{\prime}}]:=0$
11
12
13 for each node $v_{i}\in chain_{i}$ do
14
15 for each child $u^{\prime}$ of $u$ _s.t._ $val[p_{u^{\prime}}]=0$ do
16 if $C^{p}_{u^{\prime}}[i]\geq v_{i}.sid$ then $val[p_{u^{\prime}}]:=1$
17
18
19 $v^{\prime}_{i}:=v_{i}$
20 repeat
21 for each index node $v^{\prime\prime}_{i}\in L_{out}(v^{\prime}_{i})$ do
22 for each child $u^{\prime}$ of $u$ _s.t._ $val[p_{u^{\prime}}]=0$ do
23 if $C^{p}_{u^{\prime}}[v^{\prime\prime}_{i}.cid]\geq
v^{\prime\prime}_{i}.sid$ then
24 $val[p_{u^{\prime}}]:=1$
25
26
27
28 $v^{\prime}_{i}:=\textrm{next}(v^{\prime}_{i})$
29
30 until $v^{\prime}_{i}$ = null or $visited_{i}\leq v^{\prime}_{i}.sid$
31 if $f_{s}(u)$ evaluates to false with the valuation $val$ then
32 $mat(u):=mat(u)\backslash\\{v_{i}\\}$
33
34
35 $visited_{i}:=v_{i}.sid$
36
37
38 $C^{p}_{u}:=\textrm{MergePredLists}(mat(u))$
39
Procedure 6 PruneDownward
Input: 3-hop index $L_{in}$, the prime subtree
$(V^{\prime}_{t},E^{\prime}_{t})$ of a GTPQ
Output: Candidate matching nodes satisfying upward structural constraints
1 $C^{s}_{u_{root}}:=\textrm{MergeSuccLists}(mat(u_{root}))$
2 $V^{\prime}_{t}:=V^{\prime}_{t}\backslash\\{u^{\prime}|u^{\prime}\textrm{ is
a leaf node}\\}$
3 for each node $u\in V^{\prime}_{t}$ from top to bottom such that
$|mat(u)|>1$ do
4 for each child $u^{\prime}$ of $u$ such that $|mat(u^{\prime})|>1$ do
5 for each node $v\in mat(u^{\prime})$ do
6 $chain^{u^{\prime}}_{v.cid}:=chain^{u^{\prime}}_{v.cid}\cup\\{v\\}$
7 $Group_{v}:=Group_{v}\cup\\{u^{\prime}\\}$
8
9
10 merge all lists $chain^{u^{\prime}}_{i}$($u^{\prime}$ is a child of $u$)
into $chain_{i}$ for each chain $i$
11 for each $chain_{i}$ that is nonempty do
12 for each node $v_{i}\in chain_{i}$ do
13 if $C^{s}_{u}[i]\leq v_{i}.sid$ then $reach:=\textrm{true}$; break
14
15 $v^{\prime}_{i}:=v_{i}$
16 repeat
17 for each index node $v^{\prime\prime}\in L_{in}(v^{\prime}_{i})$ do
18 if $C^{s}_{u}[v^{\prime\prime}.cid]\leq v^{\prime\prime}.sid$ then
19 $reach:=\textrm{true}$; break
20
21
22 if $reach=\textrm{true}$ then break
23 $v^{\prime}_{i}:=\textrm{prev}(v^{\prime}_{i})$
24
25 until $v^{\prime}_{i}$ = null or $visited_{i}\geq v^{\prime}_{i}.sid$
26 if $reach=\textrm{false}$ then
27 for each $u^{\prime}\in Group_{v_{i}}$ do
28 $mat[u^{\prime}]:=mat[u^{\prime}]\backslash\\{v_{i}\\}$
29
30
31 else break
32 $visited_{i}:=v_{i}.sid$
33
34
35 for each non-leaf child $u^{\prime}$ of $u$ do
36 $C^{s}_{u^{\prime}}:=\textrm{MergeSuccLists}(mat(u^{\prime}))$
37
38
Procedure 7 PruneUpward
|
arxiv-papers
| 2011-09-20T13:30:32 |
2024-09-04T02:49:22.327150
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qiang Zeng and Xiaorui Jiang and Hai Zhuge",
"submitter": "Qiang Zeng",
"url": "https://arxiv.org/abs/1109.4288"
}
|
1109.4330
|
# Merging Criteria for Giant Impacts of Protoplanets
H. Genda Department of Earth and Planetary Science, The University of Tokyo,
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan genda@eps.s.u-tokyo.ac.jp E. Kokubo
Division of Theoretical Astronomy, National Astronomical Observatory of Japan,
Osawa, Mitaka, Tokyo 181-8588, Japan S. Ida Earth and Planetary Sciences,
Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan
###### Abstract
At the final stage of terrestrial planet formation, known as the giant impact
stage, a few tens of Mars-sized protoplanets collide with one another to form
terrestrial planets. Almost all previous studies on the orbital and
accretional evolution of protoplanets in this stage have been based on the
assumption of perfect accretion, where two colliding protoplanets always
merge. However, recent impact simulations have shown that collisions among
protoplanets are not always merging events, that is, two colliding
protoplanets sometimes move apart after the collision (hit-and-run collision).
As a first step towards studying the effects of such imperfect accretion of
protoplanets on terrestrial planet formation, we investigated the merging
criteria for collisions of rocky protoplanets. Using the smoothed particle
hydrodynamic (SPH) method, we performed more than 1000 simulations of giant
impacts with various parameter sets, such as the mass ratio of protoplanets,
$\gamma$, the total mass of two protoplanets, $M_{\rm T}$, the impact angle,
$\theta$, and the impact velocity, $v_{\rm imp}$. We investigated the critical
impact velocity, $v_{\rm cr}$, at the transition between merging and hit-and-
run collisions. We found that the normalized critical impact velocity, $v_{\rm
cr}/v_{\rm esc}$, depends on $\gamma$ and $\theta$, but does not depend on
$M_{\rm T}$, where $v_{\rm esc}$ is the two-body escape velocity. We derived a
simple formula for $v_{\rm cr}/v_{\rm esc}$ as a function of $\gamma$ and
$\theta$ (Eq. (16)), and applied it to the giant impact events obtained by
N-body calculations in the previous studies. We found that 40% of these events
should not be merging events.
accretion, accretion disk — planets and satellites: formation — solar system:
formation
††slugcomment: Accepted for publication in ApJ
## 1 Introduction
Planets are formed in a disk around a star called a protoplanetary disk, which
is composed of gas and dust. Terrestrial planets are formed mainly from the
dust component. Their formation process can be divided into three stages. The
first stage is the formation of a large number of kilometer sized bodies
called planetesimals by accretion among dust particles (e.g., Goldreich &
Ward, 1973; Youdin & Shu, 2002). In the second stage, these planetesimals
collide to produce a few tens of Mars-sized objects called protoplanets (e.g.,
Wetherill, 1985; Kokubo & Ida, 1998). The final stage is the formation of
terrestrial planets from protoplanets (e.g., Chambers & Wetherill, 1998; Agnor
et al., 1999). The collisions among protoplanets are referred to as giant
impacts, and thus this final stage is known as the giant impact stage.
Giant impacts have a large influence on the various features such as the
number of terrestrial planets formed, their mass and spin state (e.g., Agnor
et al., 1999; Kokubo et al., 2006). Giant impacts are highly energetic events,
and are responsible for the creation of large satellites, like the Moon (e.g.,
Canup, 2004a) and planets with extremely large cores such as Mercury (e.g.,
Benz et al., 2007). Moreover, giant impacts are closely related to the thermal
state such as a magma ocean (e.g., Tonks & Melosh, 1992), and the origins of
the terrestrial planet atmospheres (Genda & Abe, 2005).
A large number of simulations of giant impacts have been devoted to the
specific giant impact events related to the origin of the Moon or Mercury.
However, since it is recently believed that multiple giant impacts are common
during the last stage of terrestrial planet formation, several studies (Agnor
& Asphaug, 2004a; Asphaug, 2009; Marcus et al., 2009, 2010) have investigated
the giant impact simulations under various impact parameters. Agnor & Asphaug
(2004a) were the first to show that collisions of protoplanets during the
giant impact stage are not always merging events, that is, two colliding
protoplanets sometimes move apart after the collision. They called such a
collision a hit-and-run collision. Except for Kokubo & Genda (2010), all the
previous studies on the orbital and accretional evolution of protoplanets
during the giant impact stage have been based on the assumption of perfect
accretion, where two colliding protoplanets always merge. However, the hit-
and-run collisions demonstrated by Agnor & Asphaug (2004a) may have an
important influence on many of the physical characteristics of terrestrial
planets.
In order to investigate the effects of such imperfect accretion of
protoplanets on terrestrial planet formation, the merging criteria during
protoplanet collisions must be clarified. Agnor & Asphaug (2004a) performed 48
simulations of the collisions between same-sized protoplanets with masses of
$0.1M_{\oplus}$, where $M_{\oplus}$ is the Earth mass. They found that hit-
and-run collisions occurred when $v_{\rm imp}\geq 1.5v_{\rm esc}$ for
$\theta=30^{\circ}$, and $v_{\rm imp}\geq 1.2v_{\rm esc}$ for
$\theta=45^{\circ}$ or $60^{\circ}$, where $v_{\rm imp}$ is the impact
velocity, $v_{\rm esc}$ is the two-body escape velocity (equation [8]), and
$\theta$ is the impact angle. Since they varied $v_{\rm imp}$ in steps of
$0.1v_{\rm esc}$ for low-velocity collisions, the transition between the
merging and hit-and-run collisions was estimated to be $v_{\rm
imp}=1.4-1.5v_{\rm esc}$ for $\theta=30^{\circ}$, and $v_{\rm
imp}=1.1-1.2v_{\rm esc}$ for $\theta=45^{\circ}$ or $60^{\circ}$.
Subsequently, Agnor & Asphaug (2004b) investigated collisions between
different-sized protoplanets with mass ratios of 1:2 and 1:10. Their results
were presented in Asphaug (2009). The transition was able to be estimated as
follows. In the case of a 1:2 mass ratio, the transition occurs at $v_{\rm
imp}=1.4-1.5v_{\rm esc}$ for $\theta=30^{\circ}$ and $v_{\rm
imp}=1.1-1.2v_{\rm esc}$ for $\theta=45^{\circ}$ or $60^{\circ}$, whereas in a
case of 1:10 mass ratio, $v_{\rm imp}=1.5-2.0v_{\rm esc}$ for
$\theta=30^{\circ}$, $v_{\rm imp}=1.2-1.3v_{\rm esc}$ for $\theta=45^{\circ}$,
and $v_{\rm imp}=1.0-1.1v_{\rm esc}$ for $\theta=60^{\circ}$.
Marcus et al. (2009) and Marcus et al. (2010) investigated collisions of the
rocky and icy super-Earths (up to $10M_{\oplus}$), respectively. They focused
on the stripping of the rocky or icy mantle resulting from a high-velocity
impact. Although they did not perform the various simulations with low-impact
velocities, we were able to find that the transition occurs at roughly $v_{\rm
imp}=1.0-1.5v_{\rm esc}$ from their figures.
As described above, the merging criteria has already been roughly determined
for certain discrete values of the impact parameters. However, in order to
carry out N-body orbital calculations with the merging criteria for the giant
impact stage, a simple formula describing the dependence of the merging
criteria on the impact parameters is required. To achieve this, it is
necessary to determine the merging criteria over a wide range of impact
parameters. In the present study, we performed more than 1000 simulations of
giant impacts for various impact parameter sets using the smoothed particle
hydrodynamic (SPH) method in order to formulate the merging criteria.
In Section 2, we present the SPH code and initial conditions used in our giant
impact simulations. In Section 3, we show the collision outcomes, and
investigate the transition between merging and hit-and-run collisions. We also
perform a resolution test on the simulations. In Section 4, we derive the
merging criteria as a function of the impact parameters, and compare the
results with those of previous studies. Using the derived criteria, we then
discuss the merging probability of protoplanets during the giant impact stage
in Section 5.
## 2 Calculation Method
### 2.1 Numerical Code
In order to perform impact simulations for protoplanets, we used the SPH
method (e.g., Monaghan, 1992), which is a flexible Lagrangian method of
solving hydrodynamic equations, and has been widely used in previous giant
impact simulations. The SPH method can easily deal with large deformations and
shock waves. Our numerical code is based on Canup (2004b); here, we briefly
describe its essential points.
The equation of the motion for the i-th SPH particle is given by
$\frac{d\mathbf{v}_{i}}{dt}=-\sum^{\rm neighbor}_{j}\mathcal{F}_{ij}-\sum^{\rm
all}_{j}\mathcal{G}_{ij},$ (1)
where $\mathbf{v}_{i}$ is the velocity of the i-th SPH particle, $t$ is the
time, and $\mathcal{F}_{ij}$ and $\mathcal{G}_{ij}$ are the pressure gradient
and mutual gravity terms between the i-th and j-th particles, respectively.
Several forms have been used for the pressure gradient term, none of which
appear to be clearly superior to the others. In this paper, we use the
following symmetric expression,
$\mathcal{F}_{ij}=m_{j}\left(\frac{P_{i}}{\rho_{i}^{2}}+\frac{P_{j}}{\rho_{j}^{2}}+\Pi_{ij}\right)\nabla_{i}W(r_{ij},h_{ij}),$
(2)
where $m_{j}$, $P_{j}$, and $\rho_{j}$ are the mass, pressure, and density of
the j-th particle, respectively, $\Pi_{ij}$ is the artificial viscosity, $W$
is the kernel function, $r_{ij}$ is the distance between the i-th and j-th
particles, and $h_{ij}$ is the average smoothing length of the i-th and j-th
particles. For the artificial viscosity $\Pi_{ij}$, we use a Von Neumann-
Richtmyer-type viscosity with parameters of $\alpha_{\rm vis}=1.5$ and
$\beta_{\rm vis}=3.0$, as described in Monaghan (1992). For the kernel
function $W$, we use the spherically symmetric spline kernel function proposed
by Monaghan & Lattanzio (1985):
$W(r,h)=\frac{1}{\pi
h^{3}}\left\\{\begin{array}[]{ll}1-\frac{3}{2}(\frac{r}{h})^{2}+\frac{3}{4}(\frac{r}{h})^{3},&0\leq\frac{r}{h}<1,\\\
\frac{1}{4}(2-\frac{r}{h})^{3},&1\leq\frac{r}{h}<2,\\\
0,&2\leq\frac{r}{h}.\end{array}\right.$ (3)
This function satisfies $\int W(|\mathbf{r}|,h)d\mathbf{r}=1$, and has a zero
value when $r\geq 2h$. In our code, $h$ is variable for each particle and with
time, and determined to satisfy the condition that the number of neighboring
particles ($N_{\rm nei}$) within $2h$ is almost constant, $N_{\rm nei}=64\pm
2$. We used a maximum value of the smoothing length ($h_{\rm max}$) to save
computational cost when searching for neighboring particles and calculating
the pressure gradient term. The value of $h_{\rm max}$ is determined from
$\frac{m}{\rho_{\rm min}}=\frac{4\pi}{3}(2h_{\rm max})^{3},$ (4)
where $\rho_{\rm min}$ is the minimum density, which is set to $5\times
10^{-3}~{}\mathrm{kg/m^{3}}$ in our simulations.
The mutual gravity term in equation (1) can be written as
$\mathcal{G}_{ij}=G\sum_{j}\hat{m}_{j}\frac{\mathbf{r}_{i}-\mathbf{r}_{j}}{r_{ij}^{3}},$
(5)
where $G$ is the gravitational constant, and $\hat{m}_{j}$ is the effective
mass of the j-th particle toward the i-th particle defined by
$\hat{m}_{j}=\int_{0}^{r_{ij}}4\pi r^{2}m_{j}W(r,h_{j})dr.$ (6)
This equation gives $\hat{m}_{j}=m_{j}$ when $r_{ij}\geq 2h_{j}$.
The mutual gravity term between all SPH particles was directly computed using
a special-purpose computer for gravitational N-body systems named GRAPE-6A
(Fukushige et al., 2005). The GRAPE-6A can search for and produce lists of
neighboring particles while simultaneously calculating their mutual gravity.
This list of neighbors is used to compute the pressure gradient term in the
equation of motion and the time derivative of the internal energy. In a
simulation over a period of $10^{5}$ sec with 20,000 SPH particles, the
typical CPU time was about 4 hours. Thus, the GRAPE-6A allowed us to
systematically explore a wide range of impact parameters. Time integration was
performed using a PEC (predict, evaluate, and correct) scheme with variable
time steps (e.g., Serna et al., 1996), which is second-order accurate in time.
### 2.2 Pre-impact Protoplanets
Here we describe the modeling method for the pre-impact protoplanets. All the
protoplanets are assumed to be differentiated, with a 30% iron core and 70%
silicate mantle by mass. In our SPH simulations, we used the Tillotson
equation of state (Tillotson, 1962), which has been widely applied to giant
impact simulations involving shock waves (e.g., Benz et al., 1987; Canup &
Asphaug, 2001; Agnor & Asphaug, 2004a; Asphaug et al., 2006). The Tillotson
equation of state contains ten material parameters, and the pressure is
expressed as a function of the density and the specific internal energy, which
is convenient for treating fluid dynamics. We used the parameter sets of
granite for the silicate mantle and iron for the iron core, which are listed
on page 234 of Melosh (1989).
All SPH particles in the protoplanets was set to have the same mass, and the
total number of particles used for impact simulations was fixed at 20,000. For
example, in the case of a collision of protoplanets with a mass ratio of 1:9,
the smaller protoplanet consisted of 2,000 particles, and the larger one
18,000 particles. To model the pre-impact protoplanets, we placed the SPH
particles in a 3D lattice (face-centered cubic) with iron particles on the
inside, and rocky particles outside. The internal energy of the SPH particles
was set to $1.0\times 10^{6}$ J/kg.
Beginning with this configuration, we calculated vibrations of the protoplanet
until the particle velocities become slower than 100 m/s, which is much less
than the impact velocity (order of km/s). After this operation, we used these
relaxed objects as the protoplanets for impact simulation. As a first step,
the protoplanets were assumed to have no spin.
In order to set impact parameters such as the impact velocity ($v_{\rm imp}$)
and impact angle ($\theta$), we need to determine the radius of the pre-impact
protoplanets. Since the surface boundary of the protoplanets described by the
SPH particles is obscure owing to the smoothing length ($h$), we determined
the radius of the pre-impact protoplanet ($R_{\rm p}$) by the following
equation.
$\frac{4\pi}{3}R_{\rm p}^{3}=\sum_{i}\frac{m_{i}}{\rho_{i}}.$ (7)
### 2.3 Initial Conditions for Collisions
We prepared more than 1000 sets of initial conditions for the giant impact
simulations. The parameters used were the mass ratio of the protoplanets
($\gamma=M_{\rm i}/M_{\rm t}$, where $M_{\rm t}$ and $M_{\rm i}$ are the mass
of the target and impactor, respectively), the total mass of the two
protoplanets ($M_{\rm T}=M_{\rm i}+M_{\rm t}$), the impact angle ($\theta$),
and the impact velocity ($v_{\rm imp}$). We systematically varied the mass
ratio as $\gamma=$ 1, 2/3, 1/2, 1/3, 1/4, 1/6, and 1/9. For $\gamma=$ 1, 1/4,
and 1/9, we considered three different values of $M_{\rm T}$. In total, we
used 13 different combinations for the two colliding protoplanets (see table
1). For each mass combination, we varied the impact angle in the range
$\theta=0^{\circ}-75^{\circ}$ in $15^{\circ}$ steps, and the impact velocity
in the range $v_{\rm imp}=1.0-3.0v_{\rm esc}$ in $0.2v_{\rm esc}$ steps, where
$v_{\rm esc}$ is the two-body escape velocity defined as
$v_{\rm esc}=\sqrt{\frac{2GM_{\rm T}}{R_{\rm t}+R_{\rm i}}},$ (8)
where $R_{\rm t}$ and $R_{\rm i}$ are the radius of the target (larger
protoplanet) and the impactor (smaller protoplanet), respectively. To
precisely determine the transition between merging and hit-and-run collisions,
we varied $v_{\rm imp}$ with a smaller step size of 0.02 $v_{\rm esc}$ near
the transition. In total, we performed more than 1000 runs, consisting of 13
(mass combinations) $\times$ 6 (angles) $\times$ 16 (11 runs with 0.2$v_{\rm
esc}$ steps and $\sim$ 5 runs with 0.02$v_{\rm esc}$ steps).
The impact parameters $v_{\rm imp}$ and $\theta$ are defined when the two
protoplanets are in contact with each other (see Figure 1). We assumed that
the two protoplanets are mass points, and calculated backward the positions of
two mass points until their distance was apart at a distance of 3 ($R_{\rm
i}$+$R_{\rm t}$). Then, we performed the giant impact simulations over a
period of $10^{5}$ sec.
## 3 Collision Outcomes
### 3.1 Merging and Hit-and-Run Collisions
The outcomes of the collisions between the protoplanets are divided into two
types: merging and hit-and-run collisions. According to the previous studies
(Asphaug, 2010; Leinhardt et al., 2010; Leinhardt & Stewart, 2011), the
collision outcomes are subdivided into several regimes (e.g., partial
accretion, fragmentation and so on). However, we classify collision outcomes
into only two regimes of merging and hit-and-run collisions, because it is the
most essential to the evolution of protoplanets during the giant impact stage
that only one big body is left after the giant impact (i.e., merging
collision) or two big bodies are left (i.e., hit-and-run collision).
Additionally, we need to classify collision outcomes as simply as possible, in
order to incorporate those into N-body simulation.
Figure 2 shows snapshots of two typical collisions. Panels (a) to (h) in
Figure 2 show the time sequence for a relatively low-velocity collision
($v_{\rm imp}=1.3v_{\rm esc}$) of same-sized protoplanets ($M_{\rm i}=M_{\rm
t}=0.1M_{\oplus}$) with $\theta=30^{\circ}$. After the first contact, the
protoplanets become separated (see panel (e)), but remain gravitationally
bound. Although some amount of mantle material is ejected, almost all parts of
the colliding protoplanets finally merge. We refer to this type of collision
as “a merging collision”. On the other hand, a relatively high-velocity
collision leads to a completely different result. Panels (i) to (l) show the
time sequence of a collision with $v_{\rm imp}=1.5v_{\rm esc}$. The impact
angle and protoplanet masses are the same as in panels (a) to (h). After the
first contact, the protoplanets escape from each other and are no longer
gravitationally bound. We refer to this type of collision as “a hit-and-run
collision”. By examining snapshots at $t=10^{5}\mathrm{sec}$, we could easily
classify almost all the simulations performed in this study as either merging
or hit-and-run collisions. However, in some cases, the protoplanets were
separated but still gravitationally bound at $t=10^{5}~{}\mathrm{sec}$. Since
those protoplanets are expected to eventually merge, we classified such a case
as a merging collision.
### 3.2 Mass of Protoplanet after a Collision
Here, we consider the mass of the protoplanet after the collision. We briefly
describe the method to determine the mass of the gravitationally bound objects
from the SPH particle data after $t=10^{5}~{}\mathrm{sec}$. In the first step,
we roughly identify clumps of SPH particles using a friends-of-friends
algorithm (e.g., Huchra & Geller, 1982). We then iteratively check whether or
not any SPH particles not belonging to clumps are gravitationally bound to
clumps. Finally, we iteratively identify pairs of clumps that are
gravitationally bound. Such clumps should merge after
$t=10^{5}~{}\mathrm{sec}$, and we can thus regard them as a single object. We
define the mass of the largest object as $M_{1}$ and the second largest object
as $M_{2}$.
Figure 3 shows the mass of the largest object normalized by the total mass,
$M_{1}/M_{\rm T}$, as a function of the impact velocity normalized by the
escape velocity, $v_{\rm imp}/v_{\rm esc}$, for collisions of same-sized
protoplanets with $\theta=30^{\circ}$. This figure also shows the numerical
results of Agnor & Asphaug (2004a) for collisions of same-sized protoplanets
with $0.1M_{\oplus}$, which are in good agreement with our results for total
mass $M_{\rm{}_{T}}$ of $0.2M_{\oplus}$. We also performed simulations for
$M_{\rm T}=0.4M_{\oplus}$ and $1.0M_{\oplus}$. As seen in Figure 3,
$M_{1}/M_{\rm T}$ does not depend on the total mass, when the normalized
impact velocity, $v_{\rm imp}/v_{\rm esc}$, is considered. The insensitivity
to the total mass holds true to collisions between protoplanets, which has
been predicted by Asphaug (2010). If the material properties such as strength
dominates over gravity (typically collision between smaller bodies with less
than 1 km in radius), the collision outcomes would depend on the total mass
(Asphaug, 2010).
### 3.3 Transition between Merging and Hit-and-Run Collisions
Figure 3 indicates that $M_{1}/M_{\rm T}$ changes sharply around $1.4v_{\rm
esc}$. Collisions at impact velocities less than this velocity result in
almost perfect accretion (i.e., $M_{1}/M_{\rm T}\simeq 1$), and thus are
classified as merging collisions. For the case of a near head-on collision
($\theta\leq 15^{\circ}$), $M_{1}/M_{\rm T}$ gradually decreases with the
increase of $v_{\rm imp}$. Although such a collision should be classified as a
partial accretion collision or a fragmentation collision according to Asphaug
(2010) and Leinhardt & Stewart (2011), we here classify those as a merging
collision because only one large body remains after the collision. Since the
probability of a near head-on collision with high velocity is quite low during
the giant impact stage, our treatment would not become a serious problem. On
the other hand, Figure 3 indicates that the collisions at impact velocities
higher than $1.4v_{\rm esc}$ result in $M_{1}/M_{\rm T}\simeq 0.5$. Some
amount of material become stripped from the protoplanets, and the protoplanets
escape from each other. These collisions are classified as hit-and-run
collisions. We refer to the impact velocity at the transition between merging
and hit-and-run collisions as the critical impact velocity, $v_{\rm cr}$. The
normalized critical impact velocity, $v_{\rm cr}/v_{\rm esc}$, is not strongly
dependent on the total mass, $M_{\rm T}$. We obtain $v_{\rm cr}/v_{\rm esc}$ =
$1.39\pm 0.01$, $1.37\pm 0.01$, and $1.37\pm 0.01$ for $M_{\rm T}=0.2,0.4$ and
$1.0M_{\oplus}$, respectively.
The critical impact velocity is expected to depend on the impact angle. Figure
4 is similar to Figure 3, but for the collisions with $\theta=60^{\circ}$. As
is the same in the case of $\theta=30^{\circ}$, $M_{1}/M_{\rm T}$ for
$\theta=60^{\circ}$ does not depend on the total mass. However, collisions
with $\theta=60^{\circ}$ result in lower $v_{\rm cr}/v_{\rm esc}$ values than
the $\theta=30^{\circ}$ case. The calculated values are $v_{\rm cr}/v_{\rm
esc}$ = $1.11\pm 0.01$, $1.09\pm 0.01$, and $1.09\pm 0.01$ for $M_{\rm
T}=0.2,0.4$ and $1.0M_{\oplus}$, respectively. This result implies that
collisions at higher impact angles are more like to be hit-and-run collisions.
The critical impact velocity is also expected to depend on the mass ratio of
the protoplanets ($\gamma$). Figure 5 shows the results for $\gamma=1/4$ (mass
ratio of 1:4) and $\theta=30^{\circ}$. It can be seen that in this case also,
$M_{1}/M_{\rm T}$ does not depend on the total mass. This has also been
verified for $\gamma=1/9$ (mass ratio of 1:9), although the results are not
shown here. As shown in Figure 5, $M_{1}/M_{\rm T}$ changes sharply around
$1.5v_{\rm esc}$. The normalized critical impact velocities are $v_{\rm
cr}/v_{\rm esc}$ = $1.57\pm 0.01$, $1.53\pm 0.01$, and $1.53\pm 0.01$ for
$M_{\rm T}=0.5,1.0$ and $1.5M_{\oplus}$, respectively, which are larger than
for the case of collisions between same-sized protoplanets.
### 3.4 Dependence on Particle Number and Initial Internal Energy
In addition to the impact simulations with 20,000 SPH particles (standard
case), we performed simulations with 3,000 (low-resolution case), 60,000 and
100,000 particles (high-resolution cases) for certain impact parameters in
order to check the dependence of convergence on particle number. Although we
found that the critical impact velocity for the low-resolution case was
slightly different from that for the standard case, the high-resolution cases
yielded the same results. For example, for $\gamma=1$, $M_{\rm
T}=0.2M_{\oplus}$, and $\theta=30^{\circ}$, we obtained $v_{\rm cr}/v_{\rm
esc}=1.39\pm 0.01$ for both the standard and high-resolution cases. Therefore,
using 20,000 SPH particles is enough for determining the critical impact
velocity.
In subsection 2.2, the initial internal energy of the SPH particles was set to
$1.0\times 10^{6}$ J/kg. To investigate the effect of the initial thermal
state of the protoplanets, we prepared pre-impact protoplanets with an
internal energies of $1.0\times 10^{4}$ (cold-state case) and $3.0\times
10^{6}$ J/kg (hot-state case). We then performed simulations of collisions
between same-sized protoplanets with impact parameters near the transition
between merging and hit-and-run collisions. We found that although the results
for the very low-impact angle case ($\theta=15^{\circ}$) showed a very slight
dependence on the initial thermal state, for all other cases no such
dependence was observed.
## 4 Merging Criteria for Colliding Protoplanets
In the previous section, we determined the critical impact velocities for
several impact parameters. In Figure 6, we summarize $v_{\rm cr}/v_{\rm esc}$
for all parameter sets of the mass ratios ($\gamma$) and impact angles
($\theta$). For $\gamma=1$, $1/4$, and $1/9$, we performed simulations for
three sets of $M_{\rm T}$ (see Table 1). In Figure 6, we plot the average of
these three results.
The critical impact velocities estimated by Agnor & Asphaug (2004a) and Agnor
& Asphaug (2004b) are also plotted in Figure 6. Agnor & Asphaug (2004a)
investigated collisions between same-sized protoplanets with masses of
$0.1M_{\oplus}$, and found that $v_{\rm cr}/v_{\rm esc}=1.4-1.5$ for
$\theta=30^{\circ}$, and $v_{\rm cr}/v_{\rm esc}=1.1-1.2$ for
$\theta=45^{\circ}$ or $60^{\circ}$. Our results are in good agreement with
those of Agnor & Asphaug (2004a), since in our simulations, $v_{\rm cr}/v_{\rm
esc}=1.39\pm 0.01$ for $\theta=30^{\circ}$, $1.19\pm 0.01$ for
$\theta=45^{\circ}$, and $1.11\pm 0.01$ for $\theta=60^{\circ}$.
Agnor & Asphaug (2004b) investigated collisions between different-sized
protoplanets with mass ratios of 1:2 and 1:10. Based on Figure 17 in Asphaug
(2009), for a mass ratio of 1:2, $v_{\rm cr}/v_{\rm esc}=1.4-1.5$ for
$\theta=30^{\circ}$, and $1.1-1.2$ for $\theta=45^{\circ}$ or $60^{\circ}$. In
the present study, for a mass ratio of 1:2, we obtained $v_{\rm cr}/v_{\rm
esc}=1.43\pm 0.01$ for $\theta=30^{\circ}$, $1.19\pm 0.01$ for
$\theta=45^{\circ}$, and $1.09\pm 0.01$ for $\theta=60^{\circ}$, and these
values are in good agreement with those of Agnor & Asphaug (2004b). Although
we did not perform simulations for a mass ratio of 1:10, our data points for
1:9 ($\gamma=1/9$) seem to fall within the range obtained by Agnor & Asphaug
(2004b).
As shown in Figure 6, $v_{\rm cr}/v_{\rm esc}$ increases with decreasing
impact angle or mass ratio, which means that collisions with low impact angles
or low mass ratios tend to be merging events. This can be explained in terms
of the size of the overlapping volume of the colliding protoplanets. Since
this is geometrically smaller for higher impact angle, the fraction of kinetic
energy converted to thermal energy of protoplanets and kinetic energy of the
fragments is small, resulting in a hit-and-run collision. In addition, in the
case of a small impactor (i.e., small $\gamma$), most of the volume of the
impactor tends to overlap with the target. Therefore, the impactor can not
easily be ejected, which leads to be a merging collision.
In the following, we consider a simple physical model in order to express the
critical impact velocity as a function of the impact angle and mass ratio. For
two spheres with radii $R_{\rm t}$ and $R_{\rm i}$ colliding with an impact
angle $\theta$ (see Figure 7), the mass fractions of the overlapping volumes
for the target and impactor ($\beta_{\rm t}$ and $\beta_{\rm i}$,
respectively) are geometrically given by
$\beta_{\rm t}=\frac{M_{\rm t}^{\rm ov}}{M_{\rm t}}=\biggl{(}\frac{R_{\rm
t}+R_{\rm i}}{R_{\rm
t}}\biggr{)}^{2}(1-\sin\theta)^{2}\Biggl{\\{}\frac{3}{4}-\frac{1}{4}\biggl{(}\frac{R_{\rm
t}+R_{\rm i}}{R_{\rm t}}\biggr{)}(1-\sin\theta)\Biggr{\\}},$ (9) $\beta_{\rm
i}=\frac{M_{\rm i}^{\rm ov}}{M_{\rm
i}}=\left\\{\begin{array}[]{cl}\biggl{(}\frac{R_{\rm t}+R_{\rm i}}{R_{\rm
i}}\biggr{)}^{2}(1-\sin\theta)^{2}\Biggl{\\{}\frac{3}{4}-\frac{1}{4}\biggl{(}\frac{R_{\rm
t}+R_{\rm i}}{R_{\rm i}}\biggr{)}(1-\sin\theta)\Biggr{\\}},&\rm
if~{}\sin\theta\geq 1-\frac{2R_{\rm t}}{R_{\rm i}+R_{\rm t}},\\\ 1,&\rm
otherwise,\end{array}\right.$ (10)
where $M_{\rm t}^{\rm ov}$ and $M_{\rm i}^{\rm ov}$ are the masses of the
overlapping volumes for the target and impactor, respectively. A constant
density is assumed for simplicity.
As illustrated in Figure 7, we divide the spheres into overlapping parts and
non-overlapping parts, and consider the momentum exchange between the
overlapping parts. Using the parameter of the degree of the momentum exchange
($\alpha$), the post impact velocities of the overlapping parts ($v_{\rm
t,post}^{\rm ov}$ and $v_{\rm i,post}^{\rm ov}$) are expressed as
$\left\\{\begin{array}[]{l}v_{\rm t,post}^{\rm ov}=(1-\alpha)(v_{\rm t}-v_{\rm
COM}^{\rm ov})+v_{\rm COM}^{\rm ov},\\\ v_{\rm i,post}^{\rm
ov}=(1-\alpha)(v_{\rm i}-v_{\rm COM}^{\rm ov})+v_{\rm COM}^{\rm
ov},\end{array}\right.$ (11)
where $v_{\rm COM}^{\rm ov}$ is the velocity of the center of mass of the
overlapping parts, and is written as
$v_{\rm COM}^{\rm ov}=\frac{M_{\rm t}^{\rm ov}v_{\rm t}+M_{\rm i}^{\rm
ov}v_{\rm i}}{M_{\rm t}^{\rm ov}+M_{\rm i}^{\rm ov}}.$ (12)
For example, when $\alpha=1$, the velocities of the overlapping parts ($v_{\rm
t,post}^{\rm ov}$ and $v_{\rm i,post}^{\rm ov}$) become $v_{\rm COM}^{\rm ov}$
owing to the complete momentum exchange.
The post impact velocities of the entire target and impactor can be derived
based on the conservation of momentum, and written as
$\left\\{\begin{array}[]{l}M_{\rm t}v_{\rm t,post}=(M_{\rm{}_{t}}-M_{\rm
t}^{\rm ov})v_{\rm t}+M_{\rm t}^{\rm ov}v_{\rm t,post}^{\rm ov},\\\ M_{\rm
i}v_{\rm i,post}=(M_{\rm{}_{i}}-M_{\rm i}^{\rm ov})v_{\rm i}+M_{\rm i}^{\rm
ov}v_{\rm i,post}^{\rm ov},\end{array}\right.$ (13)
where $v_{\rm t,post}$ and $v_{\rm i,post}$ are the velocities for the post-
impact target and impactor, respectivery. Combining equations. (11) - (13)
gives
$\left\\{\begin{array}[]{l}v_{\rm t,post}=\biggl{(}1-\alpha\beta_{\rm
t}\beta_{\rm i}\frac{M_{\rm t}+M_{\rm i}}{\beta_{\rm t}M_{\rm t}+\beta_{\rm
i}M_{\rm i}}\biggr{)}v_{\rm t},\\\ v_{\rm i,post}=\biggl{(}1-\alpha\beta_{\rm
t}\beta_{\rm i}\frac{M_{\rm t}+M_{\rm i}}{\beta_{\rm t}M_{\rm t}+\beta_{\rm
i}M_{\rm i}}\biggr{)}v_{\rm i}.\end{array}\right.$ (14)
If the relative velocity of post-impact objects is higher than the two-body
escape velocity, a hit-and-run collision should occur. Therefore, to solve
$v_{\rm i,post}-v_{\rm t,post}=v_{\rm esc}$ using $v_{\rm cr}=v_{\rm i}-v_{\rm
t}$, the critical impact velocity is obtained as follows:
$\frac{v_{\rm cr}}{v_{\rm esc}}=\Biggl{[}1-\alpha\beta_{\rm t}\beta_{\rm
i}\frac{M_{\rm t}+M_{\rm i}}{\beta_{\rm t}M_{\rm t}+\beta_{\rm i}M_{\rm
i}}\Biggr{]}^{-1}.$ (15)
The calculated results for $\alpha=0.6$ are drawn as gray curves in Figure 8.
We could roughly reproduce the dependence of $v_{\rm cr}/v_{\rm esc}$ on
$\theta$ and $\gamma$, but detail features, especially the case for the low-
impact angle and low-mass ratio, could not be reproduced. For such a
collision, the role of fragmentation that has not been considered here may
become important. It should be also noted that $\alpha=0.6$ would not be
applied to collisions other than giant impacts between rocky protoplanets.
In addition to the above physical model, we tried to mathematically fit the
numerical data for the critical impact velocity. Of the many possibilities
available, we found that the following simple formula with five fitting
parameters, $c_{1}$ to $c_{5}$, was most effective,
$\frac{v_{\rm cr}}{v_{\rm
esc}}=c_{1}\mit\Gamma\Theta^{c_{5}}+c_{2}\Gamma+c_{3}\Theta^{c_{5}}+c_{4},$
(16)
where $\mathit{\Gamma}=(1-\gamma)/(1+\gamma)=(M_{\rm t}-M_{\rm i})/M_{\rm T}$,
and $\mathit{\Theta}=1-\sin\theta$. The fitting parameters are $c_{1}=2.43$,
$c_{2}=-0.0408$, $c_{3}=1.86$, $c_{4}=1.08$, and $c_{5}=5/2$. The fitting
curves produced by equation (16) are shown in Figure 8 as thick curves, and
are in excellent agreement with numerical results. The value of the fitting
parameters derived here is limited to the collision between protoplanets.
## 5 Merging Probability of Protoplanets
Almost all previous N-body simulations of terrestrial planet formation during
the giant impact stage have been performed based on the assumption of perfect
accretion. In order to investigate the statistical properties of fully formed
terrestrial planets, Kokubo et al. (2006) considered 10 sets of protoplanet
initial conditions, and performed 20 runs for each set under the assumption of
perfect accretion. Subsequently, to investigate the spin state of the formed
planets, Kokubo & Ida (2007) performed additional 30 runs each for 7 sets of
protoplanet initial conditions. Using the formula for the merging criteria
(equation [16]) derived in the present study, we can now determine whether
each of giant impacts was a merging or a hit-and-run event.
Figure 9 shows the normalized impact velocity as a function of impact angle
(left panel) and mass ratio (right panel) for 635 giant impact events during
50 runs under standard initial conditions (Model 1 in Kokubo & Ida (2007)).
The symbols denoted by crosses represent hit-and-run events, as determined by
equation (16). In fact, 40% of all impact events (256 out of 635) are expected
to be hit-and-run collisions. This result is consistent with a previous study
by Agnor & Asphaug (2004a), who estimated a hit-and-run probability of roughly
half.
## 6 Summary and Discussion
During the giant impact stage of terrestrial planet formation in our solar
system, a few tens of Mars-sized protoplanets collide with each other to form
terrestrial planets. Almost all previous studies on N-body calculations of the
giant impact stage have been based on the assumption of perfect accretion.
However, recent impact simulations have shown that collisions of protoplanets
are not always merging events.
As a first step towards studying the effects of such imperfect accretion on
terrestrial planet formation, we investigated the merging criteria for a
collision of rocky protoplanets. Using the SPH method, we performed more than
1000 simulations of giant impacts for various parameter sets, such as the mass
ratio of colliding protoplanets ($\gamma$), the total mass of two protoplanets
($M_{\rm T}$), the impact angle ($\theta$), and the impact velocity ($v_{\rm
imp}$). We investigated the critical impact velocity ($v_{\rm cr}$) at the
transition between merging and hit-and-run collisions. We found that the
normalized critical impact velocity, $v_{\rm cr}/v_{\rm esc}$, depends on
$\gamma$ and $\theta$, but does not depend on $M_{\rm T}$. We derived a simple
formula for $v_{\rm cr}/v_{\rm esc}$ as a function of $\gamma$ and $\theta$
(see equation [16]), and applied it to the giant impact events considered by
Kokubo et al. (2006) and Kokubo & Ida (2007). We found that 40% of these
events should not be merging events.
Kokubo & Genda (2010) was the first to performed N-body simulations of the
giant impact stage taking into account the merging criteria shown in equation
(16), and investigated the effects of imperfect accretion on terrestrial
planet formation. They found that some basic properties such as the final
number, mass, orbital elements, and growth timescale of planets did not change
very much, but the spin angular velocity of the fully formed planets was about
30$\%$ smaller than that for the perfect accretion model. They also determined
that 49% of all impact events were hit-and-run collisions, which is also
consistent with our estimate.
In this paper, we focused on the merging criteria for protoplanet collisions.
In the future, we plan to investigate additional collisional phenomena such as
mantle stripping and ejection of small particles. Using our simulation data
for more than 1000 collisional events, we can estimate the change in the core-
mantle ratio during the giant impact stage. This is highly relevant to the
formation of Mercury, and the formation probability of such planets with very
large cores will be investigated.
The ejection of small particles during each collision in the giant impact
stage may influence the orbital evolution of terrestrial planets. The ejected
material may have damped the eccentricities of the terrestrial planets to
their present low values, although it depends on the total amount of material
ejected during the giant impact stage.
acknowledgments — We thank Sarah T. Stewart for valuable comments on the
manuscript. This research was partially supported by JSPS, the Grant-in-Aid
for Young Scientists B (22740291), and MEXT, the Grant-in-Aid for Scientific
Research on Priority Areas, and the Special Coordination Fund for Promoting
Science and Technology.
## References
* Agnor & Asphaug (2004a) Agnor, C., & Asphaug, E. 2004a, ApJ, 613, L157
* Agnor & Asphaug (2004b) Agnor, C., & Asphaug, E. 2004b, American Geophysical Union, Fall Meeting, abstract #P32A-02
* Agnor et al. (1999) Agnor, C. B., Canup, R. M., & Levison, H. F. 1999, Icarus, 37, 219
* Asphaug (2009) Asphaug, E. 2009, Annu. Rev. Earth Planet. Sci., 37, 413
* Asphaug (2010) Asphaug, E. 2010, Chemie der Erde, 70, 199
* Asphaug et al. (2006) Asphaug, E., Agnor, C., & Williams, Q. 2006, Nature, 439, 155
* Benz et al. (1987) Benz, W., Slattery, W. L., & Cameron, A. G. W. 1987, Icarus, 71, 30
* Benz et al. (2007) Benz, W., Anic, A., & Whitby, J. A. 2007, Space Sci. Rev., 132, 189
* Canup & Asphaug (2001) Canup, R. M., & Asphug, E. 2001, Nature, 412, 708
* Canup (2004a) Canup, R. M. 2004a, ARA&A, 42, 441
* Canup (2004b) Canup, R. M. 2004b, Icarus, 168, 433
* Chambers & Wetherill (1998) Chambers, J. E., & Wetherill, G. W. 1998, Icarus, 136, 304
* Fukushige et al. (2005) Fukushige, T., Makino, J., & Kawai, A. 2005, PASJ, 57, 1009
* Genda & Abe (2005) Genda, H., & Abe, Y. 2005, Nature, 433, 842
* Goldreich & Ward (1973) Goldreich, P., & Ward, W. R. 1973, ApJ, 183, 1051
* Huchra & Geller (1982) Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423
* Kokubo & Genda (2010) Kokubo, E., & Genda, H. 2010, ApJ, 714, L21
* Kokubo & Ida (1998) Kokubo, E., & Ida, S. 1998, Icarus, 131, 171
* Kokubo & Ida (2007) Kokubo, E., & Ida, S. 2007, ApJ, 671, 2082
* Kokubo et al. (2006) Kokubo, E., Kominami, J., & Ida, S. 2006, ApJ, 642, 1131
* Leinhardt et al. (2010) Leinhardt, Z. M., Marcus, R. A., & Stewart, S. T. 2010, ApJ, 714, 1789
* Leinhardt & Stewart (2011) Leinhardt, Z. M., & Stewart, S. T. 2011, MNRAS, submitted
* Marcus et al. (2009) Marcus, R. A., Stewart, S. T., Sasselov, D., & Hernquist, L. 2009, ApJ, 700, L118
* Marcus et al. (2010) Marcus, R. A., Sasselov, D., Stewart, S. T., & Hernquist, L. 2010, ApJ, 719, L45
* Melosh (1989) Melosh, H. J. 1989, Impact Cratering: A Geologic Process (New York: Oxford Univ. Press)
* Monaghan (1992) Monaghan, J. J. 1992, ARA&A, 30, 543
* Monaghan & Lattanzio (1985) Monaghan, J. J., & Lattanzio, J. C. 1985, A&A, 149, 135
* Serna et al. (1996) Serna, A., Alimi, J.-M., & Chieze, J.-P. 1996, ApJ, 461, 884
* Tillotson (1962) Tillotson, J. H. 1962, Report No. GA-3216, July 18 (General Atomic, San Diego, Calfornia, 1962)
* Tonks & Melosh (1992) Tonks, W. B., & Melosh, H. J. 1992, Icarus, 100, 326
* Youdin & Shu (2002) Youdin, A. N., & Shu, F. H. 2002, ApJ, 580, 494
* Wetherill (1985) Wetherill, G. W. 1985, Science, 228, 877
Figure 1: Geometry of the collision between a larger (target) and smaller
(impactor) protoplanets with radii of $R_{\rm t}$ and $R_{\rm i}$,
respectively. Since a center of mass coordinate system is used, the impact
velocity, $v_{\rm imp}$, is given by $|v_{\rm i}-v_{\rm t}|$, where $v_{\rm
t}$ and $v_{\rm i}$ are the velocities of the larger and smaller protoplanets,
respectively. The impact angle is $\theta$. A head-on collision corresponds to
$\theta=0^{\circ}$.
Figure 2: Snapshots of two typical giant impacts between equal-mass
protoplanets with $0.1M_{\oplus}$. Panels (a) to (h) show the time sequence
for a relatively low-velocity collision of protoplanets ($v_{\rm
imp}=1.3v_{\rm esc}$) with the impact angle $30^{\circ}$. The colliding
protoplanets finally merge so that this type of collision is referred to as a
merging collision. Panels (i) to (l) show the time sequence for a relatively
high-velocity collision ($v_{\rm imp}=1.5v_{\rm esc}$). The impact angle and
mass of the protoplanets are the same as in (a) to (h), but the protoplanets
do not merge. This type of collision is referred to as a hit-and-run
collision. The dark gray circle at the top-right corner in panels (i) to (l)
indicates the size of the initial protoplanets.
Figure 3: Normalized mass of the largest gravitationally bound object,
$M_{1}/M_{\rm T}$, as a function of normalized impact velocity, $v_{\rm
imp}/v_{\rm esc}$. Data for $\theta=30^{\circ}$ impacts between two equal-mass
protoplanets with different total masses are plotted. Our collision outcomes
are very similar to those obtained by Agnor & Asphaug (2004a) who performed
simulations for collisions of same-sized protoplanets with masses of
$0.1M_{\oplus}$.
Figure 4: The same as Figure 3 but for $\theta=60^{\circ}$.
Figure 5: The same as Figure 3 but for collisions of different-sized
protoplanets with the mass ratio of 1:4 ($\gamma=1/4$). The mass of the
second-largest object, $M_{2}$, is also plotted.
Figure 6: Critical impact velocities for the cases of the various impact
angles and mass ratios of protoplanets. Note that $(M_{\rm t}-M_{\rm
i})/M_{\rm T}$ is a function of $\gamma$, i.e., $(1-\gamma)/(1+\gamma)$. Data
points represent our numerical results for $\theta=15^{\circ}$ (filled
circles), $30^{\circ}$ (filled triangles), $45^{\circ}$ (filled squares),
$60^{\circ}$ (open circles), and $75^{\circ}$ (open triangles). Bars labeled
with AA04a and AA04b are the results obtained by Agnor & Asphaug (2004a) and
Agnor & Asphaug (2004b), respectively. Figure 7: Configuration of a target and
an impactor for the simple physical model. (a) Before the impact, the
velocities of the target and the impactor are $v_{\rm t}$ and $v_{\rm i}$,
respectively. (b) During the impact, the overlapping parts (shaded areas)
exchange momentum, and the post-impact velocity of the overlapping area is
reduced to $v_{\rm t,post}^{\rm ov}$ for the target and $v_{\rm i,post}^{\rm
ov}$ for the impactor. (c) After the impact, velocities of the target and
impactor are $v_{\rm t,post}$ and $v_{\rm i,post}$, respectively. Figure 8:
Fits to the normalized critical impact velocity data. Data points are our
numerical results and the same as shown in Figure 3. Thin and thick curves
represent fits using equation (15) with $\alpha=0.6$, and equation (16),
respectively.
Figure 9: Normalized impact velocity for 635 giant impacts reported by Kokubo
et al. (2006) and Kokubo & Ida (2007) as a function of the impact angle (a)
and the mass ratio of protoplanets (b). Based on the critical impact velocity
(equation [16]), the giant impact events are distinguished as hit-and-run
collisions (cross symbols) or merging collisions (circle symbols). Although
Kokubo et al. (2006) and Kokubo & Ida (2007) assumed the perfect accretion of
protoplanets in their N-body calculations, the present study reveals that 40%
of the impact events (256 out of 635) are hit-and-run collisions.
Table 1: Parameter sets for the mass ratio of colliding protoplanets $\gamma=$ 1 | 0.1$M_{\oplus}$ vs. 0.1$M_{\oplus}$ | 0.2$M_{\oplus}$ vs. 0.2$M_{\oplus}$ | 0.5$M_{\oplus}$ vs. 0.5$M_{\oplus}$
---|---|---|---
2/3 | 0.2$M_{\oplus}$ vs. 0.3$M_{\oplus}$ | — | —
1/2 | 0.1$M_{\oplus}$ vs. 0.2$M_{\oplus}$ | — | —
1/3 | 0.1$M_{\oplus}$ vs. 0.3$M_{\oplus}$ | — | —
1/4 | 0.1$M_{\oplus}$ vs. 0.4$M_{\oplus}$ | 0.2$M_{\oplus}$ vs. 0.8$M_{\oplus}$ | 0.3$M_{\oplus}$ vs. 1.2$M_{\oplus}$
1/6 | 0.1$M_{\oplus}$ vs. 0.6$M_{\oplus}$ | — | —
1/9 | 0.05$M_{\oplus}$ vs. 0.45$M_{\oplus}$ | 0.1$M_{\oplus}$ vs. 0.9$M_{\oplus}$ | 0.2$M_{\oplus}$ vs. 1.8$M_{\oplus}$
|
arxiv-papers
| 2011-09-16T23:47:18 |
2024-09-04T02:49:22.344675
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Genda, E. Kokubo, S. Ida",
"submitter": "Hidenori Genda",
"url": "https://arxiv.org/abs/1109.4330"
}
|
1109.4426
|
††institutetext: Institute of Theoretical Physics, Lanzhou University, Lanzhou
730000, People s Republic of China
# Holographic p-wave Josephson junction
Yong-Qiang Wang 111Corresponding author. Yu-Xiao Liu and Zhen-Hua Zhao
yqwang@lzu.edu.cn, liuyx@lzu.edu.cn, zhaozhh02@gmail.com
###### Abstract
In this work we generalized holographic model for s-wave DC Josephson junction
constructed in arXiv:1101.3326[hep-th] to a holographic description for p-wave
Josephson junction. By solving numerically the coupled equations of motion of
Yang-Mills theory for a non-Abelian $SU(2)$ gauge fields in (3+1)-dimensional
AdS spacetimes, we shown that DC current of the p-wave Josephson junction is
proportional to the sine of the phase difference across the junction like the
s-wave case.
## 1 Introduction
The AdS/CFT correspondence Maldacena:1997re has attracted much attention over
the past few years. The AdS/CFT correspondence relates a gravity theory in a
weakly curved (d + 1)-dimensional AdS spacetimes to the strong coupled
d-dimensional field theory living at the AdS boundary. In the last few years,
with the application of the AdS/CFT correspondence to the superconducting
phase, ones find that black holes with charged scalar hair in AdS spacetimes
can provide a holographically dual description of superconductivity
Gubser:2008px ; Hartnoll:2008vx ; Hartnoll:2008kx . For reviews on holographic
superconductors, see Hartnoll:2009sz ; Herzog:2009xv ; Horowitz:2010gk .
Recently, a holographic model for s-wave Josephson junction with DC current
constructed in Horowitz:2011dz . Solving numerically the equations of motion
of Einstein-Maxwell-scalar model with spatial dependence $\mu$ and non-
vanishing constant $J$, ones show that the DC current of s-wave Josephson
junction is proportional to the sine of the phase difference across the
junction. Meanwhile, they also studied dependence of the maximum current on
the temperature and size of the junction, which matches precisely with the
results for condensed matter physics. This holographic model is extended to
(3+1)-dimensional Josephson junction in Wang:2011rva ; Siani:2011uj . Another
different way to construct a holographic Josephson junction based on designer
multigravity is studied in Kiritsis:2011zq .
As is well known, in condensed matter systems, SNS Josephson junction can be
made up of two p-wave superconductors. A holographic description of p-wave
superconductors has been developed in Gubser:2008zu ; Gubser:2008wv ;
Roberts:2008ns . Naturally, it would also be of interest to set up a
holographic model for p-wave Josephson junction. In this paper, we would like
to generalize the work Horowitz:2011dz to a holographic description for the
p-wave Josephson junction. We will study a non-Abelian $SU(2)$ Yang-Mills
theory in (3+1)-dimensional AdS spacetimes and solve numerically the coupled,
nonlinear partial differential equations of motion, and analyze the phase
dependence of the Josephson current between two p-wave superconductors.
The paper is organized as follows. In Sec. 2, we review non-Abelian $SU(2)$
Yang-Mills theory in (3+1)-dimensional AdS spacetimes and set up a gravity
dual of a (2+1)-dimensional p-wave Josephson junction. In Sec. 3, we show
numerical results of the EOMs and study the characteristics of the
(2+1)-dimensional holographic p-wave Josephson junctions. The last section is
devoted to conclusion.
## 2 Holographic setup of p-wave Josephson junction
Now let us concentrate on the dynamics of Yang-Mills theory for a non-Abelian
$SU(2)$ gauge fields in (3+1)-dimensional AdS spacetimes. First, we write the
action of (3+1)-dimensional AdS gravity as
$S=\int d^{4}x\sqrt{-g}(R+\frac{6}{L^{2}})\;,$ (1)
where $L$ is the curvature radius of asymptotic AdS Spacetimes. The planar
Schwarzschild black hole can be written as:
$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(dx^{2}+dy^{2})\;,\qquad
f(r)=\frac{r^{2}}{L^{2}}-\frac{M}{r}\;,$ (2)
where $M$ is the mass of the black hole. The Hawking temperature of the black
hole reads as
$T\equiv\frac{1}{4\pi}\frac{df}{dr}\bigg{|}_{r=r_{H}}=\frac{3}{4\pi}\frac{M^{1/3}}{L^{4/3}},$
(3)
where $r_{H}=M^{\frac{1}{3}}L^{\frac{2}{3}}$ is the event horizon of the black
hole. In this background of gravity, we now consider a non-Abelian $SU(2)$
gauge fields, with the action
$\displaystyle S=\int
d^{4}x\sqrt{-g}\left(-\frac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu}\right),$ (4)
where
$F_{\mu\nu}^{a}\equiv\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}$
is the field strength of the SU(2) gauge theory, and one form
$A=A_{\mu}dx^{\mu}=A_{\mu}^{a}\tau^{a}dx^{\mu}$ is gauge field, $\tau^{a}$ are
the generators of $SU(2)$ with $a=1,2,3$.
Variation of the action (1) with respect to the gauge field $A_{\mu}$ leads to
the equations of motion
$D^{\mu}F_{\mu\nu}=0\,,$ (5)
where the gauge covariant derivative is
$D_{\mu}\equiv\nabla_{\mu}+i[A_{\mu},\,\,\,\,]$.
In order to obtain a direction along which the SNS stack is arranged, we take
the gauge field ansatz as
$A=\phi(r,y)\tau^{3}\;dt+w(r,y)\;\tau^{1}dx+A_{r}(r,y)\;\tau^{3}\;dr+A_{y}(r,y)\;\tau^{3}\;dy\;,$
(6)
where $\phi$, $w$, $A_{r}$, and $A_{y}$ are real functions of $r$ and $y$.
Thus the junction can be along the $y$ direction.
With the black hole background (2) and the above ansatz (6), the equations of
SU(2) gauge theory (5) can be written as:
$\displaystyle\partial_{r}^{2}\,w+\frac{1}{r^{2}f}\partial_{yy}^{2}\,w+\frac{f^{\prime}}{f}\partial_{r}\,w+\left(\frac{\phi^{2}}{f^{2}}-A_{r}^{2}-\frac{A_{y}^{2}}{r^{2}f}\right)|\psi|$
$\displaystyle=0\;,$ (7)
$\displaystyle\partial_{r}^{2}\,\phi+\frac{1}{r^{2}f}\partial_{y}^{2}\,\phi+\frac{2}{r}\partial_{r}\phi-\frac{w^{2}}{r^{2}f}\phi$
$\displaystyle=0\;,$ (8)
$\displaystyle\partial_{y}^{2}\,A_{r}-\partial_{r}\partial_{y}\,A_{y}-w^{2}A_{r}$
$\displaystyle=0\;,$ (9)
$\displaystyle\partial_{r}^{2}\,A_{y}-\partial_{r}\partial_{y}\,A_{r}+\frac{f^{\prime}}{f}\left(\partial_{r}A_{y}-\partial_{y}A_{r}\right)-\frac{w^{2}}{r^{2}f}A_{y}$
$\displaystyle=0\;,$ (10)
$\displaystyle\partial_{r}A_{r}+\frac{1}{r^{2}f}\partial_{y}\,A_{y}+\frac{2}{w}\left(A_{r}\partial_{r}w+\frac{A_{y}}{r^{2}\,f}\partial_{y}w\right)+\frac{f^{\prime}}{f}A_{r}$
$\displaystyle=0\;.$ (11)
where a prime denotes derivative with respect to $r$. Because Eqs. (7)-(11)
are coupled nonlinear equations, one can not solve these equations
analytically. However, it is straightforward to solve them numerically.
In order to solve these coupled equations, first, we need to impose regularity
at the horizon and the boundary on the radial coordinate. Near the horizon
($r=r_{H}$), the field $\phi$ should be regular:
$\phi(r_{H})=0$ (12)
Near the AdS boundary ($r\rightarrow\infty$), the Yang-Mills fields take the
asymptotic forms
$\displaystyle w$
$\displaystyle=w^{(1)}(y)+\frac{w^{(2)}(y)}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\;,$
(13) $\displaystyle\phi$
$\displaystyle=\mu(y)-\frac{\rho(y)}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\;,$
(14) $\displaystyle A_{r}$
$\displaystyle=\mathcal{O}\left(\frac{1}{r^{3}}\right)\;,$ (15) $\displaystyle
A_{y}$
$\displaystyle=\nu(y)+\frac{J}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\;,$
(16)
here, $\mu$ is the chemical potential and $\rho$ is the charge density, $\nu$
and $J$ are the superfluid velocity and current, respectively Basu:2008bh ;
Zeng:2010fs . It is worth pointing out that current $J$ is a constant and
along the $y$ direction. For $w^{(1)}$ and $w^{(2)}$ are normalizable, one can
impose the condition either $w^{(1)}$ or $w^{(2)}$ vanishes. For simplicity,
we will adopt the constraint $w^{(1)}=0$ and interpret
$\langle\mathcal{O}\rangle=w^{(2)}$ as the condensate. Moreover, we introduce
the phase difference $\gamma=-\int A_{y}dy$, which is the gauge invariant and
can be rewritten as
$\gamma=-\int_{-\infty}^{\infty}dy\;[\nu(y)-\nu(\pm\infty)]\;.$ (17)
Furthermore, we impose the Dirichlet-like boundary condition on $A_{r}$ and
Neumann-like boundary condition on $w$, $\phi$ and $A_{y}$ at the spatial
coordinate $y=0$. At $y=\pm\infty$, the field functions are $y$-independent.
Thus the boundary conditions of the coupled equations (7)-(11) are determined
by $J$ and $\mu$.
Ones need introduce the critical temperature $T_{c}$ of the p-wave junction,
which is proportional to $\mu(\infty)=\mu(-\infty)$ :
$T_{c}=\frac{3}{4\pi}\frac{\mu(\infty)}{\mu_{c}}\;,$ (18)
where $\mu_{c}\approx 3.65$. The effective critical temperature inside the gap
can be written as
$T_{0}=\frac{3}{4\pi}\frac{\mu(0)}{\mu_{c}}\;.$ (19)
In order to describe an SNS Josephen junction, we also choose the below
$\mu(y)$ which is the same as that in Horowitz:2011dz :
$\mu(y)=\mu_{\infty}\left\\{1-\frac{1-\epsilon}{2\tanh(\frac{L}{2\sigma})}\left[\tanh\left(\frac{y+\tfrac{L}{2}}{\sigma}\right)-\tanh\left(\frac{y-\tfrac{L}{2}}{\sigma}\right)\right]\right\\}\;,$
(20)
where the chemical potential $\mu$ is proportional to
$\mu_{\infty}\equiv\mu(\infty)=\mu(-\infty)$ at $y=\pm\infty$, and $L$ is the
width of p-wave junction. The parameters $\sigma$ and $\epsilon$ control the
steepness and depth of p-wave junction, respectively.
## 3 Numerical results
In this section, we will solve the coupled equations (7)-(11) numerically with
spectral methods. In order to solve EOMs numerically , we need set the change
of variables $z=1-r_{H}/r$ and $\tilde{y}=\tanh(\frac{y}{4\sigma})$. First,
the graphs for $\phi$ and $A_{y}$ with $J/T^{2}_{c}=0.0094$,$\mu_{\infty}=5$,
$L=4$, $\epsilon=0.6$, and $\sigma=0.5$, are shown in Fig. 1.
(a) $\phi$ (b) $A_{y}$
Figure 1: The components $\phi$ and $A_{y}$ of Yang-Mills fields. The
parameters are set to $J/T^{2}_{c}=0.0094$, $\mu_{\infty}=5$, $L=3$,
$\epsilon=0.6$, $\sigma=0.5$.
Then, in Fig. 2, we show that the DC current of p-wave junction is
proportional to the sine of the phase difference across the junction, that is
to say, the red dots coming from the numerical calculations match with the the
black solid sine curve. Fitting the sine curve to the data, we can obtain the
maximum current across the junction:
$J_{\max}/T^{2}_{c}\approx 1.078.$ (21)
Figure 2: Superfluid current $J_{\max}/T^{2}_{c}$ as the function of the phase
difference $\gamma$. The black line is the sine curve. The parameters are set
to $\mu_{\infty}=5$, $L=3$, $\epsilon=0.6$, $\sigma=0.5$.
The dependence of $J_{\max}$ on the width of the gap is shown in Fig. 3. The
graph predicts an exponential decay with the growing width of the gap in
$J_{\max}$:
$\frac{J_{\max}}{T^{2}_{c}}=A_{0}\,e^{-\frac{\ell}{\xi}},$ (22)
where $\xi$ is the coherence length. In Fig. 2(c), the condensate
$\langle\mathcal{O}\rangle=\psi^{(2)}$ at zero current is shown. The graph
also predicts an exponential decay with the growing width of the gap in
$J_{\max}$:
$\qquad\frac{\langle\mathcal{O}(0)\rangle}{T^{2}_{c}}|_{J=0}=A_{1}\,e^{-\frac{\ell}{2\,\xi}}.$
(23)
Fitting Eq. (22) and Eq. (23) with the two sets of data, we can obtain
$\\{\xi,A_{0}\\}\approx\\{1.11,16.42\\}$ and
$\\{\xi,A_{1}\\}\approx\\{1.15,43.89\\}$ for Eq. (22) and Eq. (23),
respectively. The disagreement of two values of $\xi$ is about 4 percent. In
Fig. 3, we obtain the relation of $J_{\max}$ and $T$. One can verify that near
the critical temperature $T_{c}$, $J_{\max}$ can reach zero. Since
$\epsilon=0.6$, we show the region corresponds to $T/T_{c}<0.6$, which depicts
the character of a p-wave Josephson junction.
(a)
(b) (c) The curve of $J_{\max}$ and $\langle\mathcal{O}\rangle_{x=0}$ on $L$.
The parameters are set to $\mu_{\infty}=5$, $\epsilon=0.6$, $\sigma=0.5$.
Figure 3: The curve of $J_{\max}$ on $T$, The parameters are set to
$\epsilon=0.6$, $\sigma=0.5$
## 4 Conclusion
In this paper, we construct a holographic model for p-wave SNS Josephson
junction with DC current. In the background of (3+1)-dimensional gravity, we
solve a set of couple, partial differential equations of a SU(2) gauge field
numerically. By choosing spatial y-dependence $\mu$, we find the the DC
current of p-wave junction is proportional to the sine of the phase difference
across the junction. Moreover, the graph which predicts an exponential decay
with the growing width of gap in $J_{\max}$ is obtained. At last, the curve of
$J_{\max}$ and $T$ is also obtained. One can see that the model of the
holographic p-wave Josephson junction in our work can match precisely with the
results for condensed matter physics. In future, It would be interested for us
to extend to investigate the holographic d-wave Josephson junction.
## Acknowledgement
This work was supported in part by the National Natural Science Foundation of
China (No. 11005054 and No. 11075065), and the Fundamental Research Fund for
Physics and Mathematics of Lanzhou University (No. LZULL200912). Z.H. Zhao was
supported by the Scholarship Award for Excellent Doctoral Student granted by
Ministry of Education.
## References
* (1) J. M. Maldacena, The Large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200].
* (2) S. S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78, 065034 (2008) [arXiv:0801.2977 [hep-th]].
* (3) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101, 031601 (2008) [arXiv:0803.3295 [hep-th]].
* (4) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Holographic Superconductors, JHEP 0812, 015 (2008) [arXiv:0810.1563 [hep-th]].
* (5) S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26, 224002 (2009) [arXiv:0903.3246 [hep-th]].
* (6) C. P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42, 343001 (2009) [arXiv:0904.1975 [hep-th]].
* (7) G. T. Horowitz, Introduction to Holographic Superconductors, arXiv:1002.1722 [hep-th].
* (8) G. T. Horowitz, J. E. Santos and B. Way, A Holographic Josephson Junction, arXiv:1101.3326 [hep-th].
* (9) Y. Q. Wang, Y. X. Liu and Z. H. Zhao, Holographic Josephson Junction in 3+1 dimensions, arXiv:1104.4303 [hep-th].
* (10) M. Siani, On inhomogeneous holographic superconductors, arXiv:1104.4463 [hep-th].
* (11) E. Kiritsis and V. Niarchos, Josephson Junctions and AdS/CFT Networks, JHEP 1107, 112 (2011) [arXiv:1105.6100 [hep-th]].
* (12) S. S. Gubser, Colorful horizons with charge in anti-de Sitter space, Phys. Rev. Lett. 101, 191601 (2008) [arXiv:0803.3483 [hep-th]].
* (13) S. S. Gubser and S. S. Pufu, The Gravity dual of a p-wave superconductor, JHEP 0811, 033 (2008) [arXiv:0805.2960 [hep-th]].
* (14) M. M. Roberts and S. A. Hartnoll, Pseudogap and time reversal breaking in a holographic superconductor, JHEP 0808, 035 (2008) [arXiv:0805.3898 [hep-th]].
* (15) P. Basu, J. He, A. Mukherjee and H. H. Shieh, Superconductivity from D3/D7: Holographic Pion Superfluid, JHEP 0911, 070 (2009) [arXiv:0810.3970 [hep-th]].
* (16) H. B. Zeng, W. M. Sun and H. S. Zong, Supercurrent in p-wave Holographic Superconductor, Phys. Rev. D 83, 046010 (2011) [arXiv:1010.5039 [hep-th]].
|
arxiv-papers
| 2011-09-20T20:20:28 |
2024-09-04T02:49:22.354877
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong-Qiang Wang, Yu-Xiao Liu, Zhen-Hua Zhao",
"submitter": "Yongqiang Wang",
"url": "https://arxiv.org/abs/1109.4426"
}
|
1109.4465
|
# Exact Green’s Function of the reversible diffusion-influenced reaction for
an isolated pair in 2D
Thorsten Prüstel Laboratory of Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health Martin Meier-Schellersheim Laboratory of
Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health
###### Abstract
We derive an exact Green’s function of the diffusion equation for a pair of
spherical interacting particles in 2D subject to a back-reaction boundary
condition.
11footnotetext: Email: prustelt@niaid.nih.gov, mms@niaid.nih.gov
## 1 Introduction
Green’s functions (GF) for an isolated pair of molecules play an important
role in the theory of diffusion-influenced reactions for several reasons.
First, the solution for any other initial distribution can be calculated when
this GF is known [2, 3, 7]. Second, the GF can be used to derive important
other quantitities, for instance the survival probability and the time-
dependent rate coefficient [2, 3, 7]. Third, the GF may be used in particle-
based stochastic simulation algorithms to enhance the efficiency of naive
Brownian dynamics simulations, cp for instance [8]. Finally, the knowledge of
an exact analytic expression permits to validate newly devised stochastic
simulation algorithms [8].
Exact analytic expressions have been derived for the GF of an isolated pair
that can undergo a reversible reaction in 1D and 3D [2, 3, 7]. To the best of
our knowledge, no exact expression has been presented so far for the 2D case.
However, in particular with regard to cell biological applications, an
accurate theoretical treatment of two-dimensional diffusion-influenced
reactions provides the basis for a better understanding of important processes
like receptor clustering on cell membranes [4].
To derive the GF in 2D we consider an isolated pair of two spherical particles
$A$ and $B$ with diffusion constants $D_{A}$ and $D_{B}$, respectively. The
particles may associate when their separation equals the ”encounter distance”
$a$ to form a bound molecule $AB$. When bound, the molecules may dissociate
again to form an unbound pair $A+B$. Such a system may be described as the
diffusion of a point-like particle with diffusion constant $D=D_{A}+D_{B}$
around a static sphere with radius $a$. In this picture, reactions are
introduced by imposing boundary conditions at the sphere’s surface. The
association reaction is described by the radiation boundary condition that is
characterized by an intrinsic association constant $\kappa_{a}$. The radiation
boundary condition is used to describe irreversible association. To take into
account reversible reactions, i.e. dissociations also, the radiation boundary
condition has to be generalized to the back-reaction boundary condition that
involves an additional intrinsic dissociation constant $\kappa_{d}$.
We consider the probability density function $g(r,t|r_{0})$ for the
probability to find the particles at a distance equal to $r$ at time $t$ given
that the distance was inititially $r_{0}$ at time $t=0$. The time evolution of
$g(r,t|r_{0})$ is governed by the 2D diffusion equation
$\frac{\partial}{\partial t}g(r,t|r_{0})=D(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial r})g(r,t|r_{0}),\quad r\geq a$
(1.1)
The diffusion equation has to be completed by specifying boundary conditions.
Together with the following initial
$2\pi rg(r,t_{0}|r_{0},t_{0})=\delta(r-r_{0})$ (1.2)
and boundary condition
$g(r\rightarrow\infty,t|r_{0},t_{0})=0$ (1.3)
equation (1.1) (if $a=0$) is equivalent to the free-space diffusion equation
in 2D with the familiar solution
$g_{\text{free}}(r,t|r_{0})=\frac{1}{4\pi
Dt}e^{-(r^{2}+r^{2}_{0})/4Dt}I_{0}(\frac{rr_{0}}{2Dt})$ (1.4)
also known as the free-space Green’s function.
However, as described previously, the PDF we are interested in is only defined
for $r\geq a>0$ and one has to impose a boundary condition for $r=a$
specifying the behavior at the encounter distance. We will discuss the
following case [2, 7]:
$2\pi aD\frac{\partial}{\partial
r}g(r,t|r_{0})|_{r=a}=\kappa_{a}g(r=a,t|r_{0})-\kappa_{d}[1-S(t|r_{0})].$
(1.5)
Here $S(t|r_{0})$ denotes the survival probability that a pair of molecules
with initial distance $r_{0}$ survives by time $t$
$S(t|r_{0})=1-\int^{t}_{0}2\pi aD\frac{\partial}{\partial
r}g(r,t^{\prime}|r_{0})|_{r=a}dt^{\prime}$ (1.6)
Following [5], we make the following ansatz for the Laplace transform of the
Green’s function that satisfy the backreaction (BR) boundary condition
$\tilde{g}(r,q|r_{0})=\tilde{g}_{\text{free}}(r,q|r_{0})+\tilde{g}_{\text{BR}}(r,q|r_{0}).$
(1.7)
Here
$\tilde{g}_{\text{free}}(r,q|r_{0})=\frac{1}{2\pi
D}\left\\{\begin{array}[]{lr}I_{0}(qr_{0})K_{0}(qr)&\mbox{$r>r_{0}$}\\\
I_{0}(qr)K_{0}(qr_{0})&\mbox{$r<r_{0}$}\end{array}\right.$ (1.8)
is the Laplace transform of the free-space Green’s function (1.4). The
variable $q$ is defined by
$q:=\sqrt{\tfrac{p}{D}},$ (1.9)
where $p$ denotes the Laplace domain variable. The part
$\tilde{g}_{\text{BR}}$ that takes into account the boundary condition is a
solution of the Laplace transformed 2D diffusion equation
$\frac{d^{2}\tilde{g}_{\text{BR}}}{dr^{2}}+\frac{1}{r}\frac{d\tilde{g}_{\text{BR}}}{dr}-q^{2}\tilde{g}_{\text{BR}}=0.$
(1.10)
The general solution to (1.10) is $AK_{0}(qr)+BI_{0}(qr)$, where
$I_{0}(x),K_{0}(x)$ refer to the modified Bessel functions of first and second
kind, respectively, and of order zero [1]. Because we require
$\lim_{x\rightarrow\infty}\tilde{g}_{\text{BR}}\rightarrow 0$, and
$\lim_{x\rightarrow\infty}I_{0}(x)\rightarrow\infty$, the coefficient $B$ has
to vanish and hence,
$\tilde{g}_{\text{BR}}(r,q|r_{0})=A(q,r_{0})K_{0}(qr).$ (1.11)
$A(q,r_{0})$ is determined by the requirement that the complete Green’s
function (1.7) satisfies the Laplace transformed backreaction boundary
condition, cp. (1.5)
$\frac{\partial}{\partial
r}\tilde{g}_{\text{BR}}(r,q|r_{0})|_{r=a}=h\tilde{g}_{\text{BR}}(r,q|r_{0})|_{r=a}-\kappa_{d}p^{-1}\frac{\partial}{\partial
r}\tilde{g}_{\text{BR}}(r,q|r_{0})|_{r=a},$ (1.12)
where we have defined $h:=\frac{\kappa_{a}}{2\pi aD}$. Using (1.7), (1.8),
(1.11), (1.12) and
$\displaystyle I^{\prime}_{0}(x)$ $\displaystyle=$ $\displaystyle I_{1}(x),$
(1.13) $\displaystyle K^{\prime}_{0}(x)$ $\displaystyle=$ $\displaystyle-
K_{1}(x)$ (1.14)
and defining $\kappa_{D}:=\frac{\kappa_{d}}{D}$, we obtain
$\tilde{g}_{\text{BR}}(r,q|r_{0})=\frac{1}{2\pi
D}\frac{(q^{2}+\kappa_{D})I_{1}(qa)-hqI_{0}(qa)}{(q^{2}+\kappa_{D})K_{1}(qa)+hqK_{0}(qa)}K_{0}(qr_{0})K_{0}(qr).$
(1.15)
The inversion theorem for the Laplace transformation can be applied to find
the corresponding expression in the time domain
$g_{\text{BR}}(r,t|r_{0})=\frac{1}{2\pi
i}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp.$
(1.16)
To calculate the Bromwich contour integral (1.16) we first note that
$\tilde{g}_{\text{BR}}$ has a branch point at $p=0$. Therefore, we use the
contour of Figure 1 with a branch cut along the negative real axis, cp. [5].
Furthermore, because the integrand has no poles within and on the contour [6]
and because the contribution from the small circle around the origin vanishes,
we obtain
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\oint
e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp=\int^{\gamma+i\infty}_{\gamma-i\infty}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp+$
(1.17) $\displaystyle+$
$\displaystyle\int_{\mathcal{C}_{2}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp+\int_{\mathcal{C}_{4}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp.$
Thus, it remains to calculate the integrals
$\int_{\mathcal{C}_{2}},\int_{\mathcal{C}_{4}}$.
We now choose
$p=Dx^{2}e^{\pi i}$ (1.18)
and use [5]
$\displaystyle I_{n}(xe^{\pm\pi i/2})$ $\displaystyle=$ $\displaystyle e^{\pm
n\pi i/2}J_{n}(x),$ (1.19) $\displaystyle K_{n}(xe^{\pm\pi i/2})$
$\displaystyle=$ $\displaystyle\pm\frac{1}{2}\pi ie^{\mp n\pi
i/2}[-J_{n}(x)\pm iY_{n}(x)].$ (1.20)
$J_{n}(x),Y_{n}(x)$ denote the Bessel functions of first and second kind,
respectively [1]. It follows that
$\int_{\mathcal{C}_{2}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp=\frac{i}{2}\int^{\infty}_{0}e^{-Dx^{2}t}H^{(2)}_{0}(xr)H^{(2)}_{0}(xr_{0})F(x)xdx.$
(1.21)
Here $H^{(2)}_{n}(x):=J_{n}(x)-iY_{n}(x)$ denotes the Bessel function of third
kind (also referred to as Hankel function) [1] and we have defined
$F(x):=\frac{\alpha(x)[\alpha(x)+i\beta(x)]}{\alpha(x)^{2}+\beta(x)^{2}}$
(1.22)
and
$\displaystyle\alpha(x)$ $\displaystyle:=$
$\displaystyle(-x^{2}+\kappa_{D})J_{1}(xa)-hxJ_{0}(xa)$ (1.23)
$\displaystyle\beta(x)$ $\displaystyle:=$
$\displaystyle(-x^{2}+\kappa_{D})Y_{1}(xa)-hxY_{0}(xa)$ (1.24)
To evaluate the integral along the contour $\mathcal{C}_{4}$ we choose
$p=Dx^{2}e^{-i\pi}$ and after an analogous calculation one finds that
$\int_{\mathcal{C}_{2}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp=-\overline{\int_{\mathcal{C}_{4}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp}$
(1.25)
where $\overline{\int_{\mathcal{C}_{4}}}$ means complex conjugation. Thus, one
arrives at
$\displaystyle g_{\text{BR}}(r,t|r_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp$
(1.26) $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\Im\left(\int_{\mathcal{C}_{2}}e^{pt}\,\tilde{g}_{\text{BR}}(r,q|r_{0})dp\right)$
(1.27) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int^{\infty}_{0}e^{-Dx^{2}t}\frac{\alpha(x)[\alpha(x)\Omega(x)+\beta(x)\Pi(x)]}{\alpha(x)^{2}+\beta(x)^{2}}xdx$
(1.28)
where we have defined
$\displaystyle\Omega(x):=J_{0}(xr)J_{0}(xr_{0})-Y_{0}(xr)Y_{0}(xr_{0})$ (1.29)
$\displaystyle\Pi(x):=Y_{0}(xr)J_{0}(xr_{0})+J_{0}(xr)Y_{0}(xr_{0}).$ (1.30)
Next, we use the fact that the free-space Green’s function may be written as
$g_{\text{free}}(r,t|r_{0})=\frac{1}{2\pi}\int^{\infty}_{0}e^{-Dx^{2}t}J_{0}(xr)J_{0}(xr_{0})xdx$
(1.31)
to arrive at the exact Green’s function in the time domain
$g(r,t|r_{0})=\frac{1}{2\pi}\int^{\infty}_{0}e^{-Dx^{2}t}T_{0}(xr)T_{0}(xr_{0})xdx$
(1.32)
with
$T_{0}(xr):=\tfrac{J_{0}(rx)[(x^{2}-\kappa_{D})Y_{1}(xa)+hxY_{0}(xa)]-Y_{0}(rx)[(x^{2}-\kappa_{D})J_{1}(xa)+hxJ_{0}(xa)]}{\\{[(x^{2}-\kappa_{D})J_{1}(xa)+hxJ_{0}(xa)]^{2}+[(x^{2}-\kappa_{D})Y_{1}(xa)+hxY_{0}(xa)]^{2}\\}^{1/2}}.$
(1.33)
Note that the limit $\kappa_{d}\rightarrow 0$ one recovers the known Green’s
function [5] for the irreversible case with radiation boundary condition.
Figure 1$\Re$$\Im$$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\mathcal{C}_{4}$
### Acknowledgments
This research was supported by the Intramural Research Program of the NIH,
National Institute of Allergy and Infectious Diseases.
We would like to thank Bastian R. Angermann and Frederick Klauschen for
helpful and stimulating discussions.
## References
* [1] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965.
* [2] N. Agmon. J. Chem. Phys., 81:2811, 1984.
* [3] N. Agmon and A. Szabo. J. Chem. Phys., 92:5270, 1990.
* [4] B. Alberts. Molecular biology of the cell. Garland Science, New York, 2008.
* [5] H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, New York, 1986.
* [6] A. Erdelyi and W.O. Kermack. Proc. Camb. Phil. Soc., 41:74, 1945.
* [7] H. Kim and K.J. Shin. Phys. Rev. Lett., 82:1578, 1999.
* [8] J.S. van Zon and P.R. ten Wolde. Phys. Rev. Lett., 94:128103, 2005.
|
arxiv-papers
| 2011-09-21T03:05:09 |
2024-09-04T02:49:22.361275
|
{
"license": "Public Domain",
"authors": "Thorsten Pr\\\"ustel and Martin Meier-Schellersheim",
"submitter": "Thorsten Pr\\\"ustel",
"url": "https://arxiv.org/abs/1109.4465"
}
|
1109.4569
|
# Know the Star, Know the Planet. II.
Speckle Interferometry of Exoplanet Host Stars
Brian D. Mason11affiliation: Visiting Astronomer, Kitt Peak National
Observatory and Cerro Tololo Inter-American Observatory, National Optical
Astronomy Observatories, operated by Association of Universities for Research
in Astronomy, Inc. under contract to the National Science Foundation. ,
William I. Hartkopf11affiliation: Visiting Astronomer, Kitt Peak National
Observatory and Cerro Tololo Inter-American Observatory, National Optical
Astronomy Observatories, operated by Association of Universities for Research
in Astronomy, Inc. under contract to the National Science Foundation. United
States Naval Observatory
3450 Massachusetts Ave., NW, Washington, DC 20392-5420
Electronic mail: (bdm,wih)@usno.navy.mil Deepak Raghavan11affiliation:
Visiting Astronomer, Kitt Peak National Observatory and Cerro Tololo Inter-
American Observatory, National Optical Astronomy Observatories, operated by
Association of Universities for Research in Astronomy, Inc. under contract to
the National Science Foundation. Georgia State University, Dept. of Physics &
Astronomy, Atlanta, GA 30303-3083
Electronic mail: raghavan@chara.gsu.edu John P. Subasavage11affiliation:
Visiting Astronomer, Kitt Peak National Observatory and Cerro Tololo Inter-
American Observatory, National Optical Astronomy Observatories, operated by
Association of Universities for Research in Astronomy, Inc. under contract to
the National Science Foundation. Cerro Tololo Inter-American Observatory, La
Serena, Chile
Electronic mail: jsubasavage@ctio.noao.edu Lewis C. Roberts, Jr Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena CA 91109
Electronic mail: lewis.c.roberts@jpl.nasa.gov Nils H. Turner and Theo A. ten
Brummelaar Center for High Angular Resolution Astronomy, Georgia State
University, Mt. Wilson, CA 91023
Electronic mail: (nils,theo)@chara-array.org
###### Abstract
A study of the host stars to exoplanets is important to understanding their
environment. To that end, we report new speckle observations of a sample of
exoplanet host primaries. The bright exoplanet host HD 8673 (= HIP 6702) is
revealed to have a companion, although at this time we cannot definitively
establish the companion as physical or optical. The observing lists for planet
searches and for these observations have for the most part been pre-screened
for known duplicity, so the detected binary fraction is lower than what would
otherwise be expected. Therefore, a large number of double stars were observed
contemporaneously for verification and quality control purposes, to ensure the
lack of detection of companions for exoplanet hosts was valid. In these
additional observations, ten pairs are resolved for the first time and sixty
pairs are confirmed. These observations were obtained with the USNO speckle
camera on the NOAO 4m telescopes at both KPNO and CTIO from 2001 to 2010.
binaries: general — binaries: visual — binaries: orbits — techniques:
interferometry — stars:individual (HD 8673)
††slugcomment: to be submitted to The Astronomical Journal,
Version: 9/08/2011
## 1 Introduction
As discussed in Paper 1 of this series (Roberts et al. 2011) a study of the
host stars to exoplanets is essential if we wish to understand the environment
in which those planets formed. Further, the star’s luminosity, distance, mass,
and other characteristics are fundamentally related to the determination of
the planets mass and size. Determining these parameters directly for the host
star as opposed to using a template of the canonical stellar class and type
will produce more accurate and precise planetary determinations. As part of
this effort we herein report new speckle observations of a large sample of
exoplanet host primaries.
Binaries affect the formation and stability of planetary systems, as their
long-term relationship must be hierarchical. Generally speaking, based on the
precepts of Harrington (1981) if the ratio of semimajor axes is 4:1 or
greater, an exoplanet in a stellar binary is dynamically stable. Dynamically
permitted systems include the more commonly detected configuration of
planet(s) orbiting one stellar component of a sufficiently wide-orbit binary
in a hierarchical arrangement, and the harder-to-detect circumbinary
configuration of planet(s) in a wide orbit around a close stellar binary (see
Raghavan et al. 2006, especially §6.1). That said, Raghavan et al. (2010) in
their statistics updating and improving upon Duquennoy & Mayor (1991), find
that while the frequency of single stars is the same, the number of companions
has increased through instrumental and technique enhancements. Due to the
presence of stellar companions, one might imagine the environment of binary
stars to be a rich one for substellar companions. However, dynamical effects
should eject companions not found in hierarchical orbits. In any event, as
conducive as this environment might be to companions, this is not reflected in
the list of known planet hosts, however. This is due entirely to selection
effects; because of the complexities of disentangling stellar companions from
small planetary signatures, the observing lists for planet searches have for
the most part been pre-screened for known duplicity, so the detected binary
fraction is lower than what would be expected.
In addition to binary stars that are gravitationally bound, there are optical
doubles which are merely chance alignments of unrelated stars. Although they
do not contribute dynamically to the system, close optical pairs do contribute
light to the system, which should be accounted for in system analysis. While
photometric analysis of binary systems can infer $``$third” light in the
system, radial velocities or periodic variation in other astrometric
parameters (for example, the Hipparcos acceleration solutions), would not give
evidence of these companions. Optical pairs would best be found by direct
imaging or interferometric analysis.
## 2 Speckle Observations
All of these observations were obtained as part of other observing projects,
for example, analysis of white, red and subdwarfs (Jao et al. 2009), G dwarfs
(§5.3.6 of Raghavan et al. 2010), or Massive stars (Mason et al. 2009), some
of which are still in developmental and/or data collection stages. Unpublished
observations of exoplanet host stars were extracted from these data and are
presented here. The instrument used for these speckle observations was the
USNO speckle interferometer, described most recently in Mason et al. (2009).
Speckle Interferometry is a single filled-aperture interferometric technique
where the $``$speckles” of a pair of nearby stars, induced by atmospheric
turbulence, constructively interfere. Reduced by simple autocorrelation
methods, in the resulting image the binary or double star geometry is the
predominant structure when compared with the other uncorrelated pairings. It
is capable of resolving pairs to the resolution limit of the telescope in
question up to size of the observation field (typically, $\sim$1$\farcs$5), as
long as the pairs have magnitude differences of less than about three.
Calibration of the KPNO data was accomplished through the use of a double-slit
mask placed over the “stove pipe” of the 4-m telescope during observations of
a bright known-single star (as described in Hartkopf et al. 2000). This
application of the Young’s double-slit experiment allowed determination of
scale and position angle zero points without relying on binaries themselves to
determine calibration parameters. Multiple observations through the slit mask
(during five separate KPNO runs from 2001 to 2008) yielded mean errors of
0$\fdg$11 in the position angle zero point and 0.165% in the scale error.
These “internal errors” are undoubtedly underestimates of the true errors for
these observations, because these calibration tests were made on stars that
were brighter and nearer the zenith than science stars. Total errors are
likely 3–5$\times$ larger than these internal errors.
Because the slit-mask option is not available on the CTIO 4-m telescope, we
calibrated the Southern Hemisphere data using observations of numerous well-
observed wide equatorial binaries obtained at both the KPNO and CTIO
telescopes. Published orbital elements for these pairs were updated as needed,
using the recent KPNO and published measures, then predicted $\rho$ and
$\theta$ values from those orbits deemed of sufficiently high quality were
used to determine the CTIO scale and position angle zero points. The
calibration errors for these southern observations were (not surprisingly)
considerably higher than those achieved using the slit mask. Mean errors for
five CTIO runs from 2001 to 2010 (as well as a 2001 KPNO run lacking slit mask
data) were 0$\fdg$67 in position angle and 1.44% in scale. These errors are
likely overestimated, because we have assumed that the calibration binaries
have perfect orbits, and any offsets are incorporated into the errors.
## 3 Results
Following generation of the Directed Vector Autocorrelation (Bagnuolo et al.
1992), the $``$DVA” is background subtracted through boxcar subtraction and
the sharp central peak which corresponding to the zeroth order speckles
correlating with themselves is clipped. Companions in the resulting DVA are
then readily apparent as peaks several sigma above the background.
Of the 118 exoplanet hosts we observed only one, HIP 6702 showed signs of a
companion and is discussed in Section 3.1 and listed in Table 1. The null
results are listed in Table 2, a list of single star detections. In the table,
Column (1) gives the Hipparcos number, Column (2) the HD Catalog number,
Column (3) lists other common designations, Column (4) is the epoch of
observation, and Column (5) identifies the telescope (C = Cerro Tololo 4m, K =
Kitt Peak 4m). For all of these observations no companion was detected within
the ranges $\Delta$mV $<$ $3$, and $0\farcs 03~{}<~{}\rho~{}<~{}1\farcs 5$.
Table 1 lists the observations for this new detection. Column (1) gives the
precise position of the system, Column (2) is the Washington Double Star
Catalog (hereafter, WDS; Mason et al. 2001) identifier, and Column (3) lists
the discovery designation, here the WSI (= Washington Speckle Interferometry)
number. Column (4) gives the Hipparcos number of the primary as a cross-
reference. Column (5) gives the epoch of observation, and Columns (6) and (7)
provide the relative astrometry. Column (8) lists a crude estimate of the
magnitude difference of the pair in the V band (the listed number is paired
with the more reliable observation). This estimate is probably only good to
$\pm 0.5\,mag$. Column (9) provides the separation in astronomical units,
based on the Hipparcos parallax and this angular separation assuming a face-on
orbit.
The resulting multiplicity fraction is extremely low, but artificially so.
Observation of known binaries is a prime goal of the USNO speckle program and
some of these pairs had been previously published (e.g., HD 28305 in Mason et
al. 2009). Others which were known but whose motions were not especially
compelling (e.g., HD 50583 in Mason et al. 2011) were observed with our 26in
refractor in Washington and those which do have a compelling individual story
to tell unrelated to exoplanets are in preparation (Farrington et al. 2012). A
simplistic multiplicity determination of this limited result
($=~{}\frac{1}{118}$) is therefore not a meaningful number.
### 3.1 New Double Star : HIP 6702
Of all the exoplanet hosts which have been serendipitously observed, all were
unknown as close visual doubles and only one of the host stars, HIP 6702 (= HD
8673) appeared double in directed vector autocorrelations on both times it was
observed. The classification of HIP 6702 as an exoplanet is based upon
Hartmann et al. (2010) who, using iodine-cell radial velocity measurements,
detected a companion with a $Msin~{}i$ of 14.2 Mj with a period of 1634$\pm$17
days and an eccentricity of 0.723$\pm$0.016. The relative astrometric measures
of this resolved pair are provided in Table 1. Given the small number of
measures presented in Table 2, the pair, while a visual double star, is not
necessarily a binary system. Verification of physicality for the new companion
to HIP 6702 can be accomplished several ways, among them color-magnitude,
proper motion and/or kinematic analysis. The speckle interferometry observable
of relative position establishing kinematic-physical (i.e., Keplerian) motion
requires at least three measures. So, while close proximity can be a powerful
argument for physicality, it is by no means definitive (c.f., $\iota$ Ori,
§5.1 of Mason et al. 2009). Nevertheless, even a companion which is only
nearby in the angular sense should be considered in any detailed analysis of
the star, as it will contribute to the photometric signature of the examined
target. Such is the case for HIP 6702, which was recently reported as a sub-
stellar companion (Hartmann et al. 2010).
Among the possible interpretations of the new speckle companion two stand out:
first, the companion is a not-physically associated line-of-sight companion
and second, it is the companion detected in Hartmann et al. (2010).
#### 3.1.1 Physical Companion?
Hipparcos produced many types of double star solutions. The one which can be
most easily compared to other detection techniques and the most common are
those where the relative parameters ($\rho$, $\theta$) are presented. The
speckle interferometry measures presented in Table 1 are both near the
Hipparcos $``$C” code double star solution cutoff (0$\farcs$082 for HDS 446 =
HIP 27151). The other Hipparcos double star solutions may not be applicable
here. Some depend upon a priori orbital information (O code), system dynamics
in the plane of the sky (G code), variability (V code) or unknown parameters
(X and/or S code). In any event, the lack of Hipparcos detection is a
condition which is neither necessary nor sufficient to establish that the
Hartmann et al. (2010) companion is not-stellar.
However, if the two Table 1 measures represent relative measures of the
Hartmann et al. (2010) pair, the inclination must be extremely low. Assuming a
near zero inclination the mean separation of 0$\farcs$098 would approximate
the angular semi-major axis (a′′ = 0$\farcs$098$\pm$0$\farcs$011). Given this,
the Hipparcos parallax of 26.14$\pm$0.79 mas and the Hartmann et al. period of
1634$\pm$17 days, a mass sum of 2.63$\pm$0.92 M⊙ is obtained, which is not
unreasonable for two similar F dwarf stars, although the error is quite large,
primarily due to the uncertainty in a′′. The length of time between the two
speckle observations represents 1.47$\times$ the Hartmann et al. period. The
two measures of angular position represent $(0.497~{}or~{}0.503)+n$
revolutions of the system (depending on direction of rotation) which is very
similar to the Hartmann et al. period when $n=1$.
Given the estimated dynamic range ($\Delta$mV = $2.3\pm 0.5$) and assuming the
fainter limit and spectral type of the primary this would make the secondary
close to a K2V. Using the canonical mass of a K2V in $M~{}sin~{}i=14.2M_{j}$
gives an inclination of 1$\fdg$02\. Using this inclination with $a~{}sin~{}i$
from Hartmann et al. (2010) gives a semi-major axis of 0$\farcs$168 which is
consistent with the Table 1 results.
#### 3.1.2 or Optical Companion?
Since the interferometric companion to HIP 6702 has been observed so few
times, establishing the companion as optical or physical is not possible. The
proper motion of the primary is 0$\farcs$25/yr ($\alpha$=0$\farcs$236/yr,
$\delta$=$-$0$\farcs$085/yr). From the relative positions in Table 1, the
proper motion of the companion would be an even higher at 0$\farcs$276/yr. If
linear motion is assumed and reasonable errors are applied it is possible to
determine where the companion would be at some date in the future. In Figure 1
this determination is performed assuming errors slightly larger than nominal
for the two speckle interferometry measures: $\Delta\theta=1\fdg 0$,
$\Delta\rho/\rho=1.0\%$ The predicted position for 2012 through 2015 are
plotted as error boxes. Again, assuming linear motion from the two speckle
points, a separation of 0$\farcs$37 and a position angle of 255∘ is determined
for 1991.25, which would be well within the capabilities of the Hipparcos
satellite (ESA 1997).
Figure 1: A “motion characterization” system plot for HIP 6702 (= WSI 96) with
small filled circles indicating the 2001 and 2007 speckle measures from Table
1. Scales are in arcseconds, and in each figure the large shaded circle
represents the V band resolution limit of a 4m telescope. The four small error
boxes in each figure indicate the predicted location of that pair’s secondary
in 2012.0, 2013.0, 2014.0, and 2015.0, assuming the motion is linear and all
speckle measures are characterized by errors of $\Delta\theta=1\fdg 0$,
$\Delta\rho/\rho=1.0\%$. Finding the double within a box appropriate to the
observation date would be a strong indication that the relative motion of the
pair is linear (that is, just motion from an unrelated field star due to
proper motion differences). The H indicates where the companion would have
been at 1991.25, at the Hipparcos epoch.
## 4 Future Observing
Due to the relatively even distribution of targets not yet observed by speckle
interferometry, one observing run in each semester and each hemisphere will be
necessary in order to observe all remaining exoplanet host stars. However, the
target list for each of the four runs will be slight, less than one hundred
stars each. With an approximately equal number of quality control and
equatorial scale calibration pairs, each observing run could easily be
completed in 2-3 nights. Priority would obviously be given to targets not
observed before. Those observed by other speckle interferometric groups would
be next priority so they all have a common reduction algorithm. Figure 2 is an
Aitoff plot of targets from the list of known exoplanet host stars taken from
the NStED111http://nsted.ipac.caltech.edu/cgi-bin/bgServices/nph-bgExec,
extracted 12 April 2011 database and gives their observation status.
Figure 2: Aitoff projection of all 444 targets. Filled circles ($N=114$) are
those listed in Table 2. Open circles ($N=27$) are those observed by CHARA or
USNO with an ICCD and reduced with the DVA method. Asterisks ($N=11$) are
those observed by other interferometry groups, and an $``$X” ($N=292$) are
those which have yet to be observed. A valid speckle measure is only counted
if it was obtained on a 4m class telescope.
The USNO speckle interferometry program has been supported by NASA and the SIM
preparatory science program through NRA 98-OSS-007, SIM Key Project MASSIF as
well as No. NNH06AF701 issued through the Terrestrial Planet Finder Foundation
Science program. Thanks are also provided to the U.S. Naval Observatory for
their continued support of the Double Star Program. A portion of the research
in this paper was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (NASA). This research has made use of the SIMBAD
database, operated at CDS, Strasbourg, France. Grateful acknowledgment is
given to USNO interns Dean Kang, Laura Flagg and Ankit Patel for their
processing of the speckle pixel data. The telescope operators and observing
support personnel of KPNO and CTIO continue to provide exceptional support for
visiting astronomers. Thanks to Alberto Alvarez, Skip Andree, Bill Binkert,
Gale Brehmer, Bill Gillespie, Angel Guerra, Jim Hutchinson, Hillary Mathis,
Oscar Saa, Patricio Ugarte, and the rest of the KPNO and CTIO staff. We also
like to thank Richard Green of KPNO, who was able to provide us with two extra
nights on the 4-m Mayall telescope during our January 2001 run. While we were
hampered by poor weather, this additional allocation of time certainly helped
us achieve a greatly enhanced completion fraction. A special thank you is also
given to Hal Halbedel, who operated the telescope on all or part of each of
these new KPNO runs and was instrumental in the slit-mask work done at KPNO.
Table 1: New Interferometric Double Coordinates | WDS or | Discoverer | HIP | BY | $\theta$ | $\rho$ | $\Delta$m | sep
---|---|---|---|---|---|---|---|---
$\alpha$,$\delta$ (2000) | $\alpha$,$\delta$ (2000) | Designation | | 2000.0+ | (∘) | (′′) | (mag) | (AU)
01026008.780$+$34034046.9 | 01262$+$3435 | WSI | 96 | 6702 | 1.0193 | 257.3 | 0.087 | | 3.3
| | | | | 7.6049 | 078.3 | 0.109 | 2.3 | 4.2
Table 2: Stars with No Companion Detected HIP # | HD # | Common | BY | Telescope
---|---|---|---|---
| | Name | 2000.0+ |
522 | 142string | $\cdots$ | 1.5621 | C
1292 | 1237string | GJ 3021 | 1.5674 | C
1499 | 1461string | $\cdots$ | 1.5701 | C
3391 | 4113string | $\cdots$ | 1.5647 | C
3479 | 4208string | $\cdots$ | 1.5647 | C
3497 | 4308string | $\cdots$ | 1.5648 | C
5054 | 6434string | $\cdots$ | 1.5622 | C
5529 | 7199string | $\cdots$ | 1.5675 | C
5806 | 7449string | $\cdots$ | 1.5675 | C
6379 | 7924string | $\cdots$ | 1.0218 | K
7513 | 9826string | $\upsilon$ And | 1.0220 | K
7599 | 10180string | $\cdots$ | 1.5702 | C
7978 | 10647string | $\cdots$ | 1.5702 | C
8159 | 10697string | $\cdots$ | 1.0300 | K
9683 | 12661string | $\cdots$ | 1.0165 | K
10138 | 13445string | GJ 86 | 1.5676 | C
10626 | 13931string | $\cdots$ | 1.0219 | K
12186 | 16417string | $\cdots$ | 1.5676 | C
12189 | 16246string | 30 Ari | 1.0304 | K
12653 | 17051string | $\iota$ Hor | 1.5676 | C
14954 | 19994string | $\cdots$ | 1.0820 | C
15323 | 20367string | $\cdots$ | 1.0193 | K
15527 | 20782string | $\cdots$ | 1.0820 | C
16537 | 22049string | $\epsilon$ Eri | 10.0652 | C
17096 | 23079string | $\cdots$ | 1.0738 | C
20723 | 28185string | $\cdots$ | 1.0764 | C
24681 | 34445string | $\cdots$ | 1.0306 | K
25110 | 33564string | $\cdots$ | 1.0222 | K
26381 | 37124string | $\cdots$ | 1.0307 | K
26394 | 39091string | $\cdots$ | 1.0740 | C
26394 | 39091string | $\cdots$ | 6.1937 | C
26664 | 37605string | $\cdots$ | 1.0306 | K
27887 | 40307string | $\cdots$ | 1.0740 | C
27887 | 40307string | $\cdots$ | 6.1937 | C
28767 | 40979string | $\cdots$ | 1.0224 | K
29804 | 43848string | $\cdots$ | 1.0793 | C
30034 | 44627string | AB Pic | 1.0740 | C
30579 | 45364string | $\cdots$ | 1.0794 | C
30860 | 45350string | $\cdots$ | 1.0198 | K
30905 | 45652string | $\cdots$ | 1.0308 | K
31246 | 46375string | $\cdots$ | 1.0767 | C
31540 | 47186string | $\cdots$ | 1.0794 | C
32916 | 49674string | $\cdots$ | 1.0224 | K
32970 | 50499string | $\cdots$ | 1.0794 | C
33212 | 50554string | $\cdots$ | 1.0279 | K
33719 | 52265string | $\cdots$ | 1.0823 | C
36795 | 60532string | $\cdots$ | 1.0795 | C
37826 | 62509string | $\cdots$ | 1.0199 | K
38041 | 63765string | $\cdots$ | 1.0742 | C
38558 | 65216string | $\cdots$ | 1.0742 | C
40693 | 69830string | $\cdots$ | 6.1911 | C
40952 | 70642string | $\cdots$ | 1.0743 | C
43587 | 75732string | 55 Cnc | 1.0200 | K
46076 | 81040string | $\cdots$ | 1.0770 | C
47007 | 82943string | $\cdots$ | 1.0797 | C
47202 | 83443string | $\cdots$ | 1.0744 | C
48235 | 85390string | $\cdots$ | 1.0744 | C
48739 | 86226string | $\cdots$ | 1.0797 | C
49699 | 87883string | $\cdots$ | 1.0202 | K
50473 | 89307string | $\cdots$ | 1.0310 | K
50921 | 90156string | $\cdots$ | 6.1912 | C
52521 | 93083string | $\cdots$ | 1.0799 | C
53721 | 95128string | 47 UMa | 1.0202 | K
54906 | 97658string | $\cdots$ | 1.0203 | K
57172 | 101930string | $\cdots$ | 1.0746 | C
57291 | 102117string | $\cdots$ | 1.0746 | C
57370 | 102195string | $\cdots$ | 1.0311 | K
57443 | 102365string | $\cdots$ | 6.1915 | C
57443 | 102365string | $\cdots$ | 10.0659 | C
57931 | 103197string | $\cdots$ | 1.0746 | C
58451 | 104067string | $\cdots$ | 6.1915 | C
59610 | 106252string | $\cdots$ | 1.0312 | K
64295 | 114386string | $\cdots$ | 1.0827 | C
64426 | 114762string | $\cdots$ | 1.0232 | K
64457 | 114783string | $\cdots$ | 5.1915 | C
64459 | 114729string | $\cdots$ | 1.0775 | C
64459 | 114729string | $\cdots$ | 1.0802 | C
64924 | 115617string | 61 Vir | 6.1890 | C
64924 | 115617string | 61 Vir | 8.4500 | K
64924 | 115617string | 61 Vir | 10.0688 | C
65721 | 117176string | 70 Vir | 1.0232 | K
65721 | 117176string | 70 Vir | 6.1916 | C
67275 | 120136string | $\cdots$ | 1.0314 | K
67275 | 120136string | $\tau$ Boo | 6.1916 | C
71395 | 128311string | $\cdots$ | 1.5664 | C
71395 | 128311string | $\cdots$ | 6.1916 | C
72339 | 130322string | $\cdots$ | 1.0829 | C
74500 | 134987string | $\cdots$ | 1.0830 | C
74500 | 134987string | $\cdots$ | 1.5611 | C
77740 | 141937string | $\cdots$ | 1.5666 | C
78459 | 143761string | $\rho$ CrB | 8.4503 | K
79242 | 142022A | $\cdots$ | 1.5667 | C
79248 | 145675string | 14 Her | 1.4986 | K
80250 | 147018string | $\cdots$ | 1.5667 | C
80337 | 147513string | $\cdots$ | 6.1919 | C
83389 | 154345string | $\cdots$ | 1.4960 | K
83949 | 155358string | $\cdots$ | 1.4961 | K
86796 | 160691string | $\mu$ Ara | 1.5667 | C
87330 | 162020string | $\cdots$ | 1.5642 | C
88348 | 164922string | $\cdots$ | 8.4506 | K
90004 | 168746string | $\cdots$ | 1.5614 | C
90485 | 169830string | $\cdots$ | 1.5614 | C
91085 | 171238string | $\cdots$ | 1.5614 | C
94075 | 178911B | $\cdots$ | 1.4990 | K
94645 | 179949string | $\cdots$ | 1.5614 | C
96901 | 186427string | 16 Cyg | 1.4991 | K
97336 | 187123string | $\cdots$ | 1.4990 | K
97546 | 187085string | $\cdots$ | 1.5670 | C
98505 | 189733string | $\cdots$ | 8.4614 | K
98767 | 190360string | $\cdots$ | 8.4563 | K
99711 | 192263string | $\cdots$ | 5.8680 | K
99825 | 192310string | GJ 785 | 1.5616 | C
101806 | 196050string | $\cdots$ | 1.5615 | C
101966 | 196885string | $\cdots$ | 8.4481 | K
104903 | 202206string | $\cdots$ | 1.5618 | C
106006 | 204313string | $\cdots$ | 1.5618 | C
106353 | 204941string | $\cdots$ | 1.5618 | C
108375 | 208487string | $\cdots$ | 1.5672 | C
108859 | 209458string | $\cdots$ | 8.4615 | K
109378 | 210277string | $\cdots$ | 1.5645 | C
111143 | 213240string | $\cdots$ | 1.5618 | C
112441 | 215497string | $\cdots$ | 1.5646 | C
113137 | 216437string | $\cdots$ | 1.5647 | C
113238 | 216770string | $\cdots$ | 1.5646 | C
113357 | 217014string | 51 Peg | 7.5883 | K
113357 | 217014string | 51 Peg | 8.4617 | K
116727 | 222404string | $\gamma$ Cep | 1.0218 | K
116727 | 222404string | $\gamma$ Cep | 1.4993 | K
116906 | 222582string | $\cdots$ | 1.5674 | C
## References
* (1)
* (2) Bagnuolo, W.G., Jr., Mason, B.D., Barry, D.J., Hartkopf, W.I., & McAlister, H.A. 1992, AJ 103, 1399
* (3)
* (4) Duquennoy, A. & Mayor, M. 1991, A&A 248, 485
* (5)
* (6) ESA 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200
* (7)
* (8) Farrington, C.D. et al. 2012 (in preparation)
* (9)
* (10) Harrington, R.S. 1981, Planetary orbits in multiple star systems, in Life in the Uuniverse, Cambridge, MIT Press, 119
* (11)
* (12) Hartkopf, W.I. et al. 2000, AJ 119, 3084
* (13)
* (14) Hartmann, M., Guenther, E.W. & Hatzes, A.P. 2010, ApJ 717, 348
* (15)
* (16) Jao, W.-C., Mason, B.D., Hartkopf, W.I., Henry, T.J. & Ramos, S.N. 2009, AJ 137, 3800
* (17)
* (18) Mason, B.D., Hartkopf, W.I., Gies, D.R., Henry, T.J., & Helsel, J.W. 2009, AJ 137, 3358
* (19)
* (20) Mason, B.D., Hartkopf, W.I. & Wycoff, G.L. 2011, AJ (in press)
* (21)
* (22) Mason, B.D., Wycoff, G.L., Hartkopf, W.I., Douglass, G.G. & Worley, C.E. 2001, AJ 122, 3466222The Washington Double Star Catalog, published in Second USNO Double Star CD 2006.5 and available online at http://ad.usno.navy.mil/wds/wds.html.
* (23)
* (24) Raghavan, D., Henry, T.J., Mason, B.D., Subasavage, J.P., Jao, W.-C., Beaulieu, T.D. & Hambly, N.C. 2006, ApJ 646, 523
* (25)
* (26) Raghavan, D. et al. 2010, ApJS 190, 1
* (27)
* (28) Roberts, L.C., Jr., Turner, N.H., ten Brummelaar, T.A., Mason, B.D. & Hartkopf, W.I. 2011, (in progress)
* (29)
* (30) Tokovinin, A., Mason, B.D. & Hartkopf, W.I. 2010, AJ 139, 743
* (31)
## Appendix A Additional Measures of Known Pairs
Due to the high incidence of single stars among the exoplanet hosts, a
substantial number of double stars were observed contemporaneously with the
exoplanet host observations to ensure that the observing conditions and
detection capabilities given above were met. Additional measures of known or
suspected doubles were made as time permitted. Table A1 lists 549 mean
positions for 485 known systems. Column (1) is the WDS identification
(arcminute coordinate), Column (2) lists the Discovery Designation, and Column
(3) provides the Hipparcos number. Column (4) gives the epoch of observation,
and Columns (5) and (6) provide the relative astrometry. Column (7) contains
the notes for these systems. Also found in in table are ten pairs resolved for
the first time and sixty pairs which are here confirmed; estimated magnitude
differences for the new pairs (when available) are listed in the Notes column.
Table A1: Other Pairs Measured
WDS or | Discovery | HIP | BY | $\theta$ | $\rho$ | Note
---|---|---|---|---|---|---
$\alpha$,$\delta$ (2000) | Designation | | 2000.0+ | (∘) | (′′) |
00024$+$1047 | A | 1249 | AB | 190 | 7.5994 | 74.8string | 0.135string |
00026$-$0829 | A | 428 | | 210 | 7.6019 | 344.5string | 0.174string |
00055$-$1835 | RST | 3340 | | — | 1.5701 | 287.4string | 0.303string |
00095$+$1907 | COU | 247 | | 768 | 7.5992 | 259.1string | 0.294string |
00117$+$6145 | TDS | 1338 | | — | 7.6074 | 111.2string | 0.527string | 1
00167$+$3629 | STT | 4 | | 1336 | 7.6021 | 99.7string | 0.232string |
00174$+$0853 | A | 1803 | AB | 1392 | 7.6019 | 307.2string | 0.145string |
00182$+$7257 | A | 803 | | 1461 | 7.6021 | 307.9string | 0.262string |
00233$+$5132 | TDS | 1431 | | — | 7.6074 | 102.5string | 0.227string | 1
00271$-$0753 | A | 431 | | 2143 | 1.5021 | 184.5string | 0.139string |
| | | | | 1.5647 | 189.0: | 0.139: |
| | | | | 7.6019 | 13.3string | 0.201string |
00274$+$3054 | HDS | 62 | | 2170 | 7.6074 | 72.0string | 0.302string | 1
00283$+$6344 | HU | 1007 | | 2232 | 7.6021 | 89.2string | 0.449string |
00290$+$5848 | TDS | 1484 | | — | 7.6074 | 351.9string | 0.463string | 1
00310$-$1005 | BU | 1158 | BC | 2444 | 7.6019 | 310.6string | 0.325string |
00335$+$4006 | HO | 3 | | 2643 | 7.6021 | 168.2string | 0.224string |
00337$+$4310 | COU | 1354 | | — | 7.6074 | 64.9: | 0.646: | 1
00339$+$5316 | TDS | 1523 | | — | 7.6075 | 90.6string | 0.693string | 1
00362$+$7546 | TDS | 1542 | | — | 7.6075 | 338.9string | 0.721string | 1
00364$+$3330 | HDS | 84 | | 2872 | 7.6075 | 28.8string | 0.196string | 1
00373$-$2446 | BU | 395 | | 2941 | 10.5861 | 99.4string | 0.555string |
00393$+$3359 | COU | 656 | | — | 7.6075 | 150.6string | 0.821string | 1
00429$+$2047 | A | 2205 | | 3373 | 7.6019 | 8.8string | 0.344string |
00431$+$7659 | HU | 1012 | | — | 7.6021 | 124.0string | 0.446string |
00441$+$5655 | TDS | 1589 | | — | 7.6075 | 315.6string | 0.788string | 1
00456$+$5457 | TDS | 1602 | | — | 7.6075 | 70.7string | 0.409string | 1
00470$+$2315 | HU | 413 | | 3673 | 7.6019 | 309.3string | 0.365string |
00492$+$5720 | A | 921 | AB | — | 7.6021 | 183.2string | 0.168string |
00516$+$2237 | A | 1808 | | 4030 | 7.6019 | 196.7string | 0.188string |
00520$+$3154 | A | 924 | | — | 7.6021 | 318.5string | 0.261string |
00529$+$6053 | TDS | 1655 | | — | 7.6075 | 27.4string | 0.560string | 1
00533$+$0405 | A | 2307 | | 4176 | 7.6019 | 174.6string | 0.075string |
00533$-$4530 | B | 644 | | 4177 | 10.5862 | 349.2string | 0.281string |
00541$+$5701 | HDS | 118 | | 4237 | 7.6075 | 17.3string | 0.242string | 1
00542$+$4318 | COU | 1654 | | — | 7.6021 | 65.7string | 0.098string |
00552$+$4653 | HDS | 121 | | 4306 | 7.6075 | 116.6string | 0.409string | 1
00554$+$5951 | TDS | 1667 | | — | 7.6075 | 296.0string | 0.565string | 1
00564$+$5903 | HDS | 125 | | 4397 | 7.6075 | 279.1: | 0.196: | 1
00576$+$5424 | HLD | 4 | | — | 7.6049 | 36.9string | 0.300string |
00585$+$4843 | COU | 2254 | | — | 7.6075 | 160.2string | 0.447string | 1
01011$+$6022 | A | 926 | | 4755 | 7.6049 | 342.1string | 0.359string |
01014$+$3535 | COU | 854 | | 4783 | 7.6049 | 38.9string | 0.134string |
01036$+$6341 | MLR | 87 | | 4963 | 7.6049 | 30.8string | 0.246string |
01049$+$3649 | A | 1515 | | — | 7.6049 | 303.9string | 0.186string |
01112$+$4113 | A | 655 | | 5553 | 7.6049 | 346.0string | 0.322string |
01234$+$5809 | STF | 115 | AB | 6486 | 7.6069 | 161.6string | 0.290string |
01251$+$4537 | A | 939 | | 6633 | 7.6049 | 183.5string | 0.191string |
01345$+$3440 | A | 1913 | AB | 7337 | 7.6049 | 311.6string | 0.391string |
01361$-$2954 | HJ | 3447 | | 7463 | 10.5862 | 184.9string | 0.777string |
01409$+$4952 | HU | 531 | AB | 7841 | 7.6049 | 358.4string | 0.467string |
01443$+$5732 | BU | 870 | AB | 8115 | 7.6021 | 333.6string | 0.611string |
01450$+$5707 | BU | 453 | | 8166 | 7.6021 | 100.7string | 0.749string |
01450$+$2703 | COU | 750 | | — | 7.6021 | 25.8string | 0.273string |
01462$+$3343 | HU | 804 | | 8252 | 7.6021 | 67.7string | 0.285string |
01463$+$4439 | A | 948 | AB | — | 7.6021 | 133.1string | 0.302string |
01495$+$5645 | A | 950 | AB | — | 7.6021 | 278.3string | 0.222string |
01497$-$1414 | HU | 422 | | 8504 | 7.5994 | 29.8string | 0.296string |
01499$+$8053 | STT | 34 | | 8519 | 7.6075 | 284.6string | 0.510string |
01500$-$0408 | A | 2602 | | — | 7.5995 | 178.9string | 0.632string | 2
01520$+$1326 | HU | 1213 | | 8703 | 7.5995 | 84.9string | 0.105string |
01528$-$0447 | RST | 4188 | | — | 7.5994 | 31.3string | 0.419string |
01570$+$3101 | A | 819 | AB | 9087 | 7.6021 | 315.3string | 0.183string |
01559$+$0151 | STF | 186 | | 8998 | 5.8592 | 63.7string | 0.915string |
02020$+$7054 | BU | 513 | AB | 9480 | 7.6022 | 287.4string | 0.705string |
02130$+$0851 | MCA | 5 | | 10324 | 7.5995 | 84.1string | 0.104string |
02157$+$2503 | COU | 79 | | 10535 | 1.0164 | 301.6string | 0.100string | 3
| | | | | 5.8601 | 255.0string | 0.157string |
| | | | | 7.5995 | 243.6string | 0.152string |
02159$+$0638 | A | 2013 | | 10552 | 7.5995 | 256.0string | 0.172string |
02174$+$6121 | STF | 234 | AB | 10677 | 7.6021 | 223.5string | 0.719string |
02202$+$2949 | A | 961 | | 10892 | 7.6021 | 278.6string | 0.345string |
02211$+$4246 | STF | 248 | | 10952 | 7.6016 | 217.3string | 0.528string |
02231$+$7021 | MLR | 377 | AB | 11120 | 7.6022 | 140.5string | 0.674string |
02291$+$6724 | CHR | 6 | Aa,Ab | 11569 | 7.6022 | 45.8string | 0.543string |
02382$+$4604 | A | 1278 | | — | 7.6075 | 308.9string | 0.139string |
02405$-$2408 | SEE | 19 | | 12466 | 10.5864 | 241.7string | 0.117string |
02415$+$4053 | COU | 1511 | | 12552 | 7.6075 | 30.0string | 0.109string |
02442$-$2530 | FIN | 379 | Aa,Ab | 12780 | 10.5864 | 343.4string | 0.118string |
02517$+$4559 | A | 1281 | | 13336 | 7.6075 | 134.8string | 0.630string |
03175$+$6540 | STT | 52 | AB | 15309 | 7.6022 | 58.1string | 0.485string |
03184$-$2231 | SEE | 23 | | 15382 | 10.5864 | 101.4string | 0.323string |
03310$+$2937 | A | 983 | | 16376 | 7.6075 | 137.2string | 0.455string |
03318$+$0749 | A | 1931 | | — | 10.0653 | 48.2string | 0.845string |
03372$+$0121 | A | 2419 | | — | 10.0653 | 99.9string | 0.813string |
03460$+$6321 | MCA | 12 | | 17587 | 7.6075 | 14.0string | 0.058string |
03494$-$1956 | RST | 2324 | | — | 10.0646 | 213.5string | 0.257string |
03520$+$0632 | KUI | 15 | AB | 18089 | 10.0653 | 205.9string | 0.785string |
03545$+$0510 | A | 1831 | BC | 18264 | 10.0653 | 225.0string | 0.171string |
03590$+$0947 | HU | 27 | | 18618 | 10.0653 | 329.6string | 0.406string |
04024$-$2832 | DAW | 79 | | 18839 | 10.0645 | 144.7string | 0.381string |
| | | | | 10.5864 | 142.9string | 0.389string |
04049$-$3527 | I | 152 | AB | 19052 | 10.0645 | 78.0string | 1.060string |
04093$-$2025 | RST | 2333 | | — | 10.0646 | 181.0string | 0.305string |
04123$+$0939 | STT | 74 | | — | 10.0653 | 114.4string | 0.361string |
04142$-$4608 | RST | 2338 | | 19758 | 1.0791 | 197.0string | 0.135string |
| | | | | 6.1935 | 269.5string | 0.187string |
04163$-$6057 | GLE | 1 | | 19917 | 10.0677 | 201.7string | 0.319string |
04215$-$2544 | BU | 744 | AB | 20347 | 10.0645 | 189.8string | 0.297string |
04239$+$0928 | HU | 304 | | 20522 | 10.0653 | 20.7string | 0.267string |
04242$-$5704 | RMK | 4 | | 20552 | 1.0739 | 246.5string | 5.426string |
04269$-$2405 | BU | 311 | | 20765 | 10.0645 | 150.3string | 0.439string |
04275$-$1707 | HU | 440 | | 20806 | 1.0822 | 318.9string | 2.064string |
04286$+$1558 | MCA | 15 | | 20885 | 5.8688 | 350.5string | 0.151string |
04374$-$0951 | RST | 3401 | | 21536 | 10.0652 | 260.3string | 0.399string |
04395$-$4507 | I | 1489 | | 21698 | 1.0793 | 80.6string | 0.208string |
04503$+$0103 | A | 2622 | | — | 10.0653 | 252.3string | 0.294string |
04529$+$7543 | MLR | 465 | | 22703 | 1.0222 | 208.4string | 0.866string |
04545$-$0314 | RST | 5501 | | 22812 | 10.0652 | 206.4string | 0.150string |
04584$-$0344 | HDS | 644 | | 23116 | 10.0652 | 333.0string | 0.588string |
05059$-$1355 | A | 3009 | | 23716 | 1.0767 | 257.2string | 1.134string |
05079$+$0830 | STT | 98 | | 23879 | 10.0653 | 299.9string | 0.883string |
05089$+$0313 | A | 2636 | | 24227 | 10.0653 | 349.8string | 0.363string |
05135$+$0158 | STT | 517 | AB | 24349 | 10.0653 | 239.7string | 0.670string |
05167$+$4600 | ANJ | 1 | Aa,Ab | 24608 | 5.8634 | 328.8string | 0.046string | 3
05190$-$2159 | RST | 2375 | | 24800 | 10.0646 | 17.7string | 0.172string |
05234$-$3640 | I | 275 | | 25190 | 10.0680 | 220.7string | 0.788string |
05245$-$0224 | DA | 5 | AB | 25281 | 10.0652 | 77.7string | 1.788string | 2
05247$+$6323 | STF | 677 | | 25300 | 1.0223 | 127.6string | 1.074string |
05248$-$5219 | I | 345 | AB | 25303 | 10.0680 | 213.1string | 0.253string | 2
| | | | | 10.5863 | 213.6string | 0.258string |
05255$-$0033 | A | 848 | | 25365 | 10.0653 | 176.1string | 0.322string |
05387$-$0236 | BU | 1032 | AB | 26549 | 10.0652 | 89.0string | 0.256string |
05407$-$0157 | STF | 774 | AB | 26727 | 10.0653 | 165.3string | 2.450string |
05418$-$5000 | HU | 1568 | | 26830 | 10.0706 | 152.6string | 0.495string |
05455$+$5210 | WSI | 98 | | 27165 | 1.0223 | 156.4string | 0.086string | $\Delta$m = 2.0
05508$-$3945 | I | 1494 | AB | 27611 | 10.0680 | 269.4string | 0.384string |
05525$-$0217 | HDS | 787 | | 27758 | 10.0652 | 343.6string | 0.111string |
05580$-$5212 | HU | 1570 | | 28240 | 10.0656 | 345.5string | 0.944string |
06003$-$3102 | HU | 1399 | AB | 28442 | 10.0656 | 116.1string | 0.733string |
| HJ | 3823 | AC | | 10.0656 | 4.0string | 2.772string |
06041$+$2316 | KUI | 23 | AB | 28734 | 1.0280 | 174.7string | 0.263string |
| | | | | 5.8635 | 233.4string | 0.125string |
06214$+$0216 | A | 2667 | | 30217 | 10.0654 | 262.9string | 0.333string |
06274$-$2544 | B | 114 | | 30733 | 10.0656 | 3.6string | 0.654string |
06298$-$5014 | R | 65 | AB | 30953 | 10.0656 | 260.9string | 0.590string |
| HDO | 195 | CD | | 10.0656 | 155.5string | 0.394string |
06314$+$0749 | A | 2817 | | 31089 | 10.0654 | 279.2string | 0.166string |
06319$-$0938 | A | 670 | | — | 10.0656 | 174.8string | 0.358string |
06336$-$1207 | HU | 43 | | — | 10.0656 | 307.6string | 0.796string |
06380$-$6132 | I | 5 | | 31711 | 6.1937 | 122.9string | 0.187string |
06410$+$0954 | CHR | 168 | Aa,Ab | 31978 | 10.0654 | 258.3string | 0.117string |
06478$+$0020 | STT | 157 | | 32572 | 10.0654 | 168.0string | 0.520string |
06481$-$0948 | A | 1056 | | — | 10.0656 | 257.3string | 0.335string |
06545$-$2734 | B | 706 | | 33197 | 10.0656 | 291.6string | 0.725string |
07013$-$0906 | A | 671 | | — | 10.0654 | 13.4string | 0.315string |
07029$-$1313 | HDS | 980 | | 33973 | 10.0656 | 220.4string | 0.488string |
07175$-$4659 | I | 7 | | 35296 | 1.0741 | 208.3string | 0.822string |
| | | | | 6.1882 | 205.8string | 0.775string |
| | | | | 10.0656 | 204.5string | 0.745string |
07185$-$5721 | HDS | 1013 | Aa,Ab | 35374 | 10.0656 | 215.6string | 0.340string |
07305$+$0743 | A | 2869 | | 36487 | 10.0654 | 124.7string | 0.188string |
07411$-$0124 | A | 1968 | | — | 10.0654 | 314.6string | 0.301string |
07417$+$0942 | STF | 1130 | | 37484 | 10.0654 | 46.8string | 0.456string |
07430$-$4511 | WSI | 99 | | 37606 | 1.0795 | 116.2string | 0.131string | $\Delta$m = 1.0
| | | | | 6.1882 | 47.2string | 0.050string |
07573$+$0108 | STT | 185 | | 38870 | 10.0654 | 12.6string | 0.328string |
08017$-$0836 | A | 1580 | | 39264 | 10.0654 | 291.4string | 0.294string |
08088$-$7638 | RST | 1381 | | 39887 | 1.0742 | 113.5string | 0.792string |
| | | | | 10.0657 | 111.1string | 0.783string |
08122$+$1739 | STF | 1196 | AB | 40167 | 1.0200 | 78.8string | 0.857string | 3
| | | | | 1.0768 | 78.5string | 0.854string |
| | | | | 5.8636 | 54.4string | 0.983string |
| | | | | 6.1910 | 53.3string | 0.988string |
08125$-$4616 | SEE | 96 | AB | 40183 | 10.0656 | 274.6string | 0.615string |
08225$-$5931 | HDS | 1190 | | 41037 | 10.0657 | 141.1string | 0.475string |
08250$-$4246 | RST | 4888 | AB | 41250 | 10.0656 | 104.6string | 0.519string |
08251$-$4910 | RST | 321 | | 41261 | 6.1992 | 106.8string | 0.291string |
| | | | | 10.0656 | 141.9string | 0.354string |
08315$-$1935 | I | 489 | | 41817 | 10.0656 | 295.1string | 0.350string |
08331$-$2436 | BU | 205 | AB | 41949 | 10.0656 | 296.5string | 0.567string |
08380$-$0844 | HDS | 1242 | | 42345 | 10.0654 | 12.7string | 0.196string |
08391$-$2240 | BU | 208 | AB | 42430 | 1.0825 | 33.3string | 1.312string |
08421$-$5245 | B | 1624 | | 42695 | 1.0825 | 105.9string | 0.290string |
| | | | | 6.1992 | 142.0string | 0.165string |
08447$-$4238 | CHR | 238 | | 42916 | 1.0825 | 0.4string | 0.056string | 1
08447$-$5443 | I | 10 | | 42913 | 10.0658 | 303.6string | 0.491string |
08462$-$1422 | HU | 120 | | 43038 | 10.0660 | 315.3string | 0.413string |
08468$+$0625 | SP | 1 | AB | 43109 | 1.0200 | 214.6string | 0.256string | 3
| | | | | 1.0769 | 215.9string | 0.246string |
| | | | | 5.8624 | 304.0string | 0.109string |
| | | | | 6.1910 | 338.5string | 0.062string |
08542$-$0229 | A | 1754 | | — | 10.0660 | 125.1string | 0.358string |
08549$+$2612 | A | 2131 | AB | 43766 | 1.0200 | 259.1string | 0.366string |
| | | | | 5.8636 | 281.2string | 0.353string |
09001$-$1228 | HU | 225 | AB | 44190 | 10.0660 | 258.2string | 0.445string |
09100$-$2845 | B | 179 | | 45003 | 10.0659 | 159.4string | 0.409string |
09174$-$7454 | I | 12 | AB | 45581 | 10.0657 | 261.0string | 0.314string |
09191$-$4128 | CHR | 239 | | 45705 | 1.0825 | 166.6string | 0.108string | 1
09264$-$4215 | B | 1122 | | 46290 | 10.0658 | 230.8string | 0.161string |
09272$-$0913 | A | 1588 | AB | 46365 | 10.0660 | 195.7string | 0.431string |
09278$-$0604 | B | 2530 | | 46404 | 1.0825 | 153.7string | 0.362string |
09285$+$0903 | STF | 1356 | | 46454 | 1.0200 | 86.5string | 0.588string | 3
| | | | | 1.0770 | 86.5string | 0.589string |
| | | | | 1.0825 | 87.2string | 0.583string |
| | | | | 5.8637 | 95.5string | 0.663string |
| | | | | 6.1913 | 96.0string | 0.667string |
09379$+$4554 | A | 1765 | | 47250 | 1.0284 | 116.4string | 0.265string |
09414$+$3857 | STF | 1374 | AB | 47527 | 1.0201 | 307.1string | 2.851string |
09487$-$2625 | I | 205 | | 48125 | 1.0797 | 332.8string | 1.097string |
09525$-$0806 | AC | 5 | AB | 48437 | 10.0660 | 50.2string | 0.551string |
09551$-$2632 | I | 843 | AB | 48645 | 10.0659 | 130.8string | 0.929string |
10043$-$2823 | I | 292 | | 49336 | 10.0659 | 305.7string | 0.713string |
10129$-$4729 | I | 361 | | 50036 | 1.0745 | 122.6: | 5.389: |
10161$-$5954 | HU | 1597 | | 50287 | 10.0658 | 144.6string | 0.294string |
10183$-$0326 | RST | 4454 | AB | — | 10.0660 | 222.5string | 0.294string |
10196$-$5724 | FIN | 162 | | — | 10.0658 | 177.1string | 0.408string |
10205$+$0626 | STF | 1426 | AB | 50637 | 10.0660 | 311.0string | 0.942string |
10279$+$3642 | HU | 879 | | 51233 | 1.0201 | 187.6string | 0.065string |
| | | | | 1.0284 | 189.9string | 0.067string |
10361$-$2641 | BU | 411 | | 51885 | 10.0659 | 307.0string | 1.340string |
10397$+$0851 | STT | 224 | AB | 52187 | 10.0660 | 140.6string | 0.512string |
10419$-$7811 | HDS | 1530 | | 52351 | 1.0746 | 113.6string | 0.278string | 1
| | | | | 10.0657 | 125.1string | 0.163string |
10446$+$0530 | A | 2771 | | | 10.0660 | 123.0string | 0.394string |
10452$-$5944 | WSI | 56 | | 52547 | 10.0658 | 59.2string | 0.275string |
10468$-$4925 | R | 155 | | 52727 | 1.0745 | 52.5string | 2.013string |
11080$-$6457 | HDS | 1588 | | 54405 | 10.0657 | 352.3string | 0.592string |
11102$-$1122 | HDS | 1590 | | 54580 | 10.0660 | 344.5string | 0.134string |
11182$+$3132 | STF | 1523 | AB | 55203 | 1.0201 | 268.7string | 1.827string | 3
| | | | | 1.0284 | 268.8string | 1.823string |
| | | | | 1.0773 | 267.1string | 1.810string |
11239$+$1032 | STF | 1536 | AB | 55642 | 1.0203 | 109.9string | 1.726string | 3
11330$-$3151 | WSI | 102 | Aa,Ab | 56343 | 1.0800 | 14.0string | 0.055string | $\Delta$m = 2.1
11446$-$4925 | RST | 9004 | AB | 57269 | 1.0825 | 172.9string | 0.335string | 4
11532$-$1540 | A | 2579 | | 57955 | 10.2655 | 38.4string | 0.404string |
12018$-$3439 | I | 215 | | 58669 | 10.0659 | 61.1string | 0.335string |
12060$+$6842 | STF | 3123 | AB | 59017 | 8.4611 | 211.7string | 0.257string |
12060$+$2312 | WSI | 103 | | 59015 | 8.4501 | 125.8string | 0.177string | $\Delta$m = 1.5
12064$-$6543 | FIN | 367 | Aa,Ab | 59050 | 10.0657 | 302.7string | 0.140string |
12247$-$2004 | B | 1716 | | 60545 | 10.0659 | 227.9string | 0.702string |
12266$-$6306 | DUN | 252 | AB | 60718 | 10.0657 | 112.1string | 3.956string |
12283$-$6146 | CPO | 12 | A,BC | 60845 | 1.0825 | 194.8string | 2.018string |
| | | | | 6.1941 | 190.9string | 2.021string |
| RST | 4499 | BC | | 6.1941 | 246.9string | 0.229string |
12313$-$4130 | I | 82 | | 61101 | 10.0659 | 11.0string | 0.737string |
12362$-$4650 | RST | 5526 | | 61517 | 10.0658 | 125.9string | 0.420string |
12446$-$5717 | FIN | 65 | AB | 62179 | 10.0688 | 90.3string | 0.268string |
12533$+$4246 | COU | 1579 | | — | 8.4611 | 120.6string | 0.207string |
12595$+$2814 | WSI | 1 | | 63407 | 8.4501 | 112.5string | 0.147string |
13064$+$2217 | WSI | 104 | | 63951 | 8.4501 | 142.0string | 0.061string | $\Delta$m = 0.3
13137$-$6248 | HDS | 1852 | | 64537 | 10.0714 | 120.8string | 0.161string |
13138$-$6335 | WSI | 57 | | — | 10.0687 | 236.4string | 0.194string |
13147$-$6355 | WSI | 58 | Aa,Ab | 64624 | 10.0688 | 279.3string | 0.244string |
| MLO | 3 | AB | | 10.0688 | 38.2string | 1.693string |
13198$+$4747 | HU | 644 | AB | 65026 | 8.4569 | 92.9string | 1.424string | 2
13253$+$7559 | STT | 267 | | 65489 | 8.4554 | 55.1string | 0.127string |
13372$-$6142 | I | 365 | AB | 66438 | 10.0688 | 198.7string | 0.326string |
13461$+$0507 | STF | 1781 | | 67186 | 10.0660 | 189.4string | 0.934string |
13520$-$3137 | BU | 343 | | 67696 | 10.0715 | 210.8string | 0.546string |
13539$-$1910 | HU | 898 | | 67859 | 8.4609 | 285.5: | 0.310: |
13563$+$0517 | STT | 273 | AB | 68081 | 10.0660 | 112.2string | 1.019string |
14153$+$0308 | STF | 1819 | | 69653 | 10.5907 | 178.3string | 0.847string |
14165$+$3334 | HU | 1266 | | 69742 | 8.4611 | 138.2string | 0.254string |
14179$+$6914 | A | 1102 | | 69870 | 8.4611 | 84.0string | 0.463string |
14462$-$2111 | FIN | 309 | | 72217 | 10.5878 | 94.3string | 0.224string |
14581$-$4852 | WSI | 80 | | 73241 | 6.1892 | 327.3string | 0.236string | 5, $\Delta$m = 0.7
14588$+$3551 | COU | 1136 | | — | 8.4611 | 64.3string | 0.085string |
14593$+$4649 | COU | 1760 | | 73346 | 8.4480 | 244.5string | 0.219string | 2
15127$+$6008 | HU | 1159 | | 74428 | 8.4611 | 350.8string | 0.289string |
15210$+$2104 | HU | 146 | | 75117 | 8.4566 | 124.4string | 0.696string | 6
15226$-$4755 | SLR | 20 | | 75255 | 6.1998 | 27.1string | 0.783string |
15245$+$3723 | CHR | 181 | Aa,Ab | 75411 | 7.6013 | 302.7string | 0.117string |
| | | | | 8.4611 | 264.2string | 0.093string |
15258$+$8430 | MLR | 347 | | 75529 | 8.4611 | 213.5: | 0.197: |
15278$+$2906 | JEF | 1 | | 75695 | 7.6012 | 142.2string | 0.301string |
15333$+$4149 | STF | 1956 | AB | 76156 | 7.6013 | 63.7: | 0.096: |
| | | | | 8.4611 | 237.6string | 0.107string |
15347$+$2655 | COU | 798 | | — | 7.6012 | 191.8string | 0.143string |
15348$-$2808 | RST | 1847 | AB | 76275 | 10.5853 | 334.5string | 0.919string | 2
15370$+$6426 | HU | 1168 | | 76466 | 8.4611 | 297.7: | 0.539: |
15396$+$7959 | STF | 1989 | | 76695 | 7.6067 | 22.5string | 0.653string |
| | | | | 8.4611 | 23.7string | 0.657string |
15432$-$4443 | I | 403 | | 76990 | 1.5638 | 226.0string | 0.622string |
15457$+$5040 | HU | 657 | | 77188 | 7.6013 | 179.4string | 0.402string |
| | | | | 8.4611 | 181.3string | 0.419string |
15462$-$2804 | BU | 620 | AB | 77235 | 10.5853 | 173.1string | 0.618string | 2
15471$-$5107 | B | 1790 | AB | — | 10.5854 | 89.9string | 0.330string |
15493$+$6032 | HU | 912 | | 77482 | 8.4612 | 131.7string | 0.317string |
15495$+$2528 | WSI | 111 | Aa,Ab | 77504 | 7.6012 | 177.8string | 0.129string | 5, $\Delta$m = 1.5
15542$+$1659 | A | 2080 | | 77875 | 7.6012 | 98.3string | 0.365string |
| | | | | 8.4614 | 99.9string | 0.373string |
15557$-$2645 | I | 977 | | 78002 | 10.5853 | 253.4string | 0.460string | 2
16044$-$1122 | STF | 1998 | AB | 78727 | 10.5880 | 355.8string | 0.948string |
16057$-$0617 | BU | 948 | AB | 78849 | 10.5907 | 86.2string | 0.742string |
16057$-$3252 | SEE | 264 | AB | 78842 | 10.5853 | 19.4string | 0.716string |
| WSI | 84 | Ba,Bb | | 10.5853 | 58.3string | 0.105string |
16079$+$1425 | A | 1798 | | 79034 | 7.6012 | 350.4string | 0.178string |
16120$-$1928 | BU | 120 | AB | 79374 | 10.5907 | 1.0string | 1.344string |
16161$-$3037 | I | 1586 | | 79706 | 10.5853 | 204.3string | 0.337string |
16169$+$0113 | A | 2181 | | 79774 | 7.5876 | 77.1string | 0.485string |
16170$-$5342 | I | 987 | | 79787 | 10.5854 | 265.6string | 0.452string | 2
16199$+$2238 | LAW | 4 | Aa,Ab | 80003 | 7.6012 | 27.2string | 0.176string | 1
| | | | | 8.4614 | 30.6string | 0.191string |
16224$-$3220 | JSP | 691 | | 80199 | 10.5853 | 23.5string | 0.529string | 2
16229$+$3803 | COU | 1281 | | — | 7.6013 | 193.1string | 0.199string |
| | | | | 8.4613 | 195.4string | 0.198string |
16238$+$6142 | STF | 2054 | AB | 80309 | 8.4612 | 350.5string | 0.982string |
16253$-$4909 | TOK | 50 | Aa,Ab | 80448 | 10.5854 | 189.0: | 0.229: | 1
16263$-$0943 | RST | 3945 | | — | 7.6013 | 335.6string | 0.237string |
16274$-$3729 | B | 872 | | 80603 | 10.5854 | 217.2string | 0.125string |
16309$+$0159 | STF | 2055 | AB | 80883 | 10.5879 | 37.2string | 1.384string |
16318$-$0701 | STF | 3105 | | 80954 | 10.5907 | 181.9string | 0.448string |
16366$+$6948 | BU | 953 | AB | 81332 | 7.6067 | 72.2string | 0.248string |
| | | | | 8.4612 | 74.0string | 0.251string |
16385$-$5728 | TOK | 51 | Aa,Ab | 81478 | 10.5854 | 61.2string | 0.286string | 1,2
| RST | 869 | AB | | 10.5854 | 46.7string | 0.825string | 2
16413$+$3136 | STF | 2084 | | 81693 | 1.4961 | 321.9string | 0.479string | 3
| | | | | 7.6068 | 197.4string | 1.038string |
16420$+$7353 | MLR | 198 | | 81757 | 7.6067 | 73.1string | 0.196string |
| | | | | 8.4611 | 74.1string | 0.199string |
16446$+$7145 | MLR | 182 | | 81961 | 7.6067 | 290.9: | 0.290: |
| | | | | 8.4612 | 284.4string | 0.297string |
16450$+$3842 | COU | 1284 | | — | 7.6068 | 355.2string | 0.180string |
| | | | | 8.4612 | 356.1string | 0.177string |
16450$+$2928 | COU | 490 | | — | 8.4613 | 293.9string | 0.121string |
16514$+$0113 | STT | 315 | | 82480 | 10.5879 | 310.5string | 0.645string |
16555$-$0820 | KUI | 75 | AB | 82817 | 1.4988 | 135.3string | 0.224string | 3
16566$+$5711 | A | 1143 | AB | 82910 | 7.6068 | 234.4string | 0.363string |
| | | | | 8.4612 | 234.1string | 0.358string |
17066$+$0039 | BU | 823 | AB | 83716 | 10.5908 | 162.6string | 0.974string |
17075$+$3810 | COU | 1291 | | 83791 | 7.6068 | 297.8string | 0.264string |
| | | | | 8.4613 | 299.8string | 0.270string |
17080$+$3556 | HU | 1176 | AB | 83838 | 7.6021 | 308.0: | 0.065: |
| | | | | 8.4613 | 170.8string | 0.050string |
17082$-$0105 | A | 1145 | | 83853 | 10.5908 | 345.4string | 0.660string |
17083$+$5051 | CHR | 60 | Aa,Ab | 83857 | 7.6068 | 9.9string | 0.208string |
17115$-$1630 | HU | 169 | | 84092 | 7.6013 | 198.8string | 0.235string |
17136$+$1716 | A | 2087 | | 84274 | 8.4614 | 128.1string | 0.405string |
17141$+$5608 | STT | 327 | | 84300 | 7.6068 | 329.1string | 0.457string |
| | | | | 8.4612 | 330.2string | 0.453string |
17141$-$0824 | BAR | 7 | | 84303 | 10.5908 | 51.9string | 1.443string |
17156$-$3836 | FIN | 355 | | 84425 | 10.5856 | 194.5string | 0.306string |
17157$-$0949 | A | 2592 | AB | 84430 | 10.5908 | 144.2string | 0.191string |
17161$+$2316 | COU | 315 | | 84468 | 8.4613 | 132.6string | 0.125string |
17184$+$3240 | BU | 628 | | 84653 | 7.6041 | 265.7string | 0.535string |
| | | | | 8.4613 | 266.3string | 0.540string |
17191$-$3849 | WSI | 60 | | — | 10.5883 | 333.6string | 1.331string | 1
17207$-$0706 | A | 2593 | | 84866 | 7.6014 | 2.6string | 0.237string |
17215$+$2845 | KUI | 80 | AB | 84934 | 8.4613 | 169.6string | 0.640string |
17221$+$2310 | COU | 415 | | 84976 | 7.6014 | 259.9string | 0.247string |
| | | | | 8.4613 | 258.7string | 0.249string |
17240$-$0921 | RST | 3972 | | 85141 | 7.6013 | 192.7string | 0.108string |
17247$+$3802 | HSL | 1 | Aa,Ac | 85209 | 7.6041 | 236.5string | 0.202string |
17248$-$5913 | I | 385 | AB | 85216 | 10.5854 | 121.4string | 0.388string |
| WSI | 87 | AD | | 10.5854 | 92.3string | 0.257string |
17272$-$2957 | B | 340 | | 85413 | 1.5614 | 295.1string | 0.172string |
17305$-$1446 | HU | 177 | | 85679 | 7.6013 | 199.8string | 0.205string |
| | | | | 10.5908 | 184.9string | 0.228string |
17308$-$3726 | B | 912 | | 85700 | 10.5856 | 168.7string | 0.257string |
17313$+$1901 | COU | 499 | | 85740 | 7.6014 | 4.7string | 0.102string |
17335$+$5734 | MLR | 571 | | 85923 | 7.6041 | 195.9string | 0.088string |
17366$+$0723 | A | 1156 | | 86174 | 7.6014 | 348.1string | 0.205string |
17368$-$2057 | HU | 751 | | — | 10.5908 | 296.7string | 0.273string |
17375$+$2419 | CHR | 63 | | 86254 | 7.6014 | 61.8string | 0.098string |
17375$-$3747 | B | 915 | | — | 10.5856 | 312.5string | 0.267string |
17390$+$0240 | WSI | 88 | | 86374 | 8.4561 | 5.2string | 0.169string | 5, $\Delta$m = 2.5
17490$+$3704 | COU | 1145 | | 87204 | 7.6041 | 321.6string | 0.124string |
17530$+$8354 | STT | 349 | | 87534 | 7.6041 | 47.2string | 0.477string |
17533$+$2459 | A | 235 | | 87565 | 7.6021 | 280.7string | 0.749string |
17535$-$0355 | TOK | 54 | | — | 10.5908 | 137.4string | 0.126string |
17535$-$1623 | WSI | 64 | | — | 10.5883 | 101.7string | 2.096string | 1
17539$-$3445 | B | 1871 | | 87616 | 10.5857 | 273.9string | 0.263string |
17555$-$4759 | B | 1873 | | — | 10.5856 | 342.3string | 0.173string |
17563$+$0259 | A | 2189 | | 87811 | 7.6014 | 260.4: | 0.090: |
17564$+$1820 | MCA | 49 | Aa,Ab | 87823 | 7.6014 | 88.8string | 0.096string |
| STF | 2245 | AB | 87823 | 7.6014 | 290.3string | 2.626string |
17575$+$1058 | BU | 1299 | AB | 87911 | 7.6014 | 91.7string | 0.274string |
17575$-$5740 | TOK | 55 | Ba,Bb | 87914 | 10.5854 | 165.0string | 0.148string |
17591$+$3228 | HU | 1185 | | 88057 | 7.6041 | 134.6string | 0.357string |
18002$+$8000 | BAG | 6 | Aa,Ab | 88136 | 7.5959 | 330.8string | 0.101string | 2
18003$+$2154 | WSI | 106 | Aa,Ab | — | 7.5878 | 125.6string | 0.042string | 7, $\Delta$m = 0.5
| A | 1374 | AB | — | 7.5878 | 49.1string | 0.399string |
18018$+$0118 | BU | 1125 | AB | 88290 | 7.5879 | 133.6string | 0.512string |
18031$-$0811 | STF | 2262 | AB | 88404 | 7.5880 | 283.5string | 1.623string | 2
18033$+$3921 | STF | 2275 | | 88432 | 7.5878 | 294.8string | 0.254string |
18054$+$6216 | HU | 1290 | | 88592 | 7.5876 | 309.6string | 0.224string: |
18078$+$2606 | CHR | 67 | Aa,Ab | 88818 | 7.5878 | 312.9string | 0.235string |
18096$+$0400 | STF | 2281 | AB | 88964 | 10.5908 | 288.2string | 0.644string |
18112$-$1951 | BU | 132 | AB | 89114 | 10.5908 | 188.0string | 1.396string |
18177$-$1940 | BU | 246 | AB | 89647 | 10.5909 | 114.9string | 0.497string | 2
18211$+$7244 | LAB | 5 | Aa,Ab | 89937 | 7.5876 | 216.3string | 0.132string |
18218$-$1619 | CHR | 69 | | 90001 | 7.5880 | 209.3string | 0.076string |
18236$-$2610 | HO | 566 | | 90128 | 10.5909 | 144.5string | 0.535string | 2
18253$+$4846 | STT | 351 | AC | 90284 | 7.5876 | 22.6string | 0.742string | 2
| HU | 66 | BC | | 7.5876 | 27.0string | 0.962string |
18337$+$5638 | TDS | 929 | | — | 7.6069 | 78.0string | 1.188string | 1
18338$+$1744 | STF | 2239 | AB,CD | 87407 | 10.5909 | 276.5string | 1.580string |
18339$+$5221 | A | 1377 | AB | 91013 | 7.5876 | 120.8string | 0.248string |
18374$+$7741 | STT | 363 | | 91298 | 7.5876 | 336.2string | 0.446string |
18384$+$0850 | HU | 198 | | 91389 | 7.5879 | 128.0string | 0.503string |
18384$-$0312 | A | 88 | AB | 91394 | 6.2001 | 154.3string | 0.101string | 3
| | | | | 7.5879 | 63.0: | 0.063: |
| | | | | 8.4507 | 17.6string | 0.100string |
18411$+$4416 | HDS | 2647 | | 91615 | 7.6069 | 268.0string | 0.140string | 1
18443$+$3940 | STF | 2382 | AB | 91919 | 8.4479 | 346.8string | 2.354string |
| STF | 2383 | CD | 91926 | 7.5878 | 78.7string | 2.351string |
| | | | | 8.4551 | 79.8string | 2.368string |
18450$+$4239 | HDS | 2657 | | 91994 | 7.6069 | 32.2string | 0.240string | 1
18475$+$4926 | BU | 971 | AB | 92204 | 8.4479 | 34.3string | 0.450string |
18552$+$3941 | TDT | 1126 | | — | 7.6071 | 189.9string | 0.323string | 1
18558$+$0327 | A | 2192 | | 92909 | 7.5879 | 31.0string | 0.170string |
18564$+$5854 | TDS | 955 | | — | 7.6071 | 145.7string | 0.885string | 1
18567$+$6440 | TDT | 1141 | | — | 7.6069 | 153.3string | 0.450string | 1
18579$+$4215 | TDT | 1149 | | — | 7.6071 | 122.7: | 0.456: | 1
19006$+$3952 | HDS | 2696 | | 93335 | 7.6071 | 136.8string | 0.294string | 1
19025$+$5142 | TDT | 1193 | | — | 7.6071 | 90.6: | 1.168: | 1
19029$+$4930 | COU | 2626 | | — | 7.6071 | 84.5string | 0.893string | 1
19038$+$4040 | COU | 2196 | | — | 7.6071 | 330.1string | 0.490string | 1
19045$+$6245 | TDT | 1211 | | — | 7.6069 | 31.5string | 0.795string | 1
19064$-$3704 | HJ | 5084 | | 93825 | 6.1947 | 29.6string | 1.316string | 3
| | | | | 10.5909 | 8.5 | 1.341string |
19070$+$7143 | TDT | 1228 | | — | 7.6069 | 223.9string | 0.916string | 1
19089$+$3404 | COU | 1462 | | 94056 | 7.6014 | 188.9string | 0.116string |
19098$-$1948 | B | 427 | | 94144 | 7.6014 | 182.9: | 0.050: |
19106$+$5429 | A | 1391 | | 94194 | 7.6016 | 5.2string | 0.190string |
19111$+$3847 | STF | 2481 | A,BC | 94252 | 1.4990 | 20.5string | 4.578string |
| SE | 2 | BC | | 1.4989 | 74.0string | 0.400string |
| | | | | 7.6016 | 51.5string | 0.221string |
19124$-$3304 | OL | 22 | | 94367 | 10.5909 | 47.3string | 0.493string | 2
19216$+$5223 | BU | 1129 | | 95156 | 7.6016 | 342.5string | 0.269string |
19222$-$0735 | A | 102 | AB | — | 7.6014 | 242.4string | 0.155string |
19258$-$3006 | I | 1401 | AB | — | 10.5909 | 109.1string | 0.612string | 2
19296$+$1224 | A | 1653 | | 95845 | 7.6014 | 152.9string | 0.198string |
19296$-$1239 | HU | 75 | | 95847 | 7.6014 | 34.3string | 0.165string |
19302$+$5639 | A | 712 | | 95901 | 7.6016 | 85.8string | 0.116string |
19308$+$6337 | HU | 951 | | 95955 | 7.6016 | 197.2string | 0.187string |
19313$+$4729 | A | 713 | | 96011 | 7.6016 | 296.9string | 0.236string |
| | | | | 8.4562 | 300.1string | 0.230string |
19352$+$0825 | BAG | 27 | Aa,Ab | 96339 | 7.6014 | 36.6string | 0.291string | 1
19356$+$4002 | A | 1400 | | 96371 | 7.6016 | 115.1string | 0.320string |
19389$+$3514 | HU | 953 | | — | 7.6016 | 19.5string | 0.237string |
19411$+$1349 | KUI | 93 | | 96840 | 10.5909 | 310.9string | 0.168string |
19426$-$5901 | I | 119 | | 96955 | 1.5642 | 150.6string | 2.190string |
19471$-$1953 | BU | 146 | | 97348 | 10.5909 | 237.8string | 0.539string |
19474$-$0148 | A | 2993 | | 97367 | 7.6042 | 298.8string | 0.150string |
19531$-$1436 | CHR | 90 | | 97849 | 10.5909 | 20.2string | 0.266string |
19550$+$4152 | HO | 581 | | 98001 | 1.4991 | 98.0string | 0.354string | 3
19553$-$0644 | STF | 2597 | AB | 98038 | 10.5909 | 100.7string | 0.595string |
19556$+$5226 | YR | 2 | Aa,Ab | 98055 | 7.6042 | 53.1string | 0.116string |
19573$+$0513 | A | 604 | | 98203 | 7.6042 | 22.5string | 0.110string |
20012$-$3835 | HDO | 294 | | 98556 | 10.5861 | 29.4string | 1.176string |
20014$+$1045 | TOK | 34 | Aa,Ab | 98578 | 7.6042 | 339.0: | 0.378: | 1
20062$+$3520 | TDT | 1981 | | — | 7.6071 | 0.0string | 0.420string | 1
20095$+$3525 | TDT | 2034 | | — | 7.6071 | 120.1string | 0.339string | 1
20096$+$5034 | TDT | 2035 | | — | 7.6071 | 90.8string | 0.783string | 1
20124$+$5354 | HDS | 2880 | | 99573 | 7.6071 | 273.7string | 0.113string | 1
20154$+$6412 | MLR | 60 | AB | 99832 | 7.6041 | 348.8string | 0.220string |
20187$+$5823 | TDT | 2157 | | — | 7.6071 | 35.3string | 0.765string | 1
20203$+$3924 | A | 1427 | AB | 100268 | 7.6042 | 123.7string | 0.271string |
20213$+$0250 | HLD | 158 | | — | 7.6042 | 43.9string | 1.090string |
20251$+$5936 | A | 730 | | 100714 | 7.6041 | 288.2string | 0.143string |
20267$-$4334 | I | 1420 | | 100836 | 10.5860 | 346.6string | 0.447string | 2
20274$+$7245 | A | 871 | | — | 7.6041 | 183.2string | 0.305string |
20290$+$0710 | A | 610 | | 101046 | 7.6042 | 70.2string | 0.436string |
20302$+$2651 | WOR | 9 | AB | 101150 | 8.4589 | 261.3string | 0.518string |
20311$+$1548 | A | 1675 | | 101223 | 1.4991 | 339.5string | 0.078string |
20325$-$1637 | SEE | 512 | | 101357 | 7.5880 | 136.5string | 0.158string |
20329$+$1357 | L | 35 | CD | 101390 | 7.6017 | 164.2string | 0.468string |
20347$-$6319 | HU | 1615 | | 101543 | 10.5858 | 85.5string | 0.173string |
20375$+$1436 | BU | 151 | AB | 101769 | 1.4991 | 347.1string | 0.549string | 3
| | | | | 7.6017 | 7.7string | 0.499string |
20451$-$2346 | I | 666 | | — | 1.5699 | 125.1string | 0.204string |
20454$+$5735 | CIA | 1 | Aa,Ab | 102431 | 8.4508 | 205.2string | 0.093string |
20462$+$1554 | WSI | 110 | Aa,Ab | 102490 | 8.4615 | 327.8string | 0.077string | $\Delta$m = 1.0
20514$-$0538 | STF | 2729 | AB | 102945 | 7.5880 | 23.3string | 0.852string |
20537$+$5918 | A | 751 | | 103130 | 7.6018 | 69.9string | 0.176string |
20591$-$1313 | HU | 83 | | 103574 | 7.5880 | 82.4string | 0.250string |
20596$-$4746 | I | 1431 | | — | 1.5699 | 359.6: | 0.467: |
21010$+$3959 | A | 1438 | | — | 8.4508 | 254.2string | 0.299string |
21041$+$0300 | WSI | 6 | | 103987 | 7.5880 | 132.7string | 0.121string | 1
| | | | | 8.4550 | 167.5string | 0.095string |
21044$-$1951 | FIN | 328 | | 104019 | 7.5880 | 176.2string | 0.268string |
| | | | | 10.5861 | 151.6: | 0.321: |
21056$+$6030 | A | 758 | AB | 104113 | 7.5990 | 122.7: | 0.343: |
21105$+$4300 | VKI | 39 | | — | 7.6072 | 234.3string | 1.323string | 1
21115$+$2144 | COU | 227 | | 104612 | 7.6017 | 105.6string | 0.523string |
21120$+$4419 | TDT | 2816 | | — | 7.6072 | 88.6: | 0.289: | 1
21125$+$2821 | HO | 152 | | 104701 | 7.6017 | 131.0string | 0.204string |
21135$+$1559 | HU | 767 | | 104771 | 7.6017 | 164.5string | 0.174string |
21137$+$6424 | H 1 | 48 | | 104788 | 7.5990 | 238.0string | 0.276string |
21148$+$4946 | COU | 2653 | | — | 7.6072 | 65.1string | 0.298string | 1
21148$+$3803 | AGC | 13 | AB | 104887 | 7.6018 | 250.1string | 0.753string |
21198$-$2621 | BU | 271 | AB | 105312 | 10.5861 | 244.3: | 1.704: |
21200$+$5436 | TDT | 2908 | | — | 7.6073 | 295.9string | 0.301string | 1
21203$+$5354 | TDT | 2910 | | — | 7.6072 | 284.2string | 0.666string | 1
21226$+$3846 | HDS | 3044 | | 105543 | 7.6072 | 141.0string | 0.122string | 1
21239$+$8317 | HDS | 3049 | | 105644 | 7.6073 | 335.1string | 0.303string | 1
21243$+$3740 | WSI | 7 | | 105676 | 7.6018 | 101.6string | 0.157string | 1
| | | | | 8.4615 | 94.5string | 0.166string |
21243$+$0343 | A | 2288 | | 105682 | 7.5881 | 234.7string | 0.387string |
21251$+$0923 | BU | 164 | AB | 105747 | 7.5881 | 144.6string | 0.085string |
21268$+$3732 | COU | 1821 | | — | 7.6072 | 93.8string | 0.651string | 1
21310$-$3633 | B | 1008 | AB | 106224 | 10.5861 | 21.8string | 0.150string |
21373$+$6604 | HDS | 3077 | | 106733 | 7.6073 | 273.4string | 0.127string | 1
21410$+$2920 | STT | 448 | | 107060 | 7.5991 | 145.6string | 0.156string |
21424$+$4105 | KUI | 108 | | 107162 | 1.5020 | 190.8string | 0.101string |
| | | | | 8.4508 | 59.6string | 0.146string |
21434$+$2353 | HU | 374 | | 107254 | 7.5991 | 131.0string | 0.128string |
21435$+$4448 | LIN | 1 | Aa,Ab | 107256 | 8.4509 | 187.3string | 0.165string |
21446$+$2539 | BU | 989 | AB | 107354 | 7.5991 | 319.6string | 0.155string |
21459$+$1153 | A | 1223 | AB | 107461 | 7.5991 | 331.0string | 0.115string |
21477$-$1813 | CHR | 223 | | 107610 | 7.5992 | 299.8string | 0.169string | 1
21477$-$3054 | FIN | 330 | AB | 107608 | 10.5861 | 43.7string | 0.096string |
21504$-$5818 | HDS | 3109 | | 107806 | 1.5673 | 309.2string | 0.163string | 1
21511$+$6650 | HU | 972 | | 107855 | 7.5990 | 253.5string | 0.335string |
21538$-$2000 | HU | 380 | BC | 108072 | 7.5992 | 247.4string | 0.397string |
21539$+$2821 | WSI | 8 | | 108084 | 7.5991 | 258.7: | 0.115: |
21545$+$4403 | A | 620 | | 108137 | 8.4564 | 285.0string | 0.355string |
21556$+$3849 | A | 1449 | | 108231 | 7.5991 | 237.5string | 0.266string |
21593$+$4606 | COU | 2138 | | 108536 | 7.5991 | 170.4string | 0.216string |
21597$+$4907 | HU | 774 | | 108577 | 7.5991 | 174.9string | 0.196string |
22134$-$3729 | B | 2056 | | — | 10.5861 | 180.2string | 0.150string |
22241$-$0450 | BU | 172 | AB | 110578 | 10.5914 | 38.0string | 0.409string |
22307$+$1758 | COU | 234 | | 111125 | 7.5991 | 199.4string | 0.151string |
22330$+$6955 | STF | 2924 | AB | 111314 | 7.5990 | 184.3string | 0.129string |
22339$+$6550 | HU | 983 | | — | 7.5990 | 152.5string | 0.097string |
22342$-$1841 | HU | 389 | AB | 111406 | 7.5992 | 281.6string | 0.167string |
22405$+$2002 | WSI | 9 | | 111943 | 8.4615 | 133.4: | 0.175: | 1
22409$+$1433 | HO | 296 | AB | 111974 | 1.5020 | 14.0string | 0.234string | 3
| | | | | 1.5645 | 12.8string | 0.228string |
| | | | | 5.8626 | 104.7string | 0.175string |
| | | | | 7.5883 | 82.5string | 0.310string |
| | | | | 8.4617 | 77.9string | 0.357string |
22519$-$4647 | I | 683 | | 112908 | 1.5646 | 345.2string | 1.917string |
22550$+$5132 | HU | 785 | | 113166 | 7.5884 | 102.5string | 0.147string |
22562$+$7250 | STT | 484 | AB | 113273 | 7.5885 | 100.8string | 0.355string |
22586$-$4531 | HU | 1335 | | 113454 | 1.5646 | 109.3string | 0.350string |
22595$+$4355 | COU | 2146 | | 113524 | 7.6042 | 287.2string | 0.243string | 8
23026$+$4245 | BU | 1147 | AB | 113788 | 1.5020 | 354.6string | 0.318string |
23114$+$3813 | HO | 197 | AB | 114504 | 7.5884 | 286.9string | 0.197string |
23164$+$6407 | BU | 992 | | 114898 | 7.5884 | 21.3string | 0.249string |
23176$+$1818 | HU | 400 | | 115011 | 7.5883 | 83.7string | 0.342string |
23191$-$1328 | MCA | 74 | Aa,Ab | 115126 | 5.8655 | 25.1string | 0.100string | 3
23199$+$2844 | COU | 439 | | 115189 | 7.5884 | 314.4string | 0.125string |
23209$+$1643 | HEI | 88 | | 115279 | 7.5883 | 148.2string | 0.157string |
23210$+$1715 | WSI | 11 | | — | 7.5883 | 161.4string | 0.108string | 1
23277$-$1502 | HU | 295 | | 115404 | 10.5861 | 282.1string | 0.317string |
23319$+$7317 | A | 897 | | 116133 | 7.5992 | 331.4: | 0.178: |
23350$+$0136 | MEL | 9 | AC | 116384 | 7.6018 | 342.2string | 0.760string |
23374$+$0737 | FOX | 102 | AB | 116571 | 7.5994 | 41.6string | 0.173string |
23378$+$6601 | HU | 1324 | | 116598 | 7.5992 | 175.0string | 0.372string |
23382$+$5514 | A | 1493 | | 116634 | 7.5885 | 311.7string | 0.155string |
23424$+$3903 | A | 1494 | | 116952 | 7.5992 | 338.7string | 0.099string |
23498$+$2741 | A | 424 | | 117511 | 7.5992 | 152.1string | 0.159string |
23502$+$1940 | COU | 344 | | 117531 | 7.5992 | 332.4string | 0.188string |
23528$+$3841 | A | 1496 | AB | 117752 | 7.5992 | 152.4string | 0.413string | 2
23561$+$2520 | A | 426 | | 118005 | 1.5021 | 301.2: | 0.370: |
23586$-$1408 | RST | 4136 | AB | 118205 | 7.6019 | 215.0string | 0.115string |
11footnotetext: Confirming observation.
22footnotetext: Two measurements are averaged to give this mean position.
33footnotetext: Calibration system.
44footnotetext: Previously known as RST3558a.
55footnotetext: Confirmed with HRCam on SOAR 4.2m, CTIO 4m (Tokovinin et al.
2010).
66footnotetext: Three measurements are averaged to give this mean position.
77footnotetext: First measure of newly resolved pair. Primary is HD 341480.
88footnotetext: Confirming observation or more likely a new component.
|
arxiv-papers
| 2011-09-21T16:09:55 |
2024-09-04T02:49:22.369693
|
{
"license": "Public Domain",
"authors": "Brian D. Mason, William I. Hartkopf, Deepak Raghavan, John P.\n Subasavage, Lewis C. Roberts Jr., Nils H. Turner, Theo A. ten Brummelaar",
"submitter": "William Hartkopf",
"url": "https://arxiv.org/abs/1109.4569"
}
|
1109.4653
|
# Can the evolution of music be analyzed in a quantitative manner?
Vilson Vieira http://automata.cc vilson@void.cc Renato Fabbri
http://www.estudiolivre.org/el-user.php?view_user=gk renato.fabbri@gmail.com
Gonzalo Travieso gonzalo@ifsc.usp.br Instituto de Física de São Carlos,
Universidade de São Paulo (IFSC/USP) Luciano da Fontoura Costa
http://cyvision.ifsc.usp.br/~luciano/ ldfcosta@gmail.com Instituto de Física
de São Carlos, Universidade de São Paulo (IFSC/USP)
###### Abstract
We propose a methodology to study music development by applying multivariate
statistics on composers characteristics. Seven representative composers were
considered in terms of eight main musical features. Grades were assigned to
each characteristic and their correlations were analyzed. A bootstrap method
was applied to simulate hundreds of artificial composers influenced by the
seven representatives chosen. Afterwards we quantify non-numeric relations
like dialectics, opposition and innovation. Composers differences on style and
technique were represented as geometrical distances in the feature space,
making it possible to quantify, for example, how much Bach and Stockhausen
differ from other composers or how much Beethoven influenced Brahms. In
addition, we compared the results with a prior investigation on philosophy
Fabbri . Opposition, strong on philosophy, was not remarkable on music.
Supporting an observation already considered by music theorists, strong
influences were identified between composers by the quantification of
dialectics, implying inheritance and suggesting a stronger master-disciple
evolution when compared to the philosophy analysis.
music, musicology, pattern recognition, statistics
###### pacs:
89.75.Fb,05.65.+b
††preprint: AIP/123-QED
## I Introduction
In the history of music, composers developed their own styles along a
continuous search for coherence or unity. In the words of Anton Webern Webern
, “[…] ever since music has been written most great artists have striven to
make this unity ever clearer. Everything that has happened aims at this […]”.
Along this process we can identify a constant heritage of style from one
composer to another as a gradual development from its predecessor, contrasting
with the necessity for innovation. Quoting Lovelock: “[…] by experiment that
progress is possible; it is the man with the forward-looking type of mind […]
who forces man out of the rut of ‘what was good enough for my father is good
enough for me’.” Lovelock . Thus, development in music follows a dichotomy:
while composers aims on innovation, creating their own styles, their technique
is based on the works of their predecessors, in a master-apprentice tradition.
Other fields like philosophy demonstrate a well-defined trend when considering
innovation: unlike music, the quest for difference seems to drive
philosophical changes Deleuze . Recently, this observation became more evident
with the application of a quantitative method Fabbri where multivariate
statistics was used to measure non-numeric relations and to represent the
historical development as time-series. More specifically, the method consists
of scoring memorable philosophers based on some relevant characteristics. The
group of philosophers was chosen based on historical relevance. The scores
assigned to each philosopher characteristics define a state vector in a
feature space. Correlations between these characteristic vectors were
identified and principal component analysis (PCA) was applied to represent the
philosophical history as a planar space where we could identify interesting
properties. Furthermore, concepts like dialects can be modeled as mathematical
relations between the philosophical states. Here, we extend that analysis to
music.
The application of statistical analysis to music is not recent. On musicology,
statistical methods have been used to identify many musical characteristics.
Simonton Simonton1991829 ; Simonton1977791 used time-series analysis to
measure the creative productivity of composers based on their music and
popularity. Kozbelt Kozbelt01012009 ; Kozbelt01012007 also analyzed the
productivity, but based on the measure of performance time of the compositions
and investigated the relation between productivity and versatility. More
recent works Kranenburg2004 ; Kranenburg2007 use machine-learning algorithms
to recognize musical styles of selected compositions.
Differently from these works, we are not interested in applying statistical
analysis to music but on characterizing composers. Eight characteristics were
described and scored by the authors, based on the recurrent appearance of
these attributes in music pieces. We chose seven representative composers from
different periods of music history. This group was chosen purposely to model
their influence on contemporaries, represented as a group of “artificial
composers”, sampled by a bootstrap method Varian . The same statistical method
used in philosophy Fabbri was applied to this set of composers and their
characteristics, allowing us to compare the results from both fields. The
results present contrasting historical facts, recognized along the history of
music, quantified by application of distance metrics which allowed us to
formalize concepts like dialectics, innovation and opposition, resulting in
interpretations of music development which are compatible with perspectives
from musicians and theorists Webern ; Lovelock .
## II Mathematical Description
A sequence $S$ of $C$ music composers was chosen based on their relevance at
each period of the classical music history. As done for philosophers Fabbri ,
the set of $C$ measurements define a $C$-dimensional space henceforth referred
as the _musical space_. The characteristic vector $\vec{v}_{i}$ of each
composer $i$ defines a respective _composer state_ in the musical space. For
the set of $C$ composers, we defined the same relations adapted for
philosophers Fabbri , sumarized in Table 1.
Table 1: Description of mathematical relations defined for each composer $i$, $j$ and $k$ given a set of $C$ composers as a time-sequence $S$. Average state | $\vec{a}_{i}=\frac{1}{i}\sum_{k=1}^{i}\vec{v}_{k}.$
---|---
Opposite state | $\vec{r}_{i}=\vec{v}_{i}+2(\vec{a}_{i}-\vec{v}_{i})$
Opposition vector | $\vec{D}_{i}=\vec{r}_{i}-\vec{v}_{i}$
Musical move | $\vec{M}_{i,j}=\vec{v}_{j}-\vec{v}_{i}$
Opposition index | $W_{i,j}=\frac{\left<\vec{M}_{i,j},\vec{D}_{i}\right>}{||\vec{D}_{i}||^{2}}$
Skewness index | $s_{i,j}=\sqrt{\frac{|\vec{v}_{i}-\vec{v}_{j}|^{2}|\vec{a}_{i}-\vec{v}_{i}|^{2}-[(\vec{v}_{i}-\vec{v}_{j}).(\vec{a}_{i}-\vec{v}_{i})]^{2}}{|\vec{a}_{i}-\vec{v}_{i}|^{2}}}$
Counter-dialectics |
index | $d_{i\rightarrow k}=\frac{|\left<\vec{v}_{j}-\vec{v}_{i},\vec{v}_{k}\right>+\frac{1}{2}\left<\vec{v}_{i}-\vec{v}_{j},\vec{v}_{i}+\vec{v}_{j}\right>|}{|\vec{v}_{j}-\vec{v}_{i}|}$
It is important to note some details about these relations. Given a set of $C$
composers as a time-sequence $S$, the _average state_ at time $i$ is defined.
The _opposite state_ is defined as the “counterpoint” of a musical state
$\vec{v}_{i}$, considering its average state: everything running along the
opposite direction of $\vec{v}_{i}$ are understood as opposition. In other
words, any displacement from $\vec{v}_{i}$ along the direction $\vec{r}_{i}$
is a _contrary move_ , and any displacement from $\vec{v}_{i}$ along the
direction $-\vec{r}_{i}$ is an _emphasis move_. Given a musical state
$\vec{v}_{i}$ and its opposite state $\vec{r}_{i}$, we can define the
_opposition vector_ $\vec{D}_{i}$. These details are better understood
analyzing Figure 1.
Figure 1: Graphical representation of the measures derived from a _musical
move_Fabbri .
Considering the time-sequence $S$ we defined relations between pairs of
composers. The _musical move_ implied by two successive composers at time $i$
and $j$ corresponds to the $\vec{M}_{i,j}$ vector extending from $\vec{v}_{i}$
to $\vec{v}_{j}$. Given the musical move we can quantify the intensity of
opposition by the projection of $\vec{M}_{i,j}$ along the opposition vector
$\vec{D}_{i}$, normalized, yelding the _opposition index_. Considering the
same musical move, the _skewness index_ is the distance between $\vec{v}_{j}$
and the line $L_{i}$ defined by the vector $\vec{D}_{i}$, and therefore
quantifies how much the new musical state departs from the respective
opposition move.
A relationship between a triple of successive composers can also be defined.
Considering $i$, $j$ and $k$ being respectively understood as the _thesis_ ,
_antithesis_ and _synthesis_ , we defined the _counter-dialectics index_ by
the distance between the musical state $\vec{v}_{k}$ and the middle line
$ML_{i,j}$ defined by the thesis and antithesis, as shown in Figure 2. In
higher dimensional philosophical spaces, the middle-hyperplane defined by the
points which are at equal distances to both $\vec{v}_{i}$ and $\vec{v}_{j}$
should be used instead of the middle line $ML_{i,j}$. The proposed equation
for counter-dialectics scales to hyperplanes.
The counter-dialectics index is suggested and used instead of dialectics index
to maintain compatibility with the use of a distance from point to line as
adopted for the definition of skewness.
Figure 2: Graphical representation of the quantification of dialecticsFabbri .
## III Musical Characteristics
To create the musical space we derived eight variables corresponding to
distinct characteristics commonly found in music compositions. The
characteristics are related with the basic elements of music — melody,
harmony, rhythm, timbre, form and tessitura BennettHistory — and non-musical
issues like historical events that have influenced the compositions, for
example, the presence of Church. All the eight characteristics are listed
below:
Sacred - Secular ($S$-$S_{c}$): the sacred or religious music is composed
through religious influence or used for its purposes. Masses, motets and
hymns, dedicated to the Christian liturgy, are well known examples Lovelock .
Secular music has no or little relation with religion and includes popular
songs like Italian madrigals and German lieds BennettHistory .
Short duration - Long duration ($D_{s}$-$D_{l}$): compositions are quantified
having short duration when they do not have more than few minutes of
execution. Long duration compositions have at least 20 minutes of execution or
more. The same consideration was adopted by Kozbelt Kozbelt01012009 ;
Kozbelt01012007 in his analysis of time execution.
Harmony - Counterpoint ($H$-$C$): harmony regards the vertical combination of
notes, while counterpoint focuses on horizontal combinations BennettHistory .
Vocal - Instrumental ($V$-$I$): compositions using just vocals (e.g.
_cantata_) or exclusively instruments (e.g. _sonata_). It is interesting to
note the use of vocals over instruments on Sacred compositions Lovelock .
Non-discursive - Discursive ($D_{n}$-$D$): compositions based or not on verbal
discourse, like programmatic music or Baroque rhetoric, where the composer
wants to “tell a history” invoking images to the listeners mind BennettHistory
. Its contrary part is known as absolute music where the music is written to
be appreciated simply by what it is.
Motivic Stability - Motivic Variety ($M_{s}$-$M_{v}$): motivic pieces presents
equilibrium between repetition, reuse and variation of melodic motives. Bach
is noticeable by his development by variation of motives, contrasting with the
constantly inventive use of new materials by Mozart Webern .
Rhythmic Simplicity - Rhythmic Complexity ($R_{s}$-$R_{c}$): presence or not
of polyrhythms, the use of independent rhythms at the same time — also known
as rhythmic counterpointBennettHistory — a characteristic constantly found in
Romanticism and the works of 20th-century composers like Stravinsky.
Harmonic Stability - Harmonic Variety ($H_{s}$-$H_{v}$): rate of tonality
change along a piece or its stability. After the highly polyphonic development
in Renaissance, Webern regarded Beethoven as the composer who returned to the
maximum exploration of harmonic variety Webern .
## IV Results and Discussion
Memorable composers were chosen as key representatives of musical development.
This group was chosen purposely to model their influence over contemporaries,
creating a concise parallel with music history. We modeled this group of
influenced composers as new artificial samples generated by a bootstrap
method, better explained in this section.
The sequence is ordered chronologically and presented on Table 2 with each
composer related with its historical period.
Table 2: The sequence of music composers ordered chronologically with the period each represent. Composer | Movement
---|---
Monteverdi | Renaissance
Bach | Baroque
Mozart | Classical
Beethoven | Classical $\to$ Romantic
Brahms | Romantic
Stravinsky | 20th-century
Stockhausen | Contemporary
The quantification of the eight musical characteristics was performed jointly
by the authors of this article and is shown in Table 3. The scores were
numerical values between 1 and 9. Values more close of 1 reveals the composer
tended to the first element of each characteristic pair, and vice versa. We
emphasize that the focus of this work is not on the specific characteristics
used or their attributed numerical values, which can be disputed, but on the
techniques employed for the quantitative analysis.
Table 3: Quantification of the eight music characteristics for each of the seven composers. Composer | $S$-$S_{c}$ | $D_{s}$-$D_{l}$ | $H$-$C$ | $V$-$I$ | $D_{n}$-$D$ | $M_{s}$-$M_{v}$ | $R_{s}$-$R_{c}$ | $H_{s}$-$H_{v}$
---|---|---|---|---|---|---|---|---
Monteverdi | 3.0 | 8.0 | 5.0 | 3.0 | 7.0 | 5.0 | 3.0 | 7.0
Bach | 2.0 | 6.0 | 9.0 | 2.0 | 8.0 | 2.0 | 1.0 | 5.0
Mozart | 6.0 | 4.0 | 1.0 | 6.0 | 6.0 | 7.0 | 2.0 | 2.0
Beethoven | 7.0 | 8.0 | 2.5 | 8.0 | 5.0 | 4.0 | 4.0 | 7.0
Brahms | 6.0 | 6.0 | 4.0 | 7.0 | 4.5 | 6.5 | 5.0 | 7.0
Stravinsky | 8.0 | 7.0 | 6.0 | 7.0 | 8.0 | 5.0 | 8.0 | 5.0
Stockhausen | 7.0 | 4.0 | 8.0 | 7.0 | 5.0 | 8.0 | 9.0 | 6.0
This data set defines an 8-dimensional musical space where each dimension
corresponds to a characteristic that aplies to all 7 composers. Such small
data set is not adequate for statistical analysis and the imediate analysis of
this set would be highly biased by the small sample.
### IV.1 Bootstrap method for sampling _artificial composers_
To simulate a more realistic musical trajectory, we used a bootstrap method
for generating _artificial composers_ contemporaries of those seven chosen.
The bootstrap routine generated randomized scores $\vec{r}$. The values are
not totally random, following a probability distribution that models the
original $n=7$ scores, given by
$p(\vec{r})=\sum^{n}_{i=1}e^{\frac{d_{i}}{2\sigma^{2}}}$ where $d_{i}$ is the
distance between a random score $\vec{r}$ and the original score chart. For
each step a value $p(\vec{r})$ is generated and compared with a random
normalized value, characterizing the Monte Carlo Robert2011 method to choose
a set of samples. This samples simulates new randomized composers score charts
— while respecting the historical influence of the main 7 original exponents.
Higher values of $p(\vec{r})$ imply a stronger influence of the original
scores over $\vec{r}$. For the analysis we used 1000 bootstrap samples
obtained by the bootstrap process together with the original scores,
considering $\sigma=1.1$. Other values for $\sigma$ were used yelding
distributions with bootstrap samples closer to or further from the original
musical states, which does not affected the musical space substantially.
Pearson correlation coefficients between the eight musical characteristics
chosen are presented in Table 4. Emphasized coefficients have absolute values
larger than 0.5.
Table 4: Pearson correlation coefficients between the eight musical characteristics. - | $S$-$S_{c}$ | $D_{s}$-$D_{l}$ | $H$-$C$ | $V$-$I$ | $D_{n}$-$D$ | $M_{s}$-$M_{v}$ | $R_{s}$-$R_{c}$ | $H_{s}$-$H_{v}$
---|---|---|---|---|---|---|---|---
$S$-$S_{c}$ | - | -0.2 | -0.06 | 0.69 | -0.18 | 0.19 | 0.56 | -0.16
$D_{s}$-$D_{l}$ | - | - | -0.14 | -0.13 | 0.2 | -0.48 | -0.2 | 0.37
$H$-$C$ | - | - | - | -0.23 | 0.26 | 0.05 | 0.46 | 0.03
$V$-$I$ | - | - | - | - | -0.33 | 0.17 | 0.42 | -0.06
$D_{n}$-$D$ | - | - | - | - | - | -0.3 | 0.02 | -0.22
$M_{s}$-$M_{v}$ | - | - | - | - | - | - | 0.26 | -0.15
$R_{s}$-$R_{c}$ | - | - | - | - | - | - | - | -0.02
$H_{s}$-$H_{v}$ | - | - | - | - | - | - | - | -
We can identify some interesting relations between the pairs of
characteristics that reflect important facts in music history. For instance,
the Pearson correlation coefficient of 0.69 was obtained for the pairs
$S$-$S_{c}$ (Sacred or Secular) and $V$-$I$ (Vocal or Instrumental), which
indicate that sacred music tends to be more vocal than instrumental. The
coefficient of 0.56 also shows it does not commonly use polyrhythms as we can
see analysing the pairs $S$-$S_{c}$ and $R_{s}$-$R_{c}$ (Rhythmic Simplicity
or Complexity). Negative coefficients of -0.33 for the pairs $V$-$I$ and
$D_{n}$-$D$ (Non-discursive or Discursive) indicated that composers who used
just voices on their compositions also preferred to use programmatic musics
techniques like baroque rhetoric.
PCA was applied to this set of data, yielding the new variances given in Table
5 in terms of percentages of total variance. We can note the concentration of
variance along the four first PCA axes, a common effect also observed while
analyzing philosophers characteristics Fabbri . This would usualy mean that we
could consider just four dimensions but as we will see below our measurements
differs considerably with the inclusion of all eight components.
Table 5: New variances after PCA, in percentages for scores on 4. Eigenvalue | Value
---|---
$\lambda_{1}$ | 32 %
$\lambda_{2}$ | 20 %
$\lambda_{3}$ | 17 %
$\lambda_{4}$ | 14 %
$\lambda_{5}$ | 7 %
$\lambda_{6}$ | 5 %
$\lambda_{7}$ | 3 %
$\lambda_{8}$ | 3 %
### IV.2 Robustness to perturbation of the original scores
As done for philosophers analysis, we performed 1000 perturbations of the
original scores by adding to each score the values -2, -1, 0, 1 or 2 with
uniform probability. In other words, we wanted to test if scoring errors could
be sufficient to cause relevant effects on the PCA projections. Interestingly,
the values of average and standard deviation for both original and perturbed
positions listed in Table 6 show relatively small changes. It is therefore
reasonable to say that the small errors in the values assigned as scores of
composers characteristics do not affected too much its quantification.
Table 6: Averages and standard deviations of the deviations for each composer and for the 8 eigenvalues. Composers | $\mu_{\Delta}$ | $\sigma_{\Delta}$
---|---|---
Monteverdi | 3.7347 | 0.8503
Bach | 5.3561 | 0.9379
Mozart | 4.4319 | 0.8911
Beethoven | 3.4987 | 0.7851
Brahms | 3.0449 | 0.6996
Stravinsky | 3.6339 | 0.7960
Stockhausen | 4.2143 | 0.9029
Eigenvalues | $\mu_{\Delta}$ | $\sigma_{\Delta}$
$\lambda_{1}$ | -0.1759 | 0.0045
$\lambda_{2}$ | -0.0638 | 0.0026
$\lambda_{3}$ | -0.0411 | 0.0021
$\lambda_{4}$ | -0.0144 | 0.0019
$\lambda_{5}$ | 0.0578 | 0.0021
$\lambda_{6}$ | 0.0736 | 0.0023
$\lambda_{7}$ | 0.0080 | 0.0027
$\lambda_{8}$ | 0.0835 | 0.0030
### IV.3 Results
Table 7 shows the normalized weights of the contributions of each original
property on the eight axes. Most of the characteristics contribute almost
equally in defining the axes.
Table 7: Percentages of the contributions from each musical characteristic on the eight new main axes. Musical | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$ | $C_{7}$ | $C_{8}$
---|---|---|---|---|---|---|---|---
Charac.
$S$-$S_{c}$ | 19.78 | 4.04 | 10.38 | 10.60 | 17.55 | 36.60 | 4.41 | 0.63
$D_{s}$-$D_{l}$ | 13.63 | 9.21 | 19.17 | 3.55 | 3.13 | 1.65 | 25.55 | 24.05
$H$-$C$ | 1.44 | 26.62 | 8.26 | 13.97 | 21.71 | 7.76 | 13.98 | 12.20
$V$-$I$ | 18.35 | 12.82 | 9.29 | 8.02 | 9.37 | 40.95 | 2.12 | 2.03
$D_{n}$-$D$ | 6.31 | 10.73 | 15.48 | 26.29 | 4.04 | 1.86 | 25.29 | 2.35
$M_{s}$-$M_{v}$ | 16.94 | 13.28 | 15.03 | 4.84 | 32.25 | 1.70 | 2.62 | 4.37
$R_{s}$-$R_{c}$ | 14.13 | 3.26 | 15.58 | 13.80 | 7.48 | 1.88 | 1.36 | 35.99
$H_{s}$-$H_{v}$ | 9.38 | 20.00 | 6.75 | 18.88 | 4.45 | 7.56 | 24.62 | 18.36
Figure 3 presents a 2-dimensional space considering the first two main axes.
The arrows follows the time sequence along with the seven composers. Each of
these arrows corresponds to a musical move from one composer state to another
– for clarity, just the lines of the arrows are preserved. The bootstrap
samples define clusters around the original composers.
Figure 3: 2-dimensional projected musical space.
Bach is found far from the rest of composers, which suggests his key role
acknowledged by other great composers like Beethoven and Webern Webern : “In
fact Bach composed everything, concerned himself with everything that gives
food for thought!”. The greatest subsequent change takes place from Bach to
Mozart, reflecting a substantial difference in style. We can identify a strong
relationship between Beethoven and Brahms, supporting the belief by the
virtuosi Hans von Bülow Bulow when he stated the $1^{st}$ Symphony of Brahms
as, in reality, being the $10^{th}$ Symphony of Beethoven, clamming Brahms as
the true successor of Beethoven. Stravinsky is near to Beethoven and Brahms,
presumably due to his heterogeneity BennettHistory ; Lovelock . Beethoven is
also near to Mozart who deeply influenced Beethoven, mainly in his early
works. For Webern, Beethoven was the unique classicist who really came close
to the coherence found in the pieces of the Burgundian School: “Not even in
Haydn and Mozart do we see these two forms as clearly as in Beethoven. The
period and the eight-bar sentence are at their purest in Beethoven; in his
predecessors we find only traces of them” Webern . It could explain the
proximity of Beethoven to the Renaissance Monteverdi. Stockhausen is a
deviating point when compared with the others and it could present even more
detachment if we had considered vanguard characteristics — e.g. timbre
exploration by using electronic devices Lovelock — not shared by his
precursors.
To complement the analysis, Table 8 gives the opposition and skewness indices
for each of the six musical moves, showing the movements are driven by rather
small opposition and strong skewness. In other words, most musical moves seems
to seek more innovation than opposition. Dialectics is also shown in Table 9
and will play a key role in the next section.
Table 8: Opposition and skewness indices for each of the six musical moves. Musical Move | $W_{i,j}$ | $s_{i,j}$
---|---|---
Monteverdi $\to$ Bach | 1.0 | 0.
Bach $\to$ Mozart | 1.0196 | 1.9042
Mozart $\to$ Beethoven | 0.4991 | 2.8665
Beethoven $\to$ Brahms | 0.2669 | 1.7495
Brahms $\to$ Stravinsky | 0.4582 | 2.6844
Stravinsky $\to$ Stockhausen | 0.2516 | 3.1348
Table 9: Counter-dialectics index for each of the five subsequent pairs of musical moves considering the 8 components. Musical Triple | $d_{i\rightarrow k}$
---|---
Monteverdi $\to$ Bach $\to$ Mozart | 2.0586
Bach $\to$ Mozart $\to$ Beethoven | 1.2020
Mozart $\to$ Beethoven $\to$ Brahms | 1.0769
Beethoven $\to$ Brahms $\to$ Stravinsky | 0.2518
Brahms $\to$ Stravinsky $\to$ Stockhausen | 0.2549
We performed Wards hierarchical clustering Ward to complement the analysis.
This algorithm clusters the original scores taking into account their
distance. The generated dendrogram in Figure 4 shows the composers considering
their similarity. The representation supports the observations discussed
previously. It is interesting to note the cluster formed by Beethoven and
Brahms, reflecting their heritage. Stravinsky and Stockhausen forms another
cluster and Mozart remains in isolation, as like Bach and Monteverdi. Both
relations were also present in the planar space shown in Figure 3.
Figure 4: Wards hierarchical clustering of the seven composers.
## V Comparisons with Philosophers Analysis
The results of composers analysis are surprising when compared with
philosophers Fabbri . It is important to note that we preserved the number of
characteristics and performed the same bootstrap method to generate a larger
set of samples, making possible this comparison. The variances after PCA
(Table 10) concentrates in the four first new axis, similar to the variances
for composers shown at Table 5. If we compare the discussed musical space with
the philosophical one in Figure 5 we identify opposite movements along all the
philosophy history in contrast to music. This reveals a notorious
characteristic of the way philosophers seem to have evolved their ideas,
driven by opposition ($W_{i,j}$), as shown in Table 11, while composers tend
to be more influenced by their predecessors as far as their dialectics
measures are concerned ($1/d_{i\rightarrow k}$).
In general, the musical movements had minor opposition and, remembering the
beginning of this work, it reflects the master-apprentice tradition present in
music: the composers tend to build their own works confirming their precursors
legacy, resulting in a greater dialectics than the philosophers related
measures. This reveals a crucial difference considering the memory treatment
along the development of philosophy and music: using the same techniques this
article does Fabbri , we could verify that a philosopher was influenced by the
opposition of ideas from his direct predecessor, while here composers were
commonly influenced by their both predecessors. Therefore, we can argue that
philosophy presents a memory-1 state, while music presents memory-2,
considering memory-N being as number $N$ of past generations whose influence
on a philosopher or composer is being considered. Considering the linearity of
musical movements we can identify the abscissa as a “time axis” representing
the development of music along the history, with some composers like Beethoven
returning to Monteverdi and others advancing to the modern age like Stravinsky
and Stockhausen.
Table 10: New variances after PCA for philosophers scores in percentages. Eigenvalue | Value
---|---
$\lambda_{1}$ | 40 %
$\lambda_{2}$ | 23 %
$\lambda_{3}$ | 13 %
$\lambda_{4}$ | 10 %
$\lambda_{5}$ | 5 %
$\lambda_{6}$ | 4 %
$\lambda_{7}$ | 3 %
$\lambda_{8}$ | 2 %
Figure 5: 2-dimensional projected philosophical space.
The opposition and skewness indices for philosophers listed in Table 11
endorses the minor role of opposition in composers at the period considered.
We can observe strong opposition in philosophical moves contrasted to small
opposition in musical movements. Also, the dialectics presents a phase
difference suggesting knownledge and aesthetics transfer latency between each
of these human fields.
Table 11: Opposition and skewness indices for each of the six philosophical moves. Philosophical Move | $W_{i,j}$ | $s_{i,j}$
---|---|---
Plato $\rightarrow$ Aristotle | 1.0 | 0
Aristotle $\rightarrow$ Descartes | 0.8740 | 1.1205
Descartes $\rightarrow$ Espinoza | 0.9137 | 2.3856
Espinoza $\rightarrow$ Kant | 0.6014 | 1.6842
Kant $\rightarrow$ Nietzsche | 1.1102 | 2.9716
Nietzsche $\rightarrow$ Deleuze | 0.3584 | 2.4890
When comparing dialectics, other curious facts arise: the dialectics indices
for musicians in Table 9 are considerably stronger moves than for philosophers
in Table 12. Both indices are also shown in Figure 6 where we can see a
constant decrease of counter-dialectics. This makes it possible to argue that
dialectics is stronger in music where a constantly return to the origins are
clearly visible. This reveals the nature of the musical development, based on
the search for a unity. Using the words of Webern, the search for the
“comprehensibility” but always influenced by their old masters.
Table 12: Counter-dialectics index for each of the five subsequent pairs of philosophical moves, considering all components. Philosophical Triple | $d_{i\rightarrow k}$
---|---
Plato $\rightarrow$ Aristotle $\rightarrow$ Descartes | 3.0198
Aristotle $\rightarrow$ Descartes $\rightarrow$ Espinoza | 1.8916
Descartes $\rightarrow$ Espinoza $\rightarrow$ Kant | 1.1536
Espinoza $\rightarrow$ Kant $\rightarrow$ Nietzsche | 1.1530
Kant $\rightarrow$ Nietzsche $\rightarrow$ Deleuze | 0.2705
Figure 6: Comparison between composers and philosophers counter-dialectics
indices
## VI Concluding Remarks
Motivated by the understanding of how innovation evolves in music history, we
extended a quantitative method recently applied to the study of philosophical
characteristics Fabbri and compared the results. Statistical methods is
nowadays commonly used for the study of music features and composers
productivity, but analysis of composers characteristics modification along the
music history has been less explored. The method differs on the aspect of how
the characteristics concerning composers are treated: scores are assigned to
each feature common in musical works. These scores reveal not the exact
profile of composers, but a tendency of how their techniques are usually
present. To make the simulation more realistic, we considered not just the
small number of 7 composers, but derived other 1000 new “artificial composers”
through a bootstrap method. A larger data set made possible the statistical
analysis, considering not just the original scored composers, but other
samples that respect the historical presence of the formers. This other
thousand composers were modeled by a probabilistic distribution, and avoided a
biasing caused by the use of only 7 composers. In order to investigate the
relationship between this scorings we applied Pearson correlation analysis.
The results demonstrated a strong correlation between some characteristics,
which allows us to group this values, creating a reduced number of features
that summarizes the most important characteristics. PCA was also applied to
these components, reducing the complex space to a planar graph where some of
the most interesting properties can be visualized.
Historical landmarks in music are well-defined in the planar space, like the
isolation of Bach, Mozart and Stockhausen, the proximity between Beethoven and
Brahms and the distance from Bach and Mozart, the heterogeneity of Stravinsky
and the vanguard of contemporary composers like Stockhausen. Even not so
visible relations, like the trend to return to the maximum domain of polyphony
– present on Renaissance – by Beethoven could also be clearly observable,
demonstrating the chronological nature of the space.
The dichotomy between master-apprentice tradition on music and the quest for
innovation that opened this discussion could be visualized quantitatively.
Each composer demonstrated his own style, differing considerably from his
predecessor – clearly shown when analyzing pairs of subsequent composers like
Bach and Mozart, Mozart and Beethoven or Stravinsky and Stockhausen.
Otherwise, the inheritance of predecessors styles is also present when
analyzing the direct relations between Mozart and Beethoven or Beethoven and
Brahms, or indirect ones between Bach and Beethoven or Beethoven and
Monteverdi. The entire scenario presented a “continual pattern” between
composers – motivated by the influence of theirs predecessors – but also
showed a force repelling both of them: the innovation, or in the words of
William Lovelock Lovelock , the “experimentation” that makes progress
possible.
Along the analysis we noticed interesting differences when comparing composers
with philosophers. While on philosophy the innovation is notably marked by
opposition of each philosophers ideas, it is less present for music composers.
The lack of strong opposition movements and proeminent presence of dialectics
in musical space indicates the music innovation is driven by a constant
heritage of each composer from his predecessors. We represented this
characteristic referring to a memory state where philosophers shows memory-1 –
each philosopher was influenced by opposite ideas of its direct predecessor –
while composers shows memory-2 – inheriting the style of their both direct
predecessors. The analysis of both dialectics values also shown surprising
results: on philosophy the dialectics indices are arranged on a increasing
series – showing a strong influence of dialectics to philosophy development –
the dialectics indices on music exhibits the same pattern, but with an offset.
This behavior presumably indicates a constant quest for coherence by the
composers, a fact notably observed by the studies of Anton Webern Webern
should have somewhat the same kernel and a lattency between the effects.
Another result is that the quantitative methodology initially applied to the
analysis of philosophy Fabbri proved to be extensible to other fields of
knowledge – in this case music – reflecting with considerable efficiency
details concerning the specific field.
Computational analysis of music scores could be applied to automate the
quantification of composers characteristics, like identification of melodic
and harmonic patterns or the presence or not of polyrhythms, motivic and
harmonic stability Correa . More composers could be inserted in the set for
the analysis of a wider time-line, possibly including more representatives of
each music period.
We want to end this work going back to Webern, who early envisioned these
relations: “It is clear that where relatedness and unity are omnipresent,
comprehensibility is also guaranteed. And all the rest is dilettantism,
nothing else, for all time, and always has been. That’s so not only in music
but everywhere.”
###### Acknowledgements.
Luciano da F. Costa thanks CNPq (308231/03-1) and FAPESP (05/00587-5) for
sponsorship. Gonzalo Travieso thanks CNPq (308118/2010-3) for sponsorship.
Vilson Vieira and Renato Fabbri is grateful to CAPES and the Postgrad
Committee of the IFSC.
## References
* (1) Luciano da Fontoura Costa Renato Fabbri, Osvaldo N. Oliveira Jr., “Philosometrics,” (October 2010), arXiv:1010.1880v2.
* (2) Anton Webern, _The Path To The New Music_ (Theodore Presser Company, 1963).
* (3) William Lovelock, _A Concise History of Music_ (Hammond Textbooks, 1962).
* (4) G. Deleuze, _Difference and Repetition_ (Continuum, 1968).
* (5) Dean Keith Simonton, “Emergence and realization of genius: The lives and works of 120 classical composers,” Journal of Personality and Social Psychology 61, 829 – 840 (1991), ISSN 0022-3514.
* (6) Dean K. Simonton, “Creative productivity, age, and stress: A biographical time-series analysis of 10 classical composers,” Journal of Personality and Social Psychology 35, 791 – 804 (1977), ISSN 0022-3514.
* (7) Aaron Kozbelt, “Performance time productivity and versatility estimates for 102 classical composers,” Psychology of Music 37, 25–46 (2009).
* (8) Aaron Kozbelt, “A quantitative analysis of Beethoven as self-critic: implications for psychological theories of musical creativity,” Psychology of Music 35, 144–168 (2007).
* (9) Peter van Kranenburg, “Musical style recognition – a quantitative approach,” in _Proceedings of the Conference on Interdisciplinary Musicology (CIM04)_ (2004).
* (10) Peter van Kranenburg, “On measuring musical style – the case of some disputed organ fugues in the J.S. Bach (BWV) catalogue,” Computing In Musicology 15 (2007-8).
* (11) Hal Varian, “Bootstrap tutorial,” The Mathematica Journal 9 (2005).
* (12) Roy Bennett, _History of Music_ (Cambridge University Press, 1982).
* (13) Christian P. Robert, “Simulation in statistics,” in _Proceedings of the 2011 Winter Simulation Conference_ (2011) arXiv:1105.4823.
* (14) Alan Walker, _Hans von Bülow: a life and times_ (Oxford University Press, 2010).
* (15) Joe H Ward Jr, “Hierarchical grouping to optimize an objective function,” Journal of the American Statistical Association 58, 236–244 (1963).
* (16) Debora C Correa, Jose H Saito, and Luciano da F Costa, “Musical genres: beating to the rhythms of different drums,” New Journal of Physics 12 (2010).
|
arxiv-papers
| 2011-09-21T20:43:46 |
2024-09-04T02:49:22.391017
|
{
"license": "Public Domain",
"authors": "Vilson Vieira, Renato Fabbri, Gonzalo Travieso, Luciano da Fontoura\n Costa",
"submitter": "Vilson Vieira da Silva Junior",
"url": "https://arxiv.org/abs/1109.4653"
}
|
1109.4727
|
# A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth
domains
Kyeong-Hun Kim Kyeong-Hun Kim111Department of Mathematics, Korea University,
1 Anam-dong, Sungbuk-gu, Seoul, South Korea 136-701, kyeonghun@korea.ac.kr.
The research of this author was supported by Basic Science Research Program
through the National Research Foundation of Korea(NRF) funded by the Ministry
of Education, Science and Technology (20110015961)
###### Abstract
In this paper we study parabolic stochastic partial differential equations
(see equation (1.1)) defined on arbitrary bounded domain
$\mathcal{O}\subset\mathbb{R}^{d}$ allowing Hardy inequality:
$\int_{\mathcal{O}}|\rho^{-1}g|^{2}\,dx\leq
C\int_{\mathcal{O}}|g_{x}|^{2}dx,\quad\forall g\in
C^{\infty}_{0}(\mathcal{O}),$ (0.1)
where $\rho(x)=\text{dist}(x,\partial\mathcal{O})$. Existence and uniqueness
results are given in weighted Sobolev spaces
$\mathfrak{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$, where $p\in[2,\infty)$,
$\gamma\in\mathbb{R}$ is the number of derivatives of solutions and $\theta$
controls the boundary behavior of solutions (see Definition 2.5). Furthermore
several Hölder estimates of the solutions are also obtained. It is allowed
that the coefficients of the equations blow up near the boundary.
Keywords: Hardy inequality, Stochastic partial differential equation, non-
smooth domain, $L^{p}$-theory, weighted Sobolve space.
AMS 2000 subject classifications: 60H15, 35R60.
## 1 Introduction
It is a classical result that Hardy inequality holds on Lipschitz domains
([31]). There have been many other works concerning Hardy inequality. See e.g.
[3], [35] and references therein. We only mention that inequality (0.1) holds
under much weaker condition than Lipschitz condition. For instance, it holds
if $\mathcal{O}$ has plump complement, that is, there exist $b,\sigma\in(0,1]$
such that for any $s\in(0,\sigma]$ and $x\in\partial\mathcal{O}$ there exists
a point $y\in B_{s}(x)\cap\mathcal{O}^{c}$ with
$\text{dist}(y,\partial\mathcal{O})\geq bs$. For instance,
$\mathcal{O}_{\alpha}:=\\{(x,y)\in\mathbb{R}^{2}:x\in(-1,1),\,\,|x|^{\alpha}+|y|^{\alpha}<1\\}$,
where $\alpha\in(0,1)$, is a non-Lipschitz domain but satisfies the plump
complement condition.
Let $(\Omega,\mathcal{F},P)$ be a complete probability space,
$\\{\mathcal{F}_{t},t\geq 0\\}$ be an increasing filtration of $\sigma$-fields
$\mathcal{F}_{t}\subset\mathcal{F}$, each of which contains all
$(\mathcal{F},P)$-null sets. We assume that on $\Omega$ we are given
independent one-dimensional Wiener processes $w^{1}_{t},w^{2}_{t},...$
relative to $\\{\mathcal{F}_{t},t\geq 0\\}$. The main goal of this article is
to present an $L_{p}$-theory of stochastic partial differential equation
$du=(a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+f)\,dt+(\sigma^{ik}u_{x^{i}}+\mu^{k}u+g^{k})\,dw^{k}_{t}$
(1.1)
given for $t>0$ and $x\in\mathcal{O}$. Here $i$ and $j$ go from $1$ to $d$,
and $k$ runs through $\\{1,2,...\\}$ with the summation convention on $i,j,k$
being enforced. The coefficients $a^{ij},b^{i},c,\sigma^{ik},\mu^{k}$ and the
free terms $f,g^{k}$ are random functions depending on $t$ and $x$. As
mentioned in [20], such equations with a finite number of the processes
$w^{k}_{t}$ appear, for instance, in nonlinear filtering problems (estimations
of the signal by observing it when it is mixed with noises), and considering
infinitely many $w^{k}_{t}$ is instrumental in treating equations for measure-
valued processes, for instance, driven by space-time white noise (cf. [16]).
Equation (1.1) has been extensively studied by so many authors (see e.g. [4,
9, 10, 12, 14, 16, 19, 20, 28, 29, 30, 32, 36] and references therein). We
give a very brief review only on the $L_{p}$-theory of the equation. The
$L_{p}$-theory ($p\geq 2$) of equation (1.1) defined in $\mathbb{R}^{d}$ was
introduced by Krylov ([16], [19]), and later Krylov and Lototsky ([20],[21])
developed a weighted $L_{p}$-theory of the equation defined on a half space.
It turned out that for SPDEs defined on domains the Hölder space approach does
not allow one to obtain results of reasonable generality, and the Sobolev
spaces without weights are trivially inappropriate. Recently, these weighted
$L_{p}$-theory on half space were extended to equations on smooth domains
(e.g. [9, 10, 12, 11, 27]) and on (non-smooth) Lipschitz domain ([8]).
On non-smooth domains the spatial derivatives of the solution usually have
additional singularities at the boundary which are due to the shape of the
domain, see e.g. [6, 7] for the case of deterministic equations on polygonal
domains and [25] for a generalization to the stochastic setting. In the
context of numerical approximation this suggests the use of non-uniform
schemes. In [1] results of [8] are used to prove that the convergence rates of
adequate non-uniform discretization schemes are closely connected to the
regularity of the solution measured in weighted Sobolev spaces.
However, we acknowledge that there is a gap in the proof of Lemma 3.1 of [8],
and the main results of [8] are false unless stronger assumption on the range
of weights is assumed. We show this with a counterexample. In this article we
reconstruct the results in [8] under much weaker assumption on
$\partial\mathcal{O}$, but with smaller range of weights. The arguments used
in this article are slightly different from those in [8]. For instance, we do
not use any argument of flattening the boundary, which is a key tool in [8].
Most of our important steps are based just on the Hardy inequality and Iô’s
formula.
As in [8, 9, 10, 12, 11, 21, 20, 27] we prove the existence and uniqueness
results in weighted Sobolev classes
$\mathfrak{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$, where $\gamma\in\mathbb{R}$
is the number of derivatives of solutions and $\theta$ controls the boundary
behavior of solutions (see Definition 2.5). Also several (interior) Hölder
estimates of the solutions are also obtained (see Corollary 2.14).
As usual $\mathbb{R}^{d}$ stands for the Euclidean space of points
$x=(x^{1},...,x^{d})$, $\mathbb{R}^{d}_{+}=\\{x\in\mathbb{R}^{d}:x^{1}>0\\}$
and $B_{r}(x):=\\{y\in\mathbb{R}^{d}:|x-y|<r\\}$. For $i=1,...,d$, multi-
indices $\beta=(\beta_{1},...,\beta_{d})$, $\beta_{i}\in\\{0,1,2,...\\}$, and
functions $u(x)$ we set
$u_{x^{i}}=\partial u/\partial x^{i}=D_{i}u,\quad
D^{\beta}u=D_{1}^{\beta_{1}}\cdot...\cdot
D^{\beta_{d}}_{d}u,\quad|\beta|=\beta_{1}+...+\beta_{d}.$
We also use the notation $D^{m}$ for a partial derivative of order $m$ with
respect to $x$. If we write $N=N(...)$, this means that the constant $N$
depends only on what are in parenthesis. Throughout the article, for functions
depending on $\omega,t$ and $x$, the argument $\omega\in\Omega$ will be
omitted.
The author is grateful to Ildoo Kim for carefully reading the earlier version
of the article and finding several typos and to N.V. Kryolv for providing the
author an example. The author is also thankful to P.A Cioica and F. Lindner
for useful discussions regarding the numerical approximations of SPDEs on non-
smooth domains.
## 2 Main results
First we introduce some Sobolev spaces (see e.g [16], [18] and [27] for more
details). Let $p\in(1,\infty)$, $\gamma\in\mathbb{R}$ and
$H^{\gamma}_{p}=H^{\gamma}_{p}(\mathbb{R}^{d})=(1-\Delta)^{-\gamma/2}L_{p}$ be
the set of all distributions $u$ such that $(1-\Delta)^{\gamma/2}u\in L_{p}$.
Define
$\|u\|_{H^{\gamma}_{p}}=\|(1-\Delta)^{\gamma/2}u\|_{L_{p}}:=\|\mathcal{F}^{-1}[(1+|\xi|^{2})^{\gamma/2}\mathcal{F}(u)(\xi)]\|_{p},$
where $\mathcal{F}$ is the Fourier transform. It is well known that if
$\gamma$ is a nonnegative integer then
$H^{\gamma}_{p}=H^{\gamma}_{p}(\mathbb{R}^{d})=\\{u:u,Du,...,D^{\gamma}u\in
L_{p}\\}.$
Denote $\rho(x)=\text{dist}(x,\partial\mathcal{O})$ and fix a bounded
infinitely differentiable function $\psi$ defined in $\mathcal{O}$ such that
(see e.g. Lemma 4.13 in [22] or formula (2.6) in [26])
$N^{-1}\rho(x)\leq\psi(x)\leq N\rho(x),\quad\rho^{m}|D^{m}\psi_{x}|\leq
N(m)<\infty.$ (2.1)
Let $\zeta\in C^{\infty}_{0}(\mathbb{R}_{+})$ be a nonnegative function
satisfying
$\sum_{n=-\infty}^{\infty}\zeta(e^{n+t})>c>0,\quad\forall t\in\mathbb{R}.$
(2.2)
Note that any non-negative smooth function $\zeta\in
C^{\infty}_{0}(\mathbb{R}_{+})$ so that $\zeta>0$ on $[e^{-1},e]$ satisfies
(2.2). For $x\in\mathcal{O}$ and $n\in\mathbb{Z}:=\\{0,\pm 1,...\\}$ define
$\zeta_{n}(x)=\zeta(e^{n}\psi(x)).$
Then
$\text{supp}\,\zeta_{n}\subset\\{x\in\mathcal{O}:e^{-n-k_{0}}<\rho(x)<e^{-n+k_{0}}\\}=:G_{n}$
for some integer $k_{0}>0$,
$\sum_{n=-\infty}^{\infty}\zeta_{n}(x)\geq\delta>0,$ (2.3) $\zeta_{n}\in
C^{\infty}_{0}(G_{n}),\quad|D^{m}\zeta_{n}(x)|\leq N(\zeta,m)e^{mn}.$ (2.4)
For $p\geq 1$ and $\gamma\in\mathbb{R}$, by
$H^{\gamma}_{p,\theta}(\mathcal{O})$ we denote the set of all distributions
$u$ on $\mathcal{O}$ such that
$\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}^{p}:=\sum_{n\in\mathbb{Z}}e^{n\theta}\|\zeta_{-n}(e^{n}\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}}<\infty.$
(2.5)
We also use the above notation for $\ell_{2}$-valued functions
$g=(g_{1},g_{2},...)$, that is,
$\|g\|_{H^{\gamma}_{p}}=\|g\|_{H^{\gamma}_{p}(\ell_{2})}=\||(1-\Delta)^{\gamma/2}g|_{\ell_{2}}\|_{L_{p}},$
$\|g\|^{p}_{H^{\gamma}_{p}(\mathcal{O},\ell_{2})}=\sum_{n\in\mathbb{Z}}e^{n\theta}\|\zeta_{-n}(e^{n}\cdot)g(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}(\ell_{2})}.$
It is known (see Lemma 2.4) that if $\\{\bar{\zeta}_{n},n\in\mathbb{Z}\\}$ is
another set of functions satisfying (2.3) and (2.4) (such functions can be
easily constructed by mollifying the indicator functions $I_{G_{n}}$), then it
yields the same space $H^{\gamma}_{p,\theta}(\mathcal{O})$. Also if $\gamma=n$
is a nonnegative integer then
$L_{p,\theta}(\mathcal{O}):=H^{0}_{p,\theta}(\mathcal{O})=L_{p}(\mathcal{O},\rho^{\theta-d}dx),$
$H^{n}_{p,\theta}(\mathcal{O}):=\\{u:u,\rho Du,...,\rho^{n}D^{n}u\in
L_{p,\theta}(\mathcal{O})\\},$
$\|u\|^{p}_{H^{n}_{p,\theta}(\mathcal{O})}\sim\sum_{|\alpha|\leq
n}\int_{\mathcal{O}}|\rho^{|\alpha|}D^{\alpha}u|^{p}\rho^{\theta-d}\,dx.$
(2.6)
We remark that the space $H^{n}_{p,\theta}(\mathcal{O})$ is different from
$W^{n,p}(\mathcal{O},\rho,\varepsilon)$ introduced in [22], where
$W^{n,p}(\mathcal{O},\rho,\varepsilon)=\\{u:u,Du,...,D^{n}u\in
L_{p}(\mathcal{O},\rho^{\varepsilon}dx)\\}.$
Denote $\rho(x,y)=\rho(x)\wedge\rho(y)$. For $\nu\in(0,1]$ and $k=0,1,2,...$,
as in [5], define
$[f]^{(0)}_{k}=[f]^{(0)}_{k,\mathcal{O}}=\sup_{\begin{subarray}{c}x\in\mathcal{O}\\\
|\beta|=k\end{subarray}}\rho^{k}(x)|D^{\beta}f(x)|,\quad\quad[f]^{(0)}_{k+\nu}=\sup_{\begin{subarray}{c}x,y\in\mathcal{O}\\\
|\beta|=k\end{subarray}}\rho^{k+\nu}(x,y)\frac{|D^{\beta}f(x)-D^{\beta}f(y)|}{|x-y|^{\nu}},$
$|f|^{(0)}_{k}=\sum_{j=0}^{k}[f]^{(0)}_{j,\mathcal{O}},\quad\quad|f|^{(0)}_{k+\nu}=|f|^{(0)}_{k}+[f]^{(0)}_{k+\nu}.$
The above notation is used also for $\ell_{2}$ valued functions
$g=(g^{1},g^{2},\cdots)$. For instance,
$[g]^{(0)}_{k}=\sup_{\begin{subarray}{c}x\in\mathcal{O}\\\
|\beta|=k\end{subarray}}\rho^{k}(x)|D^{\beta}g(x)|_{\ell_{2}}.$
Here are some other properties of the space
$H^{\gamma}_{p,\theta}(\mathcal{O})$ taken from [27] (also see [17], [18]).
###### Lemma 2.1
$(i)$ The space $C^{\infty}_{0}(\mathcal{O})$ is dense in
$H^{\gamma}_{p,\theta}(\mathcal{O})$.
$(ii)$ Assume that $\gamma-d/p=m+\nu$ for some $m=0,1,...$ and $\nu\in(0,1]$,
and $i,j$ are multi-indices such that $|i|\leq m,|j|=m$. Then for any $u\in
H^{\gamma}_{p,\theta}(\mathcal{O})$, we have
$\psi^{|i|+\theta/p}D^{i}u\in
C(\mathcal{O}),\quad\psi^{m+\nu+\theta/p}D^{j}u\in C^{\nu}(\mathcal{O}),$
$|\psi^{|i|+\theta/p}D^{i}u|_{C(\mathcal{O})}+[\psi^{m+\nu+\theta/p}D^{j}u]_{C^{\nu}(\mathcal{O})}\leq
C\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}.$
$(iii)$ $\psi D,D\psi:H^{\gamma}_{p,\theta}(\mathcal{O})\to
H^{\gamma-1}_{p,\theta}(\mathcal{O})$ are bounded linear operators, and for
any $u\in H^{\gamma}_{p,\theta}(\mathcal{O})$
$\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq N\|\psi
u_{x}\|_{H^{\gamma-1}_{p,\theta}(\mathcal{O})}+N\|u\|_{H^{\gamma-1}_{p,\theta}(\mathcal{O})}\leq
N\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})},$
$\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq N\|(\psi
u)_{x}\|_{H^{\gamma-1}_{p,\theta}(\mathcal{O})}+N\|u\|_{H^{\gamma-1}_{p,\theta}(\mathcal{O})}\leq
N\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}.$
$(iv)$ For any $\nu,\gamma\in\mathbb{R}$,
$\psi^{\nu}H^{\gamma}_{p,\theta}(\mathcal{O})=H^{\gamma}_{p,\theta-p\nu}(\mathcal{O})$
and
$\|u\|_{H^{\gamma}_{p,\theta-p\nu}(\mathcal{O})}\leq
N\|\psi^{-\nu}u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq
N\|u\|_{H^{\gamma}_{p,\theta-p\nu}(\mathcal{O})}.$ (2.7)
$(v)$ If $\gamma\in(\gamma_{0},\gamma_{1})$ and
$\theta\in(\theta_{0},\theta_{1})$, then
$\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq\varepsilon\|u\|_{H^{\gamma_{1}}_{p,\theta}(\mathcal{O})}+N(\gamma,p,\varepsilon)\|u\|_{H^{\gamma_{0}}_{p,\theta}(\mathcal{O})},$
$\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq\varepsilon\|u\|_{H^{\gamma}_{p,\theta_{0}}(\mathcal{O})}+N(\gamma,p,\varepsilon)\|u\|_{H^{\gamma}_{p,\theta_{1}}(\mathcal{O})}.$
###### Lemma 2.2
(i) Let $s=|\gamma|$ if $\gamma$ is an integer, and $s>|\gamma|$ otherwise,
then
$\|au\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq
N(d,s,\gamma)|a|^{(0)}_{s}\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}.$
(ii) If $\gamma=0,1,2,...$, then
$\|au\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\leq
N\sup_{\mathcal{O}}|a|\|u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}+N_{0}|a|^{(0)}_{\gamma}\|u\|_{H^{\gamma-1}_{p,\theta}(\mathcal{O})}$
where $N_{0}=0$ if $\gamma=0$.
(iii) If $0\leq r\leq s$, then
$|a|^{(0)}_{r}\leq
N(d,r,s)(\sup_{\mathcal{O}}|a|)^{1-r/s}(|a|^{(0)}_{s})^{r/s}.$
The assertions also holds for $\ell_{2}$-valued functions $a$.
Proof. For (i), see Theorem 3.1 in [27]. (ii) is an easy consequence of (2.6),
and (iii) is from Proposition 4.2 in [24]. $\Box$
###### Remark 2.3
By Lemma 2.2, for any $\nu\geq 0$, $\psi^{\nu}$ is a point-wise multiplier in
$H^{\gamma}_{p,\theta}(\mathcal{O})$. Thus if $\theta_{1}\leq\theta_{2}$ then
$\|u\|_{H^{\gamma}_{p,\theta_{2}}(\mathcal{O})}\leq
N\|\psi^{(\theta_{2}-\theta_{1})/p}u\|_{H^{\gamma}_{p,\theta_{1}}(\mathcal{O})}\leq
N\|u\|_{H^{\gamma}_{p,\theta_{1}}(\mathcal{O})}.$
###### Lemma 2.4
Let $\\{\xi_{n}\\}$ be a sequence of $C^{\infty}_{0}(\mathcal{O})$ functions
such that
$|D^{m}\xi_{n}|\leq
C(m)e^{nm},\quad\text{supp}\,\xi_{n}\subset\\{x\in\mathcal{O}:e^{-n-k_{0}}<\rho(x)<e^{-n+k_{0}}\\}$
for some $k_{0}>0$. Then for any $u\in H^{\gamma}_{p,\theta}(\mathcal{O})$,
$\sum_{n}e^{n\theta}\|\xi_{-n}(e^{n}x)u(e^{n}x)\|^{p}_{H^{\gamma}_{p}}\leq
N\|u\|^{p}_{H^{\gamma}_{p,\theta}(\mathcal{O})}.$
If in addition
$\sum_{n}|\xi_{n}|^{p}>\delta>0,$
then the reverse inequality also holds.
Proof. See Theorem 2.2 in [27]. $\Box$
Let $\mathcal{P}$ be the predictable $\sigma$-field generated by
$\\{\mathcal{F}_{t},t\geq 0\\}$. Define
$\mathbb{H}^{\gamma}_{p}(T)=L_{p}(\Omega\times[0,T],\mathcal{P},H^{\gamma}_{p}),\quad\mathbb{H}^{\gamma}_{p}(T,\ell_{2})=L_{p}(\Omega\times[0,T],\mathcal{P},H^{\gamma}_{p}(\ell_{2}))\quad$
$\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)=L_{p}(\Omega\times[0,T],\mathcal{P},H^{\gamma}_{p,\theta}(\mathcal{O})),\quad\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T,\ell_{2})=L_{p}(\Omega\times[0,T],\mathcal{P},H^{\gamma}_{p,\theta}(\mathcal{O},\ell_{2})),$
$\mathbb{L}_{p,\theta}(\mathcal{O},T)=\mathbb{H}^{0}_{p,\theta}(\mathcal{O},T),\quad
U^{\gamma}_{p}=L_{p}(\Omega,\mathcal{F}_{0},H^{\gamma-2/p}_{p}),\quad
U^{\gamma}_{p,\theta}(\mathcal{O})=\psi^{-\frac{2}{p}+1}L_{p}(\Omega,\mathcal{F}_{0},H^{\gamma-2/p}_{p,\theta}(\mathcal{O})).$
That is, for instance, we say
$u\in\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$ if $u$ has a
$H^{\gamma}_{p,\theta}(\mathcal{O})$-valued predictable version $\bar{u}$
defined on $\Omega\times[0,T]$ so that
$\|u\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}=\|\bar{u}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}:=\left({\mathbb{E}}\int^{T}_{0}\|u(s,\cdot)\|^{p}_{H^{\gamma}_{p,\theta}(\mathcal{O})}dt\right)^{1/p}<\infty.$
Also by
$u\in\psi^{-\frac{2}{p}+1}L_{p}(\Omega,\mathcal{F}_{0},H^{\gamma-2/p}_{p,\theta}(\mathcal{O}))$
we mean $\psi^{2/p-1}u\in
L_{p}(\Omega,\mathcal{F}_{0},H^{\gamma-2/p}_{p,\theta}(\mathcal{O}))$, and
$\|u\|^{p}_{U^{\gamma}_{p,\theta}(\mathcal{O})}:={\mathbb{E}}\|\psi^{2/p-1}u\|^{p}_{H^{\gamma-2/p}_{p,\theta}(\mathcal{O})}.$
Below by $(u,\phi)$ we denote the image of $\phi\in
C^{\infty}_{0}(\mathcal{O})$ under a distribution $u$.
###### Definition 2.5
We write $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ if
$u\in\psi\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$, $u(0,\cdot)\in
U^{\gamma+2}_{p,\theta}(\mathcal{O})$ and for some
$f\in\psi^{-1}\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$ and
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$,
$du=f\,dt+g^{k}\,dw^{k}_{t},$ (2.8)
in the sense of distributions. In other words, for any $\phi\in
C^{\infty}_{0}(\mathcal{O})$, the equality
$(u(t,\cdot),\phi)=(u(0,\cdot),\phi)+\int^{t}_{0}(f(s,\cdot),\phi)\,ds+\sum^{\infty}_{k=1}\int^{t}_{0}(g^{k}(s,\cdot),\phi)\,dw^{k}_{s}$
holds for all $t\leq T$ with probability $1$. In this situation we write
$f=\mathbb{D}u$ and $g=\mathbb{S}u$. The norm in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ is defined by
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}=\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}+\|\psi\mathbb{D}u\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|\mathbb{S}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u(0,\cdot)\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}.$
###### Remark 2.6
(i) Remember that for any $\alpha,\gamma\in\mathbb{R}$,
$\|\psi^{\alpha}u\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\sim\|u\|_{H^{\gamma}_{p,\theta+p\alpha}(\mathcal{O})}$.
Thus the space $\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ is
independent of the choice of $\psi$.
(ii) It is easy to check (see Remark 3.2 of [16] for details) that for any
$\phi\in C^{\infty}_{0}(\mathcal{O})$ and
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$ we have
$\sum_{k=1}^{\infty}\int^{T}_{0}(g^{k},\phi)^{2}ds<\infty$, and therefore the
series of stochastic integral
$\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k},\phi)dw^{k}_{t}$ converges in
probability uniformly on $[0,T]$.
###### Theorem 2.7
Let $u_{n}\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T),n=1,2,\cdots$
and $\|u_{n}\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq K$,
where $K$ is a finite constant. Then there exists a subsequence $n_{k}$ and a
function $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ so that
(i) $u_{n_{k}},u_{n_{k}}(0,\cdot),\mathbb{D}u_{n_{k}},\mathbb{S}u_{n_{k}}$
converges weakly to $u,u(0,\cdot),\mathbb{D}u$ and $\mathbb{S}u$ in
$\mathbb{H}^{\gamma+2}_{p,\theta}(T,\mathcal{O}),U^{\gamma+2}_{p,\theta}(\mathcal{O})$,
$\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O})$ and
$\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},\ell_{2})$ respectively;
(ii) for any $\phi\in C^{\infty}_{0}(\mathcal{O})$ and $t\in[0,T]$, we have
$(u_{n_{k}}(t,\cdot),\phi)\to(u(t,\cdot),\phi)$ weakly in $L_{p}(\Omega)$.
Proof. The proof is identical to that of the proof of Theorem 3.11 in [16],
where the theorem is proved when $\mathcal{O}=\mathbb{R}^{d}$.
$\Box$
###### Theorem 2.8
For any nonnegative integer $n\geq\gamma+2$, the set
$\mathfrak{H}^{n}_{p,\theta}(\mathcal{O},T)\bigcap\bigcup^{\infty}_{k=1}L_{p}(\Omega,C([0,T],C^{n}_{0}(\mathcal{O}_{k}))),$
where $\mathcal{O}_{k}:=\\{x\in\mathcal{O}:\psi(x)>1/k\\}$, is dense in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$.
Proof. It is enough to repeat the proof of Theorem 2.9 in [20], where the
lemma is proved when $\mathcal{O}=\mathbb{R}^{d}_{+}$. $\Box$
###### Theorem 2.9
(i) Let $2/p<\alpha<\beta\leq 1$ and
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$, then
${\mathbb{E}}[\psi^{\beta-1}u]^{p}_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p,\theta}(\mathcal{O}))}\leq
NT^{(\beta-\alpha)p/2}\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)},$
where $N$ is independent of $T$ and $u$.
(ii) Let $p\in[2,\infty)$ and $T<\infty$, then
${\mathbb{E}}\sup_{t\leq
T}\|u(t)\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}\leq
N\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)},$ (2.9)
where $N=N(d,p,\gamma,\theta,\mathcal{O},T)$ is non-decreasing function of
$T$. In particular, for any $t\leq T$,
$\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}\leq
N\int^{t}_{0}\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},s)}\,ds.$
(2.10)
Proof. The theorem is proved in [8] on Lipschitz domains, and the proof works
on any arbitrary domains.
(i). Due to the definition of
${\mathbb{E}}[\psi^{\beta-1}u]^{p}_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p,\theta}(\mathcal{O}))}$
we may assume $u(0)=0$. Let $\mathbb{D}u=f$ and $\mathbb{S}u=g$. By (2.5) and
Lemma 2.1(iv),
$I:={\mathbb{E}}[\psi^{\beta-1}u]^{p}_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p,\theta}(\mathcal{O}))}$
$\leq
N\sum_{n}e^{n(\theta+p(\beta-1))}{\mathbb{E}}[u(t,e^{n}x)\zeta_{-n}(e^{n}x)]^{p}_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p})}.$
(2.11)
Denote $T_{0}:=T^{(\beta-\alpha)p/2}$. By Corollary 4.12 in [15], there exists
a constant $N>0$, independent of $T$ and $u$, so that for any $a>0$,
${\mathbb{E}}[u(t,e^{n}x)\zeta_{-n}(e^{n}x)]^{p}_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p})}\leq
NT_{0}a^{\beta-1}(a\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}$
$+a^{-1}\|f(t,e^{n})\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}+\|g(t,e^{n})\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})}).$
Take $a=e^{-np}$, then (2.11) yields
$\displaystyle I$ $\displaystyle\leq$ $\displaystyle
NT_{0}(\sum_{n}e^{n(\theta-p)}\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}+\sum_{n}e^{n(\theta+p)}\|f(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}$
$\displaystyle+$
$\displaystyle\sum_{n}e^{n\theta}\|g(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})})$
$\displaystyle=$ $\displaystyle
NT_{0}\left(\|u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta-p}(\mathcal{O},T)}+\|f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta+p}(\mathcal{O},T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}\right)\leq
NT_{0}\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}.$
Thus (i) is proved.
(ii). If $p>2$, (ii) follows from (i). But for the case $p=2$, we prove this
differently. Obviously
${\mathbb{E}}\sup_{t\leq
T}\|u(t)\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}\leq
N\sum_{n}e^{n\theta}{\mathbb{E}}\sup_{t\leq
T}\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{H^{\gamma+1}_{p}}.$
Note that $u_{0}\in U^{\gamma+2}_{p,\theta}(\mathcal{O})\subset
L_{p}(\Omega,H^{\gamma+1}_{p,\theta}(\mathcal{O}))$ since $p\geq 2$. By Remark
4.14 in [15] with $\beta=1$ there, for any $a>0$,
${\mathbb{E}}\sup_{t\leq
T}\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{H^{\gamma+1}_{p}}\leq
N(a\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}$
$+a^{-1}\|f(t,e^{n})\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}+\|g(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})}++{\mathbb{E}}\|u_{0}(e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{H^{\gamma+1}_{p}}).$
Take $a=e^{-np}$ to get
$\displaystyle{\mathbb{E}}\sup_{t\leq
T}\|u(t)\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}$ $\displaystyle\leq$
$\displaystyle
N(\sum_{n}e^{n(\theta-p)}\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}+\sum_{n}e^{n(\theta+p)}\|f(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}$
$\displaystyle+\sum_{n}e^{n\theta}\|g(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T)}+{\mathbb{E}}\sum_{n}e^{n\theta}\|u_{0}(e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{H^{\gamma+1}_{p}})$
$\displaystyle=$ $\displaystyle
N(\|u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta-p}(\mathcal{O},T)}+\|f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta+p}(\mathcal{O},T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O})}+{\mathbb{E}}\|u_{0}\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})})$
$\displaystyle\leq$ $\displaystyle
N\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}.$
Finally,
$\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}={\mathbb{E}}\int^{t}_{0}\|u\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}ds\leq\int^{t}_{0}({\mathbb{E}}\sup_{r\leq
s}\|u(r)\|^{p}_{H^{\gamma+1}_{p,\theta}(\mathcal{O})})ds\leq
N\int^{t}_{0}\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},s)}ds.$
The theorem is proved. $\Box$
Fix a nonnegative constant $\varepsilon_{0}=\varepsilon(\gamma)\geq 0$ so that
$\varepsilon_{0}>0$ only if $\gamma$ is not integer, and define
$\gamma_{+}=|\gamma|$ if $\gamma$ is an integer, and
$\gamma_{+}=|\gamma|+\varepsilon_{0}$ otherwise. Now we state our assumptions
on the coefficients.
###### Assumption 2.10
(i) For each $x$, the coefficients $a^{ij}(t,x)$, $b^{i}(t,x)$ $c(t,x)$,
$\sigma^{ik}(t,x)$ and $\mu^{k}(t,x)$ are predictable functions of
$(\omega,t)$.
(ii) The coefficients $a^{ij},\sigma^{i}$ are uniformly continuous in $x$,
that is, for any $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$
such that
$|a^{ij}(t,x)-a^{ij}(t,y)|+|\sigma^{i}(t,x)-\sigma^{i}(t,y)|_{\ell_{2}}\leq\varepsilon$
for each $\omega,t$, whenever $x,y\in\mathcal{O}$ and $|x-y|\leq\delta$.
(iii) There exist constant $\delta_{0},K>0$ such that for any $\omega,t,x$ and
$\lambda\in\mathbb{R}^{d}$,
$\delta_{0}|\lambda|^{2}\leq\bar{a}^{ij}(t,x)\lambda^{i}\lambda^{j}\leq
K|\lambda|^{2},$ (2.12)
where $\bar{a}^{ij}=a^{ij}-\frac{1}{2}(\sigma^{i},\sigma^{j})_{\ell_{2}}$.
(iv) For any $\omega,t$
$|a^{ij}(t,\cdot)|^{(0)}_{\gamma_{+}}+|\psi
b^{i}(t,\cdot)|^{(0)}_{\gamma_{+}}+|\psi^{2}c(t,\cdot)|^{(0)}_{\gamma_{+}}+|\sigma^{i}(t,\cdot)|^{(0)}_{(\gamma+1)_{+}}+|\psi\mu(t,\cdot)|^{(0)}_{(\gamma+1)_{+}}\leq
K,$ (2.13)
and if $\gamma=0$, then for some $\varepsilon>0$,
$|\sigma^{i}(t,\cdot)|^{(0)}_{1+\varepsilon}+|\psi\mu|^{(0)}_{1+\varepsilon}\leq
K,\quad\quad\forall\omega,t.$ (2.14)
(v) There is a control on the behavior of $b^{i},c$ and $\mu^{k}$ near
$\partial\mathcal{O}$, namely,
$\lim_{\begin{subarray}{c}\rho(x)\to
0\end{subarray}}\sup_{\omega,t}\left(\rho(x)|b^{i}(t,x)|+\rho^{2}(x)|c(t,x)|+\rho(x)|\mu(t,x)|_{\ell_{2}}\right)=0.$
(2.15)
###### Remark 2.11
Conditions (2.13) and (2.15) allow the coefficients $b^{i},c$ and $\nu$ to be
unbounded and to blow up near the boundary. In particular, (2.15) is satisfied
if for some $\varepsilon,N>0$,
$|b^{i}(t,x)|+|\mu(t,x)|_{\ell_{2}}\leq
N\rho^{-1+\varepsilon}(x),\quad|c(t,x)|\leq N\rho^{-2+\varepsilon}(x).$
The proof of following theorem is given in section 4.
###### Theorem 2.12
Let $p\in[2,\infty),\gamma\in[0,\infty),T<\infty$ and Assumption 2.10 be
satisfied. Then there exists $\beta_{0}=\beta_{0}(p,d,\mathcal{O})>0$ so that
if
$\theta\in(p-2+d-\beta_{0},p-2+d+\beta_{0})$ (2.16)
then for any $f\in\psi^{-1}\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$,
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)$ and $u_{0}\in
U^{\gamma+2}_{p,\theta}(\mathcal{O})$ equation (1.1) with initial data $u_{0}$
admits a unique solution $u$ in the class
$\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$, and for this solution
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq C(\|\psi
f\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}),$
(2.17)
where $C=C(d,p,\gamma,\theta,\delta_{0},K,T,\mathcal{O})$.
###### Remark 2.13
Note that Theorem 2.12 is proved only for $\gamma\geq 0$. However the theorem
can be extended for any $\gamma\in\mathbb{R}$ by using results for $\gamma\geq
0$ and arguments used e.g. in the proof of Theorem 2.16 of [12] (cf. [11,
20]). One difference is that, in place of Theorem 2.8 of [18], one has to use
the corresponding version on bounded domains (Theorem 5.1 of [27]).
Lemma 2.1(ii) and Theorem 2.9 easily yield the following result.
###### Corollary 2.14
Let $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},\tau)$ be the solution
in Theorem 2.12 (or in Theorem 2.15 below).
(i) If $\gamma+2-d/p=m+\nu$ for some $m=0,1,...$, $\nu\in(0,1]$, and $i,j$ are
multi-indices such that $|i|\leq m,|j|=m$, then for each $\omega,t$
$\psi^{|i|-1+\theta/p}D^{i}u\in
C(\mathcal{O}),\quad\psi^{m-1+\nu+\theta/p}D^{j}u\in C^{\nu}(\mathcal{O}).$
In particular,
$|\psi^{|i|}D^{i}u(x)|\leq N\psi^{1-\theta/p}(x).$
(ii) Let
$2/p<\alpha<\beta\leq 1,\quad\gamma+2-\beta-d/p=k+\varepsilon$
where $k\in\\{0,1,2,...\\}$ and $\varepsilon\in(0,1]$. Denote
$\delta=\beta-1+\theta/p$. Then for any multi-indices $i$ and $j$ such that
$|i|\leq k$ and $|j|=k$, we have
${\mathbb{E}}\sup_{t,s\leq\tau}|t-s|^{-(p\alpha/2-1)}(|\psi^{\delta+|i|}D^{i}(u(t)-u(s))|^{p}_{C(\mathcal{O})}$
$+[\psi^{\delta+|j|+\varepsilon}D^{j}(u(t)-u(s))]^{p}_{C^{\varepsilon}(\mathcal{O})})<\infty.$
Note that if $p=2$ then (2.16) is $\theta\in(d-\beta_{0},d+\beta_{0})$, but it
is not clear whether $d$ is included in the interval in (2.16) if $p>2$
because $\beta_{0}$ depends also on $p$. Below we give positive answer if $p$
is close to $2$ and negative one for large $p$. For instance, if $\theta=d=2$
then in general Theorem 2.12 do not hold for all $p>4$.
The proof of following theorem is given in section 5.
###### Theorem 2.15
There exists $p_{0}>2$ so that if $p\in[2,p_{0})$ then there exists
$\beta_{1}>0$ so that the assertion of Theorem 2.12 holds for any
$\theta\in(d-\beta_{1},d+\beta_{1})$.
###### Remark 2.16
Since
$H^{1}_{p,d-p}(\mathcal{O})=\stackrel{{\scriptstyle\circ}}{{W^{1}_{p}}}(\mathcal{O}):=\\{u:u,u_{x}\in
L_{p}(\mathcal{O})\,\,\text{and}\,\,u|_{\partial\mathcal{O}}=0\\}$, Theorem
2.15 with $\gamma\geq-1$ and $\theta\in(d-\beta_{1},d]$ implies that there
exists a unique solution $u\in
L_{p}(\Omega\times[0,T],\stackrel{{\scriptstyle\circ}}{{W^{1}_{p}}}(\mathcal{O}))$
for any $p\in[2,p_{0})$.
The following example is due to N.V. Krylov and shows that Theorem 2.12 can
not hold unless $\theta$ is sufficiently large and that in general Theorem
2.15 is false for all large $p$.
###### Example 2.17
Let $\alpha\in(1/2,1)$ and denote
$G_{\alpha}=\\{z=x+iy:|\text{arg}\,z|<\frac{\pi}{2\alpha}\\},\quad\mathcal{O}_{\alpha}=G_{\alpha}\cap\\{z:|z|<10\\},$
where $\text{arg}\,z$ is defined as a function taking values in so that
$[\pi,-\pi)$. Define
$v(z)=v(x,y)=\text{Re}\,z^{\alpha}=|z|^{\alpha}\cos\alpha\theta$, where
$\tan\theta=y/x$. Then $\Delta v=0$ in $G_{\alpha}$ and $v=0$ on $\partial
G_{\alpha}$. We claim that for some $N=N(\alpha)>1$,
$N^{-1}|z|^{\alpha-1}\leq|\rho^{-1}v|\leq N|z|^{\alpha-1},\quad|Dv|+|\rho
D^{2}v|\leq N|z|^{\alpha-1}.$
Since the second assertion is easy to check we prove the first one. If
$|\text{arg}\,z|<\frac{\pi}{2\alpha}-\frac{\pi}{2}$ then $\rho(z)=|z|$,
$|z|^{\alpha}\cos(\frac{\pi}{2}-\alpha\frac{\pi}{2})\leq|v|\leq|z|^{\alpha}$
and the claim is obvious. Also if
$\frac{\pi}{2\alpha}-\frac{\pi}{2}\leq|\text{arg}\,z|<\frac{\pi}{2\alpha}$,
then $\rho(z)=|z||\sin(\frac{\pi}{2\alpha}-\theta)|$ and
$\cos\alpha\theta/|\sin(\frac{\pi}{2\alpha}-\theta|$ is comparable to $1$ in
$\\{z:\frac{\pi}{2\alpha}-\frac{\pi}{2}\leq|\text{arg}\,z|<\frac{\pi}{2\alpha}\\}$.
It follows that
$\int_{\mathcal{O}_{\alpha}}\left(|\rho^{-1}v|^{p}+|Dv|+|\rho
D^{2}v|\right)\rho^{\theta-2}dx<\infty\quad\Leftrightarrow\quad\theta>p(1-\alpha),$
and
$\int_{\mathcal{O}_{\alpha}}(|\rho v_{x}|^{p}+|\rho
v|^{p})\rho^{\theta-2}dx<\infty,\quad\forall\,\,\theta>0.$
Now choose a smooth function $\xi\in C^{\infty}_{0}(B_{2}(0))$ so that $\xi=1$
on $B_{1}(0)$, and define $u(t,x,y):=t\xi(x,y)v(x,y)$. Then
$du=(\Delta u+f)dt,$ (2.18)
where $f:=t(-2\xi_{x^{i}}v_{x^{i}}-v\Delta\xi)+\xi v$. Above calculations show
that $\rho f\in\mathbb{L}_{p,\theta}(\mathcal{O}_{\alpha},T)$ for any
$\theta>0$ and that $u\in\mathfrak{H}^{2}_{p,p}(\mathcal{O}_{\alpha},T)$. By
Theorem 2.12 we conclude that $u$ is the unique solution of the above equation
in $\mathfrak{H}^{2}_{p,p}(\mathcal{O}_{\alpha},T)$. It also follows that the
existence result of Theorem 2.12 in
$\mathfrak{H}^{2}_{p,\theta}(\mathcal{O}_{\alpha},T)$ fails whenever
$\theta\leq p(1-\alpha),$
because if there is any solution
$w\in\mathfrak{H}^{2}_{p,\theta}(\mathcal{O}_{\alpha},T)$ then
$w\in\mathfrak{H}^{2}_{p,p}(\mathcal{O}_{\alpha},T)$ and therefore due to the
uniqueness result in $\mathfrak{H}^{2}_{p,p}(\mathcal{O}_{\alpha},T)$, we get
$u=w$. But this is not possible since
$\|\rho^{-1}u\|_{\mathbb{L}_{p,\theta}(\mathcal{O}_{\alpha},T)}=\infty$. In
particular, if $\theta=d=2$ and $p>4$ we can choose $\alpha$ close to $1/2$ so
that $2\leq p(1-\alpha)$, and consequently this leads to the fact that in
general Theorem 2.12 does not holds if $p>4$.
## 3 A priori estimate
In this section we develop some estimations of solutions of equation (1.1).
First, we introduce a result on SPDEs defined on entire space
$\mathbb{R}^{d}$.
###### Lemma 3.1
Let $a^{ij}$ and $\sigma^{ij}$ be independent of $x$. Also suppose that
$f\in\mathbb{H}^{\gamma}_{p}(T)$, $g\in\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})$,
$u_{0}\in U^{\gamma+2}_{p}$ and $u\in\mathbb{H}^{\gamma+1}_{p}(T)$ is a
solution of
$du=(a^{ij}u_{x^{i}x^{j}}+f)+(\sigma^{ik}u_{x^{i}}+g^{k})dw^{k}_{t}\quad
u(0,\cdot)=u_{0}.$ (3.1)
Then $u\in\mathbb{H}^{\gamma+2}_{p}(T)$, and
$\|u\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}\leq
N(\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T)}+\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p}}),$
(3.2)
where $N$ depends only on $d,p,\delta_{0},K$ (not on $T$).
Proof. This is a well known result. By Theorem 4.10 in [16],
$\|u_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}\leq
C(d,p)(\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p}}).$
This and the relation
$\|u\|_{H^{\gamma+2}_{p}}=\|(1-\Delta)u\|_{H^{\gamma}_{p}}\leq(\|u\|_{H^{\gamma}_{p}}+\|u_{xx}\|_{H^{\gamma}_{p}})$
certainly prove (3.2). $\Box$
In the following lemma there is no restriction on $\theta,\gamma$ and
$\partial\mathcal{O}$, that is $\theta,\gamma\in\mathbb{R}$ and $\mathcal{O}$
is any arbitrary domain.
###### Lemma 3.2
Let $a^{ij}$ and $\sigma^{ik}$ be independent of $x$. Suppose
$f\in\psi^{-1}\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$,
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$, $u_{0}\in
U^{\gamma+2}_{p,\theta}(\mathcal{O})$ and
$u\in\mathfrak{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)$ is a solution of the
equation
$du=(a^{ij}u_{x^{i}x^{j}}+f)dt+(\sigma^{ik}u_{x^{i}}+g^{k})dw^{k}_{t},\quad\quad
u(0,\cdot)=u_{0}.$
Then $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\Omega,T)$, and
$\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N(\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}+\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}).$
(3.3)
Proof. We just repeat the arguments used in [8] on Lipschitz domains. Remember
that by Lemma 2.1 we have
$\|\psi^{-1}u\|_{H^{\gamma+2}_{p,\theta}(\mathcal{O})}\sim\|u\|_{H^{\gamma+2}_{p,\theta-p}(\mathcal{O})}$.
Thus,
$\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N\sum_{n=-\infty}^{\infty}e^{n(\theta-p)}\|u(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(T)}$
$=N\sum_{n=-\infty}^{\infty}e^{n(\theta-p+2)}\|v_{n}\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(e^{-2n}T)},$
(3.4)
where $v_{n}(t,x):=u(e^{2n}t,e^{n}x)\zeta_{-n}(e^{n}x)$. Note that since
$v_{n}$ has compact support in $\mathbb{R}^{d}$ and can be regarded as
distribution defined on $\mathbb{R}^{d}$. Thus we conclude
$v_{n}\in\mathbb{H}^{\gamma+1}_{p}(e^{-2n}T)$. Also note that it satisfies
$dv_{n}=(a^{ij}(e^{2n}t)v_{nx^{i}x^{j}}+f_{n})dt+(\sigma^{ik}(e^{2n}t)v_{nx^{i}}+g^{k}_{n})dw^{k}(n)_{t},\quad
v_{n}(0)=u_{0}(e^{n}x)\zeta_{-n}(e^{n}x),$
where $w^{k}(n)_{t}:=e^{-n}w^{k}_{e^{n}t}$ are independent Wiener processes,
$\displaystyle f_{n}(t,x)$ $\displaystyle=$
$\displaystyle-2e^{n}a^{ij}(e^{2n}t,x)u_{x^{i}}(e^{2n}t,e^{n}x)e^{n}\zeta_{-nx^{j}}(e^{n}x)-a^{ij}u(e^{2n}t,e^{n}x)e^{2n}\zeta_{-nx^{i}x^{j}}(e^{n}x)$
$\displaystyle+$ $\displaystyle e^{2n}f(e^{2n}t,e^{n}x)\zeta_{-n}(e^{n}x),$
and
$g^{k}_{n}=-\sigma^{ik}(e^{2n}t)u(e^{2n}t,e^{n}x)e^{n}\zeta_{-nx^{i}}(e^{n}x)+e^{n}g^{k}(e^{2n}t,e^{n}x)\zeta_{-n}(e^{n}x).$
Since $\zeta_{-n}$ has compact support in $\mathcal{O}$ and
$u\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)$, we easily check that
$f_{n}\in\mathbb{H}^{\gamma}_{p}(e^{-2n}T),\quad
g_{n}\in\mathbb{H}^{\gamma+1}_{p}(e^{-2n}T,\ell_{2}).$
Thus by Lemma 3.1, we have $v_{n}\in\mathbb{H}^{\gamma+2}_{p}(e^{2n}T)$ and
$\|v_{n}\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(e^{2n}T)}\leq
N(\|v_{n}\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(e^{2n}T)}+\|f_{n}\|^{p}_{\mathbb{H}^{\gamma}_{p}(e^{2n}T)}+\|g_{n}\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(e^{2n}T,\ell_{2})}+\|v_{n}(0)\|^{p}_{U^{\gamma+2}_{p}}),$
where $N=N(d,p,\gamma,\delta_{0},K)$ is independent of $n$ and $T$. Next we
apply Lemma 2.4 with $\xi_{n}=e^{-n}\zeta_{nx^{i}}$ or
$\xi_{n}=e^{-2n}\zeta_{nx^{i}x^{j}}$ and get
$\displaystyle\sum_{n=-\infty}^{\infty}e^{n(\theta-p+2)}\|f_{n}\|^{p}_{\mathbb{H}^{\gamma}_{p}(e^{-2n}T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n}e^{n\theta}\|u_{x^{i}}(t,e^{n}x)e^{n}\zeta_{-nx^{j}}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}$
$\displaystyle+$ $\displaystyle
N\sum_{n}e^{n(\theta-p)}\|u(t,e^{n}x)e^{2n}\zeta_{-nx^{i}x^{j}}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}$
$\displaystyle+$ $\displaystyle
N\sum_{n}e^{n(\theta+p)}\|f(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(T)}$
$\displaystyle\leq$ $\displaystyle
N\|u_{x}\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+N\|u\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta-p}(\mathcal{O},T)}+N\|f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta+p}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}+N\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}.$
Similarly,
$\displaystyle\sum_{n=-\infty}^{\infty}e^{n(\theta-p+2)}\|g_{n}\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(e^{-2n}T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n}e^{n(\theta-p)}\|u(t,e^{n}x)e^{n}\zeta_{-nx}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T)}+N\sum_{n}e^{n\theta}\|g(t,e^{n}x)\zeta_{-n}(e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(T,\ell_{2})}$
$\displaystyle\leq$ $\displaystyle
N\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}+N\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}.$
Also,
$\sum_{n=-\infty}^{\infty}e^{n(\theta-p+2)}\|v_{n}(0)\|^{p}_{U^{\gamma+2}_{p}}\leq
N\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}.$
Thus the lemma is proved.
$\Box$
###### Remark 3.3
Let $\gamma\geq 0$. By (3.3) and the inequality (see Lemma 2.1(v))
$\|\psi^{-1}u\|_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}\leq\varepsilon\|\psi^{-1}u\|_{H^{\gamma+2}_{p,\theta}(\mathcal{O})}+N(\varepsilon)\|\psi^{-1}u\|_{L_{p,\theta}(\mathcal{O})},$
we easily get
$\|\psi^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N(\|\psi^{-1}u\|^{p}_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}).$
(3.5)
This shows that to estimate
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O})}$ it is enough to
estimate $\|\psi^{-1}u\|^{p}_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}$.
In the following lemma we estimate
$\|\psi^{-1}u\|^{p}_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}$ when
$\theta=d-2+p$ using the Hardy inequality.
###### Lemma 3.4
Let $a^{ij}$ and $\sigma^{ik}$ be independent of $x$. Then for any
$u\in\mathfrak{H}^{2}_{p,d-2+p}(\mathcal{O},T)$, we have
$\displaystyle\|u\|_{\mathfrak{H}^{2}_{p,d-2+p}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\|\psi(\mathbb{D}u-a^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T)}$
(3.6) $\displaystyle+$ $\displaystyle
N\|\mathbb{S}u-\sigma^{i\cdot}u_{x^{i}}\|_{\mathbb{H}^{1}_{p,d-2+p}(\mathcal{O},T,\ell_{2})}+N\|u(0)\|_{U^{2}_{p,d-2+p}(\mathcal{O})},$
where $N=N(d,p,\mathcal{O})$.
Proof. Step 1. First assume that $u\in
L_{p}(\Omega,C([0,T],C^{2}_{0}(\mathcal{O}_{k})))$ for some $k$, where
$\mathcal{O}_{k}:=\\{x\in\mathcal{O}:\psi(x)>1/k\\}$, so that $u$ is
sufficiently smooth in $x$ and vanishes near the boundary
$\partial\mathcal{O}$. Denote
$f=\mathbb{D}u-a^{ij}u_{x^{i}x^{j}},\quad
g=\mathbb{S}u-\sigma^{ik}u_{x^{i}},\quad u_{0}=u(0).$
Then for each $x\in\mathcal{O}$,
$u(t,x)=u_{0}(x)+\int^{t}_{0}(a^{ij}u_{x^{i}x^{j}}+f)ds+\int^{t}_{0}(\sigma^{ik}u_{x^{i}}+g^{k})dw^{k}_{t},$
for all $t\leq T$ (a.s.). Applying Itô’s formula to $|u(t,x)|^{p}$,
$\displaystyle|u(T)|^{p}$ $\displaystyle=$
$\displaystyle|u_{0}|^{p}+p\int^{T}_{0}|u|^{p-2}u(a^{ij}u_{x^{i}x^{j}}+f)\,dt+\int^{T}_{0}p|u|^{p-2}u(\sigma^{ik}u_{x^{i}}+g^{k})dw^{k}_{t}$
$\displaystyle+$
$\displaystyle\frac{1}{2}p(p-1)\int^{T}_{0}|u|^{p-2}\sum_{k=1}^{\infty}(\sigma^{ik}u_{x^{i}}+g^{k})^{2}dt.$
Note that
$\frac{1}{2}p(p-1)|u|^{p-2}\sum_{k=1}^{\infty}(\sigma^{ik}u_{x^{i}}+g^{k})^{2}=p(p-1)|u|^{p-2}\left(\alpha^{ij}u_{x^{i}}u_{x^{j}}+u_{x^{i}}(\sigma^{i},g)_{\ell_{2}}+\frac{1}{2}|g|^{2}_{\ell_{2}}\right),$
where $\alpha^{ij}=\frac{1}{2}(\sigma^{i},\sigma^{j})_{\ell_{2}}$. Taking
expectation, integrating over $\mathcal{O}$ and doing integration by parts
(that is,
$\int_{\mathcal{O}}p|u|^{p-2}ua^{ij}u_{x^{i}x^{j}}dx=-p(p-1)\int_{\mathcal{O}}a^{ij}|u|^{p-2}u_{x^{i}}u_{x^{j}}dx$),
we get
$\displaystyle
p(p-1){\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}\bar{a}^{ij}|u|^{p-2}u_{x^{i}}u_{x^{j}}dxdt$
$\displaystyle\leq$
$\displaystyle{\mathbb{E}}\int_{\mathcal{O}}|u_{0}|^{p}dx+p{\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}|u|^{p-1}|f|dxdt$
$\displaystyle+$ $\displaystyle
p(p-1)\int^{t}_{0}\int_{\mathcal{O}}|u|^{p-2}(u_{x^{i}}(\sigma^{i},g)_{\ell_{2}}+\frac{1}{2}|g|^{2}_{\ell_{2}})dxdt.$
Note that for each $\omega,t$ we have $v:=|u|^{p/2}\in\\{f:f,f_{x}\in
L_{2}(\mathcal{O}),f|_{\partial\mathcal{O}}=0\\}$, and
$v_{x}=\frac{p}{2}|u|^{p/2-2}uu_{x}$. Thus by Hardy Inequality (see (0.1)),
$\int_{\mathcal{O}}|\psi^{-1}u|^{p}\psi^{p-2}dx=\int_{\mathcal{O}}|\psi^{-1}v|^{2}dx\leq
N\int_{\mathcal{O}}|v_{x}|^{2}dx\leq
N\int_{\mathcal{O}}|u|^{p-2}|u_{x}|^{2}dx.$ (3.7)
Also note that
$\displaystyle\int_{\mathcal{O}}|u|^{p-1}|f|dx=\int_{\mathcal{O}}|\psi^{-1}u|^{p-1}|\psi
f|\psi^{p-2}dx$ $\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mathcal{O}}|\psi^{-1}u|^{p}\psi^{p-2}dx+N(\varepsilon)\int_{\mathcal{O}}|\psi
f|^{p}\psi^{p-2}dx,$
$\displaystyle\int_{\mathcal{O}}|u|^{p-2}u_{x^{i}}(\sigma^{i},g)_{\ell_{2}}dx$
$\displaystyle\leq$ $\displaystyle
N|\sigma|_{\ell_{2}}\int_{\mathcal{O}}|u|^{p-2}|u_{x}||g|_{\ell_{2}}dx$
$\displaystyle=$ $\displaystyle
N|\sigma|_{\ell_{2}}\int_{\mathcal{O}}|\psi^{-1}u|^{p-2}|u_{x}||g|_{\ell_{2}}\psi^{p-2}dx$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mathcal{O}}|\psi^{-1}u|^{p}\psi^{p-2}dx+\varepsilon\int_{\mathcal{O}}|u_{x}|^{p}\psi^{p-2}dx+N(\varepsilon)\int_{\mathcal{O}}|g|^{p}_{\ell_{2}}\psi^{p-2}dx.$
Similarly,
$\int_{\mathcal{O}}|u|^{p-2}|g|^{2}_{\ell_{2}}dx\leq\varepsilon\int_{\mathcal{O}}|\psi^{-1}u|^{p}\psi^{p-2}dx+N(\varepsilon)\int_{\mathcal{O}}|g|^{p}_{\ell_{2}}\psi^{p-2}dx.$
Since $(\bar{a}^{ij})\geq\delta_{0}I$, we have
$\delta|u|^{p-2}|Du|^{2}\leq\bar{a}^{ij}|u|^{p-2}u_{x^{i}}u_{x^{j}}$, and
therefore from above calculations
$\displaystyle(1-N_{0}\varepsilon){\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}|\psi^{-1}u|^{p}\psi^{p-2}dxdt$
$\displaystyle\leq$ $\displaystyle
N{\mathbb{E}}\int_{\mathcal{O}}|\psi^{\frac{2}{p}-1}u(0)|^{p}\psi^{p-2}\,dx+N\varepsilon{\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}|u_{x}|^{p}\psi^{p-2}\,dx\,dt$
$\displaystyle+$ $\displaystyle
N(\varepsilon){\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}|\psi
f|^{p}\psi^{p-2}dxdt+N(\varepsilon){\mathbb{E}}\int^{T}_{0}\int_{\mathcal{O}}|g|^{p}_{\ell_{2}}\psi^{p-2}dxdt.$
Thus for any $\varepsilon>0$ so that $\varepsilon N_{0}<1/2$, we have
$\displaystyle\|\psi^{-1}u\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\|u_{0}\|_{U^{1}_{p,d-2+p}}+N\varepsilon\|u_{x}\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T)}$
(3.8) $\displaystyle+$ $\displaystyle N(\varepsilon)\|\psi
f\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T)}+N(\varepsilon)\|g\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T,\ell_{2})}.$
This and (3.5) easily lead to (3.6).
Step 2. General case. We use Theorem 2.8. Take a sequence
$u^{n}\in\mathfrak{H}^{2}_{p,d-2+p}(\mathcal{O},T)$ so that $u^{n}\to u$ in
$\mathfrak{H}^{2}_{p,d-2+p}(\mathcal{O},T)$ and each $u^{n}\in
L_{p}(\Omega,C([0,T],C^{2}_{0}(G_{k})))$ for some $k=k(n)$. By Step 1, we have
(3.6) with $u^{n}$ in place of $u$. Now it is enough to let $n\to\infty$.
$\Box$
The following lemma virtually says that if Theorem 2.12 holds for some
$\theta_{0}\in\mathbb{R}$, then it also holds for all $\theta$ near
$\theta_{0}$.
###### Lemma 3.5
Suppose that there exists a $\theta_{0}\in\mathbb{R}$ so that for any
$u\in\mathfrak{H}^{2}_{p,\theta_{0}}(\mathcal{O},T)$ we have
$\|u\|_{\mathfrak{H}^{2}_{p,\theta_{0}}(\mathcal{O},T)}\leq
N\left(\|\psi\mathbb{D}u-\psi
a^{ij}u_{x^{i}x^{j}}\|_{\mathbb{L}_{p,\theta_{0}}(\mathcal{O},T)}+\|\mathbb{S}u-\sigma^{i}u_{x^{i}}\|_{\mathbb{H}^{1}_{p,\theta_{0}}(\mathcal{O},T,\ell_{2})}+\|u(0)\|_{U^{2}_{p,\theta_{0}}(\mathcal{O})}\right).$
(3.9)
Then there exists $\varepsilon_{0}=\varepsilon_{0}(N,\theta_{0},p)>0$ so that
for any $\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})$ and
$v\in\mathfrak{H}^{2}_{p,\theta}(\mathcal{O},T)$ it holds that
$\|v\|_{\mathfrak{H}^{2}_{p,\theta}(\mathcal{O},T)}\leq
N\left(\|\psi\mathbb{D}v-\psi
a^{ij}v_{x^{i}x^{j}}\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+\|\mathbb{S}v-\sigma^{i}v_{x^{i}}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|v(0)\|_{U^{2}_{p,\theta}(\mathcal{O})}\right).$
Proof. Let $v\in\mathfrak{H}^{2}_{p,\theta}(\mathcal{O},T)$. Denote
$\nu=(\theta_{0}-\theta)/{p}$ and $u:=\psi^{\nu}v$, then by (2.7),
$u\in\mathfrak{H}^{2}_{p,\theta_{0}}(\mathcal{O},T)$. Also it is easy to check
that $\mathbb{D}u=\psi^{\nu}\mathbb{D}v$, $\mathbb{S}u=\psi^{\nu}\mathbb{S}v$
and
$\mathbb{D}u-a^{ij}u_{x^{i}x^{j}}=\psi^{\nu}(\mathbb{D}v-a^{ij}v_{x^{i}x^{j}})-2a^{ij}v_{x^{i}}(\psi^{\nu})_{x^{j}}-a^{ij}v(\psi^{\nu})_{x^{i}x^{j}},$
$\mathbb{S}u-\sigma^{i}u_{x^{i}}=\psi^{\nu}(\mathbb{S}v-\sigma^{i}v_{x^{i}})-\sigma^{i}v(\psi^{\nu})_{x^{i}}.$
Note, since $\psi_{x}$ and $\psi\psi_{xx}$ are bounded, if $\nu\leq 1$ then
$|(\psi^{\nu})_{x^{j}}|=\nu|\psi^{\nu-1}\psi_{x^{i}}|\leq
N\nu\psi^{\nu-1},\quad\quad|(\psi^{\nu})_{x^{i}x^{j}}|\leq N\nu\psi^{\nu-2}.$
(3.10)
By assumption (see (3.9)) and (3.10))
$\displaystyle\|\psi^{\nu}v\|_{\mathfrak{H}^{2}_{p,\theta_{0}}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\|\psi^{\nu}\psi(\mathbb{D}v-a^{ij}v_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta_{0}}(\mathcal{O},T)}+N\|\psi^{\nu}(\mathbb{S}v-\sigma^{i}v_{x^{i}})\|_{\mathbb{H}^{1}_{p,\theta_{0}}(\mathcal{O},T,\ell_{2})}$
$\displaystyle+$ $\displaystyle
N\nu(\|\psi^{\nu}\psi^{-1}v\|_{\mathbb{H}^{1}_{p,\theta_{0}}(\mathcal{O},T)}+\|\psi^{\nu}v_{x}\|_{\mathbb{L}_{p,\theta_{0}}(\mathcal{O},T)})+N\|\psi^{\nu}v(0)\|_{U^{2}_{p,\theta_{0}}(\mathcal{O})}.$
This certainly implies (see (2.7))
$\displaystyle\|v\|_{\mathfrak{H}^{2}_{p,\theta}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\|\psi(\mathbb{D}v-a^{ij}v_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+N\|\mathbb{S}v-\sigma^{i}v_{x^{i}}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T,\ell_{2})}$
$\displaystyle+$ $\displaystyle
N_{1}\nu\left(\|\psi^{-1}v\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}+\|v_{x}\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}\right)+N\|v(0)\|_{U^{2}_{p,\theta}(\mathcal{O})}.$
It follows that the claim of the lemma holds for all sufficiently small $\nu$,
that is for any $\theta$ so that $N_{1}|\theta_{0}-\theta|/p<1$. The lemma is
proved. $\Box$
Remark 3.3, Lemma 3.4 and Lemma 3.5 obviously lead to the following result.
###### Corollary 3.6
Suppose that $\gamma\geq 0$ and the coefficients $a^{ij},\sigma^{ik}$ are
independent of $x$. Then there exists $\beta_{0}=\beta_{0}(d,p,\mathcal{O})>0$
so that if $\theta\in(d-2+p-\beta_{0},d-2+p+\beta_{0})$,
$f\in\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$,
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$, $u_{0}\in
U^{\gamma+2}_{p,\theta}(\mathcal{O})$ and
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ is a solution of
(3.1), then we have
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq N\left(\|\psi
f\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right),$
(3.11)
where $N=N(d,p,\theta,\delta_{0},K,\mathcal{O},T)$.
Now we prove a priori estimate for solutions of the equation
$du=(a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+f)\,dt+(\sigma^{ik}u_{x^{i}}+\mu^{k}u+g^{k})\,dw^{k}_{t},\quad
u(0)=u_{0}.$ (3.12)
###### Theorem 3.7
Suppose $\gamma\geq 0$, $\theta\in(d-2+p-\beta_{0},d-2+p+\beta_{0})$ and
Assumption 2.10 are satisfied. Also let
$f\in\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)$,
$g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$ and $u_{0}\in
U^{\gamma+2}_{p,\theta}(\mathcal{O})$. Then estimate (3.11) holds given that
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ is a solution of
(3.12).
Proof. Step 1. Assume
$|a^{ij}(t,x)-a^{ij}(t,y)|+|\sigma^{i}(t,x)-\sigma^{i}(t,y)|_{\ell_{2}}+|\psi(x)b^{i}(t,x)|+|\psi^{2}(x)c(t,x)|+|\psi\mu|_{\ell_{2}}\leq\kappa,\quad\forall\omega,t,x,y.$
We prove that there exists
$\kappa_{0}=\kappa_{0}(d,\gamma,\theta,\delta_{0},K)>0$ so that the assertion
of the theorem holds if $\kappa\leq\kappa_{0}$. Fix $x_{0}\in\mathcal{O}$ and
denote $a^{ij}_{0}(t,x)=a^{ij}(t,x_{0})$ and
$\sigma^{ik}_{0}(t,x)=\sigma^{ik}(t,x_{0})$. Then $u$ satisfies
$du=(a^{ij}_{0}u_{x^{i}x^{j}}+f_{0})\,dt+(\sigma^{ik}_{0}u_{x^{i}}+g^{k}_{0})\,dw^{k}_{t},\quad
u(0)=u_{0},$
where
$f_{0}=(a^{ij}-a^{ij}_{0})u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+f,\quad
g^{ik}_{0}=(\sigma^{ik}-\sigma^{ik}_{0})u_{x^{i}}+\mu^{k}u+g^{k}.$
By Corollary 3.6,
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq N\left(\|\psi
f_{0}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g_{0}\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right).$
(3.13)
If $\gamma$ is not integer, then by Lemma 2.2(iii) with some
$\nu\in(0,1-\frac{\gamma}{\gamma_{+}})$ (e.g.
$\nu=\frac{1}{2}(1-\frac{\gamma}{\gamma_{+}})$),
$\|(a^{ij}-a^{ij}_{0})\psi
u_{x^{i}x^{j}}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}\leq
N\sup|a^{ij}-a^{ij}_{0}|^{\nu}\|\psi
u_{x^{i}x^{j}}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\Omega,T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\Omega,T)},$
$\|\psi b^{i}u_{x^{i}}+\psi
cu\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}\leq N\sup|\psi
b^{i}|^{\nu}\|u_{x}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+N\sup|\psi^{2}c|\|\psi^{-1}u\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)},\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\Omega,T)},$
and similarly
$\|(\sigma^{i}-\sigma^{i}_{0})u_{x}\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}\leq
N\sup|\sigma^{i}-\sigma^{i}_{0}|^{\nu}_{\ell_{2}}\|u_{x}\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)},$
$\|\mu^{k}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}\leq
N\sup|\psi\mu|^{\nu}_{\ell_{2}}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}.$
By these and (3.13),
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}+N\left(\|\psi
f\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right).$
(3.14)
Thus it is enough to take $\kappa_{0}$ so that $N\kappa^{\nu}<1/2$ for all
$\kappa\leq\kappa_{0}$.
If $\gamma=0$, then obviously
$\displaystyle\|\psi(a^{ij}-a^{ij}_{0})u_{x^{i}x^{j}}+\psi b^{i}u_{x^{i}}+\psi
cu\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}$ $\displaystyle\leq$
$\displaystyle\sup|a^{ij}-a^{ij}_{0}|\|\psi
u_{xx}\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+\sup|\psi
b|\|u_{x}\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+\sup|\psi^{2}c|\|\psi^{-1}u\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\kappa\|\psi^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(\mathcal{O},T)},$
and by Lemma 2.2 (also see (2.14)) with $\nu=\varepsilon/(1+\varepsilon)$,
$\|(\sigma^{i}-\sigma^{i}_{0})u_{x}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N|\sigma^{i}-\sigma^{i}_{0}|^{(0)}_{1}\|u_{x}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N\sup|\sigma^{i}-\sigma^{i}_{0}|^{\nu}\|u_{x}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(\mathcal{O},T)},$
$\|\mu u\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N|\psi\mu|^{(0)}_{1}\|\psi^{-1}\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N\sup|\psi\mu|^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa^{\nu}\|\psi^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(\mathcal{O},T)}.$
These lead to (3.14) for $\gamma=0$.
If $\gamma=1,2,3,...$, then by Lemma 2.2(ii)
$\displaystyle\|(a^{ij}-a^{ij}_{0})\psi
u_{x^{i}x^{j}}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\Omega,T)}$
$\displaystyle\leq$ $\displaystyle N\sup|a^{ij}-a^{ij}_{0}|\|\psi
u_{x^{i}x^{j}}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\Omega,T)}+N|a^{ij}|^{(0)}_{\gamma}\|\psi
u_{xx}\|_{\mathbb{H}^{\gamma-1}_{p,\theta}(\mathcal{O},T)}$
$\displaystyle\leq$ $\displaystyle
N\kappa\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}+N\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)},$
and similarly,
$\|\psi b^{i}u_{x^{i}}+\psi
cu\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}\leq
N\kappa\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}+N\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)},$
$|(\sigma^{i}-\sigma^{i}_{0})u_{x}\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}\leq
N\kappa\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}+N\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}.$
Thus if $\kappa_{1}$ is sufficiently small and $\kappa\leq\kappa_{1}$, then
$\displaystyle\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N\|\psi^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}+N\left(\|\psi
f\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right).$
This and the inequality
$\|\psi^{-1}u\|_{H^{\gamma+1}_{p,\theta}(\mathcal{O})}\leq\varepsilon\|\psi^{-1}u\|_{H^{\gamma+2}_{p,\theta}(\mathcal{O})}+N(\varepsilon)\|\psi^{-1}u\|_{H^{2}_{p,\theta}(\mathcal{O})},$
yield
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)}\leq
N\|\psi^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(\mathcal{O},T)}+N\left(\|\psi
f\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right).$
Take $\kappa_{0}=\kappa_{0}(0)$ chosen in the above when $\gamma=0$. Then it
suffices to take $\kappa_{0}=\kappa_{0}(\gamma)$ so that
$\kappa_{0}<\kappa_{0}(0)\wedge\kappa_{1}$.
Step 2. We generalize the result of Step 1 by summing up the local estimations
of $u$.
Let $x_{0}\in\partial\mathcal{O}$. Fix a nonnegative function $\eta\in
C^{\infty}_{0}(B_{1}(0))$ so that $\eta(x)=1$ for $|x|\leq 1/2$ and define
$\eta_{n}(x)=\eta(n(x-x_{0}))$,
$a^{ij}_{n}(t,x)=a^{ij}(t,x)\eta_{n}(x)+(1-\eta_{n}(x))a^{ij}(t,x_{0})=a^{ij}(t,x_{0})+\eta_{n}(x)(a^{ij}(t,x)-a^{ij}(t,x_{0})),$
$\sigma^{ik}_{n}(t,x)=\eta_{n}(x)\sigma^{ik}(t,x)+(1-\eta_{n}(x))\sigma^{ik}(t,x_{0}),$
$b^{i}_{n}=b^{i}\eta_{n},\quad c_{n}=c\eta_{n},\quad\mu_{n}=\eta_{n}\mu.$
Then
$|a^{ij}_{n}(t,x)-a^{ij}_{n}(t,y)|\leq 2\sup_{x\in
supp\,\,\eta_{n}}\eta_{n}(x)|a^{ij}(t,x)-a^{ij}(t,x_{0})|,$
$|\sigma^{ij}_{n}(t,x)-\sigma^{ik}_{n}(t,y)|_{\ell_{2}}\leq 2\sup_{x\in
supp\,\,\eta_{n}}\eta_{n}(x)|\sigma^{ik}(t,x)-\sigma^{ik}(t,x_{0})|_{\ell_{2}}$
and for any multi-index $\alpha$,
$\sup_{n}\sup_{x\in\mathcal{O}}\psi^{|\alpha|}|D^{\alpha}\eta_{n}|<N(|\alpha|,\eta)<\infty.$
Indeed, for instance, if $x$ is in the support of $\eta_{n}$, then
$\rho(x)\leq 1/n$ and thus
$|\rho(x)D\eta_{n}(x)|=n\rho(x)|\eta_{x}(n(x-x_{0}))|\leq\sup_{x}|\eta_{x}|$.
Using this one can easily check that the coefficients
$a^{ij}_{n},b^{i}_{n},\cdots,\mu^{k}_{n}$ satisfy (2.12), (2.13) and (2.14)
with some constant $K_{0}$, which is independent of $n$.
Take $\kappa_{0}$ from Step 1 corresponding to $d,\gamma,\delta_{0},K,K_{0}$
and $\theta$. We fix $n$ large enough so that
$|a^{ij}_{n}(t,x)-a^{ij}_{n}(t,y)|+|\sigma^{i}_{n}(t,x)-\sigma^{i}_{n}(t,y)|_{\ell_{2}}+|\psi
b^{i}_{n}(t,x)|+|\psi^{2}c_{n}(t,x)|+|\psi\mu_{n}|_{\ell_{2}}<\kappa_{0}\quad\forall\omega,t,x,y.$
This is possible due to the uniform continuity of $a^{ij},\sigma^{i}$ and
condition (2.15).
Now we denote $v=u\eta_{2n}$. Then since $\eta_{n}=1$ and e.g.
$a^{ij}_{n}=a^{ij}$ on the support of $v$, $v$ satisfies
$dv=(a^{ij}_{n}v_{x^{i}x^{j}}+b^{i}_{n}v_{x^{i}}+c_{n}v+\bar{f})dt+(\sigma^{ik}_{n}v_{x^{i}}+\mu^{k}_{n}v+\bar{g}^{k}_{n})dw^{k}_{t},\quad
v(0)=u_{0}\eta_{2n},$
where
$\bar{f}:=-2a^{ij}u_{x^{i}}\eta_{2nx^{j}}-a^{ij}u\eta_{2nx^{i}x^{j}}-b^{i}u\eta_{nx^{i}}+\eta_{2n}f,\quad\bar{g}^{k}=-\sigma^{ik}u\eta_{2nx^{i}}+\eta_{2n}g^{k}.$
By the result of Step 1, for each $t\leq T$,
$\displaystyle\|v\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\Omega,t)}$
$\displaystyle\leq$ $\displaystyle
N(\|\psi\bar{f}\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+\|\bar{g}\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+\|u_{0}\eta_{2n}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})})$
$\displaystyle\leq$ $\displaystyle N\|\psi
u_{x}\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+N\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+N(\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})})$
$\displaystyle\leq$ $\displaystyle
N\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+N(\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}),$
where $N$ is independent of $t$, and the second inequality is due to Lemma 2.2
and the following:
$|a^{ij}\eta_{2nx}|^{(0)}_{\gamma_{+}}+|\psi
a^{ij}\eta_{2nxx}|^{(0)}_{\gamma_{+}}+|\psi
b^{i}\eta_{2nx}|^{(0)}_{\gamma_{+}}+|\sigma^{i}\eta_{2nx}|^{(0)}_{(\gamma+1)_{+}}\leq
N<\infty.$
Now to estimate $u$, one introduces a partition of unity
$\zeta_{i},i=0,1,...,N$ (remember we assume $\mathcal{O}$ is bounded) so that
$\zeta_{0}\in C^{\infty}_{0}(\mathcal{O})$ and $\zeta_{i}=\eta(2n(x-x_{i}))$,
$x_{i}\in\partial\mathcal{O}$ for $i\geq 1$. Then by the above result, for
each $i\geq 1$ and $t\leq T$,
$\|\zeta_{i}u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\Omega,t)}\leq
N(\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t,\ell_{2})}+\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}).$
(3.15)
Note that since $\zeta_{0}$ has compact support in $\mathcal{O}$, for any
$h\in H^{\gamma}_{p,\theta}(\mathcal{O})$ we have $\zeta_{0}h\in
H^{\gamma}_{p}$. Moreover for any $\nu\in\mathbb{R}$,
$\|\psi^{\nu}\zeta_{0}h\|_{H^{\gamma}_{p,\theta}(\mathcal{O})}\sim\|\psi^{\nu}\zeta_{0}h\|_{H^{\gamma}_{p}}\sim\|\zeta_{0}h\|_{H^{\gamma}_{p}}.$
(3.16)
Write down the equation for $\zeta_{0}u$ and apply Theorem 5.1 of [16] to get
$\displaystyle\|\zeta_{0}u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},t)}\sim\|\zeta_{0}u\|^{p}_{\mathcal{H}^{\gamma+2}_{p}(t)}$
$\displaystyle\leq$ $\displaystyle
N\|-2a^{ij}u_{x}\zeta_{0x}-a^{ij}u\zeta_{0xx}-b^{i}u\zeta_{0x}+\zeta_{0}f\|^{p}_{\mathbb{H}^{\gamma}_{p}(t)}$
$\displaystyle+N\|\sigma^{i}u\zeta_{0x^{i}}+\zeta_{0}g\|^{p}_{\mathbb{H}^{\gamma+1}_{p}(t)}+N\|\zeta_{0}u_{0}\|^{p}_{U^{\gamma+2}_{p}}.$
Actually the smoothness condition on the coefficients in Theorem 5.1 of [16]
is different from ours since there the coefficients are assumed to be in
standard Hölder spaces. But since $\zeta_{0}$ has compact support, one can
replace these coefficients with
$\bar{a}^{ij},\bar{b}^{i},\cdots,\bar{\mu}^{k}$ having finite standard Hölder
norms without hurting the equation. By (3.16),
$\|\bar{a}^{ij}u_{x}\zeta_{0x}\|_{\mathbb{H}^{\gamma}_{p}(t)}\leq
N\|u_{x}\zeta_{0x}\|_{\mathbb{H}^{\gamma}_{p}(t)}\leq N\|\psi
u_{x}\zeta_{0x}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}\leq N\|\psi
u_{x}\|_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}\leq
N\|u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}.$
Similar calculus easily shows $\zeta_{0}u$ also satisfies (3.15). By summing
all these estimates and using (2.10) we get, for $t\leq T$
$\displaystyle\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},t)}$
$\displaystyle\leq$ $\displaystyle
N\|u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+N\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},t)}+N\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},t)}+N\|u_{0}\|^{p}_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}$
$\displaystyle\leq$ $\displaystyle
N\int^{t}_{0}\|u\|^{p}_{\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},s)}ds+N\left(\|\psi
f\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T)}+\|g\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}(\mathcal{O})}\right).$
Thus estimate (3.11) follows from this and Gronwall’s inequality. $\Box$
## 4 Proof of Theorem 2.12
Due to the method of continuity and a priori estimate (3.11) (see e.g. the
proof of Theorem 5.1 of [16] for details), to finish the proof, we only show
that for any
$f\in\psi^{-1}\mathbb{H}^{\gamma}_{p,\theta}(\mathcal{O},T),g\in\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T)$
and $u_{0}\in U^{\gamma+2}_{p,\theta}(\mathcal{O})$, the equation
$du=(\Delta u+f)\,dt+g^{k}dw^{k}_{t},\quad u(0)=0$ (4.1)
has a solution $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$. We can
approximate $g=(g^{1},g^{2},...)$ with functions having only finite nonzero
entries, and smooth functions with compact support are dense in
$H^{\nu}_{p,\theta}(\mathcal{O})$. Therefore it follows from a priori estimate
(3.11) that, to prove existence of solution, we may assume that $g$ has only
finite nonzero entries and is bounded on $\Omega\times[0,T]\times\mathcal{O}$
along with each derivative in $x$ and vanishes if $x$ is near
$\partial\mathcal{O}$. Indeed, let $g^{n}\to g$ in
$\mathbb{H}^{\gamma+1}_{p,\theta}(\mathcal{O},T,\ell_{2})$ where $g^{n}$
satisfy the above mentioned conditions, and assume that equation (4.1) with
$g^{n}$ in place of $g$ has a solution $u^{n}$, then using (3.11) applied for
$u^{n}-u^{m}$ one easily finds that $\\{u^{n}\\}$ is a Cauchy sequence in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$ and $u_{n}\to u$ for some
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\mathcal{O},T)$. Obviously the limit
$u$ becomes a solution of (4.1) (see Theorem 2.7).
Under such assumed conditions on $g$,
$v(t,x):=\int^{t}_{0}g^{k}(s,x)dw^{k}_{s}$
is infinitely differentiable in $x$ and vanishes near $\partial\mathcal{O}$.
Therefore we conclude $v\in\mathfrak{H}^{\nu}_{p,\theta}(\mathcal{O},T)$ for
any $\nu\in\mathbb{R}$. Observe that equation (4.1) can be written as
$d\bar{u}=(\Delta\bar{u}+f+\Delta v)dt,$
where $\bar{u}:=u-v$. Thus we reduced the case to the case in which $g\equiv
0$. The same argument shows that we may further assume that $f,u_{0}$ are
bounded along each derivative in $(t,x)$ and vanish near
$\partial\mathcal{O}$. Furthermore by considering $u-u_{0}$, we find that we
also may assume $u_{0}=0$.
First, we consider the case $\theta\geq d-2+p$.
###### Lemma 4.1
Let $\theta\geq d-2+p$, $f\in\mathbb{L}_{p,d}(\mathcal{O},T)$ vanish near
$\partial\mathcal{O}$, say $f(t,x)=0$ for
$x\not\in\mathcal{O}_{k}:=\\{x\in\mathcal{O}:\psi(x)>1/k\\}$ for some $k>0$.
Also assume that the first derivatives of $f$ in $x$ exist and are bounded.
Then the equation
$du=(\Delta u+f)\,dt,\quad u(0)=0$ (4.2)
has a solution $u\in\mathfrak{H}^{1}_{p,\theta}(\mathcal{O},T)$.
Proof. By Lemma 3.2 we only need to prove that there exists a solution
$u\in\psi\mathbb{L}_{p,d-2+p}(\mathcal{O},T)$. Let $n>k$. Since
$\partial\mathcal{O}_{n}\in C^{\infty}$, by Theorem 2.10 in [11] (c.f. Theorem
IV 5.2 in [23]), there is a unique (classical) solution
$u^{n}\in\mathfrak{H}^{2}_{p,d}(\mathcal{O}_{n},T)$ of
$du^{n}=(\Delta u^{n}+f)dt,\quad u^{n}(0,\cdot)=0,$
such that $u^{n}|_{\partial\mathcal{O}_{n}}=0$ and $Du^{n},D^{2}u^{n}$ are
bounded in $[0,T]\times\mathcal{O}_{n}$. Extend $u^{n}(x)=0$ for
$x\not\in\mathcal{O}_{n}$, then $u^{n}$ is Lipschitz continuous in
$\mathcal{O}$. Since for any $q\geq 2$,
$(|u|^{q})_{t}=q|u|^{q-2}uu_{t}=q|u|^{q-2}u(\Delta u+f)$, for each
$x\in\mathcal{O}_{n}$,
$|u^{n}(T,x)|^{q}=q\int^{T}_{0}|u^{n}|^{q-2}u^{n}(\Delta u^{n}+f)dt.$
Integrate this over $\mathcal{O}_{n}$ and do integration by parts to get
$\int^{T}_{0}\int_{\mathcal{O}_{n}}|u^{n}|^{q-2}|Du^{n}|^{2}\,dx\,dt\leq
1/(q-1)\int^{T}_{0}\int_{\mathcal{O}_{n}}|\psi^{-1}u^{n}|^{q-1}|\psi
f|\psi^{q-2}dx$
$\leq\varepsilon\int^{T}_{0}\int_{\mathcal{O}}|\psi^{-1}u^{n}|^{q}\psi^{q-2}\,dx\,dt+N(\varepsilon,q)\int^{T}_{0}\int_{\mathcal{O}}|\psi
f|^{q}\psi^{q-2}\,dx\,dt.$ (4.3)
Taking $q=2$ and using Hardy inequality, we get
$\sup_{n}(\|\psi^{-1}u^{n}\|_{\mathbb{L}_{2,d}(\mathcal{O},T)}+\|Du^{n}\|_{\mathbb{L}_{2,d}(\mathcal{O},T)})<\infty.$
Now we choose $\zeta^{n}\in C^{\infty}_{0}(\mathcal{O}_{n})$ such that
$\zeta^{n}=1$ on $\mathcal{O}_{k}$, $\psi\zeta^{n}_{x},\psi^{2}\zeta^{n}_{xx}$
are bounded in $\mathcal{O}$ uniformly in $n$, and $\zeta^{n}(x)\to 1$ for
$x\in\mathcal{O}$ as $n\to\infty$. Then
$u^{n}\zeta^{n}\in\mathfrak{H}^{2}_{2,d}(\mathcal{O},T)$ satisfies
$(u^{n}\zeta^{n})_{t}=\Delta(u^{n}\zeta^{n})-2u^{n}_{x^{i}}\zeta^{n}_{x^{i}}-u^{n}\Delta\zeta^{n}+f.$
By a priori estimate (3.11)
$\|u^{n}\zeta^{n}\|_{\mathfrak{H}^{2}_{2,d}(\mathcal{O},T)}\leq
N\|u^{n}_{x^{i}}\psi\zeta^{n}_{x^{i}}-\psi^{-1}u^{n}\psi^{2}\Delta\zeta^{n}\|_{\mathbb{L}_{2,d}(\mathcal{O},T)}+N\|\psi
f\|_{\mathbb{L}_{2,d}(\mathcal{O},T)}.$
By dominated convergence theorem,
$\|u^{n}_{x^{i}}\psi\zeta^{n}_{x^{i}}-\psi^{-1}u^{n}\psi^{2}\Delta\zeta^{n}\|_{\mathbb{L}_{2,d}(\mathcal{O},T)}\to
0\quad\text{as}\quad n\to\infty.$
Denote $v^{n}=u^{n}\zeta^{n}\in\mathfrak{H}^{1}_{2,d}(\mathcal{O},T)$, then
$\\{v^{n}\\}$ is a bounded sequence in
$\mathfrak{H}^{1}_{2,d}(\mathcal{O},T)$. By Theorem 2.7 there exists
$u\in\mathfrak{H}^{1}_{2,d}(T)$ so that $v^{n}$ and $\mathbb{D}u^{n}$
converges weakly to $u$ and $\mathbb{D}u$ respectively, and for any $\phi\in
C^{\infty}_{0}(\mathcal{O})$ and $t\in[0,T]$ we have
$(v^{n}(t),\phi)\to(u(t),\phi)$ weakly in $L_{2}(\Omega)$. Since $v^{n}\to u$
weakly in $\mathbb{H}^{1}_{2,d-2}(\mathcal{O},T)$, we have $\Delta v^{n}\to v$
in $\mathbb{H}^{-1}_{2,d+2}(\mathcal{O},T)$. These and the fact that
$(-2u^{n}_{x^{i}}\zeta^{n}_{x^{i}}-u^{n}\zeta^{n}_{x^{i}x^{j}},\phi)=0$ for
all large $n$ show that $u$ satisfies (4.2) in the sense of distribution.
Also, (4.3) with $q=p$ and (3.7) certainly show that
$\sup_{n}\|\psi^{-1}u^{n}\|_{\mathbb{L}_{p,d-2+p}(\mathcal{O},T)}<\infty$. It
follows that
$\psi^{-1}u\in\mathbb{L}_{p,d-2+p}(\mathcal{O},T)\subset\mathbb{L}_{p,\theta}(\mathcal{O},T)$.
The lemma is proved. $\Box$
To finish the proof, we only need to show that there exists $\beta_{1}>0$ so
that $\theta>d-2+p-\beta_{1}$, then equation (4.2) has a solution
$u\in\mathbb{L}_{p,\theta-p}(\mathcal{O},T)$. As before we assume $f$ is
sufficiently smooth and vanishes near the boundary. Take $\kappa_{0}$ from
Step 1 of the proof of Theorem 3.7. We already proved that if $|\psi
b^{i}|+|\psi^{2}c|\leq\kappa_{0}$ and $\theta=d-2+p$, the equation
$dv=(\Delta v+b^{i}v_{x^{i}}+cv+\psi^{\beta}f),\quad v(0)=0$ (4.4)
has a unique solution $v\in\mathfrak{H}^{1}_{p,\theta}(\mathcal{O},T)$ for any
$\beta$. Since $\psi_{x}$ and $\psi\psi_{xx}$ are bounded we can fix $\beta>0$
so that for
$b^{i}:=2\psi^{\beta}(\psi^{-\beta})_{x^{i}}=-2\beta\psi^{-1}\psi_{x^{i}},$
$c:=\psi^{\beta}\Delta(\psi^{-\beta})=\beta(\beta-1)\psi^{-2}|\psi_{x}|^{2}-\beta\psi^{-1}\Delta\psi$
the inequality $|\psi b^{i}|+|\psi^{2}c|\leq\kappa_{0}$ holds, and thus (4.4)
has a solution $v\in\mathfrak{H}^{1}_{p,d-2+p}$. Now it is enough to check
that $u:=\psi^{-\beta}v$ satisfies (4.2) and
$u\in\mathfrak{H}^{1}_{p,d-2+p-\beta
p}(\mathcal{O},T)\subset\mathfrak{H}^{1}_{p,\theta}(\mathcal{O},T)$ for any
$\theta\geq d-2+p-\beta p$. The theorem is proved.
## 5 Proof of Theorem 2.15
Our previous proofs (see e.g. Lemma 3.5) show that we only need to consider
case $\theta=d$ with equation (3.1) having coefficients independent of $x$.
First observe that inclusion
$\mathfrak{H}^{\gamma+2}_{p,d}(\mathcal{O},T)\subset\mathfrak{H}^{\gamma+2}_{p,d-2+p}(\mathcal{O},T)$
gives the uniqueness result for free. Also Remark 3.3 shows that we only need
to show there is a solution $u\in\mathbb{L}_{p,d-p}(\mathcal{O},T)$, so that
$\|\psi^{-1}u\|_{\mathbb{L}_{p,d}(\mathcal{O},T)}\leq N\left(\|\psi
f\|_{\mathbb{L}_{p,d}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{1}_{p,d}(\mathcal{O},T,\ell_{2})}+\|u_{0}\|_{U^{2}_{p,d}(\mathcal{O})}\right).$
For simplicity, assume $u_{0}=0$. Denote
$\mathcal{F}_{p,\theta}=\\{(f,g):\|(f,g)\|_{\mathcal{F}_{p,\theta}}=\|\psi
f\|_{\mathbb{L}_{p,\theta}(\mathcal{O},T)}+\|g\|_{\mathbb{H}^{1}_{p,\theta}(\mathcal{O},T,\ell_{2})}<\infty\\}.$
Fix $q>2$ and $\beta\in(0,\beta_{0})$, where
$\beta_{0}=\beta_{0}(d,\delta_{0},K)$. Then by Theorem 2.12, the map
$\mathcal{R}:(f,g)\to\psi^{-1}u$, where $u$ is the solution of equation (3.1)
is a bounded operator from $\mathcal{F}_{2,d-\beta}$ to
$\mathbb{L}_{2,d-\beta}(\mathcal{O},T)$, and from $\mathcal{F}_{q,d-2+q}$ to
$\mathbb{L}_{q,d-2+q}(\mathcal{O},T)$. Choose $\nu\in(0,1)$ and $p\in(2,q)$ so
that $d=(1-\nu)(d-\beta)+\nu(d-2+q)$ and $1/p=(1-\nu)/2+\nu/q$. Then $F_{p,d}$
(resp. $\mathbb{L}_{p,d}(\mathcal{O},T)$) becomes a complex interpolation
space of $F_{2,d-\beta}$ and $F_{q,d-2+q}$ (resp.
$\mathbb{L}_{2,d-\beta}(\mathcal{O},T)$ and
$\mathbb{L}_{q,d-2+q}(\mathcal{O},T)$), that is,
$\mathcal{F}_{p,d}=[F_{2,d-\beta},F_{q,d-2+q}]_{\nu},\quad\mathbb{L}_{p,d}(\mathcal{O},T)=[\mathbb{L}_{2,d-\beta}(\mathcal{O},T),\mathbb{L}_{q,d-2+q}(\mathcal{O},T)]_{\nu}.$
(See Proposition 2.4 of [27] and Theorem 5.1.2 of [2] for details). It follows
from the interpolation theory that $\mathcal{R}$ is a bounded linear map from
$\mathcal{F}_{p,d}$ to $\mathbb{L}_{p,d}(\mathcal{O},T)$ (see Theorem (a) on
Page 59 of [34]). This proves the claim for above fixed $p$. Now for $2\leq
p^{\prime}\leq p$, it is enough to notice that for $\nu^{\prime}$ so that
$1/p^{{}^{\prime}}=(1-\nu^{\prime})/2+\nu^{\prime}/p$,
$\mathcal{F}_{p^{\prime},d}=[F_{2,d},F_{p,d}]_{\nu^{\prime}},\quad\mathbb{L}_{p^{\prime},d}(\mathcal{O},T)=[\mathbb{L}_{2,d}(\mathcal{O},T),\mathbb{L}_{p,d}(\mathcal{O},T)]_{\nu^{\prime}}.$
It follows that $\mathcal{R}$ is a bounded linear map from
$\mathcal{F}_{p^{\prime},d}$ to $\mathbb{L}_{p^{\prime},d}(\mathcal{O},T)$.
The theorem is proved.
## References
* [1] P.A. Cioica, S. Dahlke, N. Döhring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling, Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains, DFG SPP-1324 Preprint 66 (2010) URL: http://www.dfg-spp1324.de/download/preprints/preprint066.pdf.
* [2] J. Bergh and J. Löfström, Interpolation spaces, Grundlerhren der Mathematischen Wissenschafter, No. 223. Springer-Verlag, Berlin-New York, 1976.
* [3] D.E. Edmunds and R. Hurri-syrjänen, Weighted Hardy inequalities, J.Math.Anal.Appl. 310 (2005), 423-435.
* [4] F. Flandoli, Dirichlet boundary value problem for stochastic parabolic equations: compatibility relation and regularity of solutions, Stochastics Stochastics Rep. 29(3) (1990), 331-357.
* [5] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2d ed., Springer Verlag, Berlin, 1983.
* [6] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, 1985.
* [7] P. Grisvard, Singularities in boundary value problems, Springer, 1992.
* [8] K. Kim, An $L_{p}$-theoryof SPDEs on Lipschitz domains, Potential Anal. 29 (2008), 303-329.
* [9] K. Kim, On $L_{p}$-theory of SPDEs of divergence form in $C^{1}$ domains, Probability theory and related fields, 130 (2004), no.4, 473-492.
* [10] K. Kim, On stochastic partial differential equations with variable coefficients in $C^{1}$ domains, Stochastic processes and their applications, 112 (2004), no.2, 261-283.
* [11] K. Kim and N.V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^{1}$ domains, SIAM J. Math. Anal. 36 (2004), 618-642.
* [12] K. Kim and N.V. Krylov, On SPDEs with variable coefficients in one space dimension, Potential Anal., 21 (2004), no.3, 203-239.
* [13] N.V. Krylov, Some properties of traces for stochastic and determistic parabolic weighted Sobolev spaces, Journal of Functional Analysis 183 (2001), 1-41.
* [14] N.V. Krylov, SPDEs in $L_{q}(\text{\,\raise 0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$},L_{p})$ spaces, Electronic Journal of Probability, 5 (2000), no.13, 1-29.
* [15] N.V. Krylov, Some properties of Traces for Stochastic and Deterministic Parabolic Weighted Sobolev Spaces, Journal of Functional Analysis 183 (2004), 1-41.
* [16] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs 64 (1999), AMS, Providence, RI.
* [17] N.V. Krylov, Some properties of weighted Sobolev spaces in $\mathbb{R}^{d}_{+}$, Annali Scuola Normale Superiore di Pisa, Sci.Fis.Mat., Serie 4 28 (1999), Fasc. 4, 675-693.
* [18] N.V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space, Comm. in PDEs 23 (1999), no.9-10, 1611-1653.
* [19] N.V. Krylov, On $L_{p}$-theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 27 (1996), no.2. 313-340.
* [20] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal. 31 (1999), no.1, 19-33.
* [21] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., Vol. 30 , No. 2 (1999), 298-325.
* [22] Alois Kufner, Weighted Sobolev spaces, John Wiley and Sons Inc, 1984.
* [23] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Utal’stseva, Linear and quasilinear parabolic equations, Am. Math.Soc. (1968).
* [24] G. Lieberman, Second order parabolic differential equations, World Scientific, Singapore-New Jersey-London-Hong Kong, 1996.
* [25] F. Lindner, Approximation and regularity of stochastic PDEs (doctoral thesis), Shaker, 2011.
* [26] S.V. Lototsky, Linear stochastic parabolic equations, degenerating on the boundary of a domain, Electronic Journal of Probability, 6 (2001), no.24, 1-14.
* [27] S.V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods and Applications of Analysis 1 (2000), no.1, 195-204.
* [28] S.V. Lototksy, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics and Stochastics Reports, 68 (1999), no.1-2, 145-175.
* [29] R. Mikulevicius and B. Rozovskii, A note on Krylov’s $L_{p}$-theory for systems of SPDEs, Electron. J. Probab., 6 (2001), no. 12, 35 pp.
* [30] R. Mikulevicius and H. Pragarauskas, On Cauchy-Dirichlet problem in half-space for parabolic SPDEs in weighted Holder spaces, Stochastic Process. Appl. 106 (2003), no. 2, 185-222.
* [31] J. Ne$\check{c}$as, Sur une m$\acute{e}$thode pour r$\acute{e}$soudre les $\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles du type elliptique, voisine de la variationelle, Ann. Scuola Norm. Sup.Pisa Ser. 16 (1962) , 305-326.
* [32] Pardoux,E. (1979). Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3, 127-167.
* [33] B.L. Rozovskii, Stochastic evolution systems, Kluwer, Dordrecht, 1990.
* [34] H. Triebel, Interpolation theory, function spaces, differential operators. Elsevier North-Holland, Inc., 1978.
* [35] A. Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), 85-95.
* [36] H. Yoo, $L_{p}$-estimate for stochastic PDEs with discontinuous coefficients, Stochastic Anal. Appl., 17 (1999), no.4, 678-711.
|
arxiv-papers
| 2011-09-22T08:40:09 |
2024-09-04T02:49:22.401329
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kyeong-Hun Kim",
"submitter": "Kyeong-Hun Kim",
"url": "https://arxiv.org/abs/1109.4727"
}
|
1109.4788
|
arxiv-papers
| 2011-09-22T12:23:40 |
2024-09-04T02:49:22.413624
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fernand Pelletier",
"submitter": "Fernand Pelletier",
"url": "https://arxiv.org/abs/1109.4788"
}
|
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1109.4919
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# A brief overview on the BioPAX and SBML standards for formal presentation of
complex biological knowledge
Leo Lahti
University of Helsinki
leo.lahti@iki.fi
(2007)
Copyright ©2007-2011 Leo Lahti
Some Rights Reserved.
http://www.iki.fi/Leo.Lahti (leo.lahti@iki.fi)
This work is licensed under the terms of Creative Commons Attribution 3.0
Unported license available from http://www.creativecommons.org/. Accordingly,
you are free to copy, distribute, display, perform, remix, tweak, and build
upon this work even for commercial purposes, assuming that you give the
original author credit. See the licensing terms for details.
## 1 Background
The quest for systems-level understanding of cellular mechanisms is a major
paradigm in modern biology [1]. Development of various high-throughput
technologies for measuring cell-biological phenomena has accelerated systems
biology research through rapid accumulation of biological data in public and
private repositories.
Various databases focus on biological pathways, structured representations of
cellular processes. Pathways are idealized descriptions of molecular events
within the cell, and represent the current state of knowledge of these
systems. Multiple proteins and other compounds are typically involved in a
hierarchical manner through various molecular mechanisms, forming a complex
biological network. Pathway data exhibits many details of the well-studied
parts of the system, and can potentially be used to guide the search for novel
cellular mechanisms when combined with high-throughput measurements such as
gene expression, metabolomics or proteomics data (see e.g. [2]). Computational
modeling can also highlight aspects of the existing networks that have been
left unnoticed by more traditional methods.
Utilizing the huge data collections to create new biological knowledge is a
challenge where novel computational tools play a key role [3]. To take full
advantage of the possibilities opened up by large-scale community databases,
efficient standards for representing biological information are needed. This
will enhance the sharing and evaluation of knowledge, and development
computational tools.
A good standard is independent of the software environment where the data is
created and analyzed, and ensures the usability of the data beyond the
lifetime of any single software. Perhaps surprisingly, the standards for
representing biological network data have emerged only recently. In the
beginning of 2000’s, the field was lacking proper standards, and the
integration of network data from different databases required considerable
effort as each database was using its own internal representation and analysis
tools.
Various frameworks for the representation of network information were proposed
to alleviate these problems at the time. In this work, I present an overview
of two such proposals, BioPAX and SBML. These two approaches have gained
popularity and are promising candidates to become established as widely
adopted future standards in the field.
BioPAX and SBML are advancing the construction of databases that can be widely
used to model, simulate, and visualize network data in various programming
languages. They are not competing standards but rather complementary
approaches to describe related phenomena. SBML is targeted at modeling systems
of dynamic biochemical reaction networks that are described by reaction
equations. BioPAX assumes a static view on the network model. It does not
contain mathematical formulas but provides more detailed information
concerning the individual molecules and interactions.
This work is mainly based on the technical manuals of BioPAX (2.1) and SBML
(2.3 prerelease), their respective websites111http://www.biopax.org,
http://sbml.org, and the comparative review of these representation schemes by
Strömbäck and Lambrix [4]. The work is organized as follows. First, I will
illustrate the nature of biological network data through a simple example.
Then BioPAX and SBML are considered in more detail. The general scope and the
overall structure of both standards is described. Finally, various aspects of
the two approaches, and the differences between them are discussed.
The aim of this work is to give a good overall picture of these standards.
Hence I have skipped many details of the two models in this short
introduction.
## 2 Biological network models
Biological network models describe complex molecular systems that are active
in certain biological conditions. Knowledge of these systems typically arises
from various individual studies on different parts of the network.
Traditionally, biochemical networks have been summarized in schematic two-
dimensional maps. Until recently, textbook figures of such maps have been the
only general description of the system for many networks. For large-scale
computational studies, such illustrations are not sufficient, and the need for
computer-readable representations is evident.
### 2.1 A simple example of a biological network model
The BioModels database222http://www.ebi.ac.uk/biomodels/ is one of the several
public databases that currently provide information on biological networks.
The minimal cascade model for the mitotic oscillator involving cyclin and cdc2
kinase [5] picked from this database is a simple example of a molecular
network model333http://www.ebi.ac.uk/compneur-srv/biomodels-main/publ-
model.do?mid=BIOMD0000000003 (Figure 1).
In this example, as in other typical illustrations of a network model, nodes
represent the involved biomolecules, and edges indicate the interactions
between these molecules. A database may contain additional information
concerning the individual molecules and the nature of the interactions,
although such information may not be visualized in simple interaction maps. A
complex dynamical system, for example, cannot be illustrated within one static
figure, whereas a carefully designed database structure can make this
possible. Hyperlinking within and between databases is also made possible by
current database technologies, and allows easy navigation of the network data
for application-oriented users.
The network models in the BioModels database can be exported in both BioPAX
and SBML format. These describe the same network but provide complementary
information of it. The XML-based implementation of both models are given for
the cascade network of Figure 1 in the Appendix.
Even for describing such a simple network, somewhat lengthy code is apparently
needed to capture any details of the model. The code is here offered for
demonstration purposes and is not considered in detail; the intention of this
work is to provide a general overview of the two network modeling standards-
not to single out details of a specific biological network. The general
characteristics of the BioPAX and SBML schemes, and their differences are
detailed below.
Figure 1: Visualization of the network model BIOMD0000000003 from the
BioModels database.
## 3 Standards for biological network information
### 3.1 Biological Pathways Exchange Language (BioPAX 2)
BioPAX [6] is a specific data type for representing and sharing information
concerning biomolecular networks444http://www.biopax.org. BioPAX does not
present mathematical relations, but provides many details concerning the
molecular interactions within the cell. The current release of BioPAX (Level
2) can be used to represent metabolic networks and molecular interaction
networks. Support for gene and DNA interactions, genetic interactions, and
signal transduction are planned for upcoming releases.
A number of popular network databases support the BioPAX format, including
BioCyc555http://biocyc.org, Reactome666http://reactome.org/, BioModels
Database, and Pathway Interaction Database777http://pid.nci.nih.gov. Various
network analysis tools for BioPAX are available as
well888http://biopaxwiki.org/cgi-
bin/moin.cgi/Biological_software_supporting_BioPAX, many of them open source.
One of the most popular tools for viewing and editing BioPAX ontologies is
Protege999http://protege.stanford.edu, a user-friendly software with a
graphical user interface. For more computationally oriented people, some tools
for low-level analysis of the BioPAX ontology are offered by the Rredland
package101010http://bioconductor.org/packages/2.0/bioc/html/Rredland.html of
the R/BioConductor project, for example. It is worth emphasizing that the
scope of these software environments is not restricted to any single database
but can handle data from any database supporting the BioPAX format.
#### 3.1.1 A brief characterization of the BioPAX network model
BioPAX describes its objects in an inheritance class hierarchy. The basic
building blocks of a BioPAX model are the Physical Entity, Interaction and
Pathway classes. Pathway represents a set of interactions that together form a
network model. Each Interaction describes relationships between Physical
Entities. This structure is the core of a BioPAX network model.
Details concerning the interactions and entities are given by subclasses that
describe various interaction and physical entity types. Physical entities are
various objects that may interact, including molecular complexes, proteins,
DNA, RNA, and small molecules. Each interaction type limits the possible roles
and the number of potentially interacting entities. Transport, catalysis and
modulation are examples of possible interactions.
In summary, BioPAX specifies the type of the interacting molecules and details
of their interactions in a pre-defined class hierarchy. Such representation
enables efficient sharing of the information between databases. To obtain
sufficient generality in the model for such purposes, many details have to be
omitted in the model hierarchy. However, additional information and hyperlinks
to external sources can be assigned to BioPAX objects. It is also possible to
make application-specific additions to the BioPAX standard model, although
this is not recommended.
BioPAX code for the network model in Figure 1 is given in the Appendix. The
source code contains all information encoded into the model but it is not
meant to be human-readable as such. Appropriate visualization and analysis
tools help in the interpretation of the data. The example code illustrates the
internal structure of a BioPAX model and is self-explanatory for people with
previous experience on XML-based database programming.
#### 3.1.2 Rredland tool for R/BioConductor
I briefly tested the R/BioConductor tool for BioPAX (Rredland) to see if I
could utilize it in my own bioinformatics research. Unfortunately, this
specific package still seems to be in an initial stage, and only simple
queries are possible. However, the package is under active development and the
release of a sufficient collection of low-level analysis routines can be
expected soon. Currently, methods for reading the data and doing basic queries
are available.
Following is a very simple example of BioPAX hierarchy for an interaction
event in our example model in Figure 1. I picked the ’creation of cyclin’
event from the model. This appears to have the following attributes:
type | conversion
---|---
NAME | creation of cyclin
RIGHT | reaction1_RIGHT_C
Accordingly, ’creation of cyclin’ is a ’conversion’ reaction, and this
reaction has a relationship ’RIGHT’ to another subject, ’reaction1_RIGHT_C’.
Now we can proceed in inspecting the model and check the ’reaction1_RIGHT_C’
subject in more detail. It appears to have the following attributes:
type | physicalEntityParticipant
---|---
PHYSICAL-ENTITY | C
CELLULAR-LOCATION | cell
Now we see that the subject ’reaction1_RIGHT_C’ is of the type
’physicalEntityParticipant’, the corresponding ’PHYSICAL-ENTITY’ being ’C’.
Also cellular location ’cell’ is given although it is in this case very
general type and hence rather uninformative. Further checking the attributes
for ’C’, we find that it is of ’physicalEntity’ type with the name ’Cyclin’.
In summary, ’creation of cyclin’ is a conversion reaction that involves
’Cyclin’ as its ’RIGHT’ participant, denoting that Cyclin is the end product
from this reaction. In the contrary, ’LEFT’ would denote the input molecules
for the reaction. For this specific event, additional details have not been
stored in the model.
This short query illustrates the basic structure of the BioPAX hierarchy from
low-level analysis perspective. A typical user does not need to consider such
raw representation schemes; application-oriented users can browse the network
and related information with simple visualization tools. Further properties of
our example network can be retrieved from the source code in the Appendix.
### 3.2 Systems Biology Markup Language (SBML 2)
SBML is another alternative to represent biological network information. The
current version of SBML [7] (level 2) is targeted at analysing and simulating
basic biochemical reaction networks111111http://sbml.org. Examples of such
networks include cell signaling, metabolic networks and gene regulation. SBML
can be used to describe both quantitative and qualitative aspects of the
reactions that modify or transport the entities.
The SBML standard is supported by many popular network databases including
KEGG121212http://www.genome.jp/kegg, Reactome, and the BioModels Database.
Analysis of SBML data is supported by more than 110 software systems.
Computational tools are available in several languages, including C++, MATLAB,
Java and
R/BioConductor131313http://cran.r-project.org/src/contrib/Descriptions/rsbml.html.
In addition to the widely adopted XML format, the implementation of SBML model
utilizes certain other standards, including
MathML141414http://www.w3.org/TR/2003/REC-MathML2-20031021 and the
CellML151515http://www.cellml.org. In fact, the SBML standard is similar to
the independent representation scheme CellML [8]. According to the SBML
website, the editors of SBML and CellML have collaborated to fuse the two
models to create a single standard in the future.
#### 3.2.1 A brief characterization of the SBML model
An SBML model consists of a set of chemical species that are linked by
reactions. Chemical species include simple ions, simple molecules (e.g.
glucose), and large molecules such as RNAs or proteins. Each species is
located in a compartment which is describes the reaction environment. The
reactions can transform one entity into another, transport entities between
compartments, or describe the binding of the species. This is the general
structure of an SBML model.
A number of more detailed features can be assigned to the objects of an SBML
model. Chemical species can be assigned a spatial size and charge. In a given
interaction, the participating species have particular roles such as reactant,
product or modifier. It is also possible to specify whether a reaction is
reversible. Mathematical description of the reactions are used to describe the
reaction dynamics. The overall model can be provided with initial parameters,
including the concentrations of the original species and constraints for the
model parameters. Interestingly, SBML allows modeling of sudden discrete
changes in the model under specified circumstances. An SBML model can even
represent interactions between the interaction networks and other phenomena.
SBML does not impose a class hierarchy into the model as in BioPAX. Instead,
it focuses on mathematical description of the interactions. In general, SBML
is not designed to store all potentially conflicting aspects of the network
but to encode a coherent view of a biological reaction network that can be
used in computational simulations.
References to external information sources can be given in an annotation
field. SBML also provides limited options for adding data that does not fit
into the standard format.
The SBML source code for the network model in Figure 1 is given in the
Appendix for demonstration purposes, and can be compared with the BioPAX code
describing the same network. Based on the documentation, the R package for
analyzing SBML (RSBML) seems to be more complete than the R-based analysis
tools for BioPAX. I personally need a general static description of the
biomolecule relationships that I could combine with other high-throughput data
such as gene expression measurements, for example. SBML is focused on dynamic
simulations, and the biomolecular relationships in SBML are generally too
complex for my current purposes. Therefore, I have not tested the RSBML
package.
## 4 Discussion
Modern high-throughput technologies and standardized databases advance the
accumulation of biological knowledge. This knowledge is often highly
structured. Biological network data exemplifies the complex nature of
biological data. Representation of dynamic biochemical reactions and the
relationships between various compounds and molecular complexes in its full
richness is a challenging task beyond a mere listing of events.
BioPAX and SBML are general frameworks to represent network information. These
standards can be used to present complex systems-level biological knowledge
that extends beyond listing sets of individual interactions between
biomolecules, and allow the automated analysis of such data. This is different
from standards like PSI-MI [9] that offer detailed information on individual
interactions but cannot be used to describe the more general system arising
from these interactions. Various protein-protein databases that offer pairwise
interaction data between biomolecules are reviewed in [10]. Although SBML and
BioPAX could be used for similar purposes, their focus is on describing a
larger system of molecular mechanisms, and less attention is paid to
individual interactions. If necessary, schematic pathway maps representing the
pairwise interactions of a molecular network can be automatically generated
from the network information provided by SBML or BioPAX.
BioPAX seems to combine the general representation of the network with
sufficient details of the network objects better than SBML. Due to the
hierarchical nature of BioPAX, a detailed representation of interactions and
biomolecules can be obtained while allowing queries at a more general level.
For example, the substructure of molecular complexes can be described in
BioPAX while such option is only planned for the next level of SBML.
The networks in BioPAX are static, and no information concerning reaction
dynamics within the network is provided. On the contrary, SBML supports
quantitative models and is a better format for dynamical simulations of
biochemical networks. The increased mathematical complexity of SBML is
balanced by a simplified representation of individual molecules and
interactions. While BioPAX utilizes hierarchical representation to encode the
the relationships between molecules in a biological network, SBML uses
mathematical equations to define these relationships. SBML models are less
structured than BioPAX models, but they give more quantitative details
concerning the interactions and the biochemical reactions arising from these
interactions.
The utility of specialized standards depends on the fact that they restrict
the representation to commonly agreed schemes. User-specific definitions and
structures can cause problems for others handling the data, and do not fit in
the idea of standardization. However, the possibility of including additional
information is often essential in biomedical studies. Both standards offer
limited possibilities for application-specific model extension. They also
recommend that the standard names of biomolecules are used in the
representations, but neither of them enforces the use of standard names.
The implementation of both BioPAX and SBML is based on XML, a widely used
text-based format for representing hierarchical information schemes. The
advantage of well-established ontology semantics is that it these models can
be easily published in electronic form as supplementary data for peer-reviewed
journal articles, for example. The use of standardized representations for
systems-level biological knowledge will certainly advance the utilization of
this knowledge in large-scale computational studies where the availability of
such data in a suitable format is often a bottleneck.
Both BioPAX and SBML have gained popularity in the bioinformatics community in
the recent years, and are promising standards for sharing biological network
information. The development of BioPAX and SBML is community-based. New
features are added according to the needs of the bioinformatics community that
uses these representation schemes. We are likely to see remarkable development
and extensions in the two standards and related analysis tools as the user
base grows and more people are contributing to the testing and development of
these standards in one way or other.
As the two standards are complementary rather than competing, it is not
expected that one of them would override the other in the near future.
However, as BioPAX and SBML are used to describe closely related aspects of
biological networks, their future levels are likely to become more closely
related. Currently, BioPAX offers a more detailed description of static
molecular relationships within the network whereas SBML is better suited for
computational simulations of network dynamics.
## References
* [1] Francis S. Collins, Eric D. Green, Alan E. Guttmacher, and Mark S. Guyer. A vision for the future of genomics research. Nature, 422:835–847, 2003.
* [2] Barbara A. Novak and Ajay N. Jain. Pathway recognition and augmentation by computational analysis of microarray expression data. Bioinformatics, 22:233–241, 2006.
* [3] Joel E. Cohen. Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better. PLoS Biology, 2:e439, 2004.
* [4] Lena Strömbäck and Patrick Lambrix. Representations of molecular pathways: an evaluation of SBML, PSI MI and BioPAX. Bioinformatics, 21:4401–4407, 2005.
* [5] A. Goldbeter. A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proceedings of the National Academy of Sciences, USA, 88:9107–9111, 1991.
* [6] BioPAX workgroup. BioPAX - Biological Pathways Exchange Language, 2005. Level 2, Version 1.0 Documentation.
* [7] M. Hucka, A. Finney, H.M. Sauro, H. Bolouri, J.C. Doyle, Kitano, H. and the rest of the SBML Forum, A.P. Arkin, B. J. Bornstein, D. Bray, A. Cornish-Bowden, A.A. Cuellar, S. Dronov, E. D. Gilles, M. Ginkel, V. Gor, I. I. Goryanin, W. J. Hedley, T. C. Hodgman, J.-H. Hofmeyr, P. J. Hunter, N. S. Juty, J. L. Kasberger, A. Kremling, U. Kummer, N. Le Novere, L. M. Loew, D. Lucio, P. Mendes, E. Minch, E. D. Mjolsness, Y. Nakayama, M. R. Nelson, P. F. Nielsen, T. Sakurada, J. C. Schaff, B. E. Shapiro, T.S. Shimizu, H.D. Spence, J. Stelling, K. Takahashi, M. Tomita, J. Wagner, and J. Wang. The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics, 19(4):524–531, 2003.
* [8] Catherine M. Lloyd, Matt D. B. Halstead, and Poul F. Nielsen. CellML: its future, present and past. Progress in Biophysics and Molecular Biology, 85:433–450, 2004\.
* [9] H. Hermjakob, L. Montecchi-Palazzi, G. Bader, R. Wojcik, L. Salwinski, A. Ceol, S. Moore, S. Orchard, U. Sarkans, C. von Mering, B. Roechert, S. Poux, E. Jung, H. Mersch, P. Kersey, M. Lappe, Y.X. Li, R. Zeng, D. Rana, M. Nikolski, H. Husi, C. Brun, K. Shanker, S.G.N. Grant, C. Sander, P. Bork, W.M. Zhu, A. Pandey, A. Brazma, B. Jacq, M. Vidal, D. Sherman, P. Legrain, G. Cesareni, L. Xenarios, D. Eisenberg, B. Steipe, C. Hogue, and R. Apweiler. The HUPOPSI’s molecular interaction format - a community standard for the representation of protein interaction data. Nature Biotechnology, 22:177–183, 2004.
* [10] Suresh Mathivanan, Balamurugan Periaswamy, TKB Gandhi, Kumaran Kandasamy, Shubha Suresh, Riaz Mohmood, YL Ramachandra, and Akhilesh Pandey. An evaluation of human protein-protein interaction data in the public domain. BMC Bioinformatics, 7(S5):S19, 2006.
## Appendix
### BioPAX
BioPAX source code for the network model in Figure 1.
<rdf:RDF xml:base="http://www.ebi.ac.uk/biomodels/biopax">-<owl:Ontology rdf:about="">
<owl:imports rdf:resource="http://www.biopax.org/release/biopax-level2.owl"/>
</owl:Ontology>
-<bp:openControlledVocabulary rdf:ID="cell">
<bp:TERM>cell</bp:TERM></bp:openControlledVocabulary>-<bp:physicalEntity rdf:ID="C">
<bp:NAME>Cyclin</bp:NAME></bp:physicalEntity>-<bp:physicalEntity rdf:ID="M">
<bp:NAME>CDC-2 Kinase</bp:NAME></bp:physicalEntity>
-<bp:physicalEntity rdf:ID="X">
<bp:NAME>Cyclin Protease</bp:NAME></bp:physicalEntity>
-<bp:conversion rdf:ID="conversion_reaction1">
<bp:NAME>creation of cyclin</bp:NAME>
-<bp:RIGHT>-<bp:physicalEntityParticipant rdf:ID="reaction1_RIGHT_C">
<bp:PHYSICAL-ENTITY rdf:resource="#C"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:RIGHT>
</bp:conversion>-<bp:conversion rdf:ID="conversion_reaction2">
<bp:NAME>default degradation of cyclin</bp:NAME>
-<bp:LEFT>-<bp:physicalEntityParticipant rdf:ID="reaction2_LEFT_C">
<bp:PHYSICAL-ENTITY rdf:resource="#C"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:LEFT>
</bp:conversion>-<bp:conversion rdf:ID="conversion_reaction3">
<bp:NAME>cdc2 kinase triggered degration of cyclin</bp:NAME>
-<bp:LEFT>-<bp:physicalEntityParticipant rdf:ID="reaction3_LEFT_C">
<bp:PHYSICAL-ENTITY rdf:resource="#C"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:LEFT>
</bp:conversion>-<bp:control rdf:ID="control_reaction3">
-<bp:CONTROLLER>-<bp:physicalEntityParticipant rdf:ID="reaction3_CONTROLLER_X">
<bp:PHYSICAL-ENTITY rdf:resource="#X"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:CONTROLLER>
<bp:CONTROLLED rdf:resource="#conversion_reaction3"/>
</bp:control>-<bp:conversion rdf:ID="conversion_reaction4">
<bp:NAME>activation of cdc2 kinase</bp:NAME>
-<bp:RIGHT>-<bp:physicalEntityParticipant rdf:ID="reaction4_RIGHT_M">
<bp:PHYSICAL-ENTITY rdf:resource="#M"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:RIGHT>
</bp:conversion>-<bp:conversion rdf:ID="conversion_reaction5">
<bp:NAME>deactivation of cdc2 kinase</bp:NAME>
-<bp:LEFT>-<bp:physicalEntityParticipant rdf:ID="reaction5_LEFT_M">
<bp:PHYSICAL-ENTITY rdf:resource="#M"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:LEFT>
</bp:conversion>-<bp:conversion rdf:ID="conversion_reaction6">
<bp:NAME>activation of cyclin protease</bp:NAME>-<bp:RIGHT>
-<bp:physicalEntityParticipant rdf:ID="reaction6_RIGHT_X">
<bp:PHYSICAL-ENTITY rdf:resource="#X"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:RIGHT>
</bp:conversion>-<bp:conversion rdf:ID="conversion_reaction7">
<bp:NAME>deactivation of cyclin protease</bp:NAME>
-<bp:LEFT>-<bp:physicalEntityParticipant rdf:ID="reaction7_LEFT_X">
<bp:PHYSICAL-ENTITY rdf:resource="#X"/><bp:CELLULAR-LOCATION rdf:resource="#cell"/>
</bp:physicalEntityParticipant></bp:LEFT></bp:conversion></rdf:RDF>
### SBML
SBML source code for the network model in Figure 1.
<?xml version="1.0" encoding="UTF-8"?>
<sbml xmlns="http://www.sbml.org/sbml/level2" metaid="_180324" level="2" version="1">
<model metaid="_180340" id="GMO" name="Goldbeter1991_MinMitOscil">
<notes><body xmlns="http://www.w3.org/1999/xhtml">
<p><h2><center>A Simple Mitotic Oscillator</center></h2></p>
<p>Reference:Goldbeter A (1991)<i>A minimal cascade model for the mitotic oscillator involving cyclin
Ψand cdc2 kinase</i>, PNAS 88:9107-9111<br></br>Web Reference:
Ψ<a href="http://www.pnas.org/cgi/content/abstract/88/20/9107">
Ψhttp://www.pnas.org/cgi/content/abstract/88/20/9107</a></p>
<p style="font-size:x-small;">This is a Systems Biology Markup Language (SBML) file, generated by
ΨMathSBML 2.4.6 (14-January-2005) 14-January-2005 18:33:39.806932. SBML is a form of XML, and most
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<p>This model originates from BioModels Database: A Database of Annotated Published Models.
ΨIt is copyright (c) 2005-2006 The BioModels Team.<br/> For more information see the
Ψ<a href="http://www.ebi.ac.uk/biomodels/legal.html">terms of use</a>.</p></body></notes>
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|
arxiv-papers
| 2011-09-22T19:15:57 |
2024-09-04T02:49:22.420811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leo Lahti",
"submitter": "Leo Lahti",
"url": "https://arxiv.org/abs/1109.4919"
}
|
1109.4928
|
arxiv-papers
| 2011-09-22T19:46:02 |
2024-09-04T02:49:22.427730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leo Lahti, Laura L. Elo, Tero Aittokallio, Samuel Kaski",
"submitter": "Leo Lahti",
"url": "https://arxiv.org/abs/1109.4928"
}
|
|
1109.4973
|
# On Operator-valued Semicircular Random Variables
###### Abstract.
In this paper, we discuss some special properties of operator-valued
semicircular random variables including representation of the Cauchy transform
of a compactly supported probability measure in terms of their operator-valued
Cauchy transforms and existence of nonzero discrete part of their associated
distributions.
Mohsen Soltanifar111Mohsen Soltanifar
Department of Mathematics and Statistics, University of Saskatchewan, 106
Wiggins Road, Saskatoon, S7N 5E6, Canada
e-mail: mohsen.soltanifar@usask.ca
Keywords Semicircular distributions, Operator-Valued Non-Commutative
Probability, Cauchy-Stieltjes transform, Continued Fractions
Mathematics Subject Classification (2010). Primary 60B05, 30E20; Secondary
30B70, 47B80
## 1\. Introduction
In his 1995 paper on operator-valued free probability theory, which became one
of the main pillars of free probability , Voiculescu [11] introduced the
operator-valued free central limit theorem and operator-valued semicircular
random variables as operator-valued free analogous of the classical central
limit theorem and normal random variables, respectively. Similar to normal
random variables in classical probability theory, operator-valued semicircular
random variables play key roles in many areas of operator-valued free
probability and related fields. Following Wigner [12] and Voiculescu [10],
Shlyakhtenko showed in [7] that operator-valued semicircular random variables
also play an important role in random matrix theory, as asymptotic
distributional limits of Gaussian band matrices. Later this result found
several applications. We shall mention here the paper of Rashidi Far and
others [6] which shows the importance of matrix-valued semicircular random
variables in communication theory.
This paper deals with exploring some other aspects of operator-valued
semicircular random variables. Our investigation originates from a question of
Speicher on the existence of a discrete part in the spectrum of a matrix-
valued semicircular random variable. Following well-established methods in
noncommutative probability, we approach this problem through the study of
distributions of such a random variable with respect to convenient positive
linear functionals. In our work we were led to question roughly which
probability measures can occur as scalar-valued distributions of operator-
valued semicircular random variables. It turns out that the operator-valued
semicircular random variables have a certain universal property: For given
arbitrary compactly supported probability measure, its associated Cauchy
transform is the composition of an extremal state and the operator-valued
Cauchy transform of an infinite dimensional matrix-valued semicircular random
variable (Theorem 3.1). Moreover, our proof gives a constructive method to
find this semicircular random variable based on the continued fraction
representation of the Cauchy transform of the given probability measure. Our
second result directly concerns the discrete part of the distribution of a
matrix-valued semicircular random variable with respect to the linear
functional induced by the composition of the expectation with the matrix
trace. It was noted by Speicher (private communications with Belinschi) that
an $M_{2}(\mathbb{C})-$valued semicircular random variable with variance
$\eta\left(\begin{array}[]{cc}z&v\\\
y&w\end{array}\right)=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}z&v\\\
y&w\end{array}\right)\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right)$
has a purely discrete distribution equal to
$\mu=\frac{1}{4}\delta_{+1}+\frac{1}{2}\delta_{0}+\frac{1}{4}\delta_{-1}.$
Motivated by this observation and the proof of the Proposition 3.1, we were
led to show in Theorem 4.1 that any matrix-valued semicircular random variable
with nilpotent variance has an atom at zero. Finally, by combining the methods
used in the proofs of the previous results, it is shown that the semicircular
distributions of finite dimensional matrix-valued semicircular random
variables can cover finitely supported probability measures in a special
sense. This paper is divided into three sections including preliminaries,
representation of the Cauchy transform using semicircular random variables and
a discussion of atoms of distributions of matrix-valued semicircular random
variables.
## 2\. Preliminaries
The reader who has studied the concepts of Cauchy or Cauchy-Stieltjes
transform, continued fraction, operator-valued noncommutative random
variables, and semicircular distributions is well acquainted with the
following definitions and results. For an essential account of the mentioned
concepts, see [4], [8], and [11]. Throughout this paper it is assumed that the
probability measure $\mu$ on $\mathbb{R}$ is compactly supported. We begin
with a definition:
###### Definition 2.1.
Let $\mu$ be a probability measure on the Borel $\sigma$-algebra of
$\mathbb{R}$ . The associated Cauchy transform $G_{\mu}$ to $\mu$ is defined
by:
$G_{\mu}(\xi)=\int_{\mathbb{R}}\frac{d\mu(t)}{\xi-t}\ \ \ \ Im\xi\neq 0.$
Some properties of $G_{\mu}$ are listed in the following proposition. We
denote by $\mathbb{C}^{+}=\\{\xi\in\mathbb{C}:Im\xi>0\\},$
$Supp(\mu)=\mathbb{R}\setminus\bigcup\\{U\subseteq\mathbb{R}|U\
open,\mu(U)=0\\},$ and
$\Gamma_{\alpha}(r)=\\{\xi\in\mathbb{C}^{+}:|Re(\xi)-r|<\alpha Im(\xi)\\}\ \
(-\infty<r<\infty).$
###### Proposition 2.1.
Let $G=G_{\mu}$ be the Cauchy transform of a probability measure $\mu$ on
$\mathbb{R}.$ Then:
(i) $G$ is analytic on $\mathbb{C}\setminus Supp(\mu),$
(ii) If $\xi\in\mathbb{C}^{+}(\mathbb{C}^{-}),$ then
$G(\xi)\in\mathbb{C}^{-}(\mathbb{C}^{+}),$
(iii) $\lim_{\Gamma_{\alpha}(0)\ni\xi\rightarrow\infty}\xi.G(\xi)=1,$ for any
fixed $0<\alpha<\infty.$
Note that any function can be identified as a Cauchy transform of a
probability measure on $\mathbb{R}$ by possessing the three mentioned
properties in the Proposition above [8]. In addition, it is straightforward to
check that:
(1) $\displaystyle\lim_{\Gamma_{\alpha}(r)\ni\xi\rightarrow
r}(\xi-r)G_{\mu}(\xi)$ $\displaystyle=$ $\displaystyle\mu(\\{r\\})\ \
(-\infty<r<\infty).$
###### Definition 2.2.
A probability measure $\mu$ on $\mathbb{R}$ is said to have finite moment of
order $m\geq 1$ if $\int_{\mathbb{R}}|t|^{m}d\mu(t)<\infty,$ and in this case
the $m^{th}$ moment of $\mu$ is defined by
$M_{m}=\int_{\mathbb{R}}t^{m}d\mu(t).$
We denote the set of all Borel probability measures on $\mathbb{R}$ having
finite moments of all orders by $\mathcal{B}_{fm}(\mathbb{R}).$ It is trivial
that any compactly supported probability measure $\mu$ is in this set, and as
a corollary of the Carleman’s moment test [4, Theorem 1.36] it is a solution
of the determinate moment problem. Next, using the idea of associated Gram-
Schmidt orthonormal polynomials to a probability measure $\mu$, one can prove
the existence of the so-called Jacobi coefficients of the given probability
measure $\mu\in\mathcal{B}_{fm}(\mathbb{R}),$ [4, Theorem 1.44]:
###### Theorem 2.3.
Let $\\{p_{m}(t)\\}_{m=0}^{\infty}$ be the Gram-Schmidt orthonormal
polynomials associated with given $\mu\in\mathcal{B}_{fm}(\mathbb{R}).$ Then
there exists a pair of sequences
$\\{\alpha_{m}\\}_{m=1}^{\infty}\subseteq\mathbb{R}$ and
$\\{\omega_{m}\\}_{m=1}^{\infty}\subseteq\mathbb{R}^{+}$ uniquely determined
by:
$\displaystyle p_{0}(t)$ $\displaystyle=$ $\displaystyle 1,$ (2)
$\displaystyle p_{1}(t)$ $\displaystyle=$ $\displaystyle t-\alpha_{1},$
$\displaystyle tp_{m}(t)$ $\displaystyle=$ $\displaystyle
p_{m+1}(t)+\alpha_{m+1}p_{m}(t)+\omega_{m}p_{m-1}(t)\ m\geq 1,$
where in which, if $|Supp(\mu)|=\infty,$ both
$\\{\alpha_{m}\\}_{m=1}^{\infty},\\{\omega_{m}\\}_{m=1}^{\infty}$ are infinite
sequences, and if $|Supp(\mu)|=m_{0}<\infty,$ we have
$\\{\alpha_{m}\\}_{m=1}^{\infty}=\\{\alpha_{m}\\}_{m=1}^{m_{0}}$ and
$\\{\omega_{m}\\}_{m=1}^{\infty}=\\{\omega_{m}\\}_{m=1}^{m_{0}-1}$ with
$p_{m_{0}}=0.$
Note that as a corollary of the equations (2.3), for the compactly supported
probability measure $\mu\in\mathcal{B}_{fm}(\mathbb{R})$ we have:
(3) $\displaystyle\sup_{m}(|\alpha_{m}|+\omega_{m})$ $\displaystyle\leq$
$\displaystyle 2\sup_{t\in Supp(\mu)}|t|<\infty.$
The following theorem gives a continued fraction representation of the
associated Cauchy transform of any probability measure
$\mu\in\mathcal{B}_{fm}(\mathbb{R}),$ [4, Theorem 1.97]:
###### Theorem 2.4.
Let $\mu\in\mathcal{B}_{fm}(\mathbb{R})$ and
$(\\{\omega_{n}\\}_{n=1}^{\infty},\\{\alpha_{n}\\}_{n=1}^{\infty})$ be its
Jacobi coefficients. If $\mu$ is the solution of the determinate moment
problem, then the Cauchy transform of it is expanded into a continued fraction
$G_{\mu}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}}}}}}}\
\ \ \ Im(\xi)\neq 0.$
Before introducing the distribution of operator-valued semicircular random
variables, we remind the following essential definitions and results from the
operator-valued noncommutative probability theory:
###### Definition 2.5.
(1) Let $A$ be a unital $*$-algebra and let $B$ denote a fixed unital
$*$-subalgebra of $A$ over $\mathbb{C}$. A linear map $E_{B}:A\rightarrow B$
is called a conditional expectation if it satisfies the following conditions:
(i) $E_{B}(b_{1}ab_{2})=b_{1}E_{B}(a)b_{2}$ for all $a\in A,\ b_{1},b_{2}\in
B$, and $E_{B}(1)=1,$
(ii) $E_{B}(a^{*}a)\geq 0,$ for all $a\in A,$
(2) A triple $(A,E_{B},B)$ as in part (1) is called a $B$-valued non-
commutative probability space. An element $a\in A$ is called a $B$-valued
random variable,
(3) Let $(A,E_{B},B)$ be as in part (2), and $B\subseteq A_{i}\subseteq A(i\in
I)$ be subalgebras. The family $\\{A_{i}\\}_{i\in I}$ is called free over $B$
if
$E_{B}(a_{i_{1}}a_{i_{2}}\cdots a_{i_{n}})=0$
whenever $i_{1}\neq i_{2},i_{2}\neq i_{3},\cdots,i_{n-1}\neq
i_{n},a_{i_{j}}\in A_{i_{j}}$ and $E_{B}(a_{i_{j}})=0,(1\leq j\leq n).$
We call the family $\\{X_{i}\\}_{i\in I}$ of subsets of A (elements
$\\{a_{i}\\}_{i\in I}$ of A) free if the corresponding family of subalgebras
$\\{\langle X_{i}\cup B\rangle\\}_{i\in I}$ ($\\{\langle\\{a_{i}\\}\cup
B\rangle\\}_{i\in I}$) is free.
Given an operator-valued noncommutative probability space $(A,E_{B},B)$ and a
$B$-valued random variable $a\in A,$ the associated moments of $a$ are, by
definition, the multilinear functionals $\\{m_{n}\\}_{n=0}^{\infty}$ defined
by:
$\displaystyle m_{n}:B^{n}\rightarrow B,$ $\displaystyle
m_{n}(b_{1},b_{2},\cdots,b_{n})=E_{B}(ab_{1}ab_{2}\cdots ab_{n}a),$
in which the quantity $m_{0}=E_{B}(a)\in B$ is called the first moment, the
map $b_{1}\mapsto E_{B}(ab_{1}a)$ is called the second moment, and in general
the map $(b_{1},b_{2},\cdots,b_{n})\mapsto E_{B}(ab_{1}ab_{2}\cdots ab_{n}a)$
is called the $(n+1)^{th}$ moment. Next, let $B$ be a Banach algebra, and
$a,a_{1},\cdots,a_{m},\cdots$ be a sequence of of $B$-valued random variables
in $A$ with associated sequences of moments
$\\{m_{n}\\}_{n=0}^{\infty},\\{m_{n}^{(1)}\\}_{n=0}^{\infty},\cdots,\\{m_{n}^{(m)}\\}_{n=0}^{\infty},\cdots,$
respectively. We say that the sequence $\\{a_{m}\\}_{m=1}^{\infty}$
convergence to $a$ in moments if
$\lim_{m\rightarrow\infty}\|m_{n}^{(m)}(b_{1},b_{2},\cdots,b_{n})-m_{n}(b_{1},b_{2},\cdots,b_{n})\|=0,$
for all $(b_{1},b_{2},\cdots,b_{n})\in B^{n},$ and $n\geq 0.$
Having the same assumptions as above, we recall that any $a\in A$ can be
written as $a=Re(a)+i.Im(a)$ where $Re(a)=\frac{a+a^{*}}{2}$ and
$Im(a)=\frac{a-a^{*}}{2i}$ are self-adjoint elements. We define
$\mathbb{H}^{+}(A)=\\{a\in A|Im(a)>0\\},$ where $Im(a)>0$ means
$Im(a)>\epsilon.1$ for some $\epsilon>0$, and similarly $\mathbb{H}^{+}(B).$
Then the operator-valued Cauchy transform of $a\in A$ is an analytic map
$G_{a}$ defined via:
$\displaystyle G_{a}:\mathbb{H}^{+}(B)\rightarrow\mathbb{H}^{-}(B)$
$\displaystyle G_{a}(b)=E_{B}((b-a)^{-1})$ $\displaystyle\ \ \ \ \ \ \ \
=\sum_{n=0}^{\infty}b^{-1}E_{B}((ab^{-1})^{n})\ \
b\in\mathbb{H}^{+}(B),\|b^{-1}\|<\|a\|^{-1}.$
Next, the operator-valued $R-$transform of $a\in A,$ $R_{a}:B\rightarrow B,$
can be defined by the relation
$bG_{a}(b)=1+R_{a}(G_{a}(b)).G_{a}(b)\ \ (b\in B),$
[11, Theorem 4.9.]. The following central limit theorem for operator-valued
random variables is due to Voiculescu, [11, Theorem 8.4.]:
###### Theorem 2.6.
(Free Central Limit Theorem) Let $B$ be a Banach algebra and
$a_{1},a_{2},\cdots,a_{m},\cdots$ be a sequence of free $B$-valued random
variables in the non-commutative operator valued probability space
$(A,E_{B},B)$such that:
(i)$E_{B}(a_{m})=0,\ (m\in\mathbb{N}),$
(ii) there is a bounded linear map $\eta:B\rightarrow B$ such that
$\lim_{n\rightarrow\infty}\frac{\sum_{m=1}^{n}E_{B}(a_{m}ba_{m})}{n}=\eta(b),\
(b\in B),$
(iii) there are constants $C_{k}\ (k\in\mathbb{N})$ such that
$\sup_{m\in\mathbb{N}}\|E_{B}(a_{m}b_{1}a_{m}\cdots b_{k}a_{m})\|\leq
C_{k}\|b_{1}\|\cdots\|b_{k}\|\ (k\in\mathbb{N}).$
Then the sequence $S_{m}=\frac{\sum_{k=1}^{m}a_{k}}{\sqrt{m}}\
(m\in\mathbb{N})$ converges in moments.
The central limit in the above theorem which we shall denote it by $s$ is
called a $B$-valued semicircular element in the context of operator-valued
noncommutative probability, and, as in classical probability theory it is
uniquely determined by its first two moments. Indeed, its $R-$ transform is of
the form
$R_{s}(b)=D+\eta(b),\ (b\in B)$
where $D=E_{B}(s)\in B$ is a self-adjoint element and $\eta:B\rightarrow B$ is
a completely positive map given by $\eta(b)=E_{B}(sbs)-E_{B}(s)bE_{B}(s)\
(b\in B).$ Furthermore, for any completely positive map $\eta:B\rightarrow B$
there exists a $B-$valued semicircular random variable $s$ such that
$\eta(b)=E_{B}(sbs)-E_{B}(s)bE_{B}(s)\ (b\in B),$ [9, Theorem 4.3.1.].
The following result of Helton, Rashidi Far, and Speicher, shows that any
operator-valued semicircular random variable can be uniquely determined by a
functional equation involving only its first two moments, [3]:
###### Theorem 2.7.
Let $A$ be a unital $C^{*}$-algebra, $B$ a $C^{*}$-subalgebra of $A$, and
$s\in A$ be a self-adjoint $B$-valued semicircular random variable with first
moment $D\in B$ and variance $\eta:B\rightarrow B.$ Then its associated
operator-valued Cauchy transform
$G_{s}:\mathbb{H}^{+}(B)\rightarrow\mathbb{H}^{-}(B)$ is the unique solution
of the functional equation
(4) $\displaystyle b.G_{s}(b)=1+(D+\eta(G_{s}(b))).G_{s}(b),\ \ \
(b\in\mathbb{H}^{+}(B)),$
together with asymptotic condition
(5) $\displaystyle\lim_{b^{-1}\rightarrow 0}b.G_{s}(b)=1.$
Let $A$ be a unital $C^{*}$-algebra, $B$ be a unital $C^{*}$-subalgebra of
$A$, $E_{B}:A\rightarrow B$ be a conditional expectation, and
$G_{a}:\mathbb{H}^{+}(B)\rightarrow\mathbb{H}^{-}(B)$ be the operator-valued
Cauchy transform of the self-adjoint random variable $a\in A.$ Let
$\Phi:B\rightarrow\mathbb{C}$ be a given state on the $C^{*}$-algebra $B$ (In
the case of $B=M_{n}(\mathbb{C})$, for some fixed $n\geq 1,$ we can take
$\Phi=tr_{n},$ the normalized trace.), and define a map
$G\colon\mathbb{C}^{+}\rightarrow\mathbb{C}^{-}$ via:
$G(\xi)=(\Phi\circ G_{a})(\xi.1)\ \ \ \xi\in\mathbb{C}^{+}.$
Referring to the note after the Proposition 2.1, it follows that there is a
probability measure $\mu=\mu_{a}\in\mathcal{B}_{fm}(\mathbb{R})$ on
$\mathbb{R},$ which we call the distribution of $a$, such that:
(6) $\displaystyle
G(\xi)=G_{\mu_{a}}(\xi)=\int_{\mathbb{R}}\frac{d\mu_{a}(t)}{\xi-t}\ \ \
\xi\in\mathbb{C}^{+}.$
Note that the atoms of $\mu_{a}$ are determined via Equation (1).
## 3\. Representation of the Cauchy Transform Using Semicircular Random
Variables
The proof of the following results are based on the Theorem 2.7 of Section 2.
Indeed, for given $D$ and $\eta$ as in that theorem, we shall verify the
conditions (4) and (5) for $b=\xi.1\ \ \xi\in\mathbb{C}^{+}$ and $G_{s}(b)$ a
diagonal matrix of complex analytic functions. Then, Theorem 2.7 of Section 2
will guarantee us that there is a semicircular random variable $s$ with first
moment $D$ and variance $\eta$ so that $G_{s}(b)$ is the restriction of the
operator-valued Cauchy transform of $s$ to $\mathbb{C}^{+}.1.$ Next, our
variances $\eta$ will be explicitly constructed as $\eta(b)=v^{*}bv$, [5,
Theorem 4.1.] with $v$ obtained from the Jacobi coefficients of the given
compactly supported probability measure.
The first result deals with the finite dimensional matrix-valued
representations of the Cauchy transform. Here, we consider $\ell_{2}^{n}$ as
$\mathbb{C}^{n}$ with its canonical orthonormal basis $\\{e_{k}\\}_{k=1}^{n}.$
###### Proposition 3.1.
Let $\mu$ be a probability measure with compact support in $\mathbb{R}.$ Then
there exist two sequences $s_{n}^{(1)}$ and $s_{n}^{(2)}\ (n\geq 1)$of self-
adjoint operator valued semicircular random variables with associated
operator-valued Cauchy transforms
$G_{s_{n}^{(1)}}:\mathbb{H}^{+}(M_{n}(\mathbb{C}))\rightarrow\mathbb{H}^{-}(M_{n}(\mathbb{C}))$
and
$G_{s_{n}^{(2)}}:\mathbb{H}^{+}(M_{n}(\mathbb{C}))\rightarrow\mathbb{H}^{-}(M_{n}(\mathbb{C}))\
(n\geq 1)$ such that the Cauchy transform
$G_{\mu}:\mathbb{C}^{+}\rightarrow\mathbb{C}^{-}$ is represented as:
(7) $\displaystyle G_{\mu}(\xi)=\lim_{n\rightarrow\infty}\langle
G_{s_{n}^{(1)}}(\xi.1_{n})e_{1},e_{1}\rangle_{\ell_{2}^{n}}\ \
\xi\in\mathbb{C}^{+},$
and
(8) $\displaystyle G_{\mu}(\xi)=\lim_{n\rightarrow\infty}\langle
G_{s_{n}^{(2)}}(\xi.1_{n})e_{n},e_{n}\rangle_{\ell_{2}^{n}}\ \
\xi\in\mathbb{C}^{+}.$
###### Proof.
Let
$G_{\mu}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}-\displaystyle{\frac{\omega_{n-1}}{\xi-\alpha_{n}-\displaystyle{\frac{\omega_{n}}{\xi-\alpha_{n+1}-}}_{\displaystyle{\ddots}}}}}}}}}}$
be the continued fraction representation of $G_{\mu}$ as in Theorem 2.4. To
prove Equation (7), fix positive integer $n\geq 1,$ then define $b=\xi.1_{n},$
$D_{n}^{(1)}=(\alpha_{k}\delta_{kl})_{k,l=1}^{n}$ and the completely positive
map $\eta_{n}^{(1)}$ via :
$\displaystyle\eta_{n}^{(1)}:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$
$\displaystyle\eta_{n}^{(1)}\big{(}(a_{kl})_{k,l=1}^{n}\big{)}=(\omega_{k}^{\frac{1}{2}}\delta_{(k+1)l})_{k,l=1}^{n}(a_{kl})_{k,l=1}^{n}(\omega_{k-1}^{\frac{1}{2}}\delta_{k(l+1)})_{k,l=1}^{n}.$
Then, for the self-adjoint semicircular element $s_{n}^{(1)}$ with operator-
valued Cauchy transform $G_{s_{n}^{(1)}}$ satisfying the functional equation
(4) of the form:
$bG_{s_{n}^{(1)}}(b)=1+(D_{n}^{(1)}+\eta_{n}^{(1)}(G_{s_{n}^{(1)}}(b)))G_{s_{n}^{(1)}}(b),$
we have $G_{s_{n}^{(1)}}(b)=(g_{n,n-k+1}(\xi)\delta_{kl})_{k,l=1}^{n}$ where :
$g_{n,n-k+1}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}-\displaystyle{\frac{\omega_{n-k+1}}{\xi-\alpha_{n-k+1}}}}}}}}}\
\ 1\leq k\leq n,\ \xi\in\mathbb{C}^{+},$
which can be identified as $(n-k+1)^{th}$ convergent of $G_{\mu}.$
Accordingly:
$\displaystyle
G_{\mu}(\xi)=\lim_{n\rightarrow\infty}g_{n,n}(\xi)=\lim_{n\rightarrow\infty}\langle
G_{s_{n}^{(1)}}(\xi.1_{n})e_{1},e_{1}\rangle_{\ell_{2}^{n}}\ \
\xi\in\mathbb{C}^{+}.$
The proof of Equation (8) is analogous by considering a fixed positive integer
$n\geq 1,$ then defining $b=\xi.1_{n},$
$D_{n}^{(2)}=(\alpha_{n+1-k}\delta_{kl})_{k,l=1}^{n}$ and the completely
positive map $\eta_{n}^{(2)}$ via :
$\displaystyle\eta_{n}^{(2)}:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$
$\displaystyle\eta_{n}^{(2)}\big{(}(a_{kl})_{k,l=1}^{n}\big{)}=(\omega_{n-k+1}^{\frac{1}{2}}\delta_{k(l+1)})_{k,l=1}^{n}(a_{kl})_{k,l=1}^{n}(\omega_{n-k}^{\frac{1}{2}}\delta_{(k+1)l})_{k,l=1}^{n}.$
∎
The following theorem deals with the infinite dimensional matrix valued
representation of the Cauchy transform. Here, we denote by
$B(\ell_{2}(\mathbb{N}))$ the space of bounded operators on the separable
Hilbert space $\ell_{2}(\mathbb{N}),$ and we consider the orthonormal basis
$\\{e_{n}\\}_{n=1}^{\infty}$ of $\ell_{2}(\mathbb{N})$ given by
$e_{n}=\\{\delta_{mn}\\}_{m=1}^{\infty}\ (n\geq 1).$
###### Theorem 3.1.
Let $\mu$ be a compactly supported probability measure in $\mathbb{R}$ with
Jacobi coefficients
$(\\{\omega_{n}\\}_{n=1}^{\infty},\\{\alpha_{n}\\}_{n=1}^{\infty})$. Then
there exist a self-adjoint $B(\ell_{2}(\mathbb{N}))-$valued semicircular
random variable $s$ with first moment
$D=(\alpha_{k}\delta_{kl})_{k,l=1}^{\infty}\in B(\ell_{2}(\mathbb{N})),$ and
variance
$\displaystyle\eta:B(\ell_{2}(\mathbb{N}))\rightarrow B(\ell_{2}(\mathbb{N}))$
$\displaystyle\eta((a_{kl})_{k,l=1}^{\infty})=(\omega_{k}^{\frac{1}{2}}\delta_{(k+1)l})_{k,l=1}^{\infty}(a_{kl})_{k,l=1}^{\infty}(\omega_{k-1}^{\frac{1}{2}}\delta_{k(l+1)})_{k,l=1}^{\infty},$
and an state $\rho:B(\ell_{2}(\mathbb{N}))\rightarrow\mathbb{C}$ such that the
Cauchy transform $G_{\mu}:\mathbb{C}^{+}\rightarrow\mathbb{C}^{-}$ is
represented as:
$G_{\mu}(\xi)=(\rho\circ G_{s})(\xi.1)\ \ \xi\in\mathbb{C}^{+}.$
###### Proof.
Let
$G_{\mu}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}-\displaystyle{\frac{\omega_{n-1}}{\xi-\alpha_{n}-\displaystyle{\frac{\omega_{n}}{\xi-\alpha_{n+1}-}}_{\displaystyle{\ddots}}}}}}}}}}$
be the continued fraction representation of $G_{\mu}$ as in Theorem 2.4 and
consider given $D$ and $\eta$ where using inequality (3) both of them are
bounded in their corresponding norms. Now, as in proof of Proposition 3.1, we
observe that the diagonal matrix
$G_{s}(b)=(x_{kl}\delta_{kl})_{k,l=1}^{\infty},\ \ \ b=\xi.1\in
B(\ell_{2}(\mathbb{N}))$
with entries $x_{nn}$ of the form
$x_{nn}=\frac{1}{\xi-\alpha_{n}-\displaystyle{\frac{\omega_{n}}{\xi-\alpha_{n+1}-\displaystyle{\frac{\omega_{n+1}}{{\xi-\alpha_{n+2}}-\displaystyle{\frac{\omega_{n+2}}{\xi-\alpha_{n+3}-\displaystyle{\frac{\omega_{n+3}}{\xi-\alpha_{n+4}-}}_{\displaystyle{\ddots}}}}}}}}}\
n\geq 1,$
satisfies the equation (4) with condition (5). Consequently, for the state
$\rho:B(\ell_{2}(\mathbb{N}))\rightarrow\mathbb{C}$ defined by:
$\rho(T)=\langle T(e_{1}),e_{1}\rangle_{\ell_{2}(\mathbb{N})}$
the assertion follows. ∎
## 4\. Atoms of Distributions of Matrix-Valued Semicircular Random Variables
In this section, the existence of atoms of distributions of finite dimensional
matrix-valued semicircular random variables is discussed. First of all, we
give a sufficient condition on the variance of a centered semicircular random
variable so that its associated probability measure has atom.
###### Theorem 4.1.
Let
$G_{s}:\mathbb{H}^{+}(M_{n}(\mathbb{C}))\rightarrow\mathbb{H}^{-}(M_{n}(\mathbb{C}))$
be the operator valued Cauchy transform of a $M_{n}(\mathbb{C})$-valued
semicircular random variable $s$ satisfying the functional equation (4),
$b\in\mathbb{H}^{+}(M_{n}(\mathbb{C})),$ where $D=0$ and
$\eta:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$ is a nilpotent
completely positive map. Then the associated probability measure $\mu_{s}$ to
$G_{s}$ has at least one atom .
###### Proof.
Let $\eta,\cdots,\eta^{m-1}\neq 0$ and $\eta^{m}=0,$ for some $m\geq 1.$
Writing the functional equation in the form of
$b-{G_{s}(b)}^{-1}=\eta(G_{s}(b))$ for $\|b^{-1}\|<<\infty$ it follows that:
$\eta^{m-1}(b-{G_{s}(b)}^{-1})=\eta^{m}(G_{s}(b))=0\hskip 28.45274ptIm(b)>0,\
\ \|b^{-1}\|<<\infty.$
Now, if $\ker(\eta^{m-1})=\LARGE 0,$ then $G_{s}(b)=b^{-1}$ and using
Equations (1) and (6) it follows that
$\mu_{s}(\\{0\\})=1,$
proving the assertion. Hence, we may assume $\ker(\eta^{m-1})\neq\LARGE 0.$
Pick $0\neq c\in\ker(\eta^{m-1})\cap M_{n}^{+}(\mathbb{C})$ with $\|c\|=1.$
Then by Schwarz inequality for completely positive maps [5, p. 40], it follows
that:
$\eta^{m-1}(c^{\frac{1}{2^{n}}})^{*}\eta^{m-1}(c^{\frac{1}{2^{n}}})\leq\|\eta^{m-1}(1)\|\eta^{m-1}(c^{\frac{1}{2^{n-1}}})\hskip
28.45274ptn\geq 1,$
and by induction we conclude that $\eta^{m-1}(c^{\frac{1}{2^{n}}})=0\ (n\geq
1).$ By defining:
$p:=^{s.o.t}\lim_{n\rightarrow\infty}c^{\frac{1}{2^{n}}},$
it follows that $p$ is a projection in $\ker(\eta^{m-1}).$
Claim (1): There exists a unique projection $1\neq q\in\ker(\eta^{m-1})$ such
that for any projection $p\in\ker(\eta^{m-1})$ we have: $p\leq q.$
We showed that there is at least one projection $p$ in $\ker(\eta^{m-1}).$ Let
$p_{1},p_{2}$ be two projections in $\ker(\eta^{m-1}).$ Then :
$\eta^{m-1}((p_{1}+p_{2})^{\frac{1}{2^{n}}})^{*}\eta^{m-1}({(p_{1}+p_{2})}^{\frac{1}{2^{n}}})\leq\|\eta^{m-1}(1)\|\eta^{m-1}((p_{1}+p_{2})^{\frac{1}{2^{n-1}}})\
\ n\geq 1,$
and by induction it follows that
$\eta^{m-1}((p_{1}+p_{2})^{\frac{1}{2^{n}}})=0\ (n\geq 1).$ Now, define
$p_{3}:=^{s.o.t}\lim_{n\rightarrow\infty}(p_{1}+p_{2})^{\frac{1}{2^{n}}},$
then $p_{3}$ is a projection in $\ker(\eta^{m-1}).$ On the other hand
$(p_{1}+p_{2})^{\frac{1}{2^{n}}}\geq p_{1},p_{2}\ \ \ (n\geq 1),$
yielding $p_{3}\geq p_{1},p_{2}.$ Next, using the maximality argument and by
repeating this process there will be a unique maximal projection $q$ in
$\ker(\eta^{m-1})$ such that for any other projection $p$ in it, we have
$p\leq q.$ Finally, if $q=1,$ then using the same Schwarz inequality as above,
and the canonical decomposition of elements of $M_{n}(\mathbb{C})$ into its
positive elements it follows that $\eta^{m-1}=0,$ in contradiction to our
hypothesis.
Claim(2): For any $0\neq c\in\ker(\eta^{m-1})\cap M_{n}^{+}(\mathbb{C}),$ we
have $cq=qc=c.$
Indeed, since $c\in M_{n}^{+}(\mathbb{C}),$ by spectral theorem we have:
$c=\sum_{k=1}^{N}\lambda_{k}p_{k}$
where the projections $p_{k}$’s satisfy $\sum_{k=1}^{N}p_{k}=1,\ \
p_{k_{1}}p_{k_{2}}=0\ (1\leq k_{1}\neq k_{2}\leq N)$ and $\lambda_{k}\geq 0\
(1\leq k\leq N).$ Now, define:
$r:=^{s.o.t}\lim_{n\rightarrow\infty}c^{\frac{1}{2^{n}}}.$
Then it follows that $r=\sum_{\lambda_{k}\neq 0}p_{k},$ yielding $p_{k}\leq r\
(\lambda_{k}\neq 0).$ On the other hand, by definition of $q$ we have $r\leq
q$ and hence $p_{k}\leq q\ (\lambda_{k}\neq 0).$ But all of $p_{k}\leq q\
(\lambda_{k}\neq 0),$ and $q$ are projections and, consequently,
$p_{k}q=qp_{k}=p_{k}\ (\lambda_{k}\neq 0).$ Now, by multiplying all sides by
$\lambda_{k}\ (1\leq k\leq N),$ and taking summation the claim is proved.
Next, take $c_{0}=\frac{1}{i}({G_{s}(b)}^{-1}-b)$ where $b=iy.1\ (y>0).$ Then
using the fact that $s$ is centered, $E_{B}(s^{2m-1})=0\ \ m\geq 1,$ by
$\displaystyle Re(G_{s}(b))$ $\displaystyle=$ $\displaystyle
Re\Big{(}\sum_{m=0}^{\infty}b^{-1}E_{B}((sb^{-1})^{m})\Big{)}=Re\Big{(}\sum_{m=0}^{\infty}i^{m+1}(-y)^{m+1}E_{B}(s^{m})\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}(-y^{2})^{m}E_{B}(s^{2m-1})=0,\ \ \ y>\|s\|$
it follows that
${G_{s}(b)}^{-1}-b=\Big{(}Re(G_{s}(b))+i.Im(G_{s}(b))\Big{)}^{-1}-b=i.\Big{(}-{Im(G_{s}(b))}^{-1}-\frac{b}{i}\Big{)},$
and, hence $c_{0}=\frac{1}{i}({G_{s}(b)}^{-1}-b)=Im(({G_{s}(b)}^{-1}-b))\geq
0.$ Now, by claim (2) for $c=c_{0}$ we have that:
$G_{s}(b)(1-q)=b^{-1}(1-q)=(1-q)G_{s}(b)\ \ \textrm{and}\ \
G_{s}(b)q=qG_{s}(b),$
and hence:
$\displaystyle G_{s}(b)$ $\displaystyle=$
$\displaystyle(1-q)G_{s}(b)(1-q)+qG_{s}(b)q$ $\displaystyle=$ $\displaystyle
b^{-1}(1-q)+qG_{s}(b)q\hskip 28.45274ptb=iy.1\ (y>0),\|b^{-1}\|<<\infty.$
Now, applying analytic continution for the complex function $tr_{n}\circ
G_{s}|_{\mathbb{C}^{+}.1}:\mathbb{C}^{+}\rightarrow\mathbb{C}^{-}$ we conclude
that:
$(tr_{n}\circ G_{s})(\xi.1)=tr_{n}(\xi^{-1}(1-q)+qG_{s}(\xi.1)q)\ \ \
\xi\in\mathbb{C}^{+},$
and, consequently, by another application of the Equations (1) and (6):
$\mu_{s}(\\{0\\})=tr_{n}(1-q)>0,$
completing the proof. ∎
Before stating a more concrete and special case of the above theorem, using M.
D. Choi’s representation of a completely positive map from a matrix algebra to
another matrix algebra [1], we have:
###### Lemma 4.2.
Let $\eta$ be a completely positive map defined by
$\displaystyle\eta:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$
$\displaystyle\eta(a)=\sum_{j=1}^{n^{2}}a_{j}aa_{j}^{*},\ \ a_{j}\in
M_{n}(\mathbb{C})\ (1\leq j\leq n^{2}).$
Then:
(i) if the map $\eta$ is nilpotent, then all matrices $a_{j}\ (1\leq j\leq
n^{2})$ are nilpotent,
(ii) if all matrices $a_{j}\ (1\leq j\leq n^{2})$ are nilpotent and commute
with each other, then the map $\eta$ is nilpotent.
###### Proof.
(i) Let $\eta^{m}=0$ for some $m\geq 1.$ Then, it follows that:
$\displaystyle\sum_{1\leq j_{1},\cdots,j_{m}\leq n^{2}}(a_{j_{1}}\cdots
a_{j_{m}}a)(a_{j_{1}}\cdots a_{j_{m}}a)^{*}$ $\displaystyle=$
$\displaystyle\sum_{1\leq j_{1},\cdots,j_{m}\leq n^{2}}(a_{j_{1}}\cdots
a_{j_{m}})aa^{*}(a_{j_{m}}^{*}\cdots a_{j_{1}}^{*})$ $\displaystyle=$
$\displaystyle\sum_{j_{1}=1}^{n^{2}}\cdots\sum_{j_{m}=1}^{n^{2}}(a_{j_{1}}\cdots
a_{j_{m}})aa^{*}(a_{j_{m}}^{*}\cdots a_{j_{1}}^{*})$ $\displaystyle=$
$\displaystyle\sum_{j_{1}=1}^{n^{2}}a_{j_{1}}\Big{(}\cdots\Big{(}\sum_{j_{m}=1}^{n^{2}}a_{j_{m}}aa^{*}a_{j_{m}}^{*}\Big{)}\cdots\Big{)}a_{j_{1}}^{*}$
$\displaystyle=$ $\displaystyle\eta^{m}(aa^{*})$ $\displaystyle=$
$\displaystyle 0$
for all $a\in M_{n}(\mathbb{C}).$ Consequently, by positivity of all elements
of the form $(a_{j_{1}}\cdots a_{j_{m}}a)(a_{j_{1}}\cdots a_{j_{m}}a)^{*}$ it
follows that
$(a_{j_{1}}\cdots a_{j_{m}}a)(a_{j_{1}}\cdots a_{j_{m}}a)^{*}=0\ \ (1\leq
j_{1},\cdots,j_{m}\leq n^{2}),$
for all $a\in M_{n}(\mathbb{C}).$ On the other hand , $M_{n}(\mathbb{C})$ is a
$C^{*}$-algebra and hence:
$a_{j_{1}}\cdots a_{j_{m}}a=0\ \ (1\leq j_{1},\cdots,j_{m}\leq n^{2}),$
for all $a\in M_{n}(\mathbb{C}),$ or equivalently:
$a_{j_{1}}\cdots a_{j_{m}}=0\ \ (1\leq j_{1},\cdots,j_{m}\leq n^{2}).$
Now, take $j_{1}=\cdots=j_{m}=j$ where $1\leq j\leq n^{2}$ and the desired
result is proved.
(ii) Since $a_{j}\ \ (1\leq j\leq n^{2})$ are nilpotent, it follows that
$a_{j}^{n}=0\ \ (1\leq j\leq n^{2}).$ Put $m=n^{3},$ then by commutativity of
the these matrices it follows that:
$\displaystyle a_{j_{1}}\cdots a_{j_{m}}$ $\displaystyle=$
$\displaystyle\prod_{p=0}^{n-1}\prod_{q=1}^{n^{2}}a_{j_{pn^{2}+q}}=a_{j_{(j_{1},\cdots,j_{m})}}^{n}\prod\prod_{j_{pn^{2}+q}\neq
j_{(j_{1},\cdots,j_{m})}}a_{j_{pn^{2}+q}}=0,$
for all $1\leq j_{1},\cdots,j_{m}\leq n^{2}$. Consequently,
$\displaystyle\eta^{m}(a)$ $\displaystyle=$ $\displaystyle\sum_{1\leq
j_{1},\cdots,j_{m}\leq n^{2}}(a_{j_{1}}\cdots a_{j_{m}})a(a_{j_{m}}^{*}\cdots
a_{j_{1}}^{*})=0$
for all $a\in M_{n}(\mathbb{C}).$ ∎
Considering above lemma, for a category of nilpotent $\eta^{\prime}s$ we have:
###### Corollary 4.3.
Let
$G_{s}:\mathbb{H}^{+}(M_{n}(\mathbb{C}))\rightarrow\mathbb{H}^{-}(M_{n}(\mathbb{C}))$
be the operator-valued Cauchy transform of a $M_{n}(\mathbb{C})$-valued
semicircular random variable $s$ satisfying the functional equation (4),
$b\in\mathbb{H}^{+}(M_{n}(\mathbb{C})),$ $D=0$ and the completely positive map
$\eta$ is given by:
$\displaystyle\eta:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$
$\displaystyle\eta(a)=\sum_{j=1}^{n^{2}}a_{j}aa_{j}^{*},\ \ a_{j}^{n}=0\
(1\leq j\leq n^{2}),\ \ a_{j_{1}}a_{j_{2}}=a_{j_{2}}a_{j_{1}}\ (1\leq
j_{1},j_{2}\leq n^{2}).$
Then the associated probability measure $\mu_{s}$ to $G_{s}$ has at least one
atom at $x=0.$
A few examples of interest are discussed in connection to the Theorem 4.1. :
###### Remarks 4.4.
(i) The converse of the assertion of the Theorem 4.1. does not hold. To see
this, let $n=2$, $0\neq|\alpha|\neq|\beta|\neq 0$ and define:
$a_{1}=\left(\begin{array}[]{cc}0&\alpha\\\ \beta&0\end{array}\right),\ \
a_{j}=0\ (2\leq j\leq 4).$
Under these conditions, considering $D=0$ and $\eta(a)=a_{1}aa_{1}^{*}$ in the
Equation (4) with the restriction condition (5), and solving it for $G_{s}(b)$
yields:
$tr_{2}(G_{s}(b))=\frac{(|\alpha|^{2}+|\beta|^{2})\Big{(}\xi^{2}+\sqrt{(-|\alpha|^{2}+|\beta|^{2}-\xi^{2})^{2}-4.|\alpha|^{2}\xi^{2}}\Big{)}-(|\alpha|^{2}-|\beta|^{2})^{2}}{4\xi|\alpha|^{2}|\beta|^{2}},$
and, by considering the Equations (1) and (6) it follows that:
$\mu_{s}(\\{r\\})=0\ \ \textrm{if}\ \ \ r\neq 0,\
\frac{1}{2}\Big{(}1-\left|\frac{\beta}{\alpha}\right|^{2sgn(1-|\frac{\beta}{\alpha}|)}\Big{)}\
\ \ \textrm{if}\ \ r=0.$
(ii) If the assumption of nilpotency of the map $\eta$ is violated in the
statement of the Theorem 4.1, then the associated distribution can have no
atom. To see this, let $n=2$, $|\alpha|=|\beta|\neq 0$ and define:
$a_{1}=\left(\begin{array}[]{cc}0&\alpha\\\ \beta&0\end{array}\right),\ \
a_{j}=0\ (2\leq j\leq 4).$
Under these conditions, considering $D=0$ and $\eta(a)=a_{1}aa_{1}^{*}$ in the
Equation (4) with the restriction condition (5), and solving it for $G_{s}(b)$
yields:
$tr_{2}(G_{s}(b))=\frac{\xi-\sqrt{\xi^{2}-4|\alpha|^{2}}}{2|\alpha|^{2}}$
Now, by considering the Equation (6), it follows that this is the Cauchy
transform of the standard semicircular law of Wigner which has no atom.
(iii) In the Theorem 4.1 , for given nilpotent map $\eta$ the associated
probability measure to the operator-valued Cauchy transform $G_{s}(b)$ may not
be purely atomic. To see this, let $n=3$ and define :
$a_{1}=\left(\begin{array}[]{ccc}0&1&1\\\ 0&0&1\\\ 0&0&0\end{array}\right),\ \
a_{j}=0\ (3\leq j\leq 9).$
Under these conditions, considering $D=0$ and $\eta(a)=a_{1}aa_{1}^{*}$ in the
Equation (4) with the restriction condition (5), and solving it for $G_{s}(b)$
with the Groebner basis method [2] in Mathematica, it follows that:
$tr_{3}(G_{s}(b))=\frac{1}{3}\Big{(}\frac{1}{\xi}+\frac{\xi-\sqrt{\xi^{2}-4}}{2}+(\xi^{2}-1)\big{(}\frac{\xi-\sqrt{\xi^{2}-4}}{2}\big{)}^{3}\Big{)},$
and, by considering the Equations (1) and (6) it follows that:
$\mu_{s}(\\{r\\})=0\ \ \textrm{if}\ \ r\neq 0,\ \frac{1}{3}\ \textrm{if}\ \
r=0,$
showing that $\mu_{s}$ is not purely atomic.
We end this section by mentioning a “covering property” of distributions of
matrix-valued semicircular random variables. Before that, we need a
definition:
###### Definition 4.5.
Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}$. We shall say
that $\nu$ is a component of $\mu$ if there exists a finite family
$\\{\nu_{1},\dots,\nu_{n}\\}$ of probability measures so that
$\nu\in\\{\nu_{1},\dots,\nu_{n}\\}$ and
$\mu=\sum_{j=1}^{n}\alpha_{j}\nu_{j}$
for some $\alpha_{1},\dots,\alpha_{n}\in[0,1]$ satisfying
$\alpha_{1}+\cdots+\alpha_{n}=1$.
###### Proposition 4.1.
Any probability measure $\mu$ on $\mathbb{R}$ whose support is a finite set
can be realized as a component of a semicircular distribution $\mu_{s}$ of
some matrix-valued semicircular random variable $s$ with nilpotent variance.
###### Proof.
Assume $|Supp(\mu)|=n<\infty$, and let
$G_{\mu}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}\displaystyle{\frac{\omega_{n-1}}{\xi-\alpha_{n}}}}}}}}}$
be the continued fraction representation of $G_{\mu}$ as in Theorem 2.4. Then
as in the proof of the Proposition 3.1, define $b=\xi.1_{n},\
D_{n}=(\alpha_{k}\delta_{kl})_{k,l=1}^{n}$ and the nilpotent completely
positive map $\eta_{n}$ via:
$\displaystyle\eta_{n}:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C})$
$\displaystyle\eta_{n}\big{(}(a_{kl})_{k,l=1}^{n}\big{)}=(\omega_{k}^{\frac{1}{2}}\delta_{(k+1)l})_{k,l=1}^{n}(a_{kl})_{k,l=1}^{n}(\omega_{k-1}^{\frac{1}{2}}\delta_{k(l+1)})_{k,l=1}^{n}.$
Then for the self-adjoint semicircular element $s=s_{n}$ with operator-valued
Cauchy transform $G_{s}$ satisfying the functional equation (4) in the form:
$bG_{s}(b)=1+(D_{n}+\eta_{n}(G_{s}(b)))G_{s}(b),$
we have
$G_{\mu}(\xi)=\langle G_{s}(\xi.1_{n})e_{1},e_{1}\rangle_{\ell_{2}^{n}}\ \ \
\xi\in\mathbb{C}^{+}.$
Next, let $\mu_{k}\ \ (1\leq k\leq n)$ be a finitely supported probability
measure on $\mathbb{R}$ with associated Cauchy transform:
$G_{\mu_{n-(k-1)}}(\xi)=\frac{1}{\xi-\alpha_{1}-\displaystyle{\frac{\omega_{1}}{\xi-\alpha_{2}-\displaystyle{\frac{\omega_{2}}{{\xi-\alpha_{3}-}_{\displaystyle{\ddots}\displaystyle{\frac{\omega_{k-1}}{\xi-\alpha_{k}}}}}}}}}\
\ (1\leq k\leq n).$
Note that $\mu=\mu_{1}.$ Then by proof of the Proposition 3.1, we have
(9) $\displaystyle G_{\mu_{k}}(\xi)=\langle
G_{s}(\xi.1_{n})e_{k},e_{k}\rangle_{\ell_{2}^{n}}\ \ \ (1\leq k\leq n),\
\xi\in\mathbb{C}^{+}.$
Consequently, by Equations (6) and (9), we have:
$G_{\mu_{s}}(\xi)=tr_{n}(G_{s}(\xi.1))=\frac{1}{n}\sum_{k=1}^{n}\langle
G_{s}(\xi.1)e_{k},e_{k}\rangle_{\ell_{2}^{n}}=\frac{1}{n}\sum_{k=1}^{n}G_{\mu_{k}}(\xi)\
\ \ \xi\in\mathbb{C}^{+},$
and by the Equation (1) it follows that:
$\mu_{s}=\frac{1}{n}\sum_{k=1}^{n}\mu_{k},$
proving the desired result. ∎
## Acknowledgement
The author would like to express his thanks to Prof. Serban Belinschi for his
guidance and support on this paper.
## References
* [1] Choi, M. D. (1975). Completely Positive Linear Maps on Complex Matrices, Linear Algebra and Its Applications, 10, pp. 285-290,
* [2] Cox, D, Little, J, and O’shea, D. (2007). Ideals, Varieties, and Algorithms, 3rd edistion, Springer, pp. 96-98,
* [3] Helton, W, Rashidi Far, R, and Speicher, R. (2007). Operator-Valued Semicircular Elements: Solving a Quadratic Matrix Equation with Positivity Constraints, IMRN , Vol. 2007, Article ID rnm086, 15 pages,
* [4] Hora, A, and Obata, N. (2007). Quantum Probability and Spectral Analysis of Graphs, Springer, pp. 51-61,
* [5] Paulsen, V. (2003). Completely Bounded Maps and Operator Algebras, Cambridge University Press,
* [6] Rashidi Far, R, Oraby, T, Bryc, W, and Speicher, R. (2008). On Slow-fading MIMO Systems with Nonseparable Correlation, IEEE Trans. Inform. Theory No 2, Vol 54, pp. 544-553,
* [7] Shlyakhtenko, D. (1996). Random Guassian Band Matrices and Freeness with Amalgamation, IMRN, No 20, pp. 1013-1025,
* [8] Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments, American Mathematical Society, pp. 23-26,
* [9] Speicher, R. (1998). Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory, Memories of Amer. Math. Soc. 132,
* [10] Voiculescu, D. V. (1991). Limit Laws for Random Matrices and Free Products, Invent. Math, No 1, Vol 104, pp. 201-220,
* [11] Voiculescu, D. V. (1995). Operations on Certain Non-Commutative Operator Valued Random Variables, Astérisque, No 2, pp. 243-275,
* [12] Wigner E. P. (1958). On the Distributions of the Roots of Certain Symmetric Matrices, Ann. of Math, No 2, Vol 67, pp. 325-327
|
arxiv-papers
| 2011-09-22T23:44:15 |
2024-09-04T02:49:22.433434
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohsen Soltanifar",
"submitter": "Mohsen Soltanifar",
"url": "https://arxiv.org/abs/1109.4973"
}
|
1109.5133
|
# Analytical form of light-ray tracing in invisibility cloaks
Ruo-Yang Zhang zhangruoyang@gmail.com Theoretical Physics Division, Chern
Institute of Mathematics, Nankai University, Tianjin, 300071, P.R. China Qing
Zhao qzhaoyuping@bit.edu.cn Department of Physics, College of Science,
Beijing Institute of Technology, Beijing, 100081, P.R. China Mo-Lin Ge
geml@nankai.edu.cn Theoretical Physics Division, Chern Institute of
Mathematics, Nankai University, Tianjin, 300071, P.R. China Department of
Physics, College of Science, Beijing Institute of Technology, Beijing, 100081,
P.R. China
###### Abstract
In this paper, we review the methodology of transformation optics, which can
construct invisibility cloak through the transformation of coordinates based
on the form invariance of Maxwell’s equations. Three different ways to define
the components of electromagnetic fields are compared for removing some
ambiguities. The analytical expressions of light-ray and wave-normal ray are
derived in spherical and cylindrical ideal invisibility cloaks created with
any continuous radial transformation functions, and their physical
interpretation is also given. Using the duality principle in anisotropic
media, we prove that light-ray vector satisfies “ray-vector eikonal equation”
corresponding to the usual “wave-vector eikonal equation”. The results
interpret why the wave vector maps to the ray vector transferring from the
virtual space to the physical space, but not the wave vector. As an
application, we investigate the special transformation functions which make
the light-ray function satisfy harmonic equation.
## I Introduction
Recently, invisibility cloak has attracted much attention and widely study
Pendry _et al._ (2006); Leonhardt (2006); Greenleaf _et al._ (2003); Schurig
_et al._ (2006); Leonhardt and Philbin (2006); Milton _et al._ (2006);
Leonhardt and Philbin (2009); Chen (2009); Chen _et al._ (2010); Cummer _et
al._ (2006); Chen _et al._ (2007); Ruan _et al._ (2007); Zhang _et al._
(2007); Luo _et al._ (2008a, b); Cai _et al._ (2007); Niu _et al._ (2009);
Yao _et al._ (2008); Chen and Chan (2008). Its theory is based on the form
invariance of 3-dimensional Maxwell’s equations under a coordinate
transformation Pendry _et al._ (2006); Schurig _et al._ (2006); Leonhardt
and Philbin (2006); Milton _et al._ (2006). Briefly, the property of
material, such as $\varepsilon$, $\mu$ in a vacuum Euclidean space with an
original orthogonal coordinate, will be transformed into a different form
through a coordinate transformation. However, the form of Maxwell’s equations
is invariant. The transformed $\varepsilon$ and $\mu$ could be alternatively
interpreted as a new different set of material properties in the original
orthogonal coordinate system. Based on this point, the developments have
generated a new subject of transformation optics Leonhardt and Philbin (2009);
Chen (2009); Chen _et al._ (2010). Meanwhile, a conformal mapping method has
been applied to design a medium which can achieve invisibility in the
geometric limit Leonhardt (2006). The validation of the invisible cloak
designed with transformation optics method has been verified through several
ways, including ray tracing approach Schurig _et al._ (2006); Niu _et al._
(2009), full-wave simulations Cummer _et al._ (2006), and analysis based on
Mie scattering [11-16]. The early studies in the invisibility cloak were
mostly based on linear transformation proposed by Pendry et al. Pendry _et
al._ (2006); Schurig _et al._ (2006); Chen _et al._ (2007); Ruan _et al._
(2007); Zhang _et al._ (2007). Then different transformations have also been
discussed Luo _et al._ (2008a); Cai _et al._ (2007). Luo et al. proved that
the permittivity and permeability created with a most general form of
transformation can realize spherical invisibility cloaks and give the
analytical calculation of electromagnetic fields for any spherical cloak using
Mie scattering method Luo _et al._ (2008b).
In the pioneering work of Pendry et al.Pendry _et al._ (2006), it had been
noted that the Poynting vectors acting as the tangent vectors of light should
propagate to surround the hidden area in the cloak layer. They used the
canonical equations of a double anisotropic material optical system to
describe the light propagation Schurig _et al._ (2006). According to the
analogy between the curved space and double anisotropic media, Ref. Niu _et
al._ (2009) described the ray traces with the geodesic equation in the virtual
curved system, and gave the proof that it is equivalent to canonical
equations. Since the original Euclidean space (virtual space) is a vacuum, the
Poynting vectors and wave vectors of a beam of light traveling in it are
parallel to each other. Whereas, when mapping to physical space where a cloak
is placed, Poynting vectors and wave vectors are not paralleling any more,
which has been pointed out and discussed by Leonhardt in Leonhardt and Philbin
(2010). In this paper, however, we will expound this point from a different
perspective by using “ray-vector eikonal equation” and corresponding
relations.
The paper is organized as follows. At the beginning, we compare three common
ways to define the components of electromagnetic fields in 3-D curvilinear
coordinates, which are named Minkowski’s definition, Landau’s definition, and
the components in normalized bases. To do this is benefit to clarify some
ambiguities caused by the mistiness of definition. After that, We repeat the
fundamentals of transformation optics, which contain the corresponding
relations between the curvilinear coordinate system of virtual space (S
system) and real physical space (P space), and emphasize that the validity of
transformation optics is not only based on the form of Maxwell’s equations,
but also based on the correspondence of boundary conditions between the two
systems.
After explication of the fundamentals, the analytical solution of
electromagnetic fields is derived in spherical and cylindrical cloaks with
arbitrary radial transformation functions. The Poynting vectors and wave
vectors are further calculated, and the analytical expressions of light rays
and wave-normal rays are obtained, respectively. Then, we apply the
expressions to four types of cloak made by four different transformation
functions. Furthermore, we provide the physical interpretation to the
analytical solution of light-rays, and verify it satisfies the geodesic
equation in S system discussed in Niu _et al._ (2009). Through duality
principle of anisotropic material, we obtain “ray-vector eikonal equation”
corresponding to the usual “wave-vector eikonal equation”. And through the
corresponding relations, we prove that the covariant component of wave vector
$k_{i}$ in S system maps to wave vector
$k_{{\scriptscriptstyle(\mathrm{P})}i}$ in P space, however, its contravariant
component $k^{i}$ maps to ray vector
$s_{{\scriptscriptstyle(\mathrm{P})}}^{i}$ in P space. This is the precise
reason that wave vector and Poynting vector are split in P space.
Finally, as an application, we find out the general form of transformation
functions that makes the light-ray expression satisfy harmonic equation. It is
just the conformal transformation function, which Leonhardt applied to design
invisibility device in the geometric limit Leonhardt (2006). In this case,
Poynting vectors and wave vectors have neither divergence nor curl.
## II Coordinate transformation of electromagnetic quantities
### II.1 Definition of electromagnetic fields in 3-D orthogonal curvilinear
coordinates
For an arbitrary 3-D curvilinear coordinate system $\\{x^{i}\\}$ to describe
spatial part of a flat spacetime, its bases
$\vec{e}_{i}=\partial\vec{r}/\partial x^{i}$ are often called coordinate bases
or holonomic bases Misner _et al._ (1973), and its spatial metric is
$\gamma_{ij}=\vec{e}_{i}\cdot\vec{e}_{j}$. The components of a tensor
$\matr{T}$ are defined by $\matr{T}=T^{ij}\vec{e}_{i}\vec{e}_{j}$.
The component forms of Maxwell’s equations in any 3-D curvilinear coordinate
system are:
$\nabla_{i}\bar{B}^{i}\ =\
\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}\bar{B}^{i})\
=0,\quad\quad\quad\frac{\partial\bar{B}^{i}}{\partial
t}+\epsilon^{ijk}\partial_{j}\bar{E}_{k}\ =\
\frac{\partial\bar{B}^{i}}{\partial
t}+\frac{1}{\sqrt{\gamma}}e^{ijk}\partial_{j}\bar{E}_{k}\ =\ 0,$ (1a)
$\nabla_{i}\bar{D}^{i}\ =\
\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}\bar{D}^{i})\
=\rho,\quad-\frac{\partial\bar{D}^{i}}{\partial
t}+\epsilon^{ijk}\partial_{j}\bar{H}_{k}\ =\
-\frac{\partial\bar{D}^{i}}{\partial
t}+\frac{1}{\sqrt{\gamma}}e^{ijk}\partial_{j}\bar{H}_{k}\ =\ 0,$ (1b)
where $\nabla_{i}$ is 3-D covariant derivative, $\gamma$ is the determinate of
$\gamma_{ij}$, $e^{ijk}$ is 3-order completely antisymmetric symbol, and
$\epsilon^{ijk}=e^{ijk}/\sqrt{\gamma}$ is 3-order Levi-Civita tensor. We use
Latin indices $i,j,\cdots$ to denote 1 to 3 for three spatial components. The
components of electromagnetic vectors in these equations are resolved in
holonomic bases, and this mode of components is named Landau’s definition (see
Appendix A for general Landau’s definition in 4-D curved spacetime). We add a
bar on these components to discriminate Minkowski’s definition, which will be
introduced below.
In this representation, $\vec{E},\ \vec{H},\ \vec{D},\ \vec{B}$ are all
treated as 3-D spatial vectors, which imply their components satisfy the law
of vector transformation $T^{i}=(\partial x^{i}/\partial x^{\prime
k^{\prime}})T^{\prime k^{\prime}}$, meanwhile $\matr{\varepsilon}$ and
$\matr{\mu}$ are treated as 3-D spatial tensors, and their components satisfy
the law of transformation
$\bar{\varepsilon}^{ij}=\Lambda^{i}_{i^{\prime}}\Lambda^{j}_{j^{\prime}}\;\bar{\varepsilon}^{\prime
i^{\prime}j^{\prime}},\
\bar{\mu}^{ij}=\Lambda^{i}_{i^{\prime}}\Lambda^{j}_{j^{\prime}}\;\bar{\mu}^{\prime
i^{\prime}j^{\prime}},$ where $\Lambda^{i}_{i^{\prime}}=\partial
x^{i}/\partial x^{\prime i^{\prime}}$ is the element of Jacobian matrix. In
vacuum,
$\bar{\varepsilon}^{ij}/\varepsilon_{0}=\bar{\mu}^{ij}/\mu_{0}=\gamma^{ij}$.
In transformation optics, a different way to define electromagnetic fields in
curved coordinate is established in order to hold the expressions of Maxwell’s
equations in Cartesian coordinate Milton _et al._ (2006); Leonhardt and
Philbin (2009); Ward and Pendry (1996). If we define electromagnetic
quantities as
$D^{i}\ =\ \sqrt{\gamma}\bar{D}^{i},\qquad\ B^{i}\ =\
\sqrt{\gamma}\bar{B}^{i},\\\ $ (2a)
$\varepsilon^{ij}=\sqrt{\gamma}\bar{\varepsilon}^{ij},\qquad\
\mu^{ij}=\sqrt{\gamma}\bar{\mu}^{ij}.\\\ $ (2b) $\hat{\rho}\ =\
\sqrt{\gamma}\rho,\qquad\ \ \ \hat{j}^{i}\ =\ \sqrt{\gamma}j^{i},$ (2c)
substituting them into Eqs. (1), the form of Maxwell’s equations in curved
coordinates changes to the form appearing in Cartesian coordinates. As a
result, the form of Maxwell’s equations expressed by ordinary derivative is
invariant from coordinate transformation. We name this mode of definition as
Minkowski’s definition (see Appendix A for general Minkowski’s definitions).
Note that the above components $D^{i},\ B^{i},\ \varepsilon^{ij},\ \mu^{ij}$
are not the real components of the corresponding vectors or tensors in the
curvilinear coordinate, but they can be regarded as the components of
$\sqrt{\gamma}\vec{D},\ \sqrt{\gamma}\vec{B},\
\sqrt{\gamma}\matr{\varepsilon},\ \sqrt{\gamma}\matr{\mu}$ respectively. These
quantities are all vector or tensor density, which satisfy the law of
transformation Schurig _et al._ (2006)
$D^{i}\ =\
\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{i}_{i^{\prime}}\
D^{\prime i^{\prime}},\qquad B^{i}\ =\
\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{i}_{i^{\prime}}\
B^{\prime i^{\prime}},\\\ $ (3a) $\varepsilon^{ij}\ =\
\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{i}_{i^{\prime}}\Lambda^{j}_{j^{\prime}}\;\varepsilon^{\prime
i^{\prime}j^{\prime}},\quad\mu^{ij}\ =\
\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{i}_{i^{\prime}}\Lambda^{j}_{j^{\prime}}\;\mu^{\prime
i^{\prime}j^{\prime}}.$ (3b)
In vacuum, the permittivity and permeability have the simple forms as
$\varepsilon^{ij}=\varepsilon_{0}\sqrt{\gamma}\gamma^{ij},\quad\mu^{ij}=\mu_{0}\sqrt{\gamma}\gamma^{ij}.$
(4)
The components resolved by unit basis are also commonly used to expressing
tensors in curved coordinates. The set of normalized bases is defined as
$\hat{e}_{i}=\vec{e}_{i}/\|\vec{e}_{i}\|=\vec{e}_{i}/\sqrt{\gamma_{ii}}$, yet
these bases are anholonomic in that we usually can not find a set of
coordinates to satisfy $\hat{e}_{i}=\partial\vec{r}/\partial x^{i}$. The
components in orthonormal bases are also defined by $\matr{T}=T_{\langle
ij\rangle}\,\hat{e}_{i}\hat{e}_{j}$, and there is no distinction between
covariant and contravariant components when the coordinates are orthogonal.
For a 3-D orthogonal curvilinear coordinate system the set of orthonormal
bases is corresponding to the tetrad in 4-D spacetime (see Appendix A). Since
all the components of a vector in unit bases have the same dimension, they act
as the measurement of the vector in each direction. In terms of the relation
of holonomic and anholonomic bases, we can get
$\varepsilon^{\langle ij\rangle}\ =\ h_{i}h_{j}\,\bar{\varepsilon}^{ij}\ =\
\frac{h_{i}h_{j}}{\sqrt{\gamma}}\,\varepsilon^{ij},\qquad\mu^{\langle
ij\rangle}\ =\ h_{i}h_{j}\,\bar{\mu}^{ij}\ =\
\frac{h_{i}h_{j}}{\sqrt{\gamma}}\,\mu^{ij},$ (5)
where $h_{i}=\sqrt{\gamma_{ii}}$. For orthogonal coordinates,
$\sqrt{\gamma}=h_{1}\,h_{2}\,h_{3}$ . In Eqs. (5), we have canceled the
Einstein summation convention. For vacuum, $\varepsilon_{\langle
ij\rangle}/\varepsilon_{0}=\mu_{\langle ij\rangle}/\mu_{0}=\delta_{ij}$.
### II.2 Three coordinate systems in transformation optics
The method of transformation optics starts from a virtual 3-D flat Euclidean
space with medium of vacuum. An orthogonal coordinate system (Cartesian or
curvilinear coordinate), S′ system, is employed to describe the virtual space.
In S′, the coordinates are written as $x^{\,\prime\,i\,^{\prime}}$ and the
metric is $\gamma\,^{\prime}_{\,i\,^{\prime}j\,^{\prime}}$. However, another
coordinate system S with coordinates $x^{i}$ and metric $\gamma_{\ ij}$ can be
established to describe it. The transformation of coordinates is
$x^{i}=x^{i}(x\,^{\prime\,i\,^{\prime}})$.
If we investigate electromagnetic fields propagating in the virtual space,
Eqs. (4) give the Minkowski’s form of permittivity and permeability describing
vacuum either in S or in S′. And Eqs. (3) present the transformation law of
Minkowski’s form between S and S′ system.
Due to the invariance of Maxwell’s equations under Minkowski’s definition, we
can alternatively interpret the Minkowski’s form of permittivity and
permeability in S system as the properties of a real material in physical
space (P space). The P space is also an Euclidean space and described by the
same orthogonal coordinates as S′. We use
${x_{\scriptscriptstyle(\mathrm{P})}}^{i}$ to denote the coordinates of P, and
also add the subscript “$(\mathrm{P})$” to other physical quantities in P
space for definitude. Since we employ the same coordinate system as S′ system,
the spatial metric ${\gamma_{\scriptscriptstyle(\mathrm{P})}}_{ij}$ of P space
hold an identical form with $\gamma\,^{\prime}_{\,i\,^{\prime}j\,^{\prime}}$.
On the other hand, the Minkowski’s form of electromagnetic quantities in P
space takes the same formula as that in S system of virtual space (see Fig.
1).
In previous articles, the S system of virtual space is often called a curved
space because of its nontrivial metric. In mathematics, it is certainly a
metric space or manifold. However, considering its zero curvature, we only
treat it as a curvilinear coordinate system of virtual flat space. In this
paper. we only call a system space when they have a real physical background
of spacetime, such as virtual space and P space. But for the coordinate
systems used to describe a space, we only call them system but not space.
Nevertheless, we do not distinguish the P space itself and its coordinate
system, because we only set one system to describe it.
The corresponding relations of electromagnetic quantities between physical
space and S system are shown in Table 1. There are several points to
emphasize. First, corresponding means the Minkowski’s components of these
quantities regarded as functions of coordinates have same mathematical forms
in the two systems. Second, for an individual quantity, only the covariant (or
contravariant) components hold the same form in the two systems, yet the other
kind of components isn’t congruent. For instance,
$D_{\scriptscriptstyle(\mathrm{P})}^{i}(x_{\scriptscriptstyle(\mathrm{P})}^{i})$
and $D^{i}(x^{i})$ have the same mathematical form, but the forms of
${D_{\scriptscriptstyle(\mathrm{P})}}_{i}(x_{\scriptscriptstyle(\mathrm{P})}^{i})$
and $D_{i}(x^{i})$ are different. Third, the identity is only valid for the
components defined in Minkowski’s method, however the components in Landau’s
definition and in the unit bases often take the different form in the two
systems, unless the three kinds of components reduce to equivalent under
Cartesian coordinate of physical space.
Figure 1: Relations among Physical space, S′ coordinate system and S coordinate system of virtual space. Table 1: Corresponding relations. | Electromagnetic quantities
---|---
Physical space : | $D_{\scriptscriptstyle(\mathrm{P})}^{i}$ | ${E_{\scriptscriptstyle(\mathrm{P})}}_{i}$ | $B_{\scriptscriptstyle(\mathrm{P})}^{i}$ | ${H_{\scriptscriptstyle(\mathrm{P})}}_{i}$ | ${\varepsilon_{\scriptscriptstyle(\mathrm{P})}}^{ij}$ | ${\mu_{\scriptscriptstyle(\mathrm{P})}}^{ij}$
S system : | $D^{i}$ | $E_{i}$ | $B^{i}$ | $H_{i}$ | $\varepsilon^{ij}$ | $\mu^{ij}$
Now our discussion is restricted on the condition that S system is an
orthogonal coordinate system with $\gamma_{ij}={h_{i}}^{2}\delta_{ij}$. Based
on the corresponding relations, Eqs. (3), and (5), we derive the permittivity
and permeability in physical space:
$\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle
ij\rangle}/\varepsilon_{0}\ =\ \mu_{{\scriptscriptstyle(\mathrm{P})}\langle
ij\rangle}/\mu_{0}\ =\
\sum_{i^{\prime}}\frac{h^{\prime}_{1}\,h^{\prime}_{2}\,h^{\prime}_{3}}{h_{{\scriptscriptstyle(\mathrm{P})}1}\,h_{{\scriptscriptstyle(\mathrm{P})}2}\,h_{{\scriptscriptstyle(\mathrm{P})}3}}\frac{h_{{\scriptscriptstyle(\mathrm{P})}i}\,h_{{\scriptscriptstyle(\mathrm{P})}j}}{(h^{\prime}_{i^{\prime}})^{2}}\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{i}_{i^{\prime}}\Lambda^{j}_{i^{\prime}}.$
(6)
Here we take the transformation matrix as a function of $x^{i}$ and $x^{\prime
i^{\prime}}$, $i.e.$
$\Lambda^{i}_{i^{\prime}}=\Lambda^{i}_{i^{\prime}}(x^{i},\;x^{\prime
i^{\prime}})$, meanwhile the independent variables should be
$x_{\scriptscriptstyle(\mathrm{P})}^{i}$ and $x^{\prime i^{\prime}}$ in the
above formula.
For spherical invisibility cloak, if we select S′ system as spherical
coordinate system, define the inverse transformation from S′ to S system as
$r^{\prime}=f(r),\ \theta^{\prime}=\theta,\ \phi^{\prime}=\phi$, and insert it
into Eq. (6), the constitutive parameters become Luo _et al._ (2008b)
$\big{(}\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle
ij\rangle}/\varepsilon_{0}\big{)}\ =\
\big{(}\mu_{{\scriptscriptstyle(\mathrm{P})}\langle ij\rangle}/\mu_{0}\big{)}\
=\
\mathrm{diag}\left(\frac{f^{2}(r_{\scriptscriptstyle(\mathrm{P})})}{{r_{\scriptscriptstyle(\mathrm{P})}}^{2}f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})})},\
\ f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})}),\ \
f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})})\right).$ (7)
For arbitrary transformation function, it can be completely invisible, as long
as $f(a)=0$ and $f(b)=b$,where $a,\ b$ are the inner and outer radii of the
cloak layer, even if $f(r)$ is continuous but non-differentiable Luo _et al._
(2008b). Similarly, the constitutive parameters for cylindrical cloak obtained
by the mere radial transformation $r^{\prime}=f(r),\ \theta^{\prime}=\theta,\
z^{\prime}=z$ are Cai _et al._ (2007)
$\big{(}\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle
ij\rangle}/\varepsilon_{0}\big{)}\ =\
\big{(}\mu_{{\scriptscriptstyle(\mathrm{P})}\langle ij\rangle}/\mu_{0}\big{)}\
=\
\mathrm{diag}\left(\frac{f(r_{\scriptscriptstyle(\mathrm{P})})}{r_{\scriptscriptstyle(\mathrm{P})}f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})})},\
\
\frac{r_{\scriptscriptstyle(\mathrm{P})}f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})})}{f(r_{\scriptscriptstyle(\mathrm{P})})},\
\
\frac{f(r_{\scriptscriptstyle(\mathrm{P})})f^{\prime}(r_{\scriptscriptstyle(\mathrm{P})})}{r_{\scriptscriptstyle(\mathrm{P})}})\right),$
(8)
where the components are described by cylindrical coordinates. The arbitrary
$f(r)$ will achieve completely invisible only if it satisfies the boundary
condition $f(a)=0$ and $f(b)=b$ ( see the proof in Appendix B).
### II.3 Boundary conditions for perfect invisibility cloak
The congruity of Maxwell’s equations with the corresponding relations between
S system and P space is regarded as the foundation of transformation optics.
However, despite of the existence on the same form of Maxwell’s equations,
they won’t give the same solution unless the boundary conditions of the two
systems are also identical.
For example, the realization of spherical and cylindrical invisibility cloak
requires the inverse transformation function satisfies $f(a)=0$ and $f(b)=b$.
The $f(a)=0$ causes the domain $r^{\prime}<a$ in S′ system to shrink into its
original point in S, so that the coordinates with radial component $r<a$ would
never appear in the expression of electromagnetic fields in S system. It leads
to decoupling of fields between the hidden region
$r_{\scriptscriptstyle(\mathrm{P})}<a$ and cloak layer in the P space. Pendry
et al. pointed out that the condition $f(b)=b$ makes the constitutive
parameters satisfy perfectly matched layer (PML) conditions
$\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle\theta\theta\rangle}=\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle\phi\phi\rangle}=1/\varepsilon_{{\scriptscriptstyle(\mathrm{P})}\langle
rr\rangle}$ and
$\mu_{{\scriptscriptstyle(\mathrm{P})}\langle\theta\theta\rangle}=\mu_{{\scriptscriptstyle(\mathrm{P})}\langle\phi\phi\rangle}=1/\mu_{{\scriptscriptstyle(\mathrm{P})}\langle
rr\rangle}$ at the outer surface of the cloak
($r_{\scriptscriptstyle(\mathrm{P})}=b$)Pendry _et al._ (2006), therefore the
cloak is reflectiveness. However there is a more profound reason produced by
the corresponding relation of the boundary conditions between S′ system and P
space.
First, we consider a spherical cloak designed by general transformation
$x^{i}(r^{\prime},\theta^{\prime},\phi^{\prime})$ from S′ to S system. In P
space, the boundary conditions at the interface $r=b$ are
$(\vec{E}_{\scriptscriptstyle(\mathrm{P})}\big{|}_{r=b^{+}}-\vec{E}_{\scriptscriptstyle(\mathrm{P})}\big{|}_{r=b^{-}})\times\hat{e}_{r}\
=\
0,\qquad(\vec{D}_{\scriptscriptstyle(\mathrm{P})}\big{|}_{r=b^{+}}-\vec{D}_{\scriptscriptstyle(\mathrm{P})}\big{|}_{r=b^{-}})\cdot\hat{e}_{r}\
=\ 0,$ (9)
where $\hat{e}_{r}$ acts as the normal vector of interface. The boundary
conditions indicate the tangential component of $\vec{E}$ and normal component
of $\vec{D}$ are continuous across the interface, and the two boundary
conditions can completely determine
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}|_{r=b^{-}}$, if
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}|_{r=b^{+}}$ is given. Since the
metric of P space is continuous, the equality of components is always tenable
no matter what definitions. Thus, we have the continuous relations in
Minkowski’s form
$E_{{\scriptscriptstyle(\mathrm{P})}\theta}\big{|}_{r=b^{+}}=E_{{\scriptscriptstyle(\mathrm{P})}\theta}\big{|}_{r=b^{-}},\quad
E_{{\scriptscriptstyle(\mathrm{P})}\phi}\big{|}_{r=b^{+}}=E_{{\scriptscriptstyle(\mathrm{P})}\phi}\big{|}_{r=b^{-}},\quad
D_{\scriptscriptstyle(\mathrm{P})}^{r}\big{|}_{r=b^{+}}=D_{\scriptscriptstyle(\mathrm{P})}^{r}\big{|}_{r=b^{-}}.$
(10)
Non-reflection means the electric field outside the cloak only has the
incident wave. According to the corresponding relations in Table 1, the
electric field in S system should also satisfy the relations
$E^{\prime}_{\theta^{\prime}}\big{|}_{r^{\prime}=b^{+}}=E_{\theta}\big{|}_{r=b^{+}}=E_{\theta}\big{|}_{r=b^{-}}=\Lambda^{i^{\prime}}_{\theta}\,E^{\prime}_{i^{\prime}}\big{|}_{x^{\prime
i^{\prime}}(b^{-},\theta,\phi)},$ (11a)
$E^{\prime}_{\phi^{\prime}}\big{|}_{r^{\prime}=b^{+}}=E_{\phi}\big{|}_{r=b^{+}}=E_{\phi}\big{|}_{r=b^{-}}=\Lambda^{i^{\prime}}_{\phi}\,E^{\prime}_{i^{\prime}}\big{|}_{x^{\prime
i^{\prime}}(b^{-},\theta,\phi)},$ (11b) $D^{\prime
r^{\prime}}\big{|}_{r^{\prime}=b^{+}}=D^{r}\big{|}_{r=b^{+}}=D^{r}\big{|}_{r=b^{-}}=\left|\det\left(\Lambda^{i}_{i^{\prime}}\right)^{-1}\right|\Lambda^{r}_{i^{\prime}}\,D^{\prime
i^{\prime}}\big{|}_{x^{\prime i^{\prime}}(b^{-},\theta,\phi)},$ (11c)
where $E^{\prime}_{\theta^{\prime}}|_{r^{\prime}=b^{+}}=E_{\theta}|_{r=b^{+}}$
since S′ and S system hold the same coordinates in the domain $r>b$. Note that
the boundary conditions Eqs. (9) certainly satisfy in virtual space. However
the metric is discontinuous at the interface $r=b$ in S system, therefore the
relation of components $E_{\theta}|_{r=b^{+}}=E_{\theta}|_{r=b^{-}}$ is not
trivial, but a strong limit to the metric.
By comparing the first and last term, we note that the electric field at
$r^{\prime}=b$ is demanded to have a fixed relation to the field at $x^{\prime
i^{\prime}}(b^{-},\theta,\phi)$ in S′ system. Because of the locality of
electromagnetic fields, a certain relation of fields between different
positions can not be established. Therefore $(b,\,\theta,\,\phi)$ and
$x^{\prime i^{\prime}}(b^{-},\theta,\phi)$ must be the same point:
$r^{\prime}(b^{-},\theta,\phi)\,=\,b,\quad\theta^{\prime}(b^{-},\theta,\phi)\,=\,\theta,\quad\phi^{\prime}(b^{-},\theta,\phi)\,=\,\phi.$
(12)
This is the requirement of non-reflection. Thus, we have
$\left(\frac{\partial\theta^{\prime}}{\partial\theta}\,=\,\frac{\partial\phi^{\prime}}{\partial\phi}\right)\Big{|}_{r=b^{-}}=\,1,\
\left(\frac{\partial r^{\prime}}{\partial\theta}\,=\,\frac{\partial
r^{\prime}}{\partial\phi}\,=\,\frac{\partial\theta^{\prime}}{\partial\phi}=\,\frac{\partial\phi^{\prime}}{\partial\theta}\right)\Big{|}_{r=b^{-}}=\,0.$
Substituting it into Eqs. (11), the equalities are directly satisfied. A
similar study to the boundary conditions of magnetic field can give the
identical result.
In fact, the interface at $r=b$ in S system generally doesn’t correspond to
the interface $r^{\prime}=b$ in S′ system for an arbitrary transformation.
Thus the metric is usually discontinuous across the interface, and we even
hardly measure the element of length on the interface. For example, if the
inverse transformation is simply $r^{\prime}=f(r),\ \theta^{\prime}=\theta,\
\phi^{\prime}=\phi$ as $r<b$, the elements of length on the inner and outer
surface of the interface are respectively
$\mathrm{d}l^{2}\big{|}_{r=b^{-}}=\
f(b)^{2}\,\big{(}\mathrm{d}\theta^{2}\,+\,\sin^{2}\theta\mathrm{d}\phi^{2}\big{)},\quad\mathrm{d}l^{2}\big{|}_{r=b^{+}}=\
b^{2}\,\big{(}\mathrm{d}\theta^{2}\,+\,b^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\big{)}.$
Nothing but $f(b)=b$ gives
$\mathrm{d}l^{2}|_{r=b^{-}}=\mathrm{d}l^{2}|_{r=b^{+}}$ with arbitrary
$\mathrm{d}\theta$ and $\mathrm{d}\phi$. It is also available for cylindrical
cloak by a similar analysis that non-reflection requires invariance of
coordinates on the interface before and after transformation.
## III Analytical expressions of light-ray and wave-normal ray
### III.1 Spherical cloak
Considering a spherical cloak, the cloak hides a homogenous-media sphere, and
its center is at the origin of a spherical coordinate system. A plane wave
$\vec{E}^{\mathrm{in}}_{\scriptscriptstyle(\mathrm{P})}=E_{0}\mathrm{e}^{\mathrm{i}k_{0}r\cos\theta}\hat{e}_{x}$
with $x$ polarization is incident to the cloak. It has been proved that both
the fields $\vec{E}^{\mathrm{int}}_{\scriptscriptstyle(\mathrm{P})}$ inside
the internal area ($r<a$), and the scattering fields
$\vec{E}^{\mathrm{sc}}_{\scriptscriptstyle(\mathrm{P})}$ outside cloak ($r>b$)
varnish if the constitutive parameters of cloak satisfy the formula Eq. (7)
and the boundary conditions $f(a)=0,\ f(b)=b$. Meanwhile the electromagnetic
fields in cloak layer ($a<r<b$) have been given in Luo _et al._ (2008b)
$\displaystyle\vec{E}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=E_{0}\left(f^{\prime}(r)\sin\theta\cos\phi\hat{e}_{r}+\frac{f(r)}{r}\cos\theta\cos\phi\hat{e}_{\theta}-\frac{f(r)}{r}\sin\phi\hat{e}_{\phi}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta},$
(13a) $\displaystyle\vec{D}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=\varepsilon_{0}E_{0}\frac{f(r)}{r}\left(\frac{f(r)}{r}\sin\theta\cos\phi\hat{e}_{r}+f^{\prime}(r)\cos\theta\cos\phi\hat{e}_{\theta}-f^{\prime}(r)\sin\phi\hat{e}_{\phi}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta},$
(13b) $\displaystyle\vec{H}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=\frac{E_{0}}{\eta_{0}}\left(f^{\prime}(r)\sin\theta\sin\phi\hat{e}_{r}+\frac{f(r)}{r}\cos\theta\sin\phi\hat{e}_{\theta}+\frac{f(r)}{r}\cos\phi\hat{e}_{\phi}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta},$
(13c) $\displaystyle\vec{B}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=\frac{E_{0}f(r)}{cr}\left(\frac{f(r)}{r}\sin\theta\sin\phi\hat{e}_{r}+f^{\prime}(r)\cos\theta\sin\phi\hat{e}_{\theta}+f^{\prime}(r)\cos\phi\hat{e}_{\phi}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta},$
(13d)
where $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$ and $k_{0}=\omega/c$ are the
impedance and wave number of vacuum respectively. Since only fields in P space
are discussed in this section, we omit the subscript “($\mathrm{P}$)” in all
the expressions of coordinate, but still retain it in the expressions of
electromagnetic quantities. According to above fields , we obtain the time-
averaged Poynting vector
$\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle\ =\
\frac{1}{2}\mathrm{Re}(\vec{E}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\times\vec{H}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})})\
=\
\frac{{E_{0}}^{2}f(r)}{2\eta_{0}r}\left(\frac{f(r)}{r}\cos\theta\hat{e}_{r}-f^{\prime}(r)\sin\theta\hat{e}_{\theta}\right),$
(14)
It can be verified that the Poynting vector satisfies energy conservation
$\nabla\cdot\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle=0$
Born and Wolf (1999). Supposing the medium of cloak is non-dispersive, the
time-averaged energy density of electromagnetic fields has the simple form
$\langle W^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle\ =\
\frac{1}{2}\mathrm{Re}\left(\frac{1}{2}\vec{E}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\cdot\vec{D}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}+\frac{1}{2}\vec{H}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\cdot\vec{B}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\right)\
=\ \frac{\varepsilon_{0}{E_{0}}^{2}f(r)^{2}f^{\prime}(r)}{2r^{2}}.$ (15)
Thus, the ray (energy) velocity in cloak is given as Born and Wolf (1999);
Chen (1985)
$\vec{v}_{{\scriptscriptstyle(\mathrm{P})}ray}=\
\frac{\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle}{\langle
W^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle}\ =\
c\left(\frac{1}{f^{\prime}(r)}\cos\theta\hat{e}_{r}-\frac{r}{f(r)}\sin\theta\hat{e}_{\theta}\right).$
(16)
The ray vector is Chen (1985)
$\vec{s}_{\scriptscriptstyle(\mathrm{P})}=\
\frac{\vec{v}_{{\scriptscriptstyle(\mathrm{P})}ray}}{\omega}\ =\
{k_{0}}^{-1}\left(\frac{1}{f^{\prime}(r)}\cos\theta\hat{e}_{r}-\frac{r}{f(r)}\sin\theta\hat{e}_{\theta}\right).$
(17)
In terms of the eikonal $\psi=k_{0}f(r)\cos\theta$, the geometric wave-front,
$\psi=\mbox{constant}$, is
$f(r)\cos\theta\ \ =\ \ \mbox{const}.$ (18)
Then, we can get the wave vector $\vec{k}_{\scriptscriptstyle(\mathrm{P})}$,
which is orthogonal to the wave-front, in the cloak
$\vec{k}_{\scriptscriptstyle(\mathrm{P})}\ =\ \nabla\psi\ =\
k_{0}\left(f^{\prime}(r)\cos\theta\hat{e}_{r}-\frac{f(r)}{r}\sin\theta\hat{e}_{\theta}\right).$
(19)
In the anisotropic media, the ray vector ( time-averaged Poynting vector ) is
tangent to light-ray, however the wave vector is tangent to wave-normal ray,
where $\vec{s}$ and $\vec{k}$ are generally not parallel to each other. The
parametric equations of light-ray can be written as
$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\ \
{s_{\scriptscriptstyle(\mathrm{P})}}^{r}\ \ =\ \
\frac{1}{k_{0}f^{\prime}(r)}\cos\theta,$ (20a)
$\displaystyle\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\
\ {s_{\scriptscriptstyle(\mathrm{P})}}^{\theta}\ \ =\ \
-\frac{1}{k_{0}f(r)}\sin\theta,$ (20b)
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\ \
{s_{\scriptscriptstyle(\mathrm{P})}}^{\phi}\ \ =\ \ 0,$ (20c)
where $\lambda$ is the parameter of light-ray parametric equation, and the
components of ray vector employed above are contravariant in holonomic
spherical coordinate bases. The $\phi$ maintains its initial value $\phi_{0}$
through out a light-ray in terms of Eq. (20c). By eliminating $\lambda$ from
Eq. (20a) and Eq. (20b) , we have
$\int{\frac{f^{\prime}(r)}{f(r)}\ \mathrm{d}r}\ \ =\ \
-\int\frac{\cos\theta}{\sin\theta}\ \mathrm{d}\theta.$
Then the analytical expression of light-ray is obtained
$f(r)\sin\theta\ \ =\ \ \mbox{const},\qquad\phi\ \ =\ \ \phi_{0}.$ (21)
Here, the integral constant and $\phi_{0}$ depend on the incident point on the
outer surface of the cloak.
A similar analysis is used to get the parametric equations for the wave-normal
ray
$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\ \
{k_{\scriptscriptstyle(\mathrm{P})}}^{r}\ \ =\ \
k_{0}f^{\prime}(r)\cos\theta,$ (22a)
$\displaystyle\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\
\ {k_{\scriptscriptstyle(\mathrm{P})}}^{\theta}\ \ =\ \
-k_{0}\frac{f(r)}{r^{2}}\sin\theta,$ (22b)
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}\lambda}$ $\displaystyle\ \ =\ \
{k_{\scriptscriptstyle(\mathrm{P})}}^{\phi}\ \ =\ \ 0.$ (22c)
Thus, the wave-normal ray equation is
$\exp\left[\int\frac{f(r)}{r^{2}f^{\prime}(r)}\mathrm{d}r\right]\sin\theta\ \
=\ \ \mbox{const},\qquad\phi\ \ =\ \ \phi_{0}.$ (23)
According to Eq. (16), when $r$ closes to $a$, $f(r)$ tends to 0, and the
modulus of ray velocity increases beyond vacuum light speed $c$ and tends to
infinity which is against the law of causality. The contradiction indicates
that the assumption of non-dispersive cloak is unpractical. In the presence of
dispersion, the time-averaged energy density has a more complicated formula.
It includes the derivative of constitutive parameters with respect to $\omega$
Chen (1985), and then the superluminal result should disappear. More detailed
discussion to this point can be found in Yao _et al._ (2008); Chen and Chan
(2008). Nevertheless, the expressions in Eqs. (16, 17) still exactly represent
the direction of the “real ray velocity”, therefore we will retain the
expressions of $\vec{v}_{{\scriptscriptstyle(\mathrm{P})}ray}$ and
$\vec{s}_{\scriptscriptstyle(\mathrm{P})}$, but limit them only to stand for
the tangent vector of light-ray.
### III.2 Cylindrical cloak
Considering a cylindrical cloak with indefinite length, its axis is $z$ axis
of a cylindrical coordinate system. The incident wave is a transverse-
electric(TE) wave
$\vec{E}^{\mathrm{in}}_{\scriptscriptstyle(\mathrm{P})}=E_{0}\mathrm{e}^{\mathrm{i}k_{0}r\cos\theta}\hat{e}_{z}$
with $z$ polarization propagating towards $x$ direction. If the cloak hold the
perfect invisible parameters expressed in Eq. (8), the calculation in Appendix
B gives $\vec{E}^{\mathrm{int}}_{\scriptscriptstyle(\mathrm{P})}=0$ in the
inner hidden area ($r<a$), and the scattering
$\vec{E}^{\mathrm{sc}}_{\scriptscriptstyle(\mathrm{P})}=0$ outside the cloak
($r>b$). Meanwhile the fields in the cloak layer are
$\displaystyle\vec{E}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=E_{0}\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta}\hat{e}_{z},$
(24a) $\displaystyle\vec{D}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=\varepsilon_{0}E_{0}\frac{f^{\prime}(r)f(r)}{r}\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta}\hat{e}_{z},$
(24b) $\displaystyle\vec{H}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=-\frac{E_{0}}{\eta_{0}}\left(f^{\prime}(r)\sin\theta\hat{e}_{r}+\frac{f(r)}{r}\cos\theta\hat{e}_{\theta}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta},$
(24c) $\displaystyle\vec{B}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}$
$\displaystyle=-\frac{E_{0}}{c}\left(\frac{f(r)}{r}\sin\theta\hat{e}_{r}+f^{\prime}(r)\cos\theta\hat{e}_{\theta}\right)\mathrm{e}^{\mathrm{i}k_{0}f(r)\cos\theta}.$
(24d)
Thus, the time-averaged Poynting vector is
$\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle\ =\
\frac{{E_{0}}^{2}}{2\eta_{0}}\left(\frac{f(r)}{r}\cos\theta\hat{e}_{r}-f^{\prime}(r)\sin\theta\hat{e}_{\theta}\right).$
(25)
and the time-averaged energy density of electromagnetic fields under the
assumption of non-dispersion is
$\langle W^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle\ =\
\frac{\varepsilon_{0}{E_{0}}^{2}f(r)f^{\prime}(r)}{2r}.$ (26)
By calculation, the ray vector and wave vector in the cylindrical cloak have
the identical form in the spherical cloak as shown in Eq. (17) and Eq. (19).
Whereas, the variables $r$ and $\theta$ in the spherical cloak are spherical
coordinates, while in cylindrical cloak they represent cylindrical
coordinates.
Thus, following the same calculation as in spherical cloak, we derive the
light-ray expression in cylindrical cloak
$f(r)\sin\theta\ \ =\ \ \mbox{const},\qquad z\ \ =\ \ z_{0},$ (27)
and the wave-normal ray equation
$\exp\left[\int\frac{f(r)}{r^{2}f^{\prime}(r)}\mathrm{d}r\right]\sin\theta\ \
=\ \ \mbox{const},\qquad z\ \ =\ \ z_{0}.$ (28)
### III.3 Examples of four types of cloak
In this section, we apply the obtained expressions of light-ray and wave-
normal ray to four cloaks constructed by different transformation functions.
Type 1 is the linear transformation function $f_{1}(r)=[b/(b-a)](r-a)$
provided in Pendry _et al._ (2006); Schurig _et al._ (2006). Type 2 is
$f_{2}(r)=b[(r-a)/(b-a)]^{2}$, and type 3 is $f_{3}(r)=b-b[(b-r)/(b-a)]^{2}$.
Both type 2 and 3 are from Luo _et al._ (2008b). Type 4 is
$f_{4}(r)=(b/2a)[2a-b+\sqrt{b^{2}-4ab+4ar}]$ provided in Cai _et al._ (2007).
They all meet the invisibility condition $f(a)=0,\ f(b)=b$ (see Fig. 2).
Figure 2: Transformation functions. The inner and outer radiuses of cloak are
$a=1\mathrm{m}$ and $b=2\mathrm{m}$ respectively.
We construct four spherical cloaks with the four transformations. Each raw in
Fig. 3 displays a type of cloak corresponding to $f_{1}(r)$ to $f_{4}(r)$ from
top to bottom. The first column displays the components of electric fields
$E_{{\scriptscriptstyle(\mathrm{P})}\langle x\rangle}$ in the $x=0$ plane. The
second column displays the light-rays and wave-fronts of each cloak. The third
column shows the wave-normal rays and wave-fronts which are always
perpendicular to each other.
From Fig. 3, some characteristics of each cloak can be concluded. In Fig.
3(b), the light-rays and wave-fronts hold equal spacing respectively, which is
due to its linear transformation
Figure 3: Lights travel in $x=0$ plane of four space where four types of cloak
are placed respectively. Field distributions of
$E_{{\scriptscriptstyle(\mathrm{P})}\langle x\rangle}$ of each cloak are shown
in (a)(d)(g)(j). In (b)(e)(h)(k), blue lines denote light-rays, red lines
denote wave-fronts. In (c)(f)(i)(l), green lines denote wave-normal rays, red
lines still denote wave-fronts.
$f_{1}(r)$. For the second cloak, because $f_{2}^{\prime}(r)\rightarrow 0$ as
$r\rightarrow a$, the $\theta$ component of energy flow decreases in
$r\rightarrow a$ so fast that the energy traveling in cloak mainly
concentrates at the outer boundary. Thereby the light-rays mostly compress
near the outer boundary shown in Fig. 3(e). For the third cloak, since
$f_{3}^{\prime}(b)=0$, the light-rays are condensed to inner boundary as shown
in Fig. 3(h). In addition, $r=b$ as
$k_{{\scriptscriptstyle(\mathrm{P})}r}\equiv 0$, that is to say, the circle
$r=b$ becomes a closed wave-normal ray, and the wave-normal rays inside cloak
are tangent to the outer interface $r=b$, as shown in Fig. 3(i). For the
fourth cloak, its $\varepsilon$ and $\mu$ are continuous to vacuum across
outer interface due to $f_{4}^{\prime}(b)=1$. Therefore, all the light-rays,
wave-normal rays and wave-fronts are smooth when they pass through the outer
boundary of cloak as shown in Fig. 4(k)(l).
### III.4 Verifying the light-ray equation is the solution of geodesic
equations
In this section, We will verify that the light-ray analytical expression (21)
in a spherical cloak satisfies the geodesic equation in S system of virtual
space, which is Niu _et al._ (2009)
$\frac{\mathrm{d}k^{i}}{\mathrm{d}\lambda}+\Gamma^{i}_{jl}k^{j}k^{l}\ =\ 0,$
(29)
where
$\Gamma^{i}_{jl}=\frac{1}{2}\gamma^{im}(\gamma_{mj,l}+\gamma_{ml,j}-\gamma_{jl,m})$
is the Christoffel symbol in 3-D space. In S system, the spatial metric is
$\gamma_{ij}\ =\ \mathrm{diag}\big{(}f^{\prime}(r)^{2},\ \ f^{2}(r),\ \
f^{2}(r)\sin^{2}\theta\big{)}.$ (30)
Substituting it into Christoffel symbol, we have
$\left\\{\begin{array}[]{lll}\Gamma^{r}_{rr}\ =\
\frac{f^{\prime\prime}(r)}{f^{\prime}(r)},&\Gamma^{r}_{\theta\theta}\ =\
-\frac{f(r)}{f^{\prime}(r)},&\Gamma^{r}_{\phi\phi}\ =\
-\frac{f(r)}{f^{\prime}(r)}\sin^{2}\theta,\\\\[4.0pt] \Gamma^{\theta}_{\theta
r}\ =\ \frac{f^{\prime}(r)}{f(r)},&\Gamma^{\theta}_{\phi\phi}\ =\
-\sin\theta\cos\theta,&\\\\[4.0pt] \Gamma^{\phi}_{\phi r}\ =\
\frac{f^{\prime}(r)}{f(r)},&\Gamma^{\phi}_{\phi\theta}\ =\
\cot\theta,&\\\\[4.0pt] \end{array}\right.$ (31)
and other components are all null. Then, the geodesic equations become
$\displaystyle\frac{\mathrm{d}k^{r}}{\mathrm{d}\lambda}+\frac{f^{\prime\prime}(r)}{f^{\prime}(r)}k^{r}k^{r}-\frac{f(r)}{f^{\prime}(r)}k^{\theta}k^{\theta}-\frac{f(r)}{f^{\prime}(r)}\sin^{\theta}k^{\phi}k^{\phi}\
=\ 0,$ (32a)
$\displaystyle\frac{\mathrm{d}k^{\theta}}{\mathrm{d}\lambda}+2\frac{f^{\prime}(r)}{f(r)}k^{\theta}k^{r}-\sin\theta\cos\theta
k^{\phi}k^{\phi}\ =\ 0,$ (32b)
$\displaystyle\frac{\mathrm{d}k^{\phi}}{\mathrm{d}\lambda}+2\frac{f^{\prime}(r)}{f(r)}k^{\phi}k^{r}+\cot\theta
k^{\theta}k^{\phi}\ =\ 0.$ (32c)
Tracing the ray path shown in Eq. (21), the differentials of coordinates
satisfy
$\mathrm{d}\theta\ \ =\ \ -\frac{f^{\prime}(r)}{f(r)}\tan\theta\
\mathrm{d}r,\qquad\mathrm{d}\phi\ \ =\ \ 0.$
Substituting them into the relation between parameter $\lambda$ and element of
length $\mathrm{d}l=(\gamma_{ij}\;\mathrm{d}x^{i}\mathrm{d}x^{j})^{1/2}$,
which is $\mathrm{d}\lambda\ \ =\ \ -(c\sqrt{-g_{00}}/\omega)\mathrm{d}l$
Landau and Lifshitz (1995), we have
$\mathrm{d}\lambda\ \ =\ \ \frac{cf^{\prime}(r)}{\omega\cos\theta}\
\mathrm{d}r.$
Therefore, the wave vector can be expressed as
$k^{r}\ =\ \frac{\mathrm{d}r}{\mathrm{d}\lambda}\ =\
k_{0}\frac{\cos\theta}{f^{\prime}(r)},\quad\ k^{\theta}\ =\
\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}\ =\
-k_{0}\frac{\sin\theta}{f(r)},\quad\ k^{\phi}\ =\ 0.$ (33)
This is the wave vector in S system, its relation with the wave vector (19) in
P space will be discussed in section 4.2. Substituting $k^{i}$ into Eqs. (32),
then the left sides of the equations reduce to zero, equal to the right side.
As a result, we find out that Eq. (21) is a particular analytical solution of
the geodesic equations.
### III.5 Physical interpretation of light-ray equation
We have verified the light-ray (21) denotes a geodesic expressed by curved
coordinates S of virtual space, thus the physical meaning of the light-ray
equation is quite clear.
Since the virtual space is a flat vacuous space with a zero curvature, the
light-ray trajectory which is also the geodesic in virtual space is just a
straight line. Fig. 4(a) shows the light-ray with a Cartesian coordinate
system (S′ system). Through the inverse transformation $r^{\prime}=f(r)$,
where $f(a)=0,\ f(b)=b$ inside the domain $r<b$ in S′ system, we obtain a new
system, S system. Through the transformation, the domain $r^{\prime}<a$ in S′
system shrinks into origin, that is, the coordinates with component $r<a$ do
not exist in S system. Fig. 4(b) shows the result in S system. The coordinate
lines, whose tangent vectors are bases, are defined by
Latitude lines : $\displaystyle\qquad y\ =\ r\sin\theta\ =\
f^{-1}(r^{\prime})\sin\theta^{\prime}\ =\ \mbox{const.}$ (34a) Longitude lines
: $\displaystyle\qquad x\ =\ r\cos\theta\ =\
f^{-1}(r^{\prime})\cos\theta^{\prime}\ =\ \mbox{const.}$ (34b)
Figure 4: Interpretation of light-rays. (a) The red line denotes a light-ray
propagating in virtual space. The meshes are the coordinate lines of S′
system. (b) The red line is still the light-ray in virtual space. The curved
meshes are the coordinate lines of S system. Two orange dashed lines are the
boundaries inside which latitude lines shrink into the origin point. (c)
Light-ray travels in P space with Cartesian coordinate meshes. (d) The mapping
of lines with the expression of geodesic in S system to P space.
where $f^{-1}(r^{\prime})$ is the inverse function of $f(r)$. We observe that
the coordinate lines cave towards the origin, especially the lines whose
$|x|<a$ or $|y|<a$ outside the interface $r=b$ cave directly into the origin
when they pass through the interface. Therefore, the origin becomes an
ambiguous point of coordinates in S system. However, the coordinate
transformation doesn’t change the curvature of virtual space, and the light
trajectories, i.e. geodesics, are still straight with the expression in S
system
$y^{\prime}\ =\ r^{\prime}sin\theta^{\prime}\ =\ f(r)\sin\theta\ =\
\mbox{const.}$ (35)
It corresponds to the light-ray equation that we have obtained previously.
The light-ray function maps to P space with the same form
$f(r_{\scriptscriptstyle(\mathrm{P})})\sin\theta_{\scriptscriptstyle(\mathrm{P})}=\mbox{const}$.
Since P space is also a flat space, the light-ray becomes convex as shown in
Fig. 4(c). Meanwhile, the constitutive parameters both in spherical and
cylindrical cloak have singular components with zero or infinite value at
inner surface for $f(a)=0$. In other words, the geometrical singularity of
coordinates turns into the singularity of material, when mapping from S system
to P space.
We use an analogy to describe the process. The virtual space can be compared
to a piece of flat elastic cloth with straight meshes on it. The meshes look
like S′ system. If we fix the boundary $r=b$, and drag the whole region $r<a$
into the center point, the meshes on the cloth are curved as the coordinate
lines of S system. Then, we draw a straight line on the cloth just as the red
line in Fig 4(b) which plays the role of light-ray in S. After that, we loose
the piece of cloth held into the center, and let it return flat. Then the
meshes also returns to straightness, but the red line we have drawn becomes
convex exactly as the ray propagating in P space.
Now we should clarify the meaning of Fig. 4(d). Sometimes the meshes in this
figure are interpreted as the coordinate of S system Schurig _et al._ (2006).
This interpretation gives rise to an illusion that S system is a real curved
space, as well as the light trajectories. However, the meshes in Fig. 4(d) are
neither coordinate lines nor geodesics no matter in S system or in P space.
The only reasonable interpretation is that they are the curves in P space
directly mapping from the straight lines in S system, and can describe the
light trajectories in P space Leonhardt and Philbin (2009); Chen _et al._
(2010).
## IV Duality principle in anisotropic media
In virtual space, light propagates in straight lines, and the directions of
wave vectors and ray vectors are always identical, however when mapping to
physical space, they are not parallel to each other any more. So what exactly
causes the split? Why does the light-ray instead of not the wave-normal ray in
P space inherit its expression in S system? In this section, we will try to
answer these questions by using duality principle.
In charge free anisotropic media, electromagnetic fields have the following
duality rule Born and Wolf (1999)
$\mbox{Duality
rule}\left\langle\begin{array}[]{cccccccc}\quad\vec{D}&\quad\vec{E}&\quad\vec{B}&\quad\vec{H}&\quad\matr{\varepsilon}&\quad\matr{\mu}&\quad\vec{k}/\omega&\quad
v_{p}\\\ \\\
\quad\vec{E}&\quad\vec{D}&\quad\vec{H}&\quad\vec{B}&\quad{\matr{\varepsilon}}^{-1}&\quad{\matr{\mu}}^{-1}&\quad\vec{s}\omega&\quad
1/v_{ray}\\\ \end{array}\right.$
where $v_{p}=\omega/k$ is the phase speed. In terms of the above systematic
interchange relations, the either side satisfies a set of formulas for plane
waves Chen (1985):
$\displaystyle\vec{D}\ =\ \matr{\varepsilon}\cdot\vec{E}\ =\
-\frac{1}{\omega}\vec{k}\times\vec{H}$ $\displaystyle\vec{E}\ =\
\matr{\varepsilon}^{-1}\cdot\vec{D}\ =\ -\omega\vec{s}\times\vec{B}$ (36)
$\displaystyle\vec{B}\ =\ \matr{\mu}\cdot\vec{H}\ =\
\frac{1}{\omega}\vec{k}\times\vec{E}$ $\displaystyle\ \ \Leftarrow\
\mbox{duality}\ \Rightarrow$ $\displaystyle\vec{H}\ =\
\matr{\mu}^{-1}\cdot\vec{B}\ =\ \omega\vec{s}\times\vec{D}$
$\displaystyle\vec{k}/\omega\ =\ \hat{k}/v_{p}\ =\
\frac{\vec{D}\times\vec{B}}{\vec{E}\cdot\vec{D}}$ $\displaystyle\vec{s}\omega\
=\ v_{ray}\hat{s}\ =\ \frac{\vec{E}\times\vec{H}}{\vec{D}\cdot\vec{E}}$
According to the left side of the formulas, the wave equation is founded as
$\vec{k}\times\left[\ \matr{\mu}^{-1}\cdot\big{(}\ \vec{k}\times\vec{E}\
\big{)}\ \right]+\omega^{2}\ \matr{\varepsilon}\cdot\vec{E}\ =\ 0.$ (37)
Concerning the special case of impedance matched material
$\matr{\varepsilon}/\varepsilon_{0}=\matr{\mu}/\mu_{0}=\matr{n}$, the
dispersion relation, which are also called wave-vector eikonal equation, is
(derived in detail in Appendix C)Schurig _et al._ (2006)
$\bar{n}^{ij}k_{i}k_{j}-\gamma\left(\frac{\omega}{c}\right)^{2}\det\left(\bar{n}^{ij}\right)\
=\
\bar{n}_{ij}k^{i}k^{j}-\frac{1}{\gamma}\left(\frac{\omega}{c}\right)^{2}\det\left(\bar{n}_{ij}\right)\
=\ 0.$ (38)
Here, there are two ambiguities to be clarify. First, the expression given in
Schurig _et al._ (2006) is based on Cartesian coordinates, so $\gamma$
reduces to unit, while the expression here is valid in any curved coordinates.
Second, $\bar{n}^{ij}$ and $\bar{n}_{ij}$ presented above express the
components in holonomic coordinate bases (Landau’s definition), however, the
dispersion relations expressed either in Minkowski’s definition or in
anholonomic unit basis are a little different from Eq. (38) (see Appendix C).
In Schurig _et al._ (2006), the three kinds of expression are congruent with
each other in Cartesian coordinates.
Using the duality principle, we can easily obtain the dual form of wave
equation
$\vec{s}\times\left[\ \matr{\mu}\cdot\big{(}\ \vec{s}\times\vec{D}\ \big{)}\
\right]+\frac{1}{\omega^{2}}\ \matr{\varepsilon}^{-1}\cdot\vec{D}\ =\ 0.$ (39)
Thus, the “ray-vector eikonal equation” is
${(\bar{n}^{\scriptscriptstyle-1})}^{ij}s_{i}s_{j}-\gamma\left(\frac{c}{\omega}\right)^{2}\det\left[(\bar{n}^{\scriptscriptstyle-1})^{ij}\right]\
=\
{(\bar{n}^{\scriptscriptstyle-1})}_{ij}s^{i}s^{j}-\frac{1}{\gamma}\left(\frac{c}{\omega}\right)^{2}\det\left[(\bar{n}^{\scriptscriptstyle-1})_{ij}\right]\
=\ 0,$ (40)
where $(\bar{n}^{\scriptscriptstyle-1})^{ij}$ is the inverse matrix of
$\bar{n}_{ij}$, and it is easy to verify that its corresponding covariant
components $(\bar{n}^{\scriptscriptstyle-1})_{ij}$ are also the inverse matrix
of $\bar{n}^{ij}$.
### IV.1 Eikonal equation in curved coordinate system S
We first consider the relation between $k^{i}$ and $s^{i}$ in curved
coordinate system S. In S system, the relative constitutive parameter is
$\bar{n}^{ij}=(\bar{n}^{\scriptscriptstyle-1})^{ij}=\gamma^{ij}$. Substituting
it into Eq. (38) and Eq. (40), we get the wave-vector eikonal equation in S
system
$\gamma^{ij}k_{i}k_{j}-\left(\frac{\omega}{c}\right)^{2}\ =\
\gamma_{ij}k^{i}k^{j}-\left(\frac{\omega}{c}\right)^{2}\ =\ 0,$ (41)
and the ray-vector eikonal equation
$\gamma^{ij}s_{i}s_{j}-\left(\frac{c}{\omega}\right)^{2}\ =\
\gamma_{ij}s^{i}s^{j}-\left(\frac{c}{\omega}\right)^{2}\ =\ 0.$ (42)
The two equations indicate that $|\vec{k}|=k_{0}$ and $|\vec{s}|=1/k_{0}$. By
considering the general relation $\vec{k}\cdot\vec{s}=\gamma_{ij}k^{i}s^{j}=1$
Chen (1985), we can conclude that the directions of $\vec{k}$ and $\vec{s}$
completely coincide, and they have the relation $k_{i}={k_{0}}^{2}s_{i}$. This
result is actually not marvelous, since the electromagnetic wave is merely
propagating in a vacuous space, despite the space is expressed in a
curvilinear coordinate system with non-unit metric $\gamma_{ij}$.
### IV.2 Eikonal equation in physical space P
In physical space, the media of the cloak are inhomogeneous and waves
traveling in it are no longer plane waves. Nevertheless, we can verify the
fields obtained both in the spherical and cylindrical cloak still satisfy Eqs.
(36, 37, 39). The relative constitutive parameter of the cloak is
${\bar{n}_{\scriptscriptstyle(\mathrm{P})}}^{ij}=\sqrt{\gamma/\gamma_{\scriptscriptstyle(\mathrm{P})}}\
\gamma^{ij}$, and its inverse is
$({\bar{n}^{\scriptscriptstyle-1}_{\scriptscriptstyle(\mathrm{P})}})_{ij}=\sqrt{\gamma_{\scriptscriptstyle(\mathrm{P})}/\gamma}\
\gamma_{ij}$. Substituting them into Eq. (38) and Eq. (40), we obtain the
wave-vector eikonal equation in P space
$\gamma^{ij}{k_{\scriptscriptstyle(\mathrm{P})}}_{i}{k_{\scriptscriptstyle(\mathrm{P})}}_{j}-\left(\frac{\omega}{c}\right)^{2}\
=\
\big{(}\gamma_{{\scriptscriptstyle(\mathrm{P})}ik}\gamma_{{\scriptscriptstyle(\mathrm{P})}il}\gamma^{kl}\big{)}{k_{\scriptscriptstyle(\mathrm{P})}}^{i}{k_{\scriptscriptstyle(\mathrm{P})}}^{j}-\left(\frac{\omega}{c}\right)^{2}\
=\ 0,$ (43)
and the ray-vector eikonal equation in P space
$\gamma_{ij}{s_{\scriptscriptstyle(\mathrm{P})}}^{i}{s_{\scriptscriptstyle(\mathrm{P})}}^{j}-\left(\frac{c}{\omega}\right)^{2}\
=\
\big{(}{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{ik}{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{jl}\gamma_{kl}\big{)}{s_{\scriptscriptstyle(\mathrm{P})}}_{i}{s_{\scriptscriptstyle(\mathrm{P})}}_{j}-\left(\frac{c}{\omega}\right)^{2}\
=\ 0.$ (44)
Comparison of Eqs. (43, 44) and Eqs. (41, 42) manifests that the wave-vector
eikonal equations expressed by covariant components $k_{i}$ have the same form
in P space and S system. However if we write the wave-vector eikonal equations
with contravariant components $k^{i}$, the expressions in P and in S are
different. Similarly, the expressions with contravariant components $s^{i}$
have the same form in the two systems, yet the expressions with covariant
components $s_{i}$ are different. Considering the relation
$\vec{k}_{\scriptscriptstyle(\mathrm{P})}\cdot\vec{s}_{\scriptscriptstyle(\mathrm{P})}=1$
simultaneously, we can get
${k_{\scriptscriptstyle(\mathrm{P})}}_{i}\ \ =\ \
{k_{0}}^{2}\;\gamma_{ij}{s_{\scriptscriptstyle(\mathrm{P})}}^{i},\qquad{s_{\scriptscriptstyle(\mathrm{P})}}^{i}\
\ =\ \ {k_{0}}^{-2}\;\gamma^{ij}{k_{\scriptscriptstyle(\mathrm{P})}}_{i}.$
(45)
This relation is easily validated according to Eq. (17) and (19). It is also
valid in S system which means $\vec{k}$ is parallel to $\vec{s}$, however this
does not make sense in invisibility cloak.
These results also can be verified with canonical equations given by Schurig
_et al._ (2006). If we define Hamiltonian by wave-vector eikonal:
$H\ =\
\gamma^{ij}{k_{\scriptscriptstyle(\mathrm{P})}}_{i}{k_{\scriptscriptstyle(\mathrm{P})}}_{j}-\left(\frac{\omega}{c}\right)^{2},$
(46)
and according to the first equation of canonical equations
$\frac{\mathrm{d}x_{\scriptscriptstyle(\mathrm{P})}^{i}}{\mathrm{d}\tau}\ =\
\frac{\partial H}{\partial
k_{{\scriptscriptstyle(\mathrm{P})}i}},\qquad\frac{\mathrm{d}k_{{\scriptscriptstyle(\mathrm{P})}i}}{\mathrm{d}\tau}\
=\ -\frac{\partial H}{\partial x_{\scriptscriptstyle(\mathrm{P})}^{i}},$ (47)
where $\tau$ is the parameter of canonical equations, we obtain the tangent
vector of light-ray
$\mathrm{d}x_{\scriptscriptstyle(\mathrm{P})}^{i}/{\mathrm{d}\tau}=2\gamma^{ij}k_{{\scriptscriptstyle(\mathrm{P})}j}.$
It reveals the tangent of light-ray is exactly towards the direction of
${s_{\scriptscriptstyle(\mathrm{P})}}^{i}$ given in Eq. (45). If we define
another Hamiltonian by ray-vector eikonal:
$\tilde{H}\ =\
\big{(}{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{ik}{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{jl}\gamma_{kl}\big{)}{s_{\scriptscriptstyle(\mathrm{P})}}_{i}{s_{\scriptscriptstyle(\mathrm{P})}}_{j}-\left(\frac{c}{\omega}\right)^{2},$
(48)
and modify the canonical equations into
$\frac{\mathrm{d}x_{\scriptscriptstyle(\mathrm{P})}^{i}}{\mathrm{d}\tau}\ =\
\frac{\partial\tilde{H}}{\partial
s_{{\scriptscriptstyle(\mathrm{P})}i}},\qquad\frac{\mathrm{d}s_{{\scriptscriptstyle(\mathrm{P})}i}}{\mathrm{d}\tau}\
=\ -\frac{\partial\tilde{H}}{\partial
x_{\scriptscriptstyle(\mathrm{P})}^{i}},$ (49)
we would have
$\frac{\mathrm{d}x_{\scriptscriptstyle(\mathrm{P})}^{i}}{\mathrm{d}\tau}\ =\
2{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{ik}{\gamma_{{\scriptscriptstyle(\mathrm{P})}}}^{jl}\gamma_{kl}{s_{\scriptscriptstyle(\mathrm{P})}}_{j}\
=\ 2{k_{0}}^{-2}k_{\scriptscriptstyle(\mathrm{P})}^{i}.$ (50)
As a result, the modified canonical equations depict the trace of wave-normal
rays.
Actually, if we express $k_{i}$ and $s^{i}$ with $D^{i},\ B^{i},\ E_{i},\
H_{i}$ in Minkowski’s definition, we have
$k_{i}\ =\ \omega\frac{e_{ijk}D^{j}B^{k}}{E_{i}D^{i}},\qquad s^{i}\ =\
\frac{1}{\omega}\frac{e^{ijk}D_{j}B_{k}}{E_{i}D^{i}}.$ (51)
The expressions apply to both S system and P space. According to the relations
in Table 1, we can get two new corresponding relations between S and P:
$k_{i}\ \Leftrightarrow\ k_{{\scriptscriptstyle(\mathrm{P})}i},\qquad s^{i}\
\Leftrightarrow\ s_{{\scriptscriptstyle(\mathrm{P})}}^{i}.$
We note that $k^{i}$ and $k_{{\scriptscriptstyle(\mathrm{P})}}^{i}$ do not
correspond to each other, and neither do $s_{i}$ and
$s_{{\scriptscriptstyle(\mathrm{P})}i}$. In other words, covariant components
of wave vector $k_{i}$ in S system map to wave vector
$k_{{\scriptscriptstyle(\mathrm{P})}i}$ in P space, yet contravariant
components of wave vector $k^{i}\ (={k_{0}}^{2}s^{i})$ map to ray vector
$s_{{\scriptscriptstyle(\mathrm{P})}}^{i}$ in P space. Although $k_{i},\
s^{i}$ denote a same orientation in S system,
$k_{{\scriptscriptstyle(\mathrm{P})}i},and\
s_{{\scriptscriptstyle(\mathrm{P})}}^{i}$ denote two different orientations in
P space. In addition, the contravariant components
$s^{i}=\mathrm{d}x^{i}/\mathrm{d}\lambda$ directly represent the directional
change of coordinates $\mathrm{d}x^{i}$, therefore light-ray equation in P
space inherits itself’s expression in S. By contrast, $k_{i}$ can not directly
evince the change of $\mathrm{d}x^{i}$, so the expression of wave-normal ray
takes a new form in P space.
According to the dispersion relation (43), we can obtain the group velocity in
the cloak through applying the formula
$\vec{v}_{g}=\nabla_{\scriptscriptstyle\vec{k}}\omega$, where
$\nabla_{\scriptscriptstyle\vec{k}}$ is the divergence operator in
$\vec{k}$–space. Supposing the medium is non-dispersive, the group velocity
takes the identical expression of ray velocity (16). It means the group
velocity and ray velocity are congruent in the cloak.
As commented in section 3.1, the contradiction of super velocity of light
takes into account the existence of dispersion. However, the dispersive
processes of $\varepsilon$ and $\mu$ are different, so the case of impedance
matching established in a particular frequency would hardly retain when
frequency changes. Therefore, the dispersion relation of impedance matched
material (38) is unavailable when we use
$\vec{v}_{g}=\nabla_{\scriptscriptstyle\vec{k}}\omega$ to calculate the group
velocity in dispersive case, because Eq. (38) can not display the differences
between the derivative of $\varepsilon$ and $\mu$ with respect to $\omega$.
## V Design of the invisibility cloak with rays satisfying harmonic function
In Ref. Leonhardt (2006), Leonhardt applied Zhukovskiæ function
$\zeta(z)=z+1/z$, where $z=r\ \mathrm{e}^{\mathrm{i}\theta}$, to a conformal
mapping, and obtained an inhomogeneous but isotropic medium which can realize
an invisibility device in geometric limit. In the media, the families of
light-rays and of wave-fronts are the imaginary and real part of $\zeta(z)$
respectively. In addition, they are both harmonic functions and orthogonal to
each other.
Using the expression of light-ray (27), we can design an invisibility cloak by
means of transformation-optics method, which gives the same light trajectories
as in Leonhardt’s device. However, we start our work with a more general
target to find the general form of transformation function $f(r)$ which causes
the light-rays in cylindrical cloak to satisfy harmonic function. Substitution
of light-ray (27) into Laplace’s equation yields
$\nabla_{\scriptscriptstyle 2-D}\ \left[\ f(r)\sin\theta\ \right]\ =\ 0.$ (52)
Then, it reduces to Euler’s differential equation,
$r^{2}f^{\prime\prime}(r)+rf^{\prime}(r)-f(r)=0$, therefore the general
solution is $f(r)=C_{1}(r^{2}-C_{2})/r$. The function has a zero point
$r={C_{2}}^{\scriptscriptstyle 1/2}>0$, which is the most important condition
to construct an invisibility cloak. If we set $C_{1}=b^{2}/(b^{2}-a^{2})$ and
$C_{2}=a^{2}$, the transformation function becomes
$f(r)\ =\ \frac{b^{2}}{b^{2}-a^{2}}\;\frac{r^{2}-a^{2}}{r}.$ (53)
Figure 5: Cloak with rays satisfying harmonic equation. (a) Blue lines denote
light-rays, orange lines denote the surfaces orthogonal to light-rays. In
cloak layer, they are painted by Eq. (54). (b) Green lines denote wave-normal
rays, red lines denote wave-fronts. In cloak layer, they are painted by Eq.
(55).
The result evinces that each solution corresponds to an invisibility cloak
with a spacial pair of inner and outer radii. In the process of derivation, we
didn’t command $f(a)=0,f(b)=b$, however the result indicates that the form of
harmonic functions $V(r,\theta)=f(r)\sin\theta$ is unique, and it always can
realize invisibility cloak. For convenience, we omit the coefficient of $f(r)$
in the following derivation which doesn’t influence the shape of rays.
$V=\mbox{const}$ represents the family of light-rays in the cloak. Let $V$ be
the imaginary part of an analytical function $F(z)$. Using Cauchy-Riemann
condition, we obtain the real part
$U(r,\theta)=[(r^{2}+a^{2})/r]\cos\theta$.The $U=\mbox{const}$ stands for a
family of curved surfaces, which is orthogonal to the energy flows. The
analytical function $F(z)$ is
$F(z)\ =\ z\ +\ \frac{a^{2}}{z}.$ (54)
If we simply set $z^{\prime}=z/a$, then $F(z)=a\zeta(z^{\prime})$, where
$\zeta(z)=z+1/z$ is precisely Zhukovskiæ function. It is exactly the conformal
mapping which Leonhardt used to design the invisibility devices Leonhardt
(2006). The complex analytical function is a compact way to describe the ray
path by the meaning of real and imaginary part explained above, as is shown in
Fig. 5(a).
Regarding to the transformation (53), the eikonal $\psi=k_{0}f(r)\cos\theta$
is a harmonic function as well. So we also can compress the wave-fronts and
wave-normal rays into another analytical function
$G(z)\ =\ z\ -\ \frac{a^{2}}{z}.$ (55)
The real part represents the family of wave fronts, and the imaginary part
represents the family of wave-normal rays exhibited in Fig. 5(b).
Owing to the energy conservation, the time-averaged Poynting vector
$\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle$ is
divergence-free. And the wave vector must be irrotational, since
$\vec{k}_{\scriptscriptstyle(\mathrm{P})}=\nabla\psi$. However, if and only if
the transformation satisfies Eq. (53), we can prove both
$\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle$ and
$\vec{k}_{\scriptscriptstyle(\mathrm{P})}$ are not only divergence-free but
also irrotational. This conclusion is consistent to the fact that both light-
rays and wave-normal rays satisfy Laplace’s equation for the particular
transformation function.
## VI Conclusion
In conclusion, we have meticulously compared the behaviors of electromagnetic
fields in P space with in S system of virtual space, and provided the
corresponding relations with electromagnetic quantities and boundary
conditions between the two systems. Then, we obtain the analytical expression
of light-rays and wave-normal rays, and give their physical interpretation.
Furthermore, we verify the expression of light-rays we obtained is exactly a
particular solution of geodesic equation in S system. In addition, using
duality principle, we show that the covariant and contravariant components of
wave vector in S system map to wave vector and ray vector respectively in P
space, and it causes the split of $k_{{\scriptscriptstyle(\mathrm{P})}i}$ and
$s_{{\scriptscriptstyle(\mathrm{P})}}^{i}$ in P space. Finally, we find out
the general form of transformation function which turns the light-ray to
harmonic, and therefore we can construct a special cloak in which
$\langle\vec{S}^{\mathrm{c}}_{\scriptscriptstyle(\mathrm{P})}\rangle$ and
$\vec{k}_{\scriptscriptstyle(\mathrm{P})}$ are both divergence-free and curl-
free.
## Acknowledgment
We thank K. Niu for the help with composing the article. Meanwhile, we also
thank C. Song, Y.-B. Hu, L. Fang, J.-N. Zhang, and J. Dai for the helpful
discussion. This work was supported in part by NSF of China (Grants No.
11075077), and by SRFDP (Grants No. 200800550015).
## Appendix A: Different way to define electromagnetic fields in curved
spacetime
In Minkowski spacetime, 3-D macroscopic Maxwell’s equations in a Lorentz frame
are
$\partial_{i}B^{i}\ =\ 0,\qquad\quad\frac{\partial B^{i}}{\partial
t}+e^{ijk}\partial_{j}E_{k}\ =\ 0,$ (A.1a) $\partial_{i}D^{i}\ =\
\rho,\qquad-\frac{\partial D^{i}}{\partial t}+e^{ijk}\partial_{j}H_{k}\ =\ 0,$
(A.1b)
and their 4-D covariant forms are
$\partial_{\lambda}F_{\mu\nu}+\partial_{\nu}F_{\lambda\mu}+\partial_{\mu}F_{\nu\lambda}\
=\ 0,$ (A.2a) $\partial_{\alpha}G^{\alpha\beta}\ =\ J^{\beta}.$ (A.2b)
The four 3-D electric and magnetic vectors, $E_{i}$, $H_{i}$, $D^{i}$ and
$B^{i}$ are defined as
$E_{i}\ =\ F_{0i},\quad B^{i}\ =\ -\frac{1}{2c}e^{ijk}F_{jk},\quad D^{i}\ =\
-\frac{1}{c}G^{0i},\quad H_{i}\ =\ -\frac{1}{2}e_{ijk}G^{jk}.\\\ $ (A.3)
Here we use Greek indices $\alpha,\beta,\cdots$ to run from 0 to 3 for time
and spatial parts, and stipulate the sign of metric takes the form $(\,-,\ +,\
+,\ +\,)$. For vacuum, $G^{\alpha\beta}\ =\ c\varepsilon_{0}F^{\alpha\beta}.$
In curved space time, 4-D microscopic Maxwell equations which have the same
form as the macroscopic Maxwell’s equations in vacuum becomeMisner _et al._
(1973); Landau and Lifshitz (1995)
$\partial_{\lambda}F_{\mu\nu}+\partial_{\nu}F_{\lambda\mu}+\partial_{\mu}F_{\nu\lambda}\
=\ 0,$ (A.4a) $c\varepsilon_{0}\nabla_{\alpha}F^{\alpha\beta}\ =\
\frac{c\varepsilon_{0}}{\sqrt{-g}}\partial_{\alpha}(\sqrt{-g}F^{\alpha\beta})\
=\ \ J^{\beta},$ (A.4b)
where $\nabla_{\alpha}$ is 4-D covariant derivative, $g$ is the determinant of
the spacetime metric $g_{\mu\nu}$.
If the excitation tensor $G^{\alpha\beta}$ is defined in the form Schurig _et
al._ (2006); Leonhardt and Philbin (2009); Post (1962); Plebanski (1960)
$G^{\alpha\beta}\ =\ c\varepsilon_{0}\sqrt{-g}F^{\alpha\beta},$ (A.5)
and the definitions of 3-D electric and magnetic vectors are retained for flat
Minkowski spacetime as in Eqs. (A.3), both 3-D and 4-D Maxwell’s equations
keep the form of flat spacetime in Eqs. (A.1) and Eqs. (A.2), except the
4-current $J^{\beta}$, charge density $\rho$, and 3-current $j^{i}$ should be
replaced with the reduced forms $\hat{J}^{\beta}=\sqrt{-g}J^{\beta},\
\hat{\rho}=\sqrt{\gamma}\rho,\ \hat{j}^{i}=\sqrt{\gamma}j^{i}$ respectively
Leonhardt and Philbin (2009), where $\gamma$ denotes the determinate of
spacial metric $\gamma_{ij}\ =\ g_{ij}-g_{0i}g_{0j}/g_{00}$. We name this set
of excitation tensor and 3-D electric and magnetic vectors as Minkowski’s
definition.
The constitutive relations in Minkowski’s definition are Leonhardt and Philbin
(2006, 2009); Plebanski (1960)
$D^{i}\ =\
\varepsilon_{0}\sqrt{\frac{\gamma}{-g_{00}}}\gamma^{ij}E_{j}+\frac{1}{cg_{00}}e^{ijk}g_{0j}H_{k},$
(A.6a) $B^{i}\ =\
\mu_{0}\sqrt{\frac{\gamma}{-g_{00}}}\gamma^{ij}H^{j}-\frac{1}{cg_{00}}e^{ijk}g_{0j}E_{k}.,$
(A.6b)
Thus, the permittivity and permeability satisfy
$\varepsilon^{ij}=\varepsilon_{0}\sqrt{-\gamma/g_{00}}\gamma^{ij},\quad\mu^{ij}=\mu_{0}\sqrt{-\gamma/g_{00}}\gamma^{ij}$,
and magnetoelectric coupling tensor is $\kappa^{ij}=e^{ijk}g_{0j}/(cg_{00})$.
If $g_{\mu\nu}$ satisfies $g_{00}=-1,g_{0i}=0$, then magnetoelectric coupling
vanishes, while $\varepsilon^{ij}$ and $\mu^{ij}$ reduce to Eqs. (4).
Another definition of 3-D electric and magnetic vectors appears in Landau’s
book Landau and Lifshitz (1995), therefore, we call this set as Landau’s
definitions. In Landau’s definition, we put a bar on each quantity, and the
excitation tensor retains its formula in flat spacetime
$\bar{G}^{\alpha\beta}\ =\ c\varepsilon_{0}F^{\alpha\beta}$, and the four
electric and magnetic vectors are redefined as Landau and Lifshitz (1995);
Bergamin (2008)
$\bar{E}_{i}\ =\ F_{0i},\quad\bar{B}^{i}\ =\
-\frac{1}{2c\sqrt{\gamma}}e^{ijk}F_{jk},\quad\bar{D}^{i}\ =\
-\frac{\sqrt{-g_{00}}}{c}\bar{G}^{0i},\quad\bar{H}_{i}\ =\
-\frac{\sqrt{-g}}{2}e_{ijk}\bar{G}^{jk}.\\\ $ (A.7)
Under this set of definitions, 4-D Maxwell’s equations have the form of
$\partial_{\lambda}F_{\mu\nu}+\partial_{\nu}F_{\lambda\mu}+\partial_{\mu}F_{\nu\lambda}\
=\ 0,$ (A.8a) $\nabla_{\alpha}\bar{G}^{\alpha\beta}\ =\ J^{\beta}.$ (A.8b)
The corresponding 3-D Maxwell’s equations are
$\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}\bar{B}^{i})\ =\
0,\qquad\quad\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial
t}(\sqrt{\gamma}\bar{B}^{i})+\frac{1}{\sqrt{\gamma}}e^{ijk}\partial_{j}\bar{E}_{k}\
=\ 0,\\\ $ (A.9a) $\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial
x^{i}}(\sqrt{\gamma}\bar{D}^{i})\ =\
\rho,\qquad-\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial
t}(\sqrt{\gamma}\bar{D}^{i})+\frac{1}{\sqrt{\gamma}}e^{ijk}\partial_{j}\bar{H}_{k}\
=\ J^{i}.$ (A.9b)
Comparing Eqs. (A.3) with Eqs. (A.7), we have
$G^{\alpha\beta}=\sqrt{-g}\bar{G}^{\alpha\beta}$, and the transformations of
other quantities between the two definitions are identical with Eqs. (2).
Assume that an observer whose 4-velocity is $u^{\alpha}$ in a curved spacetime
endowed with an electromagnetic field $F_{\mu\nu}$. Then, questions arouse
naturally. What is the measurement of electric and magnetic fields by the
observer? Is the measurement equal to one of the two definitions mentioned
above? Actually, there has been a mature way to answer this question. In the
curved spacetime, electric and magnetic field vectors are defined respected
with the frame $u^{\alpha}$ as Felice and Clarke (1990)
$\tilde{E}_{\alpha}\ =\ -F_{\alpha\beta}u^{\beta},\qquad\tilde{H}_{\alpha}\ =\
^{\ast}\bar{G}_{\alpha\beta}u^{\beta}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}\bar{G}^{\mu\nu}u^{\beta},$
(A.10)
where
${}^{\ast}\bar{G}_{\alpha\beta}=\epsilon_{\alpha\beta\mu\nu}\bar{G}^{\mu\nu}/2$
is the dual of $\bar{G}^{\mu\nu}$. Here, we put a tilde on electric and
magnetic vectors to discriminate its representation under Minkowski’s
definition and Landau’s definition, however, the definition of excitation
tensor retains its formula from Landau’s definition $\bar{G}^{\alpha\beta}\ =\
c\varepsilon_{0}F^{\alpha\beta}$, which guarantees that $\bar{G}$ is a tensor
of 4-dimension.
In terms of the definitions of Eqs. (A.10),
$\tilde{E}_{\alpha}u^{\alpha}=\tilde{H}_{\alpha}u^{\alpha}=0$, which evinces
that $\tilde{E}$ and $\tilde{H}$ are both space-like vectors for the observer
$u^{\alpha}$. In addition, if we simply let electric displacement
$\tilde{D}^{\alpha}=\varepsilon_{0}\tilde{E}^{\alpha}$ and magnetic induction
$\tilde{B}^{\alpha}=\mu_{0}\tilde{H}^{\alpha}$, i.e. what are made in vacuum
of flat spacetime, the components of $\tilde{E}$, $\tilde{H}$, $\tilde{D}$,
and $\tilde{B}$ in the tetrad
$e_{\scriptscriptstyle{\langle}\scriptstyle{\mu}\scriptscriptstyle{\rangle}}^{\
\alpha}$ of the observer, where
$e_{\scriptscriptstyle{\langle}\scriptstyle{0}\scriptscriptstyle{\rangle}}^{\
\alpha}=u^{\alpha}$, are
$\tilde{E}_{\langle i\rangle}\ =\ F_{\langle 0i\rangle},\ \ \tilde{H}_{\langle
i\rangle}\ =\ -\frac{1}{2}e_{ijk}\bar{G}^{\langle jk\rangle},\ \
\tilde{D}^{\langle i\rangle}\ =\ -\frac{1}{c}\bar{G}_{\langle 0i\rangle},\ \
\tilde{B}^{\langle i\rangle}\ =\ -\frac{c}{2}e^{ijk}F_{\langle jk\rangle}.$
(A.11)
As we see, the components in tetrad satisfy the original relation of flat
spacetime (A.3). Furthermore, the electromagnetic tensors admit the following
decomposition Felice and Clarke (1990); Sonego and Abramowicz (1998); Tsagas
(2005):
$F_{\mu\nu}\ =\
(\tilde{E}_{\mu}u_{\nu}-\tilde{E}_{\nu}u_{\mu})-c\epsilon_{\mu\nu\alpha\beta}u^{\alpha}\tilde{B}^{\beta},$
(A.12a) $\bar{G}^{\mu\nu}\ =\
c(\tilde{D}^{\mu}u^{\nu}-\tilde{D}^{\nu}u^{\mu})-\epsilon^{\mu\nu\alpha\beta}u_{\alpha}\tilde{H}_{\beta},$
(A.12b)
and a further study reveals the form of Maxwell’s equations expressed with
$\tilde{E}$ and $\tilde{H}$ Sonego and Abramowicz (1998); Tsagas (2005).
Here, we are interested in the condition that the observer is comoving. For a
comoving observer, $u^{\alpha}=(1/\sqrt{-g_{00}},\ 0,\ 0,\ 0)$, and the
transverse projector $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ reduces to the
spatial metric $\gamma_{ij}$ mentioned earlier. In this case, the relations
(A.12) lead to the result
$\left\\{\begin{array}[]{rcl}F_{0i}&=&\sqrt{-g_{00}}\tilde{E}_{i},\\\\[4.0pt]
F_{jk}&=&-c\sqrt{\gamma}e_{ijk}\tilde{B}^{i}+\frac{2}{\sqrt{-g_{00}}}\tilde{E}_{[j}g_{k]0},\\\\[5.0pt]
\bar{G}^{0i}&=&-\frac{c}{\sqrt{-g_{00}}}\tilde{D}^{i}+\frac{1}{\sqrt{g_{00}g}}e^{ijk}g_{j0}\tilde{H}_{k},\\\\[6.0pt]
\bar{G}^{jk}&=&-\frac{1}{\sqrt{\gamma}}e^{ijk}\tilde{H}_{i}.\\\\[5.0pt]
\end{array}\right.$ (A.13)
It seems that the measurement of electric and magnetic fields by an observer
conforms neither to Minkowski’s definition nor to Landau’s definition.
However, if we set $g_{00}=-1,g_{0i}=0$, the reduced relations Eqs. (A.13) are
identical with the reduced relations Eqs. (A.7) under Landau’s definition. So,
if we use artificial materials to simulate the behavior of electromagnetic
fields in curved spacetime based on the form equivalence of Maxwell’s
equations, it should be heeded that the distinction between the electric and
magnetic fields measured in materials and measured by the observer moving in
real curved spacetime.
## Appendix B: Electromagnetic fields in Cylindrical cloak
In this portion, we would only consider TE polarized wave. Substituting the
constitutive parameters Eq. (8)into wave equation (37), we get the general
scalar wave equation for all transformation $f(r)$ which is exactly the
Helmholtz equation in original S′ coordinate system of virtual space:
$\frac{1}{f}\,\frac{\partial}{\partial
f}\left(f\frac{\partial{E_{\scriptscriptstyle(\mathrm{P})}}_{z}}{\partial
f}\right)\ +\
\frac{1}{f^{2}}\,\frac{\partial^{2}{E_{\scriptscriptstyle(\mathrm{P})}}_{z}}{\partial\theta^{2}}\
+\ k^{2}\,{E_{\scriptscriptstyle(\mathrm{P})}}_{z}\ =\ 0.$ (B.1)
Through separating variables, electromagnetic fields can be represented by
series
$\displaystyle\vec{E}_{\scriptscriptstyle(\mathrm{P})}\ =\ $ $\displaystyle\
\sum_{-\infty}^{\infty}\ \left[\ a_{n}\,J_{n}(x)\ +\ b_{n}\,N_{n}(x)\
\right]\;\mathrm{e}^{\mathrm{i}n\theta}\ \hat{e}_{z},$ (B.2)
where $J_{n}(x)$ and $N_{n}(x)$ are n-order Bessel function and n-order
Neumann function respectively. These expressions are available in every area,
however $x$ takes different forms in each area. Outside the cloak ($r>b$),
$x=k_{0}r$ is in the expressions of incident wave
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{in}}$ and scattering wave
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{sc}}$. In the cloak layer
($a<r<b$), $\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{c}}$ takes
$x=k_{0}f(r)$. Inside the internal hidden area ($r<a$), $x=k_{1}r$ is in the
expression of $\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{int}}$, where
$k_{1}=\omega\sqrt{\varepsilon_{1}\mu_{1}}$ and $\varepsilon_{1},\ \mu_{1}$
are the material properties of the hidden medium. The incident wave
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{in}}=E_{0}\mathrm{e}^{\mathrm{i}k_{0}r\cos\theta}\hat{e}_{z}$
can be expanded as
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{in}}\ =\
E_{0}\sum_{-\infty}^{\infty}\
\mathrm{i}^{n}\;J_{n}(k_{0}r)\;\mathrm{e}^{\mathrm{i}n\theta}.$ (B.3)
Because $\vec{E}^{\mathrm{sc}}$ should tend to an outgoing wave as $r$
approaches to infinity, it is more convenient to expand scattering wave with
Hankel function of first kind:
$\vec{E}_{\scriptscriptstyle(\mathrm{P})}^{\mathrm{sc}}\ =\
\sum_{-\infty}^{\infty}\
a_{n}^{\mathrm{sc}}H_{n}^{\scriptscriptstyle(1)}(k_{0}r)\,\mathrm{e}^{\mathrm{i}n\theta}.$
(B.4)
In addition, $b_{n}^{\mathrm{int}}=0$ means that fields should be finite when
$r\rightarrow 0$. Then applying the continuity of $E_{z}$ and $H_{\theta}$ at
inner and outer surface of cloak, we have following equations:
$\displaystyle a_{n}^{\mathrm{int}}\,J_{n}(k_{1}a)\ $ $\displaystyle=\
a_{n}^{\mathrm{c}}\,J_{n}\big{(}k_{0}f(a)\big{)}\ +\
b_{n}^{\mathrm{c}}\,N_{n}\big{(}k_{0}f(a)\big{)},$ (B.5a) $\displaystyle
a_{n}^{\mathrm{int}}\,\frac{\mu_{0}}{\mu_{1}}k_{1}J_{n}(k_{1}a)\ $
$\displaystyle=\
\frac{k_{0}f(a)}{a}\left[a_{n}^{\mathrm{c}}\,J_{n}^{\prime}\big{(}k_{0}f(a)\big{)}\
+\ b_{n}^{\mathrm{c}}\,N_{n}^{\prime}\big{(}k_{0}f(a)\big{)}\right],$ (B.5b)
$\displaystyle E_{0}\mathrm{i}^{n}\,J_{n}(k_{0}b)\ +\
a_{n}^{\mathrm{sc}}\,H_{n}^{\scriptscriptstyle(1)}(k_{0}b)\ $ $\displaystyle=\
a_{n}^{\mathrm{c}}\,J_{n}\big{(}k_{0}f(b)\big{)}\ +\
b_{n}^{\mathrm{c}}\,N_{n}\big{(}k_{0}f(b)\big{)},$ (B.5c) $\displaystyle
E_{0}\mathrm{i}^{n}\,J_{n}^{\prime}(k_{0}b)\ +\
a_{n}^{\mathrm{sc}}\,H_{n}^{\scriptscriptstyle(1)\prime}(k_{0}b)\ $
$\displaystyle=\
\frac{f(b)}{b}\left[a_{n}^{\mathrm{c}}\,J_{n}^{\prime}\big{(}k_{0}f(b)\big{)}\
-\ b_{n}^{\mathrm{c}}\,N_{n}^{\prime}\big{(}k_{0}f(b)\big{)}\right],$ (B.5d)
Solving the group of equations, we obtain
$\displaystyle\begin{split}a_{n}^{\mathrm{c}}\ =\
&E_{0}\,\mathrm{i}^{n}\,\left[J_{n}(k_{0}b)H_{n}^{\scriptscriptstyle(1)\prime}(k_{0}b)\
-\
J_{n}^{\prime}(k_{0}b)H_{n}^{\scriptscriptstyle(1)}(k_{0}b)\right]/\Big{\\{}H_{n}^{\scriptscriptstyle(1)\prime}(k_{0}b)\left[J_{n}\big{(}k_{0}f(b)\big{)}\right.\\\
&\left.\left.+A_{n}N_{n}\big{(}k_{0}f(b)\big{)}\right]-\frac{f(b)}{b}H_{n}^{\scriptscriptstyle(1)}(k_{0}b)\left[J_{n}^{\prime}\big{(}k_{0}f(b)\big{)}+A_{n}N_{n}^{\prime}\big{(}k_{0}f(b)\big{)}\right]\right\\},\end{split}$
(B.6a) $\displaystyle b_{n}^{\mathrm{c}}\ =\ $
$\displaystyle\frac{\mu_{0}ak_{1}J_{n}\big{(}k_{0}f(a)\big{)}J_{n}^{\prime}(k_{1}a)\
-\
\mu_{1}k_{0}f(a)J_{n}^{\prime}\big{(}k_{0}f(a)\big{)}J_{n}(k_{1}a)}{\mu_{1}k_{0}f(a)N_{n}^{\prime}\big{(}k_{0}f(a)\big{)}J_{n}(k_{1}a)-\mu_{0}ak_{1}N_{n}\big{(}k_{0}f(a)\big{)}J_{n}^{\prime}(k_{1}a)}\
a_{n}^{\mathrm{c}}\ =\ A_{n}\;a_{n}^{\mathrm{c}},$ (B.6b) $\displaystyle
a_{n}^{\mathrm{int}}\ =\ $
$\displaystyle\frac{1}{J_{n}(k_{1}a)}\;\left[\,J_{n}\big{(}k_{0}f(a)\big{)}\
+\ A_{n}\,N_{n}\big{(}k_{0}f(a)\big{)}\,\right]\ a_{n}^{\mathrm{c}},$ (B.6c)
$\displaystyle a_{n}^{\mathrm{sc}}\ =\ $
$\displaystyle\frac{1}{H_{n}^{\scriptscriptstyle(1)\prime}(k_{0}b)}\left\\{\frac{f(b)}{b}\left[\,J_{n}^{\prime}\big{(}k_{0}f(b)\big{)}\
-\ A_{n}\,N_{n}^{\prime}\big{(}k_{0}f(b)\big{)}\right]a_{n}^{\mathrm{c}}\
-E_{0}\mathrm{i}^{n}\,J_{n}^{\prime}(k_{0}b)\right\\}.$ (B.6d)
Substituting invisibility condition $f(a)=0,\ f(b)=b$, we have
$a_{n}^{\mathrm{c}}=E_{0}\;\mathrm{i}^{n}$, $b_{n}^{\mathrm{c}}=0\ (\ i.e.\
A_{n}=0\ )$, therefore $a_{n}^{\mathrm{sc}}\ =\ 0$ which indicates no
scattering. However, when calculating $a_{n}^{\mathrm{int}}$, there is a term
$A_{n}\cdot N_{n}\big{(}k_{0}f(a)\big{)}$, in which $A_{n}\rightarrow 0$ while
$N_{n}\big{(}k_{0}f(a)\big{)}\rightarrow\infty$ as $f(a)\rightarrow 0$. A
meticulous calculation gives the result $a_{n}^{\mathrm{int}}\rightarrow 0$,
so waves can not spread into the hidden area. Finally, we get the expressions
of electromagnetic fields in the whole space as shown in Eqs.(24).
A detailed derivation should be identified. If we substitute the invisibility
condition $f(a)=0,\ f(b)=b$ at the beginning, the fields in the cloak layer
must have no terms of Neumann function, which means $b_{n}^{\mathrm{c}}=0$ ,
since $N_{n}\big{(}k_{0}f(a)\big{)}\rightarrow\infty$ is against the
requirement that fields should be finite. Therefore, the last terms vanish in
Eqs. (B.5a) and (B.5b). In this case, the Eq. (B.5b) still leads to
$a_{n}^{\mathrm{int}}=0$, yet the result can not satisfy Eq. (B.5a) when
$n=0$. In the more general derivation above, when $n=0$, though
$b_{n}^{\mathrm{c}}\rightarrow 0$, the product $b_{n}^{\mathrm{c}}\cdot
N_{n}\big{(}k_{0}f(a)\big{)}$ is towards to a finite quantity
$-E_{0}\mathrm{i}^{n}/J_{n}(k_{1}a)$, and it counterbalances the equality
(B.5a), however, the product only exists at the inner surface, so it no longer
has the meaning of fields but represents the magnetic surface current flowing
along the $z$ axis Zhang _et al._ (2007).
## Appendix C: Eikonal equation of different forms
Wave equation (37) can be written as the components form
$\left[\epsilon_{ipj}k^{p}{(\bar{n}^{\scriptscriptstyle-1})}^{jk}\epsilon_{kql}k^{q}+\left(\frac{\omega}{c}\right)^{2}\bar{n}_{il}\right]E^{l}\
=\ 0.$ (C.1)
The condition which protects $E^{l}$ from trivial solution is
$\det\left[\epsilon_{ipj}k^{p}{(\bar{n}^{\scriptscriptstyle-1})}^{jk}\epsilon_{kql}k^{q}+\left(\frac{\omega}{c}\right)^{2}\bar{n}_{il}\right]\
=\ 0.$ (C.2)
Expanding the determinant
$\begin{split}&\det\left[\epsilon_{ipj}k^{p}{(\bar{n}^{\scriptscriptstyle-1})}^{jk}\epsilon_{kql}k^{q}+\left(\frac{\omega}{c}\right)^{2}\bar{n}_{il}\right]\\\
=\
&e^{i_{1}i_{2}i_{3}}\cdot\prod^{3}_{\alpha=1}\left[\epsilon_{i_{\alpha}p_{\alpha}j_{\alpha}}k^{p_{\alpha}}{(\bar{n}^{\scriptscriptstyle-1})}^{j_{\alpha}k_{\alpha}}\epsilon_{k_{\alpha}q_{\alpha}\alpha}k^{q_{\alpha}}+\left(\frac{\omega}{c}\right)^{2}\bar{n}_{i_{\alpha}\alpha}\right]\\\
=\
&\sqrt{\gamma}\epsilon^{i_{1}i_{2}i_{3}}\left[\prod^{3}_{\alpha=1}M_{i_{\alpha}\alpha}+\left(\frac{\omega}{c}\right)^{2}\sum^{3}_{\alpha=1}\bar{n}_{i_{\alpha}\alpha}M_{i_{\alpha}(\alpha+1)}M_{i_{\alpha}(\alpha+2)}\right.\\\
&\left.+\left(\frac{\omega}{c}\right)^{4}\sum^{3}_{\alpha=1}M_{i_{\alpha}\alpha}\bar{n}_{i_{\alpha+1}(\alpha+1)}\bar{n}_{i_{\alpha+2}(\alpha+2)}+\left(\frac{\omega}{c}\right)^{6}\prod^{3}_{\alpha=1}\bar{n}_{i_{\alpha}\alpha}\right]\end{split}$
(C.3)
where
$M_{il}=\epsilon_{ipj}N^{pj}_{l}=\epsilon_{ipj}k^{p}{(\bar{n}^{\scriptscriptstyle-1})}^{jk}\epsilon_{kql}k^{q}$,
and the indices of $\epsilon_{i_{\alpha}j_{\alpha}k_{\alpha}}$ satisfy
$\alpha\mod\ 3$. Calculating each term in the expansion, we have
1st term:
$\displaystyle\sqrt{\gamma}\epsilon^{i_{1}i_{2}i_{3}}\prod^{3}_{\alpha=1}M_{i_{\alpha}\alpha}\
=\
\sqrt{\gamma}\epsilon^{i_{1}i_{2}i_{3}}\epsilon_{i_{1}p_{1}j_{1}}\epsilon_{i_{2}p_{2}j_{2}}\epsilon_{i_{3}p_{3}j_{3}}N_{1}^{p_{1}j_{1}}N_{2}^{p_{2}j_{2}}N_{3}^{p_{3}j_{3}}$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\epsilon_{p_{1}p_{2}j_{2}}\epsilon_{j_{1}p_{3}j_{3}}-\epsilon_{j_{1}p_{2}j_{2}}\epsilon_{p_{1}p_{3}j_{3}}\right)N_{1}^{p_{1}j_{1}}N_{2}^{p_{2}j_{2}}N_{3}^{p_{3}j_{3}}=\
0,$ (C.4) 2nd term:
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{2}\sum^{3}_{\alpha=1}\epsilon^{i_{1}i_{2}i_{3}}\bar{n}_{i_{\alpha}\alpha}M_{i_{\alpha}(\alpha+1)}M_{i_{\alpha}(\alpha+2)}$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{2}\sum^{3}_{\alpha=1}\epsilon^{i_{\alpha}i_{\alpha+1}i_{\alpha+2}}\bar{n}_{i_{\alpha}\alpha}M_{i_{\alpha}(\alpha+1)}M_{i_{\alpha}(\alpha+2)}$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{2}\sum^{3}_{\alpha=1}\bar{n}_{i_{\alpha}\alpha}\epsilon_{p_{\alpha+2}j_{\alpha+1}j_{\alpha+2}}N^{p_{\alpha+1}j_{\alpha+1}}_{\alpha+1}N^{p_{\alpha+2}j_{\alpha+2}}_{\alpha+2},$
because $\displaystyle\mbox{2nd term}\cdot\det\left(\bar{n}_{ij}\right)$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{2}\sum^{3}_{\alpha=1}\bar{n}_{i_{\alpha}\alpha}\epsilon_{p_{\alpha+2}j_{\alpha+1}j_{\alpha+2}}\cdot\left[k^{i_{\alpha}}{(\bar{n}^{\scriptscriptstyle-1})}^{j_{\alpha+1}k_{\alpha+1}}\epsilon_{k_{\alpha+1}q_{\alpha+1}(\alpha+1)}k^{q_{\alpha+1}}\right]$
$\displaystyle\cdot\left[k^{p_{\alpha+2}}{(\bar{n}^{\scriptscriptstyle-1})}^{j_{\alpha+2}k_{\alpha+2}}\epsilon_{k_{\alpha+2}q_{\alpha+2}(\alpha+2)}k^{q_{\alpha+2}}\right]\cdot\left[\sqrt{\gamma}\epsilon^{t_{1}t_{2}t_{3}}\bar{n}_{t_{1}1}\bar{n}_{t_{2}2}\bar{n}_{t_{3}3}\right]$
$\displaystyle=\ $
$\displaystyle\gamma\left(\frac{\omega}{c}\right)^{2}\left[\bar{n}_{i_{\alpha}\alpha}\bar{n}_{t_{\beta}\beta}k^{i_{\alpha}}k^{t_{\beta}}\gamma
k^{\alpha}k^{\beta}\right]\ =\
\left(\gamma\frac{\omega}{c}\bar{n}_{ij}k^{i}k^{j}\right)^{2},$ therefore we
have $\displaystyle\hskip 35.0pt\mbox{2nd term}\ =\
\left(\gamma\frac{\omega}{c}\bar{n}_{ij}k^{i}k^{j}\right)^{2}/\det\left(\bar{n}_{ij}\right),$
(C.5) 3rd term:
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{4}\sum^{3}_{\alpha=1}\epsilon^{i_{1}i_{2}i_{3}}M_{i_{\alpha}\alpha}\bar{n}_{i_{\alpha+1}(\alpha+1)}\bar{n}_{i_{\alpha+2}(\alpha+2)}$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{4}\sum^{3}_{\alpha=1}\epsilon^{i_{\alpha}i_{\alpha+1}i_{\alpha+2}}\epsilon_{i_{\alpha}p_{\alpha}j_{\alpha}}\bar{n}_{i_{\alpha+1}(\alpha+1)}\bar{n}_{i_{\alpha+2}(\alpha+2)}N^{p_{\alpha}j_{\alpha}}_{\alpha}$
$\displaystyle=\ $
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{4}\sum^{3}_{\alpha=1}\left[\bar{n}_{p_{\alpha}(\alpha+1)}\epsilon_{q_{\alpha}\alpha(\alpha+2)}-\bar{n}_{p_{\alpha}(\alpha+2)}\epsilon_{q_{\alpha}\alpha(\alpha+1)}\right]k^{p_{\alpha}}k^{q_{\alpha}}$
$\displaystyle=\ $
$\displaystyle-2\gamma\left(\frac{\omega}{c}\right)^{4}\bar{n}_{ij}k^{i}k^{j},$
(C.6) 4th term:
$\displaystyle\sqrt{\gamma}\left(\frac{\omega}{c}\right)^{6}\epsilon^{i_{1}i_{2}i_{3}}\prod^{3}_{\alpha=1}\bar{n}_{i_{\alpha}\alpha}\
=\ \left(\frac{\omega}{c}\right)^{6}\det\left(\bar{n}_{ij}\right).$ (C.7)
Substituting them into Eq. (C.3), we have
$\det\left[\epsilon_{ipj}k^{p}{(\bar{n}^{\scriptscriptstyle-1})}^{jk}\epsilon_{kql}k^{q}+\left(\frac{\omega}{c}\right)^{2}\bar{n}_{il}\right]\
=\
\frac{(\gamma\omega/c)^{2}}{\det\left(\bar{n}_{ij}\right)}\left[\bar{n}_{ij}k^{i}k^{j}-\frac{1}{\gamma}\left(\frac{\omega}{c}\right)^{2}\det\left(\bar{n}_{ij}\right)\right]^{2}\
=\ 0.$ (C.8)
This is exactly the wave-vector eikonal equation (38) expressed by Landau’s
definition. The ray-vector eikonal equation (40) can be simply obtained
through duality rules. In terms of Eq. (2b) and Eq. (5), we can get the two
kinds of eikonal equation written by Minkowski’s components and anholonomic
components.
Under Minkowski’s definition, wave-vector eikonal equation is
$n^{ij}k_{i}k_{j}-\left(\frac{\omega}{c}\right)^{2}\det\left(n^{ij}\right)\ =\
n_{ij}k^{i}k^{j}-\frac{1}{\gamma^{2}}\left(\frac{\omega}{c}\right)^{2}\det\left(n_{ij}\right)\
=\ 0,$ (C.9)
and ray-vector eikonal equation is
${(n^{\scriptscriptstyle-1})}^{ij}s_{i}s_{j}-\gamma^{2}\left(\frac{c}{\omega}\right)^{2}\det\left[(n^{\scriptscriptstyle-1})^{ij}\right]\
=\
{(n^{\scriptscriptstyle-1})}_{ij}s^{i}s^{j}-\left(\frac{c}{\omega}\right)^{2}\det\left[(n^{\scriptscriptstyle-1})_{ij}\right]\
=\ 0.$ (C.10)
The wave-vector eikonal equation expressed by the components in anholonomic
unit basis is
${(n^{\scriptscriptstyle-1})}_{\langle ij\rangle}k_{\langle
i\rangle}k_{\langle
j\rangle}-\left(\frac{\omega}{c}\right)^{2}\det\left[(n^{\scriptscriptstyle-1})_{\langle
ij\rangle}\right]\ =\ 0,$ (C.11)
and the ray-vector eikonal equation is
${(n^{\scriptscriptstyle-1})}_{\langle ij\rangle}s_{\langle
i\rangle}s_{\langle
j\rangle}-\left(\frac{c}{\omega}\right)^{2}\det\left[(n^{\scriptscriptstyle-1})_{\langle
ij\rangle}\right]\ =\ 0.$ (C.12)
## References
* Pendry _et al._ (2006) J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
* Leonhardt (2006) U. Leonhardt, Science 312, 1777 (2006).
* Greenleaf _et al._ (2003) A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
* Schurig _et al._ (2006) D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Express 14, 9794 (2006).
* Leonhardt and Philbin (2006) U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).
* Milton _et al._ (2006) G. W. Milton, M. Briane, and J. R. Willis, New J. Phys. 8, 248 (2006).
* Leonhardt and Philbin (2009) U. Leonhardt and T. G. Philbin, Prog. Optics 53, 69 (2009).
* Chen (2009) H. Chen, J. Opt. A: Pure Appl. Opt. 11, 075102 (2009).
* Chen _et al._ (2010) H. Chen, C. T. Chan, and P. Sheng, Nat. Mater. 9, 387 (2010).
* Cummer _et al._ (2006) S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phys. Rev. E 74, 036621 (2006).
* Chen _et al._ (2007) H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 063903 (2007).
* Ruan _et al._ (2007) Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, Phys. Rev. Lett. 99, 113903 (2007).
* Zhang _et al._ (2007) B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, Phys. Rev. B 76, 121101(R) (2007).
* Luo _et al._ (2008a) Y. Luo, J. Zhang, H. Chen, S. Xi, and B.-I. Wu, Appl. Phys. Lett. 93, 033504 (2008a).
* Luo _et al._ (2008b) Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, Phys. Rev. B 77, 125127 (2008b).
* Cai _et al._ (2007) W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Appl. Phys. Lett. 91, 111105 (2007).
* Niu _et al._ (2009) K. Niu, C. Song, and M.-L. Ge, Opt. Express 17, 11753 (2009).
* Yao _et al._ (2008) P. Yao, Z. Liang, and X. Jiang, Appl. Phys. Lett. 92, 031111 (2008).
* Chen and Chan (2008) H. Chen and C. T. Chan, J. Appl. Phys. 104, 033113 (2008).
* Leonhardt and Philbin (2010) U. Leonhardt and T. Philbin, _Geometry and Light: The Scicence of Invisibility_ (Dover, 2010).
* Misner _et al._ (1973) C. W. Misner, K. S. Thorne, and J. A. Wheeler, _Gravitation_ (Freeman, New York, 1973).
* Ward and Pendry (1996) A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
* Born and Wolf (1999) M. Born and E. Wolf, _Principles of Optics_ , 7th ed. (Cambridge Univ. Press, 1999).
* Chen (1985) H. C. Chen, _Theory of Electromagnetic Waves: A Coordinate Free Approach_ (McGraw-Hill, 1985).
* Landau and Lifshitz (1995) L. D. Landau and E. M. Lifshitz, _The Classical Theory of Fields_ (Butterworth-Heinemann, Oxford, 1995).
* Post (1962) E. J. Post, _Formal Structure of Electromagnetics_ (Wiley, New York, 1962).
* Plebanski (1960) J. Plebanski, Phys. Rev. 118, 1396 (1960).
* Bergamin (2008) L. Bergamin, Phys. Rev. A 78, 043825 (2008).
* Felice and Clarke (1990) F. D. Felice and C. J. S. Clarke, _Relativity on Curved Manifolds_ (Cambridge Univ. Press, 1990).
* Sonego and Abramowicz (1998) S. Sonego and M. A. Abramowicz, J. Math. Phys. 39, 3158 (1998).
* Tsagas (2005) C. G. Tsagas, Class. Quantum Grav. 22, 393 (2005).
|
arxiv-papers
| 2011-09-23T17:12:06 |
2024-09-04T02:49:22.449498
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruo-Yang Zhang, Qing Zhao, and Mo-Lin Ge",
"submitter": "Ruo-Yang Zhang",
"url": "https://arxiv.org/abs/1109.5133"
}
|
1109.5330
|
# On quasi-Prüfer and UM$t$ domains
Parviz Sahandi School of Mathematics, Institute for Research in Fundamental
Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran and Department of
Mathematics, University of Tabriz, Tabriz, Iran sahandi@ipm.ir
###### Abstract.
In this note we show that an integral domain $D$ of finite $w$-dimension is a
quasi-Prüfer domain if and only if each overring of $D$ is a $w$-Jaffard
domain. Similar characterizations of quasi-Prüfer domains are given by
replacing $w$-Jaffard domain by $w$-stably strong S-domain, and $w$-strong
S-domain. We also give new characterizations of UM$t$ domains.
###### Key words and phrases:
Strong S-domain, stably strong S-domain, Jaffard domain, quasi-Prüfer domain,
star operation
###### 2000 Mathematics Subject Classification:
Primary 13A15, 13G05, 13C15
This research was in part supported by a grant from IPM (No. 900130059)
## 1\. Introduction
The quasi-Prüfer notion was introduced in [2] for rings (not necessarily
domains). As in [9], we say that an integral domain $D$ is a _quasi-Prüfer
domain_ if for each prime ideal $P$ of $D$, if $Q$ is a prime ideal of $D[X]$
with $Q\subseteq P[X]$, then $Q=(Q\cap D)[X]$. It is well known that an
integral domain is a Prüfer domain if and only if it is integrally closed and
quasi-Prüfer [11, Theorem 19.15]. There are several different equivalent
condition for quasi-Prüfer domains (c.f. [9, 2, 3]).
On the other hand as a $t$-analogue, an integral domain $D$ is called a _UM
$t$ domain_ [12], if every upper to zero in $D[X]$ is a maximal $t$-ideal and
has been studied by several authors (see [8], [6], and [18]). UM$t$ domains
are closely related to quasi-Prüfer domains in the sense that a domain $D$ is
a UM$t$ domain if and only if $D_{P}$ is a quasi-Prüfer domain for each
$t$-prime ideal $P$ of $D$ [8, Theorem 1.5]. And the other relation is the
characterization of quasi-Prüfer domains due to Fontana, Gabelli and Houston
[8, Corollary 3.11]; a domain $D$ is a quasi-Prüfer domain if and only if each
overring of $D$ is a UM$t$-domain.
In [16] we defined and studied the $w$-Jaffard domains and proved that all
strong Mori domains (domains that satisfy the ACC on $w$-ideals) and all UM$t$
domains of finite $w$-dimension, are $w$-Jaffard domains. In [17] we defined
and studied a subclass of $w$-Jaffard domains, namely the $w$-stably strong
S-domains and showed how this notion permit studies of UM$t$ domains in the
spirit of earlier works on quasi-Prüfer domains. The aim of this paper is to
prove that, for a domain $D$ with some condition on
$w$-$\operatorname{dim}(D)$, the following statements are equivalent, which
gives new descriptions of quasi-Prüfer domains; a result reminiscent of the
well-known result of Ayache, Cahen and Echi [2] (see also [9, Theorem 6.7.8]).
* (1)
Each overring of $D$ is a $w$-stably strong S-domain.
* (2)
Each overring of $D$ is a $w$-strong S-domain.
* (3)
Each overring of $D$ is a $w$-Jaffard domain.
* (4)
Each overring of $D$ is a UM$t$ domain.
* (5)
$D$ is a quasi-Prüfer domain.
Throughout, the letter $D$ denotes an integral domain with quotient field $K$
and $F(D)$ denotes the set of nonzero fractional ideals. Let $f(D)$ be the set
of all nonzero finitely generated fractional ideals of $D$. Let $*$ be a star
operation on the domain $D$. For every $A\in F(D)$, put $A^{*_{f}}:=\bigcup
F^{*}$, where the union is taken over all $F\in f(D)$ with $F\subseteq A$. It
is easy to see that $*_{f}$ is a star operation on $D$. A star operation $*$
is called of _finite character_ if $*_{f}=*$. We say that a nonzero ideal $I$
of $D$ is a _$*$ -ideal_ of $D$, if $I^{*}=I$; a _$*$ -prime_, if $I$ is a
prime $*$-ideal of $D$. It has become standard to say that a star operation
$*$ is _stable_ if $(A\cap B)^{*}=A^{*}\cap B^{*}$ for all $A$, $B\in F(D)$.
Given a star operation $*$ on an integral domain $D$ it is possible to
construct a star operation $\widetilde{*}$ which is stable and of finite
character defined as follows: for each $A\in F(D)$,
$A^{\widetilde{*}}:=\\{x\in K|xJ\subseteq A,\text{ for some }J\subseteq D,J\in
f(D),J^{*}=D\\}.$
The $\widetilde{*}$-dimension of $D$ is defined as follows:
$\widetilde{*}\text{-}\operatorname{dim}(D)=\sup\\{\operatorname{ht}(P)\mid
P\text{ is a }\widetilde{*}\text{-prime ideal of }D\\}.$
The most widely studied star operations on $D$ have been the identity $d$, and
$v$, $t:=v_{f}$, and $w:=\widetilde{v}$ operations, where
$A^{v}:=(A^{-1})^{-1}$, with $A^{-1}:=(D:A):=\\{x\in K|xA\subseteq D\\}$.
Let $D$ be a domain and $T$ an overring of $D$. Let $*$ and $*^{\prime}$ be
star operations on $D$ and $T$, respectively. One says that $T$ is
_$(*,*^{\prime})$ -linked to_ $D$ if $F^{*}=D\Rightarrow(FT)^{*^{\prime}}=T$
for each nonzero finitely generated ideal $F$ of $D$. As in [5] we say that
$T$ is $t$-linked to $D$ if $T$ is $(t,t)$-linked to $D$. As in [6] a domain
$D$ is called _$t$ -linkative_ if each overring of $D$ is $t$-linked to $D$.
As a matter of fact $t$-linkative domains are exactly the domains such that
the identity operation coincides with the $w$-operation, that is _DW-domains_
in the terminology of [15].
If $F\subseteq K$ are fields, then $\operatorname{tr.deg.}_{F}(K)$ stands for
the _transcendence degree_ of $K$ over $F$. If $P$ is a prime ideal of the
domain $D$, then we set $\mathbb{K}(P):=D_{P}/PD_{P}$.
## 2\. $w$-Jaffard domains
First we recall a special case of a general construction for semistar
operations (see [16]). Let $D$ be an integral domain with quotient field $K$,
let $X$, $Y$ be two indeterminates over $D$ and $*$ be a star operation on
$D$. Set $D_{1}:=D[X]$, $K_{1}:=K(X)$ and take the following subset of
$\operatorname{Spec}(D_{1})$:
$\Theta_{1}^{*}:=\\{Q_{1}\in\operatorname{Spec}(D_{1})|\text{ }Q_{1}\cap
D=(0)\text{ or }(Q_{1}\cap D)^{*_{f}}\subsetneq D\\}.$
Set
$\mathfrak{S}_{1}^{*}:=D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\Theta_{1}^{*}\\})$
and:
$E^{\circlearrowleft_{\mathfrak{S}_{1}^{*}}}:=E[Y]_{\mathfrak{S}_{1}^{*}}\cap
K_{1},\text{ for all }E\in F(D_{1}).$
It is proved in [16, Theorem 2.1] that the mapping
$*[X]:=\circlearrowleft_{\mathfrak{S}_{1}^{*}}:F(D_{1})\to F(D_{1})$,
$E\mapsto E^{*[X]}$ is a stable star operation of finite character on $D[X]$,
i.e., $\widetilde{*[X]}=*[X]$. It is also proved that
$\widetilde{*}[X]=*_{f}[X]=*[X]$, $d_{D}[X]=d_{D[X]}$. If $X_{1},\cdots,X_{r}$
are indeterminates over $D$, for $r\geq 2$, we let
$*[X_{1},\cdots,X_{r}]:=(*[X_{1},\cdots,X_{r-1}])[X_{r}].$
For an integer $r$, put $*[r]$ to denote $*[X_{1},\cdots,X_{r}]$ and $D[r]$ to
denote $D[X_{1},\cdots,X_{r}]$.
Let $*$ be a star operation on $D$. A valuation overring $V$ of $D$ is called
a _$*$ -valuation overring of $D$_ provided that $F^{*}\subseteq FV$, for each
$F\in f(D)$. Following [16], the _$*$ -valuative dimension_ of $D$ is defined
as:
$*\text{-}\operatorname{dim}_{v}(D):=\sup\\{\operatorname{dim}(V)|V\text{ is
}*\text{-valuation overring of }D\\}.$
It is shown in [16, Theorem 4.5] that
$\widetilde{*}\text{-}\operatorname{dim}_{v}(D)=\sup\\{w\text{-}\operatorname{dim}(R)|R\text{
is a }(*,t)\text{-linked over }D\\}.$
It is observed in [16] that we have always the inequality
$\widetilde{*}$-$\operatorname{dim}(D)\leq\widetilde{*}$-$\operatorname{dim}_{v}(D)$.
We say that $D$ is a _$*$ -Jaffard domain_, if
$*\text{-}\operatorname{dim}(D)=*\text{-}\operatorname{dim}_{v}(D)<\infty$.
When $*=d$ the identity operation then $d$-Jaffard domain coincides with the
classical Jaffard domain (cf. [1]). It is proved in [16], that $D$ is a
$\widetilde{*}$-Jaffard domain if and only if
$*[X_{1},\cdots,X_{n}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{n}])=\widetilde{*}\text{-}\operatorname{dim}(D)+n,$
for each positive integer $n$. In [19] we gave examples to show that the two
classes of $w$-Jaffard and Jaffard domains are incomparable by constructing a
$w$-Jaffard domain which is not Jaffard and a Jaffard domain which is not
$w$-Jaffard.
We are now prepared to state and prove the first main result of this paper.
###### Theorem 2.1.
Let $D$ be an integral domain of finite $w$-dimension. Then the following
statements are equivalent:
* (1)
Each overring of $D$ is a $w$-Jaffard domain.
* (2)
$D$ is a quasi-Prüfer domain.
###### Proof.
$(1)\Rightarrow(2)$ Let $Q$ be a prime ideal of an overring $T$ of $D$, and
set $\mathfrak{q}:=Q\cap D$. Let $\tau:T_{Q}\to\mathbb{K}(Q)$ be the canonical
surjection and let $\iota:\mathbb{K}(\mathfrak{q})\to\mathbb{K}(Q)$ be the
canonical embedding. Consider the following pullback diagram:
$\textstyle{D(Q):=\tau^{-1}(\mathbb{K}(\mathfrak{q}))=D_{\mathfrak{q}}+QT_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{K}(\mathfrak{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{K}(Q).}$
Since $T_{Q}$ is quasilocal and $\mathbb{K}(\mathfrak{q})$ is a DW-domain,
then $D(Q)$ is a DW-domain by [15, Theorem 3.1(2)]. Thus the $w$-operation
coincides with the identity operation $d$ for $D(Q)$. Since by the hypothesis
$D(Q)$ is a $w$-Jaffard domain we actually have $D(Q)$ is a Jaffard domain. On
the other hand by [1, Proposition 2.5(a)] we have
$\operatorname{dim}_{v}(D(Q))=\operatorname{dim}_{v}(T_{Q})+\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q)).$
In particular
$\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q))$ and
$\operatorname{dim}_{v}(T_{Q})$ are finite numbers. Note that by [7,
Proposition 2.1(5)] we have
$\operatorname{dim}(D(Q))=\operatorname{dim}(T_{Q})$ and since
$\operatorname{dim}_{v}(D(Q))=\operatorname{dim}(D(Q))$, we obtain that
$\operatorname{dim}(T_{Q})=\operatorname{dim}_{v}(T_{Q})+\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q)).$
Since $\operatorname{dim}(T_{Q})\leq\operatorname{dim}_{v}(T_{Q})$, then
$\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q))=0$.
Consequently $D$ is a residually algebraic domain, and hence is a quasi-Prüfer
domain by [3, Corollary 2.8].
$(2)\Rightarrow(1)$ Let $T$ be an overring of $D$. We claim that $T$ is of
finite $w$-dimension. Since $D$ is a quasi-Prüfer domain, [6, Theorem 2.4]
implies that $D$ is a $t$-linkative and UM$t$ domain. Thus in particular $T$
is a $t$-linked overring of $D$. Then
$\displaystyle w\text{-}\operatorname{dim}(T)\leq$
$\displaystyle\sup\\{w\text{-}\operatorname{dim}(R)|R\text{ is }t\text{-linked
over }D\\}$ $\displaystyle=$ $\displaystyle
w\text{-}\operatorname{dim}_{v}(D)=w\text{-}\operatorname{dim}(D)<\infty,$
where the first equality is by [16, Theorem 4.5]. Finally by [17, Corollary
2.6], every UM$t$ domain of finite $w$-dimension is a $w$-Jaffard domain to
deduce that $T$ is a $w$-Jaffard domain. ∎
As an immediate corollary we have:
###### Corollary 2.2.
Let $D$ be an integral domain of finite $w$-dimension. Then the following
statements are equivalent:
* (1)
Each $t$-linked overring of $D$ is a $w$-Jaffard domain.
* (2)
$D$ is a UM$t$ domain.
###### Proof.
$(1)\Rightarrow(2)$ Let $P$ be a $t$-prime ideal of $D$, and $T$ be an
overring of $D_{P}$. Thus $T=T_{D\backslash P}$ is a $t$-linked overring of
$D$ by [5, Proposition 2.9]. Therefore $T$ is a $w$-Jaffard domain by the
hypothesis. Consequently $D_{P}$ is a quasi-Prüfer domain by Theorem 2.1. Then
$D$ is a UM$t$ domain by [8, Theorem 1.5].
$(2)\Rightarrow(1)$ Let $T$ be a $t$-linked overring of $D$. Then as the proof
of Theorem 2.1 we have
$\displaystyle w\text{-}\operatorname{dim}(T)\leq$
$\displaystyle\sup\\{w\text{-}\operatorname{dim}(R)|R\text{ is }t\text{-linked
over }D\\}$ $\displaystyle=$ $\displaystyle
w\text{-}\operatorname{dim}_{v}(D)=w\text{-}\operatorname{dim}(D)<\infty.$
By [17, Corollary 2.6] we get that $T$ is a $w$-Jaffard domain. ∎
## 3\. $w$-stably strong S-domains
Let $*$ be a star operation on $D$. Following [17] the domain $D$ is called a
_$*$ -strong S-domain_, if each pair of adjacent $*$-prime ideals
$P_{1}\subset P_{2}$ of $D$, extend to a pair of adjacent $*[X]$-prime ideals
$P_{1}[X]\subset P_{2}[X]$, of $D[X]$. If for each $n\geq 1$, the polynomial
ring $D[n]$ is a $*[n]$-strong S-domain, then $D$ is said to be an _$*$
-stably strong S-domain_. It is observed in [17] that a domain $D$ is
$*$-strong S-domain (resp. $*$-stably strong S-domain) if and only if $D_{P}$
is strong S-domain (resp. stably strong S-domain) for each $*$-prime ideal $P$
of $D$. Thus a strong S-domain (resp. stably strong S-domain) $D$ is
$*$-strong S-domain (resp. $*$-stably strong S-domain) for each star operation
$*$ on $D$. However, the converse is not true in general; i.e., for some star
operation $*$, the domain $D$ might be $*$-strong S-domain (resp. $*$-stably
strong S-domain), but $D$ is not strong S-domain (resp. stably strong
S-domain). In [14, Example 4.17] Malik and Mott gave an example of a UM$t$
domain (in fact a Krull domain) which is not strong S-domain. But a UM$t$
domain is a $w$-stably strong S-domain (and hence $w$-strong S-domain as well)
by [17, Corollary 2.6].
We observe [17, Corollary 2.3] that a finite $w$-dimensional $w$-stably strong
S-domain is a $w$-Jaffard domain.
We are now prepared to state and prove the second main result of this paper.
###### Theorem 3.1.
Let $D$ be an integral domain of finite $w$-valuative dimension. Then the
following statements are equivalent:
* (1)
Each overring of $D$ is a $w$-stably strong S-domain.
* (2)
Each overring of $D$ is a $w$-strong S-domain.
* (3)
Each overring of $D$ is a UM$t$ domain.
* (4)
$D$ is a quasi-Prüfer domain.
###### Proof.
The implication $(1)\Rightarrow(2)$ is trivial, and $(3)\Rightarrow(1)$ holds
by [17, Corollary 2.6].
$(2)\Rightarrow(4)$ Let $Q$ be a prime ideal of an overring $T$ of $D$ and set
$\mathfrak{q}:=Q\cap D$. As in the proof of Theorem 2.1 we have the following
pullback diagram:
$\textstyle{D(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{K}(\mathfrak{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{K}(Q).}$
Since $T_{Q}$ is quasilocal and $\mathbb{K}(\mathfrak{q})$ is a DW-domain,
then $D(Q)$ is a DW-domain by [15, Theorem 3.1(2)]. Thus the $w$-operation
coincides with the identity operation $d$ for $D(Q)$. Since by the hypothesis
$D(Q)$ is a $w$-strong S-domain, we actually have $D(Q)$ is a strong S-domain.
Next we claim that $D(Q)$ is of finite dimension. Indeed since $D(Q)$ is a DW-
domain it is in fact a $t$-linked overring of $D$. Then
$\displaystyle\operatorname{dim}(D(Q))=$ $\displaystyle
w\text{-}\operatorname{dim}(D(Q))$ $\displaystyle\leq$
$\displaystyle\sup\\{w\text{-}\operatorname{dim}(R)|R\text{ is }t\text{-linked
over }D\\}$ $\displaystyle=$ $\displaystyle
w\text{-}\operatorname{dim}_{v}(D)<\infty,$
where the second equality is by [16, Theorem 4.5]. On the other hand by [1,
Proposition 2.7] we have the inequality belove
$\displaystyle
1+\operatorname{dim}(T_{Q})+\min\\{\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q)),1\\}\leq$
$\displaystyle\operatorname{dim}(D(Q)[X])$ $\displaystyle=$
$\displaystyle\operatorname{dim}(D(Q))+1$ $\displaystyle=$
$\displaystyle\operatorname{dim}(T_{Q})+1.$
The first equality holds since $D(Q)$ is strong S-domain and [13, Theorem 39],
and the second one holds by [7, Proposition 2.1(5)]. Thus
$\operatorname{tr.deg.}_{\mathbb{K}(\mathfrak{q})}(\mathbb{K}(Q))=0$.
Consequently $D$ is a residually algebraic domain and hence is a quasi-Prüfer
domain by [3, Corollary 2.8].
$(4)\Rightarrow(3)$ Suppose that $D$ is a quasi-Prüfer domain and let $T$ be
an overring of $D$. Thus $T$ is also a quasi-Prüfer domain. Therefore $T$ is a
UM$t$ domain by [6, Theorem 2.4]. ∎
As an immediate corollary we have:
###### Corollary 3.2.
Let $D$ be an integral domain of finite $w$-valuative dimension. Then the
following statements are equivalent:
* (1)
Each $t$-linked overring of $D$ is a $w$-stably strong S-domain.
* (2)
Each $t$-linked overring of $D$ is a $w$-strong S-domain.
* (3)
Each $t$-linked overring of $D$ is a UM$t$ domain.
* (4)
$D$ is a UM$t$ domain.
###### Proof.
The implication $(1)\Rightarrow(2)$ is trivial.
For $(2)\Rightarrow(4)$ let $P$ be a $t$-prime ideal of $D$, and $T$ be an
overring of $D_{P}$. Thus $T=T_{D\backslash P}$ is a $t$-linked overring of
$D$ by [5, Proposition 2.9]. Therefore $T$ is a $w$-strong S-domain by the
hypothesis. Consequently $D_{P}$ is a quasi-Prüfer domain by Theorem 3.1. Then
$D$ is a UM$t$ domain by [8, Theorem 1.5].
$(4)\Rightarrow(3)$ Suppose $T$ is a $t$-linked overring of $D$. Then $T$ is a
UM$t$ domain by [18, Theorem 3.1].
$(3)\Rightarrow(1)$ Is true by [17, Corollary 2.6]. ∎
Note that the equivalence $(3)\Leftrightarrow(4)$ in Theorem 3.1 (resp.
Corollary 3.2) is well known [8, Corollary 3.11] (resp. [4, Theorem 2.6]), but
our proof is completely different.
ACKNOWLEDGMENT
I would like to thank the referee for carefully reading the first version of
this paper.
## References
* [1] D. F. Anderson, A. Bouvier, D. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domain, Expo. Math., 6, (1988), 145–175.
* [2] A. Ayache and P. Cahen and O. Echi, Anneaux quasi-Prüfériens et P-anneaux. Boll. Un. Mat. Ital 10-B, (1996), 1–24.
* [3] A. Ayache and A. Jaballah, Residually algebraic pairs of rings, Math. Z. 225 (1997), 49–65.
* [4] G.W. Chang and M. Zafrullah, The $w$-integral closure of integral domains, J. Algebra, 295, (2006), 195–210.
* [5] D. E. Dobbs, E. G. Houston, T. G. Lucas and M. Zafrullah, T-linked overrings and Prüfer v-multiplication domains, Comm. Algebra 17 (1989), 2835–2852.
* [6] D. E. Dobbs, E. G. Houston, T. G. Lucas M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20. No. 5, (1992), 1463–1488.
* [7] M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123, (1980), 331–355.
* [8] M. Fontana, S. Gabelli and E. Houston, UMT-domains and domains with Prüfer integral closure, Comm. Algebra 26, (1998), 1017–1039.
* [9] M. Fontana, J. Huckaba, and I. Papick, Prüfer domains, New York, Marcel Dekker, 1997.
* [10] M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings and related semistar operations, Comm. Algebra 31 (2003), 4775–4801.
* [11] R. Gilmer, Multiplicative ideal theory, New York, Dekker, 1972.
* [12] E. Houston and M. Zafrullah, On $t$-invertibility, II, Comm. Algebra 17 (1989), 1955–1969.
* [13] I. Kaplansky, Commutative rings, rev. ed., Univ. Chicago Press, Chicago, 1974.
* [14] S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra, 28 (1983), 249–264.
* [15] A. Mimouni, Integral domains in which each ideal is a $w$-ideal, Comm. Algebra 33, No. 5, (2005), 1345–1355.
* [16] P. Sahandi, Semistar-Krull and valuative dimension of integral domains, Ricerche Mat., 58, (2009), 219–242.
* [17] P. Sahandi, Universally catenarian integral domains, strong S-domains and semistar operations, Comm. Algebra 38, No. 2, (2010), 673–683.
* [18] P. Sahandi, Semistar dimension of polynomial rings and Prüfer-like domains, Bull. Iranian Math. Soc., to appear, arXiv:0808.1331v2 [math.AC].
* [19] P. Sahandi, W-Jaffard domains in pullbacks, J. Algebra and its Applications, to appear, arXiv:1003.1565v3 [math.AC].
|
arxiv-papers
| 2011-09-25T07:05:39 |
2024-09-04T02:49:22.466041
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Parviz Sahandi",
"submitter": "Parviz Sahandi Dr.",
"url": "https://arxiv.org/abs/1109.5330"
}
|
1109.5420
|
# Incremental Relaying for the Gaussian Interference Channel with a Degraded
Broadcasting Relay ††thanks: Manuscript received November 7, 2011; revised
August 9, 2012 and Nov 24, 2012; accepted Dec 7, 2012. Date of current version
December 12, 2012. This work was supported by the Natural Science and
Engineering Research Council (NSERC) of Canada. The material in this paper has
been presented in part at Allerton Conf. Commun., Control and Computing, Sept.
2011. ††thanks: The authors are with The Edward S. Rogers Sr. Department of
Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S
3G4 Canada (email: zhoulei@comm.utoronto.ca; weiyu@comm.utoronto.ca).
Lei Zhou, Student Member, IEEE and Wei Yu, Senior Member, IEEE
###### Abstract
This paper studies incremental relay strategies for a two-user Gaussian relay-
interference channel with an in-band-reception and out-of-band-transmission
relay, where the link between the relay and the two receivers is modelled as a
degraded broadcast channel. It is shown that generalized hash-and-forward
(GHF) can achieve the capacity region of this channel to within a constant
number of bits in a certain weak-relay regime, where the transmitter-to-relay
link gains are not unboundedly stronger than the interference links between
the transmitters and the receivers. The GHF relaying strategy is ideally
suited for the broadcasting relay because it can be implemented in an
incremental fashion, i.e., the relay message to one receiver is a degraded
version of the message to the other receiver. A generalized-degree-of-freedom
(GDoF) analysis in the high signal-to-noise ratio (SNR) regime reveals that in
the symmetric channel setting, each common relay bit can improve the sum rate
roughly by either one bit or two bits asymptotically depending on the
operating regime, and the rate gain can be interpreted as coming solely from
the improvement of the common messages rate, or alternatively in the very weak
interference regime as solely coming from the rate improvement of the private
messages. Further, this paper studies an asymmetric case in which the relay
has only a single single link to one of the destinations. It is shown that
with only one relay-destination link, the approximate capacity region can be
established for a larger regime of channel parameters. Further, from a GDoF
point of view, the sum-capacity gain due to the relay can now be thought as
coming from either signal relaying only, or interference forwarding only.
###### Index Terms:
Approximate capacity, generalized hash-and-forward (GHF), generalized degrees
of freedom, Han-Kobayashi strategy, interference channel, relay channel.
## I Introduction
Interference is a key limiting factor in modern communication systems. In a
wireless cellular network, the performance of cell-edge users is severely
limited by intercell interference. This paper considers the use of relays in
cellular networks. The uses of relays to combat channel shadowing and to
extend coverage for wireless systems have been widely studied in the
literature. The main goal of this paper is to demonstrate the benefit of
relaying for interference mitigation in the interference-limited regime.
Consider a two-cell wireless network with two base-stations each serving their
respective receivers while interfering with each other, as shown in Fig. 1.
The deployment of a cell-edge relay, which observes a linear combination of
the two transmit signals from the base-stations and is capable of
independently communicating with the receivers over a pair of relay links, can
significantly help the receivers mitigate intercell interference. This model
is often referred to as an in-band-reception and out-of-band-transmission
relay-interference channel, as the relay-to-receiver transmission can be
thought of as taking place on a different frequency band.
A particular feature of the channel model considered in this paper is that the
relay-to-receivers link is modeled as a Gaussian broadcast channel. This is
motivated by the fact that the relay’s transmission to the remote receivers
often takes place in a wireless medium. Consequently, the same relay message
can be heard by both receivers and can potentially help both receivers at the
same time. Further, it is convenient (and without loss of generality as shown
later for the achievability scheme and the converse proved in this paper) to
model the relay-to-receiver links as digital links with capacities
$\mathsf{C}_{1}$ and $\mathsf{C}_{2}$ respectively, but where one relay
message is required to be a degraded version of the other relay message, as in
a Gaussian broadcast channel. The goal of this paper is to devise an
incremental relaying strategy and to quantify its benefit for this particular
relay-interference channel.
### I-A Related Work
The classic two-user interference channel consists of two transmitter-receiver
pairs communicating in the presence of interference from each other. Although
the capacity region of the two-user Gaussian interference channel is still not
known exactly, it can be approximated to within one bit [1] using a Han-
Kobayashi power splitting strategy [2].
The use of cooperative communication for interference mitigation has received
much attention recently. For example, [3, 4, 5] studied the Gaussian
Z-interference channel with a unidirectional receiver cooperation link, and
[6, 7, 8, 9] studied the Gaussian interference channel with bi-directional
transmitter/receiver cooperation links. In addition, the Gaussian interference
channel with an additional relay node has also been studied extensively in the
literature. Depending on the types of the links between the relay and the
transmitters/receivers, the relay-interference channel can be categorized as
having in-band transmission/reception [10, 11, 12, 13, 14, 15, 16, 17], out-
of-band transmission/reception [18, 19, 20], out-of-band transmission and in-
band reception [21, 22, 23, 24], or in-band transmission and out-of-band
reception [25], the last of which is directly related to the channel model in
this paper. In the following, we review different transmission schemes and
relaying strategies that have emerged for each of these cases.
For interference channels equipped with an in-band transmission and reception
relay, the relay interacts with both transmitters and receivers in the same
frequency band. Relaying strategies that have been investigated in the
literature include decode-and-forward, compress-and-forward, and amplify-and-
forward. For example, [12, 13] show that decoding-and-forwarding either the
intended signal or the interfering signal to a receiver can both be
beneficial. The former is termed as signal relaying, the latter interference
forwarding. Decode-and-forward and half-duplex amplify-and-forward strategies
are also studied in [14, 16]. When combining decode-and-forward relaying
strategy and the Han-Kobayashi rate splitting input scheme, [17] gives an
achievable rate region that has a shape similar to the Chong-Motani-Garg (CMG)
region for the interference channel [26]. The exact capacity for this type of
relay-interference channel is in general open, but there is a special potent-
relay case [11] for which the sum capacity is known in some specific regimes.
The difficulty in establishing the capacity of the interference channel with
in-band transmission/reception relay is in part due to the fact that the
relay’s received and transmit signals intertwine with that of the underlying
interference channel. To simplify the matter, the interference channel with an
out-of-band transmission/reception relay has been studied in [18, 20, 19]. In
this channel model, the relay essentially operates on a separate set of
parallel channels. Based on signal relaying and interference forwarding
strategies, [18] identifies the condition under which the capacity region can
be achieved with separable or nonseparable coding between the out-of-band
relay and the underlying interference channel. Further, [19] studies this
channel model in a symmetric setting and characterizes the sum capacity to
within $1.15$ bits. The transmission scheme of [19] involves further splitting
of the common messages in the Han-Kobayashi scheme and a relay strategy that
combines nested lattice coding and Gaussian codes. It is shown that in the
strong interference regime, the use of structured codes is optimal.
\begin{overpic}[width=231.26378pt]{./figures/intercell_new}
\put(30.0,23.0){\small$h_{11}$} \put(40.0,22.0){\small$h_{21}$}
\put(53.0,21.0){\small$h_{12}$} \put(66.0,19.0){\small$h_{22}$}
\put(35.0,30.0){\small$g_{1}$} \put(58.0,29.5){\small$g_{2}$}
\put(40.0,11.0){\small$\mathsf{C}_{1}$}
\put(55.0,11.0){\small$\mathsf{C}_{2}$} \put(23.0,37.0){\small$P_{1}$}
\put(72.0,37.0){\small$P_{2}$} \end{overpic} Figure 1: A two-cell network with
an in-band reception and out-of-band-broadcasting relay for interference
mitigation
Another variation of the relay-interference channel involves an out-of-band
reception and in-band transmission relay. This channel is studied in [21], in
which the transmitter further splits the transmit signal according to the Han-
Kobayashi scheme; the relay decodes only part of the message depending the
capacity of the transmitter-relay links; the rest of the codewords are
transmitted directly from the sources to the destinations without the help of
the relay. With this partial decode-and-forward relaying scheme, the sum
capacity is found under a so-called strong relay-interference condition.
The interference channel with an in-band reception/out-of-band transmission
relay has been briefly discussed in [25], and studied in [22, 23] for a case
where the relay-destination links are shared between the two receivers.
Conventional decode-and-forward and compress-and-forward relay strategies are
not well matched for helping both receivers simultaneously with a common
relayed message. Thus, [22, 23] consider a generalized hash-and-forward (GHF)
strategy, which generalizes the conventional compress-and-forward scheme, and
is shown to achieve the capacity region of this channel model to within a
constant number of bits in the regime where the shared relay-destination link
rate is sufficiently small. The channel model under consideration in this
paper further extends the shared relay-destinations link to be a degraded
broadcast channel. We focus on a different weak-relay regime. The main
objective is similar: to efficiently use the relay bits to simultaneously
benefit both users and to achieve capacity to within a constant gap.
Finally, the GHF relay strategy used in this paper is essentially the same as
the noisy network coding [27, 28, 29] and the quantize-map-and-forward relay
strategies [30]. The result of this paper can be thought of as an effort in
generalizing these relay strategies to a particular case of the multiple
unicast setting, for which constant-gap result continues to hold for certain
channel-parameter regimes. Related works for the multiple unicast problem
include [31, 32, 33].
### I-B Main Contributions
This paper considers a relay-interference channel with in-band reception and
out-of-band degraded broadcasting links from the relay to the receivers. The
key features of the transmission strategy and the main results of the paper
are as follows.
#### I-B1 Incremental Relaying
This paper uses a GHF relaying strategy to take advantage of the in-band
reception link and the out-of-band broadcasting link from the relay to the
receivers. In GHF, the relay quantizes its observation, which is a linear
combination of the transmitted signals, using a fixed quantizer, then bins and
forwards the quantized observation to the receivers. This strategy of fixing
the quantization level is near optimal when a certain weak-relay condition is
satisfied, and is ideally matched to the degraded broadcasting relay-to-
receiver links with capacities $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$, because
it allows an incremental binning strategy at the relay. Assuming that
$\mathsf{C}_{1}\leq\mathsf{C}_{2}$, the relay may first bin its quantized
observation into $2^{n\mathsf{C}_{1}}$ bins and send the bin index to both
receivers, then further divide each bin into
$2^{n(\mathsf{C}_{2}-\mathsf{C}_{1})}$ sub-bins and sends the extra bin index
to receiver $2$ only. Thus, the relay message to the first receiver is a
degraded version of the message to the second receiver.
#### I-B2 Oblivious Power Splitting
The transmission scheme used in this paper consists of a Han-Kobayashi power
splitting strategy [2] at the transmitter. The common-private power splitting
ratio in such a strategy is crucial. In a study of the interference channel
with conferencing links [6], Wang and Tse used the power splitting strategy of
Etkin, Tse and Wang [1] where the private power is set at the noise level at
the receivers. This is sensible for the conferencing-receiver model considered
in [6], but not necessarily so for the interference channel with an
independent relay, unless again a certain weak-relay condition is satisfied.
This strategy of fixing the power splitting at the transmitter to be
independent of the relay is termed oblivious power splitting in [23].
Oblivious power splitting is used in this paper as well.
#### I-B3 Constant Gap to Capacity in the weak-relay Regime
The main result of this paper is that when the relay links are not unboundedly
stronger than the interfering links, i.e.,
$\max\left\\{\frac{|g_{1}|^{2}}{|h_{12}|^{2}},\frac{|g_{2}|^{2}}{|h_{21}|^{2}}\right\\}=\rho<\infty,$
(1)
for some fixed $\rho$, the capacity of the relay-interference channel with a
broadcast link can be achieved to within a constant gap, where the gap is a
function of $\rho$ but otherwise independent of channel parameters. This
operating regime is called the weak-relay regime in this paper.
The main result of this paper is motivated by the results in [22] and [23],
which studies a two-user interference channel augmented with a shared digital
relay link to the receivers of rate $R_{0}$, and obtains a constant-gap-to-
capacity result under a certain small-$R_{0}$ condition using GHF and
oblivious power splitting. The relay strategy studied in this paper goes one
step further in that the relay-to-receivers link is modeled as a degraded
broadcast channel. Moreover, the weak-relay regime studied in this paper is a
counterpart of the small-$R_{0}$ regime studied in [23], as can be visualized
in the practical setup of Fig. 1. When the mobiles are close to their
respective cell centers, the relay link capacities $\mathsf{C}_{1}$ and
$\mathsf{C}_{2}$ are small, thereby satisfying the small-$R_{0}$ condition of
[23]. In the more practically important regime where the mobile terminals are
close to the cell edge, the channel falls into the weak-relay regime of this
paper. An interesting feature of the result in this paper is that the gap to
capacity is a function of $\rho$, the relative channel strength between the
interfering channel and the channel to the relay; the gap becomes smaller as
$\rho\rightarrow 1$. In the limiting case with $\rho=1$, corresponding to the
situation where the mobiles are at the cell edge, the capacity region can be
achieved to within $\frac{1}{2}\log\frac{5+\sqrt{33}}{2}=1.2128$ bits.
A technical contribution of this paper is a particular set of capacity region
outer bounds which are established by giving different combinations of side
information (genies) to the receivers and by applying the known outer-bound
results of the Gaussian interference channel [1] and the single-input
multiple-output (SIMO) Gaussian interference channel [34]. It is shown that
there are two constraints for the individual rates $R_{1}$ and $R_{2}$, twelve
constraints for the sum rate $R_{1}+R_{2}$, six constraints for
$2R_{1}+R_{2}$, and six constraints for $R_{1}+2R_{2}$. Furthermore, the outer
bounds established in this paper hold for all channel parameters. This set of
outer bounds is tight to within a constant gap in the weak-relay regime.
To obtain insights from the performance gain brought by the relay, this paper
further investigates the improvement in the generalized degrees of freedom
(GDoF) per user for the relay-interference channel due to a broadcasting link.
In the symmetric setting, it is shown that a common broadcast link can improve
the sum capacity by two bits per each relay bit in the very weak, moderately
weak, and very strong interference regimes, but by one bit per each relay bit
in other regimes. This asymptotic behavior can be interpreted by noting that
the relay link essentially behaves like a deterministic channel in the high
signal-to-noise-ratio (SNR) regime. Further, in the symmetric setting, the
sum-capacity gain due to the relay can be thought of as solely coming from the
rate improvement of the common messages, or alternatively in a very weak
interference regime as solely coming from the rate improvement of the private
messages.
In asymmetric settings, the improvement in the sum capacity by the relay can
be interpreted in different ways. To illustrate this point, this paper
investigates a special case of the channel model, where the relay link is
available to only one but not both destinations. In this case, the relay may
forward information about both the intended signal and the interference, and
the capacity can benefit from both signal-relaying and interference-
forwarding. This paper shows that a constant-gap-to-capacity result can be
derived for this setting under a more relaxed weak-relay condition that
requires only $|g_{2}|\leq\sqrt{\rho}|h_{21}|$ (and not
$|g_{1}|\leq\sqrt{\rho}|h_{12}|$). Moreover, this paper shows that in term of
GDoF, when the relay link is above a certain threshold, the sum-capacity gain
is equivalent to that of that of a single relay link from user $1$. When the
relay link is below the threshold, the sum-capacity gain is equivalent to that
of a single relay link from user $2$.
Finally, the results of this paper show that GHF is sufficient for achieving
the approximated capacity region of an in-band reception and out-of-band
transmission Gaussian relay-interference channel in the weak-relay regime.
Thus, more recently proposed relay techniques based on compute-and-forward
[35] or lattice coding [36] is not necessary in this regime as far as constant
gap to capacity is concerned . Outside of the weak-relay regime, the optimal
relay strategies remain an open problem; lattice coding strategies may be
helpful.
### I-C Organization of This Paper
The rest of the paper is organized as follows. Section II introduces the
Gaussian relay-interference channel model, derives capacity region outer
bounds that hold for all channel parameters and an achievable rate region, and
presents the main constant-gap theorem and the GDoF analysis. Section III
deals with the relay-interference channel with a single relay link, derives
the corresponding constant-gap result, and gives a quantitative analysis on
the relation between signal relaying and interference forwarding. Section IV
concludes the paper.
## II Gaussian Relay-Interference Channel: General Case
Figure 2: Gaussian relay-interference channel with two independent digital
relay links
### II-A Channel Model and Definitions
A Gaussian relay-interference channel consists of two transmitter-receiver
pairs and an independent relay. Each transmitter communicates with the
intended receiver while causing interference to the other transmitter-receiver
pair. The relay receives a linear combination of the two transmit signals and
helps the transmitter-receiver pairs by forwarding a message to receiver $1$
and another message to receiver $2$ through rate-limited digital links with
capacities $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$ respectively. We start by
treating a channel model with independent relay links, and later show that
requiring one relay message to be a degraded version of the other is without
loss of approximate optimality. As shown in Fig. 2, $X_{1},X_{2}$ and
$Y_{1},Y_{2}$ are real-valued input and output signals, respectively, and
$Y_{R}$ is the observation of the relay. The receiver noises are assumed to be
independent and identically distributed (i.i.d.) Gaussian random variables
with variance one, i.e., $Z_{i}\sim\mathcal{N}(0,1),i=1,2$ and $R$. The input-
output relationship can be described by
$\displaystyle Y_{1}$ $\displaystyle=h_{11}X_{1}+h_{21}X_{2}+Z_{1},$ (2)
$\displaystyle Y_{2}$ $\displaystyle=h_{22}X_{2}+h_{12}X_{1}+Z_{2},$ (3)
$\displaystyle Y_{R}$ $\displaystyle=g_{1}X_{1}+g_{2}X_{2}+Z_{R},$ (4)
where $h_{ij}$ is the channel gain from transmitter $i$ to receiver $j$, and
$g_{j}$ is the channel gain from transmitter $j$ to the relay, all real
valued. The powers of the input signals are normalized to one, i.e.,
$\mathbb{E}[|X_{i}|^{2}]\leq 1,i=1,2$.
Define the signal-to-noise ratios and interference-to-noise ratios as follows:
$\displaystyle\mathsf{SNR}_{i}=|h_{ii}|^{2},\;$
$\displaystyle\quad\mathsf{SNR}_{ri}=|g_{i}|^{2},\;\;i=1,2$
$\displaystyle\mathsf{INR}_{1}=|h_{12}|^{2},$
$\displaystyle\quad\mathsf{INR}_{2}\;\;=|h_{21}|^{2}.$
Define functions $\alpha(\cdot)$ and $\beta(\cdot)$ as
$\alpha(x)=\frac{1}{2}\log(2x+2+\rho),\quad\beta(x)=\frac{1}{2}+\frac{1}{2}\log\left(1+\frac{1+\rho}{x}\right),$
(5)
where $\log(\cdot)$ is base $2$ and $\rho$ is defined as
$\rho\triangleq\max\left\\{\frac{|g_{1}|^{2}}{|h_{12}|^{2}},\frac{|g_{2}|^{2}}{|h_{21}|^{2}}\right\\}.$
(6)
This paper considers a weak-relay regime where $\rho$ is a finite constant.
### II-B Outer Bounds and Achievable Rate Region
We first present outer bounds and achievability results that are applicable to
the relay-interference channel model with two independent digital relays as
shown in Fig. 2.
###### Theorem 1 (Capacity Region Outer Bounds).
The capacity region of the Gaussian relay-interference channel as depicted in
Fig. 2 is contained in the outer bound $\overline{\mathcal{C}}$ given by the
set of $(R_{1},R_{2})$ for which
$\displaystyle R_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})$ (7)
$\displaystyle+\min\left\\{\mathsf{C}_{1},\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{r1}}{1+\mathsf{SNR}_{1}}\right)\right\\}$
$\displaystyle R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2})$ (8)
$\displaystyle+\min\left\\{\mathsf{C}_{2},\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{r2}}{1+\mathsf{SNR}_{2}}\right)\right\\}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1})$ (9)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (10)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{INR}_{2}+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{1}+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}\right)$
(12)
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})$
$\displaystyle\qquad\quad+\mathsf{SNR}_{r2}+\mathsf{INR}_{1}+\mathsf{SNR}_{r1})+\mathsf{C}_{1}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (13)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}+\mathsf{INR}_{2}\right)$
(14)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)+\mathsf{C}_{1}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}\right)$
(15)
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}$
$\displaystyle\qquad\quad+\mathsf{INR}_{2}+\mathsf{SNR}_{r2})+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1})$ (16)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}+\mathsf{INR}_{1}\right)$
(17)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}\right)$
(18)
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}$
$\displaystyle\qquad\quad+\mathsf{INR}_{1}+\mathsf{SNR}_{r1})$ $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}\right)$
(19)
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}$
$\displaystyle\qquad\quad+\mathsf{INR}_{2}+\mathsf{SNR}_{r2})$ $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}\right.$
(20)
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2}\right)$
(21)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{1}+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)+2\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}\right)$
(22)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (23)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)+\mathsf{C}_{1}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)$
(24)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}+\mathsf{INR}_{1}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{2}\right)+\mathsf{C}_{2}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (25)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)+2\mathsf{C}_{1}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (26)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{1}+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\mathsf{C}_{1}+\mathsf{C}_{2},$
and $R_{1}+2R_{2}$ bounded by (21)-(26) with indices $1$ and $2$ switched,
where $\phi_{1}^{2}$ and $\phi_{2}^{2}$ are defined as
$\phi_{1}^{2}=\left|\frac{g_{1}h_{21}}{g_{2}h_{11}}-1\right|^{2},\quad\phi_{2}^{2}=\left|\frac{g_{2}h_{12}}{g_{1}h_{22}}-1\right|^{2}.$
(27)
###### Proof:
The above outer bounds can be proved in a genie-aided approach. See Appendix
-A for details. ∎
###### Theorem 2 (Achievable Rate Region).
Let $\mathcal{P}$ denote the set of probability distributions $P(\cdot)$ that
factor as
$\displaystyle
P(q,w_{1},w_{2},x_{1},x_{2},y_{1},y_{2},y_{R},\hat{y}_{R1},\hat{y}_{R2})$ (28)
$\displaystyle=p(q)p(x_{1},w_{1}|q)p(x_{2},w_{2}|q)p(y_{1},y_{2},y_{R}|x_{1},x_{2},q)$
$\displaystyle\qquad p(\hat{y}_{R1},\hat{y}_{R2}|y_{R},q).$
For a fixed distribution $P\in\mathcal{P}$, let $\mathcal{R}(P)$ be the set of
all rate pairs $(R_{1},R_{2})$ satisfying
$\displaystyle 0\leq R_{1}$ $\displaystyle\leq$ $\displaystyle
d_{1}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{d}_{1}\right\\},$ (29)
$\displaystyle 0\leq R_{2}$ $\displaystyle\leq$ $\displaystyle
d_{2}+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{d}_{2}\right\\},$ (30)
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{2}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{a}_{1}\right\\}$
(31)
$\displaystyle+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{g}_{2}\right\\},$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{1}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{g}_{1}\right\\}$
(32)
$\displaystyle+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{a}_{2}\right\\},$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+e_{2}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{e}_{1}\right\\}$
(33)
$\displaystyle+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{e}_{2}\right\\},$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{1}+e_{2}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{a}_{1}\right\\}$
(34)
$\displaystyle+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{g}_{1}\right\\}$
$\displaystyle+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{e}_{2}\right\\},$
$\displaystyle R_{1}+2R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{2}+e_{1}+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{a}_{2}\right\\}$
(35)
$\displaystyle+\min\left\\{(\mathsf{C}_{2}-\xi_{2})^{+},\Delta{g}_{2}\right\\}$
$\displaystyle+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{e}_{1}\right\\},$
where
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1}|W_{1},W_{2},Q),$ (36) $\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle I(X_{1};Y_{1}|W_{2},Q),$ (37) $\displaystyle e_{1}$
$\displaystyle=$ $\displaystyle I(X_{1},W_{2};Y_{1}|W_{1},Q),$ (38)
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle I(X_{1},W_{2};Y_{1}|Q),$
(39) $\displaystyle\Delta{a}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};\hat{Y}_{R1}|Y_{1},W_{1},W_{2},Q),$ (40) $\displaystyle\Delta{d}_{1}$
$\displaystyle=$ $\displaystyle I(X_{1};\hat{Y}_{R1}|Y_{1},W_{2},Q),$ (41)
$\displaystyle\Delta{e}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1},W_{2};\hat{Y}_{R1}|Y_{1},W_{1},Q),$ (42) $\displaystyle\Delta{g}_{1}$
$\displaystyle=$ $\displaystyle I(X_{1},W_{2};\hat{Y}_{R1}|Y_{1},Q),$ (43)
$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
I(Y_{R};\hat{Y}_{R1}|Y_{1},X_{1},W_{2},Q),$ (44)
and
$a_{2},\Delta{a}_{2},d_{2},\Delta{d}_{2},e_{2},\Delta{e}_{2},g_{2},\Delta{g}_{2}$,
and $\xi_{2}$ are defined by (36)-(44) with indices $1$ and $2$ switched. Then
$\mathcal{R}=\bigcup_{P\in\mathcal{P}}\mathcal{R}(P)$ (45)
is an achievable rate region for the Gaussian relay-interference channel as
shown in Fig. 2.
###### Proof.
The achievable scheme consists of a Han-Kobayashi strategy at the transmitters
and a generalized hash-and-forward strategy at the relay. They are the same
strategies as adopted in [23] except that unlike the GHF relaying scheme in
[23, Theorem 2], where the relay quantizes the received signal and broadcasts
its bin index to both receivers through a shared digital link, the relay here
quantizes the received signal with two different quantization resolutions,
then sends the bin indices of the quantized signals to the receivers through
separated digital links of rates $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$. The
following is a sketch of the encoding/decoding process.
Encoding: Each transmit signal is comprised of a common message of rate
$T_{i}$ and a private message of rate $S_{i}$. The common message codewords
$W_{i}^{n}(j)$, $j=1,2,\cdots,2^{nT_{i}}$ of length $n$ are generated
according to the probability distribution $\Pi_{i=1}^{n}p(w_{i}|q)$, where
$q\sim p(q)$ serves as the time-sharing random variable. Based on the common
message codewords, user $i$ generates codewords
$X_{i}^{n}(j,k),k=1,2,\cdots,2^{nS_{i}}$ of length $n$ following the
conditional distribution $\Pi_{i=1}^{n}p(x_{i}|w_{i},q)$. Each input message
$\theta_{i}\in[1,2,\cdots,2^{S_{i}+T_{i}}],i=1,2$ is mapped to a message pair
$(s_{i},t_{i})\in[1,\cdots,2^{S_{i}}]\times[1,\cdots,2^{T_{i}}]$, then sent to
the destinations as $X_{i}^{n}(s_{i},t_{i})$. At the relay, the quantization
codebook is generated according to the probability distribution
$p(\hat{y}_{R1},\hat{y}_{R2}|y_{R},q)$. After receiving $Y_{R}^{n}$, the relay
quantizes $Y_{R}^{n}$ into $\hat{Y}_{R1}^{n}$ and $\hat{Y}_{R2}^{n}$, then
bins $\hat{Y}_{R1}^{n}$ to $2^{nC_{1}}$ bins, and bins $\hat{Y}_{R1}^{n}$ to
$2^{nC_{1}}$ bins, and sends the bin indices to the receivers through the
digital links.
Decoding: The decoding process follows the Han-Kobayashi framework:
$X_{1}^{n}$ and $W_{2}^{n}$ are decoded by receiver $1$ with the help of the
index of the relayed message $\hat{Y}_{R1}^{n}$; $X_{2}^{n}$ and $W_{1}^{n}$
are decoded by receiver $2$ with the help of the index of the relayed message
$\hat{Y}_{R2}^{n}$. To decode, receiver $1$ first constructs a list of
candidates for the relayed message $\hat{Y}_{R1}^{n}$, then jointly decodes
$X_{1}^{n}$, $W_{2}^{n}$ and $\hat{Y}_{R1}^{n}$ using typicality decoding.
Similarly, receiver $2$ jointly decodes $X_{2}^{n}$, $W_{1}^{n}$ and
$\hat{Y}_{R2}^{n}$. Following the error probability analysis in [23, Theorem
2], the rate tuple $(S_{1},T_{1},S_{2},T_{2})$ satisfying the following
constraints is achievable:
Constraints at receiver 1:
$\displaystyle S_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{1};Y_{1}|W_{1},W_{2},Q)+(\mathsf{C}_{1}-\xi_{1})^{+},$
(46) $\displaystyle\qquad\;I(X_{1};Y_{1},\hat{Y}_{R1}|W_{1},W_{2},Q)\\}$
$\displaystyle S_{1}+T_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{1};Y_{1}|W_{2},Q)+(\mathsf{C}_{1}-\xi_{1})^{+},$
(47) $\displaystyle\qquad\;I(X_{1};Y_{1},\hat{Y}_{R1}|W_{2},Q)\\}$
$\displaystyle S_{1}+T_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{1},W_{2};Y_{1}|W_{1},Q)+(\mathsf{C}_{1}-\xi_{1})^{+},$
(48) $\displaystyle\qquad\;I(X_{1},W_{2};Y_{1},\hat{Y}_{R1}|W_{1},Q)\\}$
$\displaystyle S_{1}+T_{1}+T_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{1},W_{2};Y_{1}|Q)+(\mathsf{C}_{1}-\xi_{1})^{+},$
(49) $\displaystyle\qquad\;I(X_{1},W_{2};Y_{1},\hat{Y}_{R1}|Q)\\}$
Constraints at receiver 2:
$\displaystyle S_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{2};Y_{2}|W_{1},W_{2},Q)+(\mathsf{C}_{2}-\xi_{2})^{+},$
(50) $\displaystyle\qquad\;I(X_{2};Y_{2},\hat{Y}_{R2}|W_{1},W_{2},Q)\\}$
$\displaystyle S_{2}+T_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{2};Y_{2}|W_{1},Q)+(\mathsf{C}_{2}-\xi_{2})^{+},$
(51) $\displaystyle\qquad\;I(X_{2};Y_{2},\hat{Y}_{R2}|W_{1},Q)\\}$
$\displaystyle S_{2}+T_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{2},W_{1};Y_{2}|W_{2},Q)+(\mathsf{C}_{2}-\xi_{2})^{+},$
(52) $\displaystyle\qquad\;I(X_{2},W_{1};Y_{2},\hat{Y}_{R2}|W_{2},Q)\\}$
$\displaystyle S_{2}+T_{2}+T_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{I(X_{2},W_{1};Y_{2}|Q)+(\mathsf{C}_{2}-\xi_{2})^{+},$
(53) $\displaystyle\qquad\;I(X_{2},W_{1};Y_{2},\hat{Y}_{R2}|Q)\\}$
The achievable rate region consists of all rate pairs $(R_{1},R_{2})$ such
that $R_{1}=S_{1}+T_{1}$ and $R_{2}=S_{2}+T_{2}$. Applying the Fourier-Motzkin
elimination procedure [37] gives the achievable rate region (29)-(35). ∎
We remark here that although both Theorem 1 and Theorem 2 are stated for the
digital noise-free relay-destination links, it can be easily verified that
both results continue to hold when the digital links are replaced by analog
additive Gaussian noise channels. The fact that the achievable rate region for
the analog channel is at least as large as the rate region for the digital
channel is obvious since one can always digitize the analog link. The fact
that the outer bound continues to hold can be verified by going through the
proof of that converse in Appendix -A. The outer bounds in the converse
involve terms like $I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})$, which is in turn upper
bounded by $I(X_{1}^{n};Y_{1}^{n})+n\mathsf{C}_{1}$. It is easy to show that
when the digital link $\mathsf{C}_{1}$ is replaced by an analog link with
input $X_{a1}$ and output $Y_{a1}$, the mutual information term is upper
bounded by $I(X_{1}^{n};Y_{1}^{n})+I(X_{a1}^{n};Y_{a1}^{n})$. As a result, all
the outer bounds in Theorem 1 continue to hold in the case of the analog relay
link with $\mathsf{C}_{1}$ replaced by $I(X_{a1};Y_{a1})$ and $\mathsf{C}_{2}$
replaced by $I(X_{a2};Y_{a2})$.
### II-C Constant Gap in the Weak-Relay Regime
We now specialize to the Gaussian case, and show that under the weak-relay
condition (1), the achievable rate region and the outer bounds of the Gaussian
relay-interference channel with independent relay links can be made to be
within a constant gap to each other. The relaying strategy that achieves this
capacity to within a constant gap turns out to be naturally suited for the
Gaussian relay-interference channel with a degraded broadcasting relay, thus
establishing the constant-gap result for the broadcasting-relay case as well.
Assuming Gaussian codebooks and a Gaussian quantization scheme, the key design
parameters are the choice of common-private power splitting ratio at the
transmitters and the quantization level at the relay. Our choice of design
parameters is inspired by that of Wang and Tse [6], where the capacity region
of a Gaussian interference channel with rate-limited receiver cooperation is
characterized to within a constant gap. Two key observations are made in [6].
First, the Etkin-Tse-Wang strategy [1] of setting the private power to be at
the noise level at the opposite receiver is used. Second, the relay quantizes
its observation at the private signal level in order to preserve all the
information of interest to the destinations. At the destinations, a joint
decoding (see [38, 39, 22, 30]) is performed to recover the source messages.
Consider now the optimal power splitting in a Gaussian relay-interference
channel with independent relay links. The Etkin-Tse-Wang strategy, i.e.,
setting private powers $P_{ip}$ as
$P_{1p}=\min\\{1,h_{12}^{-2}\\},\quad P_{2p}=\min\\{1,h_{21}^{-2}\\}.$ (54)
is near optimal for the Gaussian interference channel with conferencing
receivers, but is not necessarily so for relay-interference channel shown in
Fig. 2 in its most general form. Consider an extreme scenario of
$\mathsf{C}_{1},\mathsf{C}_{2}\rightarrow\infty$. In this case, the relay
fully cooperates with both receivers, so the relay-interference channel
becomes a single-input multiple-output (SIMO) interference channel with two
antennas at the receivers. Thus, the private powers at the transmitters must
be set at the effective noise level for the two-antenna output in order to
achieve capacity to within constant bits [34] [40], i.e.,
$P_{1p}=\min\\{1,(g_{1}^{2}+h_{12}^{2})^{-1}\\},\quad
P_{2p}=\min\\{1,(g_{2}^{2}+h_{21}^{2})^{-1}\\}.$ (55)
When $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$ are finite, the optimal power
splitting strategy is expected to be a function of not only $h_{12}$ and
$h_{21}$ but also $g_{1}$, $g_{2}$, $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$,
lying somewhere between (54) and (55).
This complication can be avoided, however, if we focus on the weak-relay
regime (1), namely $|g_{1}|\leq\sqrt{\rho}|h_{12}|$ and
$|g_{2}|\leq\sqrt{\rho}|h_{21}|$ for some finite constant $\rho$. In this
case, the power splittings (54) and (55) differ by at most a constant factor.
The main result of this section shows that in this weak-relay regime, the
Etkin-Tse-Wang’s original power splitting (54) is sufficient for achieving the
capacity of the Gaussian relay-interference channel to within a constant gap
(which is a function of $\rho$).
Consider next the optimization of the quantization level. Applying the insight
of [6] to the Gaussian relay-interference channel with independent relay links
shown in Fig. 2, the quantized messages for two receivers can be expressed as
$\displaystyle\hat{Y}_{R1}=g_{1}U_{1}+g_{1}W_{1}+g_{2}W_{2}+\overbrace{g_{2}U_{2}+Z_{R}}^{\text{of
no interest at $Y_{1}$}}+\eta_{1}$ (56)
$\displaystyle\hat{Y}_{R2}=g_{1}W_{1}+g_{2}U_{2}+g_{2}W_{2}+\underbrace{g_{1}U_{1}+Z_{R}}_{\text{of
no interest at $Y_{2}$}}+\eta_{2}$ (57)
where $W_{i}$ and $U_{i}$ are common message and private message respectively,
and $\eta_{i}\sim\mathcal{N}(0,\mathsf{q}_{i})$ is the quantization noise,
$i=1,2$. Therefore, a reasonable choice of the quantization levels for
receiver $1$ and receiver $2$ is
$\mathsf{q}_{1}=1+g_{2}^{2}P_{2p},\quad\mathsf{q}_{2}=1+g_{1}^{2}P_{1p}.$ (58)
Now observe that in the weak-relay regime, i.e.,
$|g_{1}|\leq\sqrt{\rho}|h_{12}|,|g_{2}|\leq\sqrt{\rho}|h_{21}|$, the above
quantization levels (with Etkin-Tse-Wang power splitting) are between $1$ and
the constant $\rho+1$. Thus, we can choose the quantization levels to be a
constant and optimize it between $1$ and $\rho+1$.
###### Theorem 3 (Constant Gap in the Weak-Relay Regime).
For the Gaussian relay-interference channel with independent relay links as
depicted in Fig. 2, in the weak-relay regime, using the generalized hash-and-
forward relaying scheme with quantization levels
$\mathsf{q}_{1}=\mathsf{q}_{2}=\frac{\sqrt{\rho^{2}+16\rho+16}-\rho}{4}$,
where $\rho$ is defined in (6), and using the Han-Kobayashi scheme with Etkin-
Tse-Wang power splitting strategy, $X_{i}=U_{i}+W_{i},i=1,2$, where $U_{i}$
and $W_{i}$ are both Gaussian distributed with the powers of $U_{1}$ and
$U_{2}$ set according to $P_{1p}=\min\\{1,h_{12}^{-2}\\}$ and
$P_{2p}=\min\\{1,h_{21}^{-2}\\}$, respectively, the achievable rate region
given in Theorem 2 is within
$\delta=\frac{1}{2}\log\left(2+\frac{\rho+\sqrt{\rho^{2}+16\rho+16}}{2}\right)$
(59)
bits of the capacity region outer bound in Theorem 1.
###### Proof:
The main step is to show that using superposition coding
$X_{i}=U_{i}+W_{i},i=1,2$, where $U_{i}\sim\mathcal{N}(0,P_{ip})$ and
$W_{i}\sim\mathcal{N}(0,P_{ic})$ with $P_{ip}+P_{ic}=1$,
$P_{1p}=\min\\{1,h_{12}^{-2}\\}$, and $P_{2p}=\min\\{1,h_{21}^{-2}\\}$, each
of the achievable rate constraints in (29)-(35) is within a finite gap to the
corresponding upper bound in (7)-(26). Specifically, it is shown in Appendix
-C that
(i) Individual rate (29) is within
$\delta_{R_{1}}=\max\left\\{\alpha(\mathsf{q}_{1}),\beta(\mathsf{q}_{1})\right\\}$
(60)
bits of the upper bound (7), where $\alpha(\cdot)$ and $\beta(\cdot)$ are as
defined in (5);
(ii) Individual rate (30) is within
$\delta_{R_{2}}=\max\left\\{\alpha(\mathsf{q}_{2}),\beta(\mathsf{q}_{2})\right\\}$
(61)
bits of the upper bound (8);
(iii) Sum rates (31), (32), and (33) are within
$\displaystyle\delta_{R_{1}+R_{2}}$ $\displaystyle=$
$\displaystyle\max\left\\{\alpha(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2}),\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{2}),\right.$
(62)
$\displaystyle\quad\left.\beta(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2}),\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})\right\\}$
bits of the upper bounds (9)-(20);
(iv) $2R_{1}+R_{2}$ rate (34) is within
$\displaystyle\delta_{2R_{1}+R_{2}}$ $\displaystyle=$
$\displaystyle\max\left\\{2\alpha(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2}),2\beta(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2}),\right.$
(63)
$\displaystyle\qquad\;\;\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2}),$
$\displaystyle\qquad\;\;2\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{2}),2\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
$\displaystyle\qquad\;\;\left.\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})\right\\}$
bits of the upper bounds (21)-(26);
(v) $R_{1}+2R_{2}$ rate (35) is within
$\displaystyle\delta_{R_{1}+2R_{2}}$ $\displaystyle=$
$\displaystyle\max\left\\{\alpha(\mathsf{q}_{1})+2\alpha(\mathsf{q}_{2}),\alpha(\mathsf{q}_{1})+2\beta(\mathsf{q}_{2}),\right.$
(64)
$\displaystyle\qquad\;\;\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})+\alpha(\mathsf{q}_{2}),$
$\displaystyle\qquad\;\;\beta(\mathsf{q}_{1})+2\alpha(\mathsf{q}_{2}),\beta(\mathsf{q}_{1})+2\beta(\mathsf{q}_{2}),$
$\displaystyle\qquad\;\;\left.\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})+\alpha(\mathsf{q}_{2})\right\\}$
bits of the upper bounds not shown explicitly in Theorem 1 but can be obtained
by switching the indices $1$ and $2$ of (21)-(26).
Since $\alpha(\cdot)$ is a monotonically increasing function and
$\beta(\cdot)$ is a monotonically decreasing function. In order to minimize
the above gaps over $\mathsf{q}_{1}$ and $\mathsf{q}_{2}$, the quantization
levels should be set such that
$\alpha(\mathsf{q}_{1}^{*})=\beta(\mathsf{q}_{1}^{*})=\alpha(\mathsf{q}_{2}^{*})=\beta(\mathsf{q}_{2}^{*}),$
(65)
which results in
$\mathsf{q}_{1}^{*}=\mathsf{q}_{2}^{*}=\frac{\sqrt{\rho^{2}+16\rho+16}-\rho}{4}$.
Substituting $\mathsf{q}_{1}^{*}$ and $\mathsf{q}_{2}^{*}$ into the above
gaps, we prove that the constant gap is $\delta$ bits per dimension, where
$\delta$ is given in (59). ∎
Note that the finite capacity gap is an increasing function of $\rho$: smaller
$\rho$ results in a smaller gap. In the case that $\rho=1$, i.e.,
$|g_{1}|\leq|h_{12}|$ and $|g_{2}|\leq|h_{21}|$, the optimal quantization
levels are $\mathsf{q}_{1}^{*}=\mathsf{q}_{2}^{*}=\frac{\sqrt{33}-1}{4}$, and
the gap to the capacity is given by
$\frac{1}{2}\log\left(\frac{5+\sqrt{33}}{4}\right)=1.2128$ bits.
### II-D Gaussian Relay-Interference Channel with a Broadcasting Relay
(a) GHF
(b) Incremental GHF with a refinement process
(c) Universal GHF
Figure 3: Evolution of the generalized hash-and-forward relay scheme
The GHF relaying scheme originally stated in Theorem 2 requires independent
relay links. As shown in Fig. 3(a), the relay observation $Y_{R}^{n}$
undergoes two separate quantization and binning processes to obtain the two
messages for the two receivers. However, in the weak-relay regime, Theorem 3
shows that using an identical quantization level for the two receivers is
without loss of approximate optimality, thus a common quantization process can
be shared between the two receivers. Further, since the same $\hat{Y}_{R}^{n}$
is binned into bins of sizes $2^{n\mathsf{C}_{1}}$ and $2^{n\mathsf{C}_{2}}$,
this is equivalent to first binning $\hat{Y}_{R}^{n}$ into
$2^{n\mathsf{C}_{1}}$ bins (assuming $\mathsf{C}_{1}\leq\mathsf{C}_{2}$) then
further binning each bin into $2^{n(\mathsf{C}_{2}-\mathsf{C}_{1})}$ sub-bins,
as shown in Fig. 3(b). The message sent to receiver 2 can be thought of as the
refinement of the message sent to receiver 1. This is exactly the incremental
relaying strategy we seek for the Gaussian interference channel with a
broadcasting relay, where the message to receiver 1 is a degraded version of
the message to receiver 2\. Finally, if
$\mathsf{C}_{1}=\mathsf{C}_{2}=\mathsf{C}$, the relay-interference channel
reduces to the universal relaying scheme studied in [23], where a digital link
is shared between the relay and the receivers, as shown in Fig. 3(c). We note
here that the outer bounds for the independent relay link case (Theorem 1)
continues to hold for the degraded broadcast relay case.
###### Corollary 1.
The constant-gap-to-capacity result stated in Theorem 3 holds also for the
Gaussian relay-interference channel with degraded broadcasting relay links,
where (assuming $\mathsf{C}_{1}\leq\mathsf{C}_{2}$) the message sent through
the link with capacity $\mathsf{C}_{1}$ must be a degraded version of the
message sent through the link with capacity $\mathsf{C}_{2}$.
### II-E Comments on the Strong-Relay Regime
The constant-gap result in this paper holds only in the weak-relay regime of
$|g_{1}|\leq\sqrt{\rho}|h_{12}|$ and $|g_{2}|\leq\sqrt{\rho}|h_{21}|$, where
$\rho$ is finite. The main difficulty in extending this result to the general
case is that both the choice of the Han-Kobayashi power splitting ratio and
the GHF relay strategy are no longer optimal in the strong-relay regime. As
mentioned earlier, the Etkin-Tse-Wang power splitting is not optimal when the
relay links $g_{i},i=1,2$ grow unboundedly stronger than the interference
links $h_{12}$ and $h_{21}$. Further, GHF may not be an appropriate relay
strategy. To see this, assume a channel model with separate relay links and
consider an extreme scenario where the relay links $g_{i},i=1,2$ go to
infinity, while all other channel parameters are kept constant. This special
case is known as the cognitive relay-interference channel [41]. The capacity
region outer bound of Theorem 1 for this case reduces to
$\displaystyle R_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})+\mathsf{C}_{1}$ (66)
$\displaystyle R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2})+\mathsf{C}_{2}$ (67)
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1})$ (68)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})$ (69)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{INR}_{2}+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{1}+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2}\right)$
(71)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{1}+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)+2\mathsf{C}_{1}+\mathsf{C}_{2}$
$\displaystyle R_{1}+2R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1}\right)$
(72)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{2}+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}+2\mathsf{C}_{2},$
which is in fact the outer bound of the underlying interference channel
expanded by $\mathsf{C}_{1}$ bits in the $R_{1}$ direction and
$\mathsf{C}_{2}$ in the $R_{2}$ direction. In this special case, a decode-and-
forward strategy can easily achieve the capacity region to within a constant
gap. This is because the relay is capable of decoding all the source messages,
so it can simply forward the bin indices of the privates messages to achieve
$(R_{1}+\mathsf{C}_{1},R_{2}+\mathsf{C}_{2})$ for any achievable rate pair
$(R_{1},R_{2})$ in the absence of the relay. Etkin-Tse-Wang power splitting
with decode-and-forward then achieves the outer bound to within a constant
gap. In contrast, GHF cannot achieve the capacity region to within a constant
gap in this case.
### II-F Generalized Degrees of Freedom
We can gain further insights into the effect of relaying on the Gaussian
interference channel by analyzing the GDoF of the sum rate in the symmetric
channel setting. Consider the case where
$\mathsf{INR}_{1}=\mathsf{INR}_{2}=\mathsf{INR}$,
$\mathsf{SNR}_{1}=\mathsf{SNR}_{2}=\mathsf{SNR}$,
$\mathsf{SNR}_{r1}=\mathsf{SNR}_{r2}=\mathsf{SNR}_{r}$, and
$\mathsf{C}_{1}=\mathsf{C}_{2}=\mathsf{C}$. In the high SNR regime, similar to
[1, 6], define
$\displaystyle\alpha$ $\displaystyle:=$
$\displaystyle\lim_{\mathsf{SNR}\rightarrow\infty}\frac{\log\mathsf{INR}}{\log\mathsf{SNR}},$
(73) $\displaystyle\beta$ $\displaystyle:=$
$\displaystyle\lim_{\mathsf{SNR}\rightarrow\infty}\frac{\log\mathsf{SNR}_{r}}{\log\mathsf{SNR}},$
(74) $\displaystyle\kappa$ $\displaystyle:=$
$\displaystyle\lim_{\mathsf{SNR}\rightarrow\infty}\frac{\mathsf{C}}{\frac{1}{2}\log\mathsf{SNR}}.$
(75)
The GDoF of the sum capacity is defined as
$d_{\mathrm{sum}}=\left.\lim_{\mathsf{SNR}\rightarrow\infty}\frac{C_{\mathrm{sum}}}{\frac{1}{2}\log\mathsf{SNR}}\right|_{\mathrm{fixed}\;\alpha,\beta,\kappa}$
(76)
As a direct consequence of the constant-gap result, $d_{\mathrm{sum}}$ can be
characterized in the weak-relay regime as follows.
###### Corollary 2.
For the symmetric Gaussian relay-interference channel in the weak-relay regime
(i.e., $\beta\leq\alpha$), the GDoF of the sum capacity is given by the
following. When $0\leq\alpha<1$
$\displaystyle d_{\mathrm{sum}}$ $\displaystyle=$
$\displaystyle\min\left\\{2-\alpha+\min\\{\beta,\kappa\\},2\max\\{\alpha,1-\alpha\\}+2\kappa,\right.$
(77) $\displaystyle\left.\qquad\;2\max\\{\alpha,1+\beta-\alpha\\}\right\\}.$
When $\alpha\geq 1$
$\displaystyle
d_{\mathrm{sum}}=\min\left\\{\alpha+\kappa,\alpha+\beta,2(1+\kappa),2\max\\{1,\beta\\}\right\\}.$
(78)
Note that when $\alpha=1$, the GDoF of the sum capacity is in fact not well
defined. This is because both $\mathsf{INR}=\gamma\mathsf{SNR}$ (where
$\gamma\neq 1$ is finite) and $\mathsf{INR}=\mathsf{SNR}$ result in the same
$\alpha=1$. However, in the case of $\mathsf{INR}=\mathsf{SNR}$, the channel
becomes ill conditioned, i.e. $\phi_{1}=\phi_{2}=0$, which results in a
$d_{\mathrm{sum}}$ other than the one in (78). In other words, multiple values
of $d_{\mathrm{sum}}$ correspond to the same $\alpha=1$. This is similar to
the situation of [6, Theorem 7.3]. Applying the similar argument that the
event $\left\\{\mathsf{INR}=\mathsf{SNR}\right\\}$ is of zero measure, we have
the GDoF of the sum capacity as shown in (78) almost surely.
When the relay links and the interference links share the same channel gain,
i.e. $\alpha=\beta$, the GDoF of the sum capacity reduces to
$\displaystyle
d_{\mathrm{sum}}=\min\left\\{2+\kappa-\alpha,2\max\\{\alpha,1-\alpha\\}+2\kappa,2\right\\}$
(79)
for $0\leq\alpha<1$, and
$d_{\mathrm{sum}}=\min\left\\{\alpha+\kappa,2(1+\kappa),2\alpha\right\\},$
(80)
for $\alpha\geq 1$. Interestingly, this is the same as the sum capacity (in
GDoF) of the Gaussian interference channel with rate-limited receiver
cooperation [6]. Therefore, the same sum capacity GDoF gain can be achieved
with either receiver cooperation or with an independent in-band-reception and
out-of-band-transmission relay assuming that the source-relay links are the
same as the interfering links of the underlying interference channel (i.e.
$\alpha=\beta$).
Fig. 4 shows the GDoF gain due to the relay for the $\alpha=\beta$ case. There
are several interesting features. When $\kappa=0.2$, the GDoF curve remains
the “W” shape for the conventional Gaussian interference channel [1]. The sum-
capacity gain is $2\kappa$ in the very and moderately weak interference
regimes (when $0.2\leq\alpha\leq 0.6$) or the very strong interference regime
($\alpha\geq 2.2$), and is $\kappa$ in other regimes
($\frac{2}{3}\leq\alpha\leq 2$). As $\kappa$ gets larger, the left “V” branch
of the “W” curve becomes smaller, and it disappears completely at the critical
point of $\kappa=0.5$. As $\kappa$ keeps increasing, the right “V” of the “W”
curve also eventually disappears. The detailed sum-capacity gains for
different values of $\alpha$ are listed in Table I.
TABLE I: Sum-capacity GDoF gain due to the relay for the symmetric Gaussian relay-interference channel for the $\alpha=\beta$ and $\kappa\leq\frac{1}{2}$ case Range of $\alpha$ | $\alpha\leq\kappa$ | $\kappa\leq\alpha\leq\frac{2-\kappa}{3}$ | $\frac{2-\kappa}{3}\leq\alpha\leq\frac{2}{3}$ | $\frac{2}{3}\leq\alpha\leq 2$ | $2\leq\alpha\leq 2+\kappa$ | $\alpha\geq 2+\kappa$
---|---|---|---|---|---|---
Gain | $2\alpha$ | $2\kappa$ | $2-3\alpha+\kappa$ | $\kappa$ | $\alpha+\kappa-2$ | $2\kappa$
Figure 4: The GDoF gain due to the relay in a symmetric Gaussian relay-
interference channel for the $\alpha=\beta$ case
### II-G Interpretation via the Deterministic Relay Channel
In the Han-Kobayashi framework, each input signal of the interference channel
consists of both a common message and a private message. The sum-capacity gain
due to the relay in the relay-interference channel therefore in general
includes improvements in both the common and the private message rates. This
section illustrates that in the asymptotic high SNR regime, the rate
improvement can be interpreted as either a private rate gain alone, or a
common rate gain alone. Further, the one-bit-per-relay-bit or the two-bits-
per-relay-bits GDoF improvement shown in the previous section can be
interpreted using a deterministic relay model. The rest of this section
illustrates this point for the symmetric Gaussian relay-interference channel
in the $\alpha=\beta$ and $\kappa\leq\frac{1}{2}$ case as an example.
#### II-G1 Very Weak Interference Regime
For the symmetric Gaussian interference channel, in the very weak interference
regime of $0\leq\alpha\leq\frac{1}{2}$, common messages do not carry any
information (although it can be assigned nonzero powers as in the Etkin-Tse-
Wang power splitting strategy). Setting $X_{1}$ and $X_{2}$ to be private
messages only is capacity achieving in terms of GDoF ([1, 42, 43, 44]).
Assigning $X_{1}$ and $X_{2}$ to be private only is also optimal for GDoF for
the symmetric Gaussian relay-interference channel in the very weak
interference regime. This is because when $X_{1}$ and $X_{2}$ are both private
messages and are treated as noises at $Y_{2}$ and $Y_{1}$ respectively, the
relay-interference channel asymptotically becomes two deterministic relay
channels in the high SNR regime. Consider the relay operation for $Y_{1}$ as
illustrated in Fig. 5(a). When noise variances of $Z_{1}$ and $Z_{R}$ go down
to zero, the observation at the relay becomes $Y_{R}=gX_{1}+gX_{2}$ and the
received signal at receiver $1$ becomes $Y_{1}=h_{d}X_{1}+h_{c}X_{2}$. In this
case, the relay’s observation is a deterministic function of $X_{1}$ and
$Y_{1}$, i.e. $Y_{R}=gX_{1}+\frac{g}{h_{c}}(Y_{1}-h_{d}X_{1})$. Thus $X_{1}$
and $Y_{1}$, along with the relay $Y_{R}$ form a deterministic relay channel
of the type studied in [45]. According to [45], the achievable rate of user
$1$ is given by
$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\min\left\\{I(X_{1};Y_{1},Y_{R}),I(X_{1};Y_{1})+\mathsf{C}\right\\}$
(81) $\displaystyle=$
$\displaystyle\min\left\\{\frac{1}{2}\log(1+h_{d}^{2}),\frac{1}{2}\log\left(1+\frac{h_{d}^{2}}{h_{c}^{2}}\right)+\mathsf{C}\right\\}$
$\displaystyle\rightarrow$ $\displaystyle\min\\{1,1-\alpha+\kappa\\},$
resulting in one-bit improvement for each relay bit in the regime
$\kappa\leq\alpha\leq\frac{1}{2}$. Similarly, as illustrated in Fig. 5(b),
$X_{2}$, $Y_{2}$, and $Y_{R}$ form another deterministic relay channel with
$X_{2}$ as the input, $Y_{2}$ as the output, and $Y_{R}$ as the relay. Thus,
the achievable rate of user $2$ is the same as user $1$, resulting in the same
one-bit-per-relay-bit improvement. Further, as shown in [45], a hash-and-
forward relay strategy achieves the capacity for deterministic relay channels.
As the hashing operation is the same for both case, the same relay bit can
therefore benefit both receivers at the same time, resulting in two-bit
increase in sum capacity for one relay bit, as first pointed out in [22].
(a) At receiver $1$, $X_{2}$ is treated as noise.
(b) At receiver $2$, $X_{1}$ is treated as noise.
Figure 5: Asymptotic deterministic relay channels in the very weak
interference regime $\kappa\leq\alpha\leq\frac{1}{2}$.
#### II-G2 Moderately Weak and Strong Interference Regimes
The above interpretation, which states that the GDoF improvement in the very
weak interference regime comes solely from the private rate gain, is not the
only possible interpretation. The rate gain can also be interpreted as
improvement in common information rate — an interpretation that applies not
only to the very weak interference regime, but in fact to all regimes (for the
symmetric rate with symmetric channels). In the following, we illustrate this
point by focusing on a two-stage Han-Kobayashi strategy, where common messages
are decoded first, then the private messages. This is the same two-stage Han-
Kobayashi scheme used in [1] for the Gaussian interference channel without the
relay.
Specifically, the relay uses the same GHF relaying strategy as in Theorem 3,
but it is now designed to help the common messages only. Here, both common
messages $W_{1}^{n}$ and $W_{2}^{n}$ are decoded and subtracted at both
receivers with the help of the GHF relay first (while treating private
messages as noise), the private messages are then decoded at each receiver
treating each other as noise. The decoding of the private message at the
second stage results in
$\displaystyle R_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{p}}{1+\mathsf{INR}_{p}}\right)$
(82) $\displaystyle\rightarrow$ $\displaystyle\max\\{0,1-\alpha\\},$
Note that the relay does not help the private rate.
In the common-message decoding stage, $W_{1}^{n}$ and $W_{2}^{n}$ are jointly
decoded at both receiver $1$ and receiver $2$ with the help of the GHF relay.
As a result, $(W_{1}^{n},W_{2}^{n},Y_{1}^{n},Y_{R}^{n})$ forms a multiple-
access relay channel at receiver $1$ with $W_{1}^{n},W_{2}^{n}$ as the inputs,
$Y_{1}^{n}$ as the output and $Y_{R}^{n}$ as the relay. The achievable rate
region of such a multiple-access channel with a GHF relay is given by
$\displaystyle R_{w1}$ $\displaystyle\leq$ $\displaystyle
I(W_{1};Y_{1}|W_{2})$
$\displaystyle+\min\left\\{(\mathsf{C}-\xi)^{+},I(W_{1};\hat{Y}_{R}|Y_{1},W_{2})\right\\}$
$\displaystyle R_{w2}$ $\displaystyle\leq$ $\displaystyle
I(W_{2};Y_{1}|W_{1})$
$\displaystyle+\min\left\\{(\mathsf{C}-\xi)^{+},I(W_{2};\hat{Y}_{R}|Y_{1},W_{1})\right\\}$
$\displaystyle R_{w1}+R_{w2}$ $\displaystyle\leq$ $\displaystyle
I(W_{1},W_{2};Y_{1})$
$\displaystyle+\min\left\\{(\mathsf{C}-\xi)^{+},I(W_{1},W_{2};\hat{Y}_{R}|Y_{1})\right\\}.$
With the Etkin-Tse-Wang input strategy (i.e.
$P_{1p}=\min\\{1,h_{12}^{-2}\\},P_{2p}=\min\\{1,h_{21}^{-2}\\}$) and the GHF
relaying scheme with
$\mathsf{q}_{1}=\mathsf{q}_{2}=\frac{\sqrt{\rho^{2}+16\rho+16}-\rho}{4}$, it
can be shown that the common-message rate region for the receiver 1 in the
high SNR regime in term of GDoF is given as follows. When $0\leq\alpha\leq 1$
$\displaystyle R_{w1}$ $\displaystyle\leq$ $\displaystyle\alpha$
$\displaystyle R_{w2}$ $\displaystyle\leq$
$\displaystyle\min\\{\alpha,\kappa+\max\\{2\alpha-1,0\\}\\}$ $\displaystyle
R_{w1}+R_{w2}$ $\displaystyle\leq$
$\displaystyle\alpha+\min\\{\alpha,\kappa\\}.$
When $\alpha\geq 1$
$\displaystyle R_{w1}$ $\displaystyle\leq$
$\displaystyle\min\\{\alpha,1+\kappa\\}$ $\displaystyle R_{w2}$
$\displaystyle\leq$ $\displaystyle\alpha$ $\displaystyle R_{w1}+R_{w2}$
$\displaystyle\leq$ $\displaystyle\alpha+\kappa.$
Due to symmetry, the rate region for the multiple-access relay channel at
receiver $2$ can be obtained by switching the indices $1$ and $2$.
Note that in suitable interference regimes, both the individual rate and the
sum rate can potentially be increased by one bit for each relay bit. This is
again a consequence of the fact that the relay operation has a deterministic
relay channel interpretation in the high SNR regime. For example, in the
strong interference regime where $1\leq\alpha\leq 2+\kappa$, the sum rate of
the multiple-access relay channel benefits by one bit for each relay bit in
the high SNR regime as shown in Fig. 6(a). In the very strong interference
regime, the interference can be decoded, subtracted or can serve as side
information, therefore the individual rate increases by one bit for each relay
bit as shown in Fig. 6(b).
(a) $Y_{1}$ decodes both $X_{1}$ and $X_{2}$.
(b) $X_{2}$ is decoded and serves as side information.
Figure 6: Asymptotic deterministic relay channels in the strong and very
strong interference regimes.
Now, the achievable rates of common messages can be obtained by intersecting
the two rate regions. Taking the achievable rates of private messages in (82)
into account, it is easy to verify that this two-stage Han-Kobayashi scheme
achieves the sum capacity in (79) and (80). As depicted in Fig. 4, the sum-
capacity gain due to the relay can be one-bit-per-bit or two-bits-per-bit. In
the following, we demonstrate in Fig. 7 how these gains are obtained by
pictorially showing the intersection of the two common-message rate regions
for different values of $\alpha$.
(a) $0\leq\alpha\leq\kappa$
(b) $\kappa\leq\alpha\leq\frac{1}{2}$
(c) $\frac{1}{2}\leq\alpha\leq\frac{2-\kappa}{3}$
(d) $\frac{2}{3}\leq\alpha\leq 1$
(e) $1\leq\alpha\leq 1+\kappa$
(f) $\alpha\geq 2+\kappa$
Figure 7: Generalized-degree-of-freedom gain due to relaying is roughly
$\kappa$ or $2\kappa$ depending on how the two common-message multiple-access
regions are intersected
* •
When $\alpha\leq\kappa$, as can be seen from Fig. 7(a), the two rate regions
are identical and are both given by
$\left\\{(R_{w1},R_{w2}):R_{w1}\leq\alpha,R_{w2}\leq\alpha\right\\}$. The
intersection of the two is the same rectangle with the top-right corner
located at $(\alpha,\alpha)$. This gives a $2\alpha$-bit gain over the
baseline, which is located at the origin.
* •
As $\alpha$ increases to $\kappa\leq\alpha\leq\frac{1}{2}$, the baseline rate
pair is still at the origin. With the help of the relay, the two common-
message rate regions become rectangles
$\left\\{(R_{w1},R_{w2}):R_{w1}\leq\alpha,R_{w2}\leq\kappa\right\\}$ and
$\left\\{(R_{w1},R_{w2}):R_{w1}\leq\kappa,R_{w2}\leq\alpha\right\\}$
respectively. As shown in Fig. 7(b), the intersection of the two gives a
square with the top-right corner located at $(\kappa,\kappa)$. As a result,
the sum-capacity gain is $2\kappa$ bits.
* •
As $\alpha$ increases to $\frac{1}{2}\leq\alpha\leq 1$, the common-message
rate regions at receivers $1$ and $2$ become pentagons. However, depending on
the value of $\alpha$, the sum rate improves by different amounts. When
$\alpha\leq\frac{2-\kappa}{3}$, as shown in Fig. 7(c), the intersection of the
two pentagon regions gives a square shape with the top-right corner located at
$(2\alpha-1+\kappa,2\alpha-1+\kappa)$. Compared with $(2\alpha-1,2\alpha-1)$
achieved without the relay, a sum-capacity gain of $2\kappa$ bits is obtained.
However, when $\alpha\geq\frac{2-\kappa}{3}$, as depicted in Fig. 7(d), the
intersection of the two rate regions is still a pentagon with the sum-capacity
limited by $R_{w1}+R_{w2}\leq 2-\alpha+\kappa$. In this case, depending on the
value of $\alpha$, the sum-rate gain is $2-3\alpha+\kappa$ bits when
$\frac{2-\kappa}{3}\leq\alpha\leq\frac{2}{3}$, and is $\kappa$ bits when
$\frac{2}{3}\leq\alpha\leq 1$. (The latter case is shown in Fig. 7(d).)
* •
When $1\leq\alpha\leq 2+\kappa$, the common-message rate regions are again
pentagons and the interpretation is similar to the
$\frac{2-\kappa}{3}\leq\alpha\leq 1$ case. Fig. 7(e) shows an example of
$1\leq\alpha\leq 1+\kappa$. In this case, the two rate regions are identical
pentagons with the sum capacity limited by $R_{w1}+R_{w2}\leq\alpha+\kappa$.
Compared with the baseline sum capacity, a $\kappa$-bits gain is obtained.
When $1+\kappa\leq\alpha\leq 2+\kappa$, the intersection of the two common-
message rate regions again gives a sum-capacity of $\alpha+\kappa$. However,
since the baseline sum capacity becomes saturated when when $\alpha\geq 2$
([1, 46, 47]), the sum-capacity gain over the baseline is $\kappa$ bits when
$1\leq\alpha\leq 2$, and is $\alpha+\kappa-2$ bits when $2\leq\alpha\leq
2+\kappa$.
* •
Finally, $\alpha\geq 2+\kappa$ falls into the very strong interference regime.
The intersection of the two common-message rate regions is a rectangle with
the top-right corner located at $(1+\kappa,1+\kappa)$ as shown in Fig. 7(f).
The sum-capacity gain is thus $2\kappa$ bits in the very strong interference
regime.
## III Gaussian Relay-Interference Channel With a Single Digital Link
The result of the previous section shows that for the symmetric channel, the
sum-capacity improvement can be thought as coming solely from the improvement
of the common message rate, or in a very weak interference regime as coming
solely from the improvement of the private message rates. Thus, the function
of the relay for the symmetric rate in symmetric channel is solely in
forwarding useful signals. This interpretation does not necessarily hold for
the asymmetric cases. In this section, we study a particular asymmetric
channel to illustrate the composition of the sum-capacity gain. We are
motivated by the fact that the relay’s observation in a relay-interference
channel is a linear combination of the intended signal and the interfering
signal. Clearly, forwarding the intended signal and the interfering signal can
both be beneficial (e.g. [12]). This section illustrates that depending on the
different channel parameters, the sum-rate gain from forwarding both intended
signal and interference signal happens to be the same as that of forwarding
intended signal only or forwarding interference signal only.
Specifically, we focus on a particular asymmetric model as shown in Fig. 8,
where the digital relay link exists only for receiver $1$, and not for
receiver $2$, i.e., $\mathsf{C}_{2}=0$. This section first derives a constant-
gap-to-capacity result for this channel. Note that this channel is a special
case of the general channel model studied in the previous section, but the
constant-gap-to-capacity result can be established in this special case for a
broader set of channels. Unlike the weak-relay assumption
$|g_{1}|\leq\sqrt{\rho}|h_{12}|$ and $|g_{2}|\leq\sqrt{\rho}|h_{21}|$ made in
the previous section, this section assumes that
$|g_{2}|\leq\sqrt{\rho}|h_{21}|$ only with no constraints on $g_{1}$ or
$h_{12}$. Under this channel setup, it can be shown that in the high SNR
regime, the sum capacity improvement can also be obtained as if only the
intended signal is forwarded or only the interference signal is forwarded.
Note that this conclusion applies to the case of a single relay-destination
link only, and not necessarily to the general case with two relay-destination
links.
### III-A Capacity Region to within Constant Gap in the Weak-Relay Regime
Since the channel model studied in Fig. 8 is a special case of the general
Gaussian relay-interference channel, we first simplify the achievable rate
region in Theorem 2 to the following corollary by setting $\mathsf{C}_{2}=0$.
The only difference in the coding scheme is that instead of performing two
quantizations as in the general relay-interference channel, the relay in Fig.
8 does one quantization of the received signal $Y_{R}$ into $\hat{Y}_{R1}$ and
sends the bin index of $\hat{Y}_{R1}$ to receiver $1$ through the digital link
$\mathsf{C}_{1}$.
###### Corollary 3.
Figure 8: Gaussian relay-interference channel with a single digital link
For the Gaussian relay-interference channel with a single digital link as
shown in Fig. 8, the following rate region is achievable:
$\displaystyle 0\leq R_{1}$ $\displaystyle\leq$ $\displaystyle
d_{1}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{d}_{1}\right\\}$ (83)
$\displaystyle 0\leq R_{2}$ $\displaystyle\leq$ $\displaystyle d_{2}$ (84)
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{2}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{a}_{1}\right\\}$
(85) $\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{1}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{g}_{1}\right\\}$
(86) $\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+e_{2}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{e}_{1}\right\\}$
(87) $\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{1}+e_{2}+\min\left\\{2(\mathsf{C}_{1}-\xi_{1})^{+},\right.$ (88)
$\displaystyle\left.\quad(\mathsf{C}_{1}-\xi_{1})^{+}+\Delta{a}_{1},\Delta{a}_{1}+\Delta{g}_{1}\right\\}$
$\displaystyle R_{1}+2R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{2}+e_{1}+\min\left\\{(\mathsf{C}_{1}-\xi_{1})^{+},\Delta{e}_{1}\right\\},$
where all the parameters are as defined in Theorem 2.
The proof follows directly from Theorem 2. Note that in (88), we apply the
fact that $\Delta{a}_{1}\leq\Delta{g}_{1}$. Likewise, the capacity region
outer bound in Theorem 1 also simplifies when $\mathsf{C}_{2}=0$. We can now
prove the following constant-gap theorem for the Gaussian relay-interference
channel with a single digital link.
###### Theorem 4.
For the Gaussian relay-interference channel with a single digital link as
depicted in Fig. 8, with the same signaling strategy as in Theorem 3, i.e. a
combination of the Han-Kobayashi scheme with Etkin-Tse-Wang power splitting
strategy and the GHF relaying scheme with the fixed quantization level
$\mathsf{q}_{1}=\frac{\sqrt{\rho^{2}+16\rho+16}-\rho}{4}$, in the weak-relay
regime of $|g_{2}|\leq\sqrt{\rho}|h_{21}|$, the achievable rate region in
Corollary 3 is within $\delta$ bits of the capacity region outer bound in
Theorem 1 (with $\mathsf{C}_{2}$ set to zero), where $\delta$ is defined in
Theorem 3.
###### Proof:
Although the signalling scheme and the constant gap result resemble those of
Theorem 3, Theorem 4 is not simply obtained by setting $\mathsf{C}_{2}=0$ in
Theorem 3, since the weak-relay condition has been relaxed. In the following,
we prove the constant-gap result by directly comparing each achievable rate
expression with its corresponding upper bound.
Applying the inequalities of Lemma 1 and following along the same lines of the
proof of Theorem 3 in Appendix -C, it is easy to show that each of the
achievable rates in (83)-(LABEL:R12R2_1_single_digital_link) achieves to
within a constant gap of its corresponding upper bound in Theorem 1 (with
$\mathsf{C}_{2}$ set to zero) in the weak-relay regime. The constant gaps are
shown as follows:
(i) Individual rate (83) is within
$\delta_{R_{1}}=\max\left\\{{\alpha}(\mathsf{q}_{1}),{\beta}(\mathsf{q}_{1})\right\\}$
(90)
bits of (7).
(ii) Individual rate (84) is within
$\delta_{R_{2}}=\frac{1}{2}$ (91)
bits of (8).
(iii) Sum rates (85), (86) and (87) are within
$\delta_{R_{1}+R_{2}}=\frac{1}{2}+\max\left\\{{\alpha}(\mathsf{q}_{1}),{\beta}(\mathsf{q}_{1})\right\\}$
(92)
bits of their upper bounds (9), (16), (10), (15),
(LABEL:sumrate_bound_3_twolinks), and (17). Specifically,
* •
The first term of (85) is within $\frac{1}{2}+{\beta}(\mathsf{q}_{1})$ bits of
(9). The second term is within $\frac{1}{2}+{\alpha}(\mathsf{q}_{1})$ bits of
(16).
* •
The first term of (86) is within $\frac{1}{2}+{\beta}(\mathsf{q}_{1})$ bits of
(10). The second term is within $\frac{1}{2}+{\alpha}(\mathsf{q}_{1})$ bits of
(15).
* •
The first term of (87) is within $\frac{1}{2}+{\beta}(\mathsf{q}_{1})$ bits of
(LABEL:sumrate_bound_3_twolinks). The second term is within
$\frac{1}{2}+{\alpha}(\mathsf{q}_{1})$ bits of (17).
Therefore, the achievable sum rate in (85)-(87) is within a constant gap of
the sum-rate upper bound specified by (9)-(20) in the weak-relay regime.
(iv) $2R_{1}+R_{2}$ rate (88) is within
$\delta_{2R_{1}+R_{2}}=\frac{1}{2}+\max\left\\{2{\alpha}(\mathsf{q}_{1}),{\alpha}(\mathsf{q}_{1})+{\beta}(\mathsf{q}_{1}),2{\beta}(\mathsf{q}_{1})\right\\}$
(93)
bits of the upper bounds (21), (26), and (24). Specifically, the first term of
(88) is within $\frac{1}{2}+2{\beta}(\mathsf{q}_{1})$ bits of (21). The second
term is within $\frac{1}{2}+{\alpha}(\mathsf{q}_{1})+{\beta}(\mathsf{q}_{1})$
bits of (26). The third term is within $\frac{1}{2}+2{\alpha}(\mathsf{q}_{1})$
bits of (24).
(v) $R_{1}+2R_{2}$ rate (LABEL:R12R2_1_single_digital_link) is within
$\delta_{R_{1}+2R_{2}}=1+\max\left\\{{\alpha}(\mathsf{q}_{1}),{\beta}(\mathsf{q}_{1})\right\\}$
(94)
bits of the upper bounds
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1}\right)$
(95)
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{INR}_{2}+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}\right)+\mathsf{C}_{1}$
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1})$ (96)
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right.$
$\displaystyle\left.\qquad\qquad+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right),$
which are not shown explicitly in Theorem 1 but can be obtained by switching
the indices $1$ and $2$ of (21) and (25) followed by setting
$\mathsf{C}_{2}=0$.
Since ${\alpha}(\cdot)$ is an increasing function and ${\beta}(\cdot)$ is a
decreasing function, to minimize the gaps above, we need
${\alpha}(\mathsf{q}_{1}^{*})={\beta}(\mathsf{q}_{1}^{*}),$ (97)
which results in the quantization level
$\mathsf{q}_{1}^{*}=\frac{\sqrt{\rho^{2}+16\rho+16}-\rho}{4}$. With this
optimal quantization level applied to the gaps above, we prove that the
achievable rate region (83)-(LABEL:R12R2_1_single_digital_link) is within
$\displaystyle\max\left\\{\frac{1}{2},\frac{1}{2}\log\left(2+\frac{\rho+\sqrt{\rho^{2}+16\rho+16}}{2}\right)\right\\}$
(98) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(2+\frac{\rho+\sqrt{\rho^{2}+16\rho+16}}{2}\right)$
bits of the capacity region. ∎
### III-B Generalized Degree of Freedom
We now derive the GDoF of the channel depicted in Fig. 8, for the case where
the underlying interference channel is symmetric, i.e.,
$\mathsf{INR}_{1}=\mathsf{INR}_{2}=\mathsf{INR}$ and
$\mathsf{SNR}_{1}=\mathsf{SNR}_{2}=\mathsf{SNR}$. In the high SNR regime,
define
$\displaystyle\beta_{i}$ $\displaystyle:=$
$\displaystyle\lim_{\mathsf{SNR}\rightarrow\infty}\frac{\log\mathsf{SNR}_{ri}}{\log\mathsf{SNR}},\;\;i=1,2,$
(99) $\displaystyle\kappa_{1}$ $\displaystyle:=$
$\displaystyle\lim_{\mathsf{SNR}\rightarrow\infty}\frac{\mathsf{C}_{1}}{\frac{1}{2}\log\mathsf{SNR}},$
(100)
Applying Theorem 4, we have the following result on the GDoF:
###### Corollary 4.
In the weak-relay regime where $\beta_{2}\leq\alpha$, the GDoF sum capacity of
the symmetric relay-interference channel with a single digital link is given
by the following. For $0\leq\alpha<1$
$\displaystyle
d_{\mathrm{sum}}=\left\\{\begin{array}[]{l}\min\\{2-\alpha,2\max(\alpha,1-\alpha)+\kappa_{1},\max(\alpha,1-\alpha)\\\
\qquad+\max(\beta_{1},1+\beta_{2}-\alpha,\alpha)\\},\qquad\qquad\beta_{1}\leq
1\\\ \min\\{2-\alpha+\kappa_{1},2\max(\alpha,1-\alpha)+\kappa_{1},\\\ \qquad
1+\beta_{1}-\alpha\\},\quad\qquad\qquad\qquad\qquad\quad\beta_{1}\geq
1\end{array}\right.$ (103)
and for $\alpha\geq 1$
$d_{\mathrm{sum}}=\min\\{\alpha,2+\kappa\\}.$ (104)
TABLE II: Sum-capacity GDoF gain due to the relay for the symmetric Gaussian relay-interference channel with a single digital relay link for $\alpha=\beta_{1}=\beta_{2}$ Range of $\alpha$ | $\alpha\leq\kappa$ | $\kappa\leq\alpha\leq\frac{2-\kappa}{3}$ | $\frac{2-\kappa}{3}\leq\alpha\leq\frac{2}{3}$ | $\frac{2}{3}\leq\alpha\leq 2$ | $2\leq\alpha\leq 2+\kappa$ | $\alpha\geq 2+\kappa$
---|---|---|---|---|---|---
Gain | $\alpha$ | $\kappa$ | $2-3\alpha$ | $0$ | $\alpha-2$ | $\kappa$
Figure 9: Impact of the relay-destination link on sum capacity
Table II and Fig. 9 illustrate the GDoF gain due to the relay where the direct
links, the interference links and the links to the relay are symmetric for
both users, and where $\alpha=\beta_{1}=\beta_{2}$. The main feature here is
that there is no gain in sum capacity for $\frac{2}{3}\leq\alpha\leq 2$. In
other regimes of $\alpha$, the sum-capacity gain is roughly one bit per relay
bit.
### III-C Signal Relaying vs. Interference Forwarding
In the relay-interference channel, the relay observes a corrupted version of
the weighted sum of two source signals $X_{1}$ and $X_{2}$, and forwards a
description to the receiver. Intuitively, the observations about both source
signals are helpful. For the receiver $1$, the observation about $X_{1}$ helps
receiver $1$ reinforce the signal intended for it; the observation about
$X_{2}$ helps receiver $1$ mitigate the interference. The former can be
thought of as signal relaying, the latter interference forwarding.
In this section, we show that the sum-capacity gain in a Gaussian interference
channel due to a single relay link is equivalent to that achievable with
signal relaying alone or with interference forwarding alone, depending on the
channel parameters. Toward this end, we first set the source-relay link from
$X_{2}$ to zero, i.e., $g_{2}=0$, and compute the GDoF of the sum capacity. In
this case, the sum-capacity gain must be solely due to forwarding intended
signal $X_{1}$. Similarly, we can also set $g_{1}=0$, and compute the GDoF of
the sum-capacity gain due solely to forwarding interference signal $X_{2}$. By
comparing these rates we show that interestingly when the relay link of user
$1$ is under certain threshold, i.e., $\beta_{1}\leq 1-\alpha+\beta_{2}$, the
sum-capacity gain is equivalent to that achievable by interference forwarding.
When $\beta_{1}\geq 1-\alpha+\beta_{2}$, the sum-capacity gain is equivalent
to that achievable by signal relaying.
More specifically, with $g_{2}=0$, the sum-capacity can be computed as
$\displaystyle
d_{SR}=\left\\{\begin{array}[]{l}\min\\{2-\alpha,2\max(\alpha,1-\alpha)+\kappa_{1},\max(\alpha,1-\alpha)\\\
\qquad+\max(\beta_{1},1-\alpha,\alpha)\\},\qquad\qquad\qquad\;\;\beta_{1}\leq
1\\\ \min\\{2-\alpha+\kappa_{1},2\max(\alpha,1-\alpha)+\kappa_{1},\\\
\qquad\;1+\beta_{1}-\alpha\\}.\qquad\qquad\qquad\qquad\qquad\beta_{1}\geq
1\end{array}\right.$ (107)
Similarly, let $g_{1}=0$. The sum-capacity GDoF obtained by forwarding
interference signal is
$\displaystyle d_{IF}$ $\displaystyle=$
$\displaystyle\min\\{2-\alpha,2\max(\alpha,1-\alpha)+\kappa_{1},\max(\alpha,1-\alpha)$
(108) $\displaystyle\qquad\;+\max(1+\beta_{2}-\alpha,\alpha)\\}.$
Comparing (103), (107), and (108), it is easy to verify that
$\displaystyle d_{sum}=\left\\{\begin{array}[]{ll}d_{IF}&{\rm\ when\ \
}\beta_{1}\leq 1+\beta_{2}-\alpha\\\ d_{SR}&{\rm\ when\ \ }\beta_{1}\geq
1+\beta_{2}-\alpha\end{array}\right..$ (111)
Therefore, we observe the following threshold effects. When the relay link
from user $1$ is weak, the sum-capacity gain is equivalent to a channel with a
single source-relay link from $X_{2}$. As the source-relay link from $X_{1}$
grows stronger and crosses a threshold $\beta_{1}\geq
1+\beta_{2}-\alpha\triangleq\beta_{1}^{*}$, the sum-capacity gain becomes
equivalent to that of a single source-relay link from $X_{1}$. Note that this
is a GDoF phenomenon in the high SNR regime. In the general SNR regime, the
sum-capacity gain contains contributions from both signal relaying and
interference forwarding.
Figure 10: Signal relaying vs. interference forwarding
To visualize the interaction of signal relaying and interference forwarding, a
numerical example is provided in Fig. 10. The channel parameters are set to
$\alpha=0.5$, $\beta_{2}=0.2$, and $\kappa_{1}=0.5$. The GDoF of the sum
capacity is plotted as a function of $\beta_{1}$. The sum capacity of the
interference channel without the relay serves as the baseline:
$d_{BL}=\min\left\\{2-\alpha,2\max(\alpha,1-\alpha)\right\\}.$ (112)
Fig. 10 shows the sum-capacity gain due to the relay. When
$\beta_{1}\leq\beta_{1}^{*}=0.7$, the gain (labeled as $\mathcal{R}_{1}$) is
equivalent to that by forwarding interference signal only. When $\beta\geq
0.7$, the gain (labeled as $\mathcal{R}_{2}$) is equivalent to that by
forwarding intended signal only.
## IV Conclusion
This paper investigates GHF as an incremental relay strategy for a Gaussian
interference channel augmented with an out-of-band broadcasting relay, in
which the relay message to one receiver is a degraded version of the message
to the other receiver. We focus on a weak-relay regime, where the transmitter-
to-relay links are not unboundedly stronger than the interfering links of the
interference channel, and show that GHF achieves to within a constant gap to
the capacity region in the weak-relay regime. Further, in a symmetric setting,
each common relay bit can be worth either one or two bits in the sum capacity
gain, illustrating the potential for a cell-edge relay in improving the system
throughput of a wireless cellular network.
Furthermore, the Gaussian relay-interference channels with a single relay link
is also studied. The capacity region is characterized to within a constant gap
for a larger range of channel parameters. It is shown that in the high SNR
regime, the sum-capacity improvement is equivalent either to that of a single
source-relay link from user $1$ or that of a single source-relay link from
user $2$.
### -A Proof of Theorem 1
Define $V_{1}^{n}$ as the output of the digital link $\mathsf{C}_{1}$, and
$V_{2}^{n}$ as the output of the digital link $\mathsf{C}_{2}$. The outer
bounds are proved as follows:
(i) Individual-rate bounds: First, the first term of (7) is the simple cut-set
upper bound for $R_{1}$. For the second term, starting from Fano’s inequality,
we have
$\displaystyle n(R_{1}-\epsilon_{n})$ $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})$ $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n},X_{2}^{n})$ $\displaystyle\leq$
$\displaystyle\frac{n}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}).$
The outer bound of $R_{2}$ in (8) can be proved in the same way.
(ii) Sum-rate bounds:
* •
First, (9)-(LABEL:sumrate_bound_3_twolinks) are obtained from Fano’s
inequalities, i.e.,
$\displaystyle n(R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle=$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+I(X_{1}^{n};V_{1}^{n}|Y_{1}^{n})$
$\displaystyle+I(X_{2}^{n};V_{2}^{n}|Y_{2}^{n})$ $\displaystyle\leq$
$\displaystyle
I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+h(V_{1}^{n})+h(V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
nC_{sum}(0)+n\mathsf{C}_{1}+n\mathsf{C}_{2},$
where $C_{sum}(0)$ is the sum capacity of the interference channel without
relay. Clearly, the sum-rate gain due to the digital relay is upper bounded by
the rates of digital links. Although the sum-rate capacity $C_{sum}(0)$ is not
known in general, its upper bound has been studied in literature [1, 34, 42,
43, 44, 48]. Applying the sum-rate outer bounds in [34], we obtain
(9)-(LABEL:sumrate_bound_3_twolinks).
* •
Second, (12)-(14) can be obtained by the following steps:
$\displaystyle n(R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n})+h(V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n}),$
where in (a) we give genie $Y_{R}^{n}$ to receiver $2$ and apply the fact that
$\hat{Y}_{R}$ is a function of $Y_{R}$. Note that
$I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n})$ is upper bounded by
the sum capacity of the SIMO interference channel with $X_{1}^{n}$ and
$X_{2}^{n}$ as the input, and $Y_{1}^{n}$ and $(Y_{2}^{n},Y_{R}^{n})$ as the
output. The sum-rate outer bound of such a SIMO interference channel has been
studied in [34], which along with $h(V_{1}^{n})\leq n\mathsf{C}_{1}$ gives the
outer bounds of (12)-(14).
* •
Third, (15)-(17) can be similarly derived following the same steps of
(12)-(14) with indices $1$ and $2$ switched.
* •
Fourth, (18)-(20) can be obtained by giving $Y_{R}^{n}$ as a genie to both
receivers, i.e.,
$\displaystyle n(R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n}),$
which is upper bounded by the sum capacity of the SIMO interference channel
with $X_{1}^{n}$ and $X_{2}^{n}$ as input, and $(Y_{1}^{n},Y_{R}^{n})$ and
$(Y_{2}^{n},Y_{R}^{n})$ as output. Applying the result in [34], we have
(18)-(20).
(iii) $2R_{1}+R_{2}$ bounds: Six upper bounds on $2R_{1}+R_{2}$.
* •
First, (21) is simply the cut-set bound, i.e.,
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
2I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})+2h(V_{1}^{n})+h(V_{2}^{n}),$
where $2I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n})$ is upper bounded by the
$2R_{1}+R_{2}$ bound of the interference channel with $X_{1}^{n}$ and
$X_{2}^{n}$ as the input, and $Y_{1}^{n}$ and $Y_{2}^{n}$ as the output, which
together with $h(V_{1}^{n})\leq n\mathsf{C}_{1}$ and $h(V_{2}^{n})\leq
n\mathsf{C}_{2}$ gives the upper bound in (21).
* •
Second, (22) can be derived by giving genie $Y_{R}^{n}$ to both receivers:
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
2I(X_{2}^{n};Y_{1}^{n},Y_{R}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n}),$
which is upper bounded by the $2R_{1}+R_{2}$ bound of the SIMO interference
channel with $X_{1}^{n}$ and $X_{2}^{n}$ as the input, and
$(Y_{1}^{n},Y_{R}^{n})$ and $(Y_{2}^{n},Y_{R}^{n})$ as the output. Applying
the result of [34], we obtain (22).
* •
Third, (23) can be obtained by giving genies $(X_{2}^{n},Y_{R}^{n},S_{1}^{n})$
to $Y_{1}^{n}$ in one of the two $R_{1}$ expressions and
$(S_{2}^{n},Y_{R}^{n})$ to $Y_{2}^{n}$, where genies $S_{1}^{n}$ and
$S_{2}^{n}$ are defined as
$S_{1}^{n}=h_{12}X_{1}^{n}+Z_{2},\qquad S_{2}^{n}=h_{21}X_{2}^{n}+Z_{1}.$
(119)
According to Fano’s inequality, we have
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n},S_{1}^{n},X_{2}^{n})+I(X_{1}^{n};Y_{1}^{n})+h(V_{1}^{n})$
$\displaystyle+I(X_{2}^{n};Y_{2},Y_{R}^{n},S_{2}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n},S_{1}^{n}|X_{2}^{n})+h(Y_{1}^{n})-h(S_{2}^{n})+n\mathsf{C}_{1}$
$\displaystyle+I(X_{2}^{n};S_{2}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n}|S_{2}^{n})$
$\displaystyle=$ $\displaystyle
I(X_{1}^{n};S_{1}^{n})+I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})+h(Y_{1}^{n})$
$\displaystyle-h(S_{2}^{n})+n\mathsf{C}_{1}+h(S_{2}^{n})-h(Z_{1}^{n})$
$\displaystyle+h(Y_{2}^{n},Y_{R}^{n}|S_{2}^{n})-h(S_{1}^{n})-h(Y_{R}^{n}|Y_{2}^{n},X_{2}^{n})$
$\displaystyle=$ $\displaystyle
h(Y_{1}^{n})-h(Z_{1}^{n})+h(Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})+n\mathsf{C}_{1}$
$\displaystyle-h(Z_{1}^{n},Z_{R}^{n})+h(Y_{2}^{n},Y_{R}^{n}|S_{2}^{n})-h(Z_{2}^{n},Z_{R}^{n})$
$\displaystyle-I(Y_{R}^{n};X_{1}^{n}|X_{2}^{n},Y_{2}^{n})$ $\displaystyle\leq$
$\displaystyle
h(Y_{1}^{n})-h(Z_{1}^{n})+h(Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})+n\mathsf{C}_{1}$
$\displaystyle-h(Z_{1}^{n},Z_{R}^{n})+h(Y_{2}^{n},Y_{R}^{n}|S_{2}^{n})-h(Z_{2}^{n},Z_{R}^{n}),$
where in (a) we use the fact that $X_{1}^{n}$ is independent with $X_{2}^{n}$.
Note that, the last inequality of (• ‣ -A) is maximized by Gaussian inputs
$X_{1}^{n}$ and $X_{2}^{n}$ with i.i.d $\mathcal{N}(0,1)$ entries, because
* –
$h(Y_{1}^{n})$ is maximized by Gaussian distributions, and
* –
$h(Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})$ and
$h(Y_{2}^{n},Y_{R}^{n}|S_{2}^{n})$ are both maximized by Gaussian inputs since
the conditional entropy under a power constraint is maximized by Gaussian
distributions.
Applying Gaussian distributions to the last inequality of (• ‣ -A), we have
(23).
* •
Fourth, (24) can be obtained by giving genie $Y_{R}^{n}$ to $Y_{1}^{n}$, i.e.,
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
2I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n})+I(X_{2}^{n};Y_{2}^{n})+h(V_{2}^{n}),$
where $2I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n})+I(X_{2}^{n};Y_{2}^{n})$ is upper
bounded by the $2R_{1}+R_{2}$ bound of the SIMO interference channel with
$X_{1}^{n}$ and $X_{2}^{n}$ as the input, and $(Y_{1}^{n},Y_{R}^{n})$ and
$Y_{2}^{n}$ as the output. Applying the result of [34] and the fact that
$h(V_{2}^{n})\leq n\mathsf{C}_{2}$, we obtain (24).
* •
Fifth, (25) can be obtained by giving genie $Y_{R}^{n}$ to $Y_{2}^{n}$, i.e.,
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
2I(X_{1}^{n};Y_{1}^{n})+2h(V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n}),$
where $2I(X_{1}^{n};Y_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},Y_{R}^{n})$ is upper
bounded by the $2R_{1}+R_{2}$ bound of the SIMO interference channel with
$X_{1}^{n}$ and $X_{2}^{n}$ as the input, and $Y_{1}^{n}$ and
$(Y_{2}^{n},Y_{R}^{n})$ as the output. Applying the result of [34] and the
fact that $h(V_{1}^{n})\leq n\mathsf{C}_{1}$, we obtain (25).
* •
Sixth, (26) can be obtained by giving genies $(X_{2}^{n},Y_{R}^{n},S_{1}^{n})$
to $Y_{1}^{n}$ in one of the two $R_{1}$ expressions, and $S_{2}^{n}$ to
$Y_{2}^{n}$, i.e.,
$\displaystyle n(2R_{1}+R_{2}-\epsilon_{n})$ $\displaystyle\leq$
$\displaystyle
2I(X_{1}^{n};Y_{1}^{n},V_{1}^{n})+I(X_{2}^{n};Y_{2}^{n},V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};Y_{1}^{n},Y_{R}^{n},S_{1}^{n},X_{2}^{n})+I(X_{1}^{n};Y_{1}^{n})+h(V_{1}^{n})$
$\displaystyle+I(X_{2}^{n};Y_{2}^{n},S_{2}^{n})+h(V_{2}^{n})$
$\displaystyle\leq$ $\displaystyle
I(X_{1}^{n};S_{1}^{n})+I(X_{1}^{n};Y_{1}^{n}Y_{R}^{n}|S_{1}^{n},X_{2}^{n})+h(Y_{1}^{n})$
$\displaystyle-h(S_{2}^{n})+I(X_{2}^{n};S_{2}^{n})+I(X_{2}^{n};Y_{2}^{n}|S_{2}^{n})$
$\displaystyle+n\mathsf{C}_{1}+n\mathsf{C}_{2}$ $\displaystyle\leq$
$\displaystyle
h(S_{1}^{n})-h(Z_{2}^{n})+h(Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})$
$\displaystyle-h(Z_{1}^{n},Z_{R}^{n})+h(Y_{1}^{n})-h(S_{2}^{n})+h(S_{2}^{n})-h(Z_{1}^{n})$
$\displaystyle+h(Y_{2}^{n}|S_{2}^{n})-h(S_{1}^{n})+n\mathsf{C}_{1}+n\mathsf{C}_{2}$
$\displaystyle=$ $\displaystyle
h(Y_{1}^{n})-h(Z_{1}^{n})+h(Y_{1}^{n},Y_{R}^{n}|S_{1}^{n},X_{2}^{n})$
$\displaystyle-h(Z_{1}^{n},Z_{R}^{n})+h(Y_{2}^{n}|S_{2}^{n})-h(Z_{2}^{n})+n\mathsf{C}_{1}+n\mathsf{C}_{2},$
which is maximized by Gaussian distributions of $X_{1}^{n}$ and $X_{2}^{n}$
with i.i.d entries following $\mathcal{N}(0,1)$. Applying Gaussian
distributions to (• ‣ -A), we obtain (26).
### -B Useful Inequalities
This appendix provides several inequalities that are useful to prove the
constant-gap theorems.
###### Lemma 1.
For
$\Delta{a}_{i},a_{i},\Delta{d}_{i},d_{i},\Delta{e}_{i},e_{i},\Delta{g}_{i},g_{i}$
and $\xi_{i},i=1,2$ as defined in (36)-(44), with $Q$ set as a constant, when
$W_{i},X_{i}$ are generated from a superposition coding of $X_{i}=U_{i}+W_{i}$
with $U_{i}\sim~{}\mathcal{N}(0,P_{ip})$ and $W_{i}\sim\mathcal{N}(0,P_{ic})$,
where $P_{ip}+P_{ic}=1$ and
$P_{1p}=\min\\{1,h_{12}^{-2}\\},P_{2p}=\min\\{1,h_{21}^{-2}\\}$, and when the
GHF quantization variables are set to
$\hat{Y}_{R1}=Y_{R}+e_{1},\hat{Y}_{R2}=Y_{R}+e_{2}$, where
$e_{1}\sim\mathcal{N}(0,\mathsf{q}_{1})$ and
$e_{2}\sim\mathcal{N}(0,\mathsf{q}_{2})$, in the weak-relay regime of
$|g_{1}|\leq\sqrt{\rho}|h_{12}|,|g_{2}|\leq\sqrt{\rho}|h_{21}|$, the mutual
information terms in (36)-(44) can be bounded as follows:
$\displaystyle a_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2},$
(124) $\displaystyle a_{1}+\Delta{a}_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1}),$
(125) $\displaystyle d_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})-\frac{1}{2},$ (126)
$\displaystyle d_{1}+\Delta{d}_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{SNR}_{r1})-\alpha(\mathsf{q}_{1}),$
(127) $\displaystyle e_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}+\mathsf{INR}_{2}\right)-\frac{1}{2},$
(128) $\displaystyle e_{1}+\Delta{e}_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right.$
(129)
$\displaystyle\left.\qquad\;\;+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1}),$
$\displaystyle g_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}$
(130) $\displaystyle g_{1}+\Delta{g}_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}\right.$
(131)
$\displaystyle\left.\qquad\;\;+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1}),$
$\displaystyle\xi_{1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1+\rho}{\mathsf{q}_{1}}\right)=\beta(\mathsf{q}_{1})-\frac{1}{2},$
(132)
and the lower bounds of
$a_{2},a_{2}+\Delta{a}_{2},d_{2},d_{2}+\Delta{d}_{2},e_{2},e_{2}+\Delta{e}_{2},g_{2},g_{2}+\Delta{g}_{2}$
and the upper bound of $\xi_{2}$ can be obtained by switching the indices of
$1$ and $2$ in (124)-(132).
###### Proof:
First, define the signal-to-noise and interference-to-noise ratios of the
private messages as
$\displaystyle\mathsf{SNR}_{1p}$
$\displaystyle=|h_{11}|^{2}P_{1p},\qquad\mathsf{SNR}_{2p}$
$\displaystyle=|h_{22}|^{2}P_{2p},$ (133) $\displaystyle\mathsf{INR}_{1p}$
$\displaystyle=|h_{12}|^{2}P_{1p},\qquad\mathsf{INR}_{2p}$
$\displaystyle=|h_{21}|^{2}P_{2p},$ (134) $\displaystyle\mathsf{SNR}_{r1p}$
$\displaystyle=|g_{1}|^{2}P_{1p},\qquad\mathsf{SNR}_{r2p}$
$\displaystyle=|g_{2}|^{2}P_{2p},$ (135)
which can be lower bounded or upper bounded as follows:
$\displaystyle\mathsf{SNR}_{1p}$ $\displaystyle=$
$\displaystyle|h_{11}|^{2}P_{1p}$ (136) $\displaystyle=$
$\displaystyle\min\left\\{|h_{11}|^{2},\frac{|h_{11}|^{2}}{|h_{12}|^{2}}\right\\}$
$\displaystyle=$
$\displaystyle\min\left\\{\mathsf{SNR}_{1},\frac{\mathsf{SNR}_{1}}{\mathsf{INR}_{1}}\right\\}$
$\displaystyle\geq$
$\displaystyle\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}},$
and
$\displaystyle 0\leq\mathsf{INR}_{1p}=\min\\{1,\mathsf{INR}_{1}\\}\leq 1,$
(137)
and
$\displaystyle\mathsf{SNR}_{r1p}$ $\displaystyle=$
$\displaystyle|g_{1}|^{2}P_{1p}$ (138) $\displaystyle=$
$\displaystyle\min\left\\{|g_{1}|^{2},\frac{|g_{1}|^{2}}{|h_{12}|^{2}}\right\\}$
$\displaystyle=$
$\displaystyle\min\left\\{\mathsf{SNR}_{r1},\frac{\mathsf{SNR}_{r1}}{\mathsf{INR}_{1}}\right\\}$
$\displaystyle\geq$
$\displaystyle\frac{\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}.$
Since $|g_{1}|\leq\sqrt{\rho}|h_{12}|$, $\mathsf{SNR}_{r1p}$ is upper bounded
by $\rho$. Therefore
$\rho\geq\mathsf{SNR}_{r1p}\geq\frac{\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}.$
(139)
Switching the indices of $1$ and $2$, we have
$\displaystyle\mathsf{SNR}_{2p}$
$\displaystyle\geq\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}},$ (140)
$\displaystyle 1\geq\mathsf{INR}_{2p}$ $\displaystyle\geq 0,$ (141)
$\displaystyle\rho\geq\mathsf{SNR}_{r2p}$
$\displaystyle\geq\frac{\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}.$ (142)
Now, starting from (124), we prove the inequalities one by one.
* •
First, (124) is lower bounded by
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1}|W_{1},W_{2})$ (143) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{1p}+\mathsf{INR}_{2p}}{1+\mathsf{INR}_{2p}}\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1p})-\frac{1}{2}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\geq}}$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2},$
where (a) holds because $0\leq\mathsf{INR}_{2p}\leq 1$ and (b) is due to the
fact that $\mathsf{SNR}_{1p}\geq\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}$.
* •
Second, (125) is lower bounded by (144),
$\displaystyle a_{1}+\Delta{a}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1}|W_{1},W_{2})+I(X_{1};\hat{Y}_{R1}|Y_{1},W_{1},W_{2})$ (144)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{(\mathsf{q}_{1}+1)(1+\mathsf{SNR}_{1p}+\mathsf{INR}_{2p})+\mathsf{SNR}_{r1p}+\mathsf{SNR}_{r2p}(1+\phi_{1}^{2}\mathsf{SNR}_{1p})}{(\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1p}+\mathsf{SNR}_{r1p})-\frac{1}{2}\log((\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1}),$
where (a) holds because
$\mathsf{SNR}_{1p}\geq\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}$,
$\mathsf{SNR}_{r1p}\geq\frac{\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}$, and
$\displaystyle\frac{1}{2}\log((\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p})$
(145) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\log((\mathsf{q}_{1}+1)(1+1)+\rho)$ $\displaystyle=$
$\displaystyle\alpha(\mathsf{q}_{1}).$
* •
Third, (126) is lower bounded by
$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle I(X_{1};Y_{1}|W_{2})$
(146) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{1}+\mathsf{INR}_{2p}}{1+\mathsf{INR}_{2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})-\frac{1}{2}.$
* •
Fourth, (127) is lower bounded by (147).
$\displaystyle d_{1}+\Delta{d}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1};Y_{1}|W_{2})+I(X_{1};\hat{Y}_{R1}|Y_{1},W_{2})$ (147) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{(\mathsf{q}_{1}+1)(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2p})+\mathsf{SNR}_{r1}+\mathsf{SNR}_{r2p}(1+\phi_{1}^{2}\mathsf{SNR}_{1})}{(\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{SNR}_{r1})-\alpha(\mathsf{q}_{1}).$
* •
Fifth, (128) is lower bounded by
$\displaystyle e_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1},W_{2};Y_{1}|W_{1})$ (148) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{1p}+\mathsf{INR}_{2}}{1+\mathsf{INR}_{2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}+\mathsf{INR}_{2}\right)-\frac{1}{2}.$
* •
Sixth, (129) is lower bounded by (149).
$\displaystyle e_{1}+\Delta{e}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1},W_{2};Y_{1}|W_{1})+I(X_{1},W_{2};\hat{Y}_{R1}|Y_{1},W_{1})$ (149)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{(\mathsf{q}_{1}+1)(1+\mathsf{SNR}_{1p}+\mathsf{INR}_{2})+\mathsf{SNR}_{r1p}+\mathsf{SNR}_{r2}(1+\phi_{1}^{2}\mathsf{SNR}_{1p})}{(\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1}).$
* •
Seventh, (130) is lower bounded by
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle I(X_{1},W_{2};Y_{1})$
(150) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{1}+\mathsf{INR}_{2}}{1+\mathsf{INR}_{2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}.$
* •
Eighth, (131) is lower bounded by (151).
$\displaystyle g_{1}+\Delta{g}_{1}$ $\displaystyle=$ $\displaystyle
I(X_{1},W_{2};Y_{1})+I(X_{1},W_{2};\hat{Y}_{R1}|Y_{1})$ (151) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{(\mathsf{q}_{1}+1)(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})+\mathsf{SNR}_{r1}+\mathsf{SNR}_{r2}(1+\phi_{1}^{2}\mathsf{SNR}_{1})}{(\mathsf{q}_{1}+1)(1+\mathsf{INR}_{2p})+\mathsf{SNR}_{r2p}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1}).$
* •
Ninth, (132) is upper bounded by
$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
I(Y_{R}:\hat{Y}_{R1}|Y_{1},X_{1},W_{2})$ (152) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1}{\mathsf{q}_{1}}\left(1+\frac{\mathsf{SNR}_{r2p}}{1+\mathsf{INR}_{2p}}\right)\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1+\rho}{\mathsf{q}_{1}}\right)$
∎
### -C Proof of Theorem 3
In this appendix, we show that using the Han-Kobayashi power splitting
strategy with the private message power set to
$P_{1p}=\min\\{1,h_{12}^{-2}\\}$ and $P_{2p}=\min\\{1,h_{21}^{-2}\\}$, all the
achievable rates in (29)-(35) are within constant bits of their corresponding
outer bounds in Theorem 1. Note that, in the following proof, inequalities in
Appendix -B are implicitly used without being mentioned.
(i) First, (29) is within constant bits of (7), and (30) is within constant
bits of (8). To see this, the first term of (29) is lower bounded by
$\displaystyle d_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1})+\mathsf{C}_{1}-\left(\frac{1}{2}+\frac{1}{2}\log\left(1+\frac{1+\rho}{\mathsf{q}_{1}}\right)\right)$
which is within $\beta(\mathsf{q}_{1})$ bits of the first term of (7).
According to Lemma 1, the second term of (29) is lower bounded by
$\displaystyle d_{1}+\Delta{d}_{1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{SNR}_{r1})-\alpha(\mathsf{q}_{1}),$
which is within $\alpha(\mathsf{q}_{1})$ bits of the second term of (7). As a
result, the gap between (29) and (7) is bounded by
$\delta_{R_{1}}=\max\left\\{\alpha(\mathsf{q}_{1}),\beta(\mathsf{q}_{1})\right\\}.$
(155)
Due to symmetry, (30) is within
$\delta_{R_{2}}=\max\left\\{\alpha(\mathsf{q}_{2}),\beta(\mathsf{q}_{2})\right\\}$
(156)
bits of the upper bound (8).
(ii) Second, (31)-(33) are within constant bits of their upper bounds
(9)-(20). To see this, inspecting the expressions of the achievable sum rates,
it is easy to see that each of (31)-(33) has four possible combinations:
having both $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$, having $\mathsf{C}_{1}$
only, having $\mathsf{C}_{2}$ only, and having none of $\mathsf{C}_{1}$ and
$\mathsf{C}_{2}$. In the following, we show that, when specialized into the
above four combinations, (31)-(33) are within constant gap to the upper bounds
(9)-(20). The constant gaps are given by
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}$,
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})}$,
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{C}_{2})}$, and
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}$ (to be defined later)
respectively, each corresponding to a specific combination.
* •
First, when having both $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$, (31)-(33)
become
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}+(\mathsf{C}_{2}-\xi_{2})^{+},$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+(\mathsf{C}_{2}-\xi_{2})^{+},$
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+e_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}+(\mathsf{C}_{2}-\xi_{2})^{+},$
which are within constant bits of (9)-(LABEL:sumrate_bound_3_twolinks)
respectively. To show this, first, according to Lemma 1, (• ‣ -C) is lower
bounded by
$\displaystyle
a_{1}+g_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}+(\mathsf{C}_{2}-\xi_{2})^{+}$ (160)
$\displaystyle\geq\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2}$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{2}+\mathsf{INR}_{1})-\frac{1}{2}$
$\displaystyle+\mathsf{C}_{1}-\xi_{1}+\mathsf{C}_{2}-\xi_{2},$
which is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}=\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
(161)
bits of the upper bound (9). Due to symmetry, (• ‣ -C) is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}$ bits of the upper
bound (10) as well. Now applying Lemma 1, (• ‣ -C) is lower bounded by
$\displaystyle
e_{1}+e_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}+(\mathsf{C}_{2}-\xi_{2})^{+}$ (162)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}+\mathsf{INR}_{2}\right)-\frac{1}{2}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}+\mathsf{INR}_{1}\right)-\frac{1}{2}$
$\displaystyle+\mathsf{C}_{1}-\xi_{1}+\mathsf{C}_{2}-\xi_{2},$
which is within $\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}$ bits
of the upper bound (LABEL:sumrate_bound_3_twolinks). Therefore, when
specialized to the form with both $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$ as
shown in (• ‣ -C)-(• ‣ -C), (31)-(33) have a gap of
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}$ bits to their upper
bounds (9)-(LABEL:sumrate_bound_3_twolinks).
* •
Second, when having $\mathsf{C}_{1}$ only, (31)-(33) become
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{2}+\Delta{g}_{2}+(\mathsf{C}_{1}-\xi_{1})^{+},$ (163) $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+\Delta{a}_{2}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+},$ (164) $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+e_{2}+\Delta{e}_{2}+(\mathsf{C}_{1}-\xi_{1})^{+},$ (165)
where (163) is lower bounded by
$\displaystyle a_{1}+g_{2}+\Delta{g}_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}$ (166)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})}=\alpha(\mathsf{q}_{2})+\beta(\mathsf{q}_{1})$
(167)
bits of the upper bound (12), and (164) is lower bounded by
$\displaystyle a_{2}+\Delta{a}_{2}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right)-\alpha(\mathsf{q}_{2})$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}+\mathsf{C}_{1}+\xi_{1},$
which is within $\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})}$ bits of
the upper bound (13), and (165) can be lower bounded by
$\displaystyle e_{1}+e_{2}+\Delta{e}_{2}+(\mathsf{C}_{1}-\xi_{1})^{+}$ (169)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}+\mathsf{INR}_{2}\right)-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within $\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})}$ bits of
the upper bound (14).
* •
Third, when having $\mathsf{C}_{2}$ only, (31)-(33) become
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+\Delta{a}_{1}+g_{2}+(\mathsf{C}_{2}-\xi_{2})^{+},$ (170) $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{1}+\Delta{g}_{1}+(\mathsf{C}_{2}-\xi_{2})^{+},$ (171) $\displaystyle
R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+\Delta{e}_{1}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+}.$ (172)
Due to the symmetry between (170)-(172) and (163)-(165), and the symmetry
between their upper bounds, we can see that (170), (171) and (172) are within
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{C}_{2})}=\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
(173)
bits of the upper bounds (15), (16), and (17) respectively.
* •
Fourth, when having none of $\mathsf{C}_{1}$ and $\mathsf{C}_{2}$, (31)-(33)
become
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+\Delta{a}_{1}+g_{2}+\Delta{g}_{2},$ (174) $\displaystyle R_{1}+R_{2}$
$\displaystyle\leq$ $\displaystyle a_{2}+\Delta{a}_{2}+g_{1}+\Delta{g}_{1},$
(175) $\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
e_{1}+\Delta{e}_{1}+e_{2}+\Delta{e}_{2},$ (176)
where (174) is lower bounded by
$\displaystyle a_{1}+\Delta{a}_{1}+g_{2}+\Delta{g}_{2}$ (177)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}\right.$
$\displaystyle\qquad\quad\;\;\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}=\alpha(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2})$
(178)
bits of the upper bound (18). Due to symmetry, (175) is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}$ bits of the upper bound (19)
as well. Further, (176) can be lower bounded by
$\displaystyle e_{1}+\Delta{e}_{1}+e_{2}+\Delta{e}_{2}$ (179)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right.$
$\displaystyle\qquad\quad\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within $\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}$ bits of the
upper bound (20). Therefore, when specialized into the form with none of
$\mathsf{C}_{1}$ and $\mathsf{C}_{2}$, (31)-(33) is within
$\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}$ bits of their upper bounds
(18)-(20).
Overall, the gap between the achievable sum-rates (31)-(33) and the upper
bounds in (9)-(20) is upper bounded as follows:
$\delta_{R_{1}+R_{2}}=\max\left\\{\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})},\delta_{R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})},\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{C}_{2})},\delta_{R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}\right\\}.$
(180)
(iii) Third, the achievable rate (34) is within constant bits of upper bounds
(21)-(26). To see this, note that (34) has $8$ different forms as follows:
$\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+},$
(181) $\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+\Delta{g}_{1}+e_{2}+\Delta{e}_{2},$ (182)
$\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+\Delta{e}_{2},$
(183) $\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+\Delta{g}_{1}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+},$
(184) $\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+\Delta{e}_{2},$
(185) $\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+},$
(186) $\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+\Delta{g}_{1}+e_{2}+\Delta{e}_{2},$
(187) $\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+\Delta{g}_{1}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+},$
(188)
where (187) is redundant compared with (183) and (188) is redundant compared
with (186) due to the fact that $\Delta{g}_{1}\geq\Delta{a}_{1}$. Therefore,
there are six active rate constraints in total. In the following, we prove
that all active achievable rates of $2R_{1}+R_{2}$ in (181)-(186) are within
constant bits of their corresponding upper bounds in (21)-(26).
* •
First, (181) is lower bounded by
$\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}+\mathsf{INR}_{1}\right)-\frac{1}{2}+\mathsf{C}_{2}-\xi_{2},$
which is within
$\delta_{2R_{1}+R_{2}}^{(2\mathsf{C}_{1},\mathsf{C}_{2})}=2\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
(190)
bits of the upper bound (21).
* •
Second, (182) is lower bounded by
$\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+g_{1}+\Delta{g}_{1}+e_{2}+\Delta{e}_{2}$ (191)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}\right.$
$\displaystyle\qquad\quad\;\;\left.+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within
$\delta_{2R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})}=2\alpha(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2})$
(192)
bits of the upper bound (22).
* •
Third, (183) is lower bounded by
$\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+\Delta{e}_{2}$
(193) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\left.\qquad\qquad+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within
$\delta_{2R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})}=\alpha(\mathsf{q}_{1})+\alpha(\mathsf{q}_{2})+\beta(\mathsf{q}_{1})$
(194)
bits of the upper bound (23).
* •
Fourth, (184) is lower bounded by
$\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+\Delta{g}_{1}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\mathsf{SNR}_{1}(1+\phi_{1}^{2}\mathsf{SNR}_{r2})+\mathsf{SNR}_{r1}\right.$
$\displaystyle\left.\qquad\quad\;\;+\mathsf{INR}_{2}+\mathsf{SNR}_{r2}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}+\mathsf{INR}_{1}\right)-\frac{1}{2}+\mathsf{C}_{2}-\xi_{2},$
which is within
$\delta_{2R_{1}+R_{2}}^{(\mathsf{0},\mathsf{C}_{2})}=2\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
(196)
bits of the upper bound (24).
* •
Fifth, (185) is lower bounded by
$\displaystyle
a_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+\Delta\tilde{e}_{2}$
(197) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}}{1+\mathsf{INR}_{1}}\right)-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}(1+\phi_{2}^{2}\mathsf{SNR}_{r1})+\mathsf{SNR}_{r2}}{1+\mathsf{INR}_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\mathsf{INR}_{1}+\mathsf{SNR}_{r1}\right)-\alpha(\mathsf{q}_{2}),$
which is within
$\delta_{2R_{1}+R_{2}}^{(2\mathsf{C}_{1},\mathsf{0})}=\alpha(\mathsf{q}_{2})+2\beta(\mathsf{q}_{1})$
(198)
bits of the upper bound (25).
* •
Sixth, (186) is lower bounded by
$\displaystyle
a_{1}+\Delta{a}_{1}+g_{1}+(\mathsf{C}_{1}-\xi_{1})^{+}+e_{2}+(\mathsf{C}_{2}-\xi_{2})^{+}$
(199) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{1}+\mathsf{SNR}_{r1}}{1+\mathsf{INR}_{1}}\right)-\alpha(\mathsf{q}_{1})$
$\displaystyle+\frac{1}{2}\log(1+\mathsf{SNR}_{1}+\mathsf{INR}_{2})-\frac{1}{2}+\mathsf{C}_{1}-\xi_{1}$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{SNR}_{2}}{1+\mathsf{INR}_{2}}+\mathsf{INR}_{1}\right)$
$\displaystyle-\frac{1}{2}+\mathsf{C}_{2}-\xi_{2},$
which is within
$\delta_{2R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}=\alpha(\mathsf{q}_{1})+\beta(\mathsf{q}_{1})+\beta(\mathsf{q}_{2})$
(200)
bits of the upper bound (26).
Therefore, the gap between the achievable rate (34) and the corresponding
upper bounds (21)-(26) is bounded by the following constant
$\displaystyle\delta_{2R_{1}+R_{2}}$ $\displaystyle=$
$\displaystyle\max\left\\{\delta_{2R_{1}+R_{2}}^{(2\mathsf{C}_{1},\mathsf{C}_{2})},\delta_{2R_{1}+R_{2}}^{(\mathsf{0},\mathsf{0})},\delta_{2R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{0})},\delta_{2R_{1}+R_{2}}^{(\mathsf{0},\mathsf{C}_{2})},\right.$
(201)
$\displaystyle\left.\qquad\;\;\delta_{2R_{1}+R_{2}}^{(2\mathsf{C}_{1},\mathsf{0})},\delta_{2R_{1}+R_{2}}^{(\mathsf{C}_{1},\mathsf{C}_{2})}\right\\}.$
Due the the symmetry between (35) and (34), and the symmetry between their
corresponding upper bounds, it is easy to see that (35) is also within
constant gap to the upper bounds. The constant gap $\delta_{R_{1}+2R_{2}}$ can
be obtained by simply switching indices of $1$ and $2$ in
$\delta_{2R_{1}+R_{2}}$.
## References
* [1] R. Etkin, D. N. C. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 12, pp. 5534–5562, Dec. 2008.
* [2] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” _IEEE Trans. Inf. Theory_ , vol. 27, no. 1, pp. 49–60, Jan. 1981.
* [3] L. Zhou and W. Yu, “Gaussian Z-interference channel with a relay link: Achievability region and asymptotic sum capacity,” _IEEE Trans. Inf. Theory_ , vol. 58, no. 4, pp. 2413–2426, Apr. 2012.
* [4] L. Zhou and W. Yu, “Gaussian Z-Interference channel with a relay link: Achievable rate region and asymptotic sum capacity,” in _Proc. Int. Symp. Inf. Theory and Its App. (ISITA)_ , Dec. 2008, pp. 1–6.
* [5] W. Yu and L. Zhou, “Gaussian Z-interference channel with a relay link: Type II channel and sum capacity bound,” in _Proc. Inf. Theory and App. (ITA) Workshop_ , Feb. 2009, pp. 439–446.
* [6] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limited receiver cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2913–2940, May 2011.
* [7] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limited transmitter cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2941–2965, May 2011.
* [8] V. M. Prabhakaran and P. Viswanath, “Interference channels with destination cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 1, pp. 187–209, Jan. 2011.
* [9] V. M. Prabhakaran and P. Viswanath, “Interference channels with source cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 1, pp. 156–186, Jan. 2011.
* [10] O. Sahin and E. Erkip, “Achievable rates for the Gaussian interference relay channel,” in _Proc. Global Telecommun. Conf. (Globecom)_ , Nov. 2007, pp. 1627–1631.
* [11] Y. Tian and A. Yener, “The Gaussian interference relay channel: Improved achievable rates and sum rate upper bounds using a potent relay,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2865–2879, May 2011.
* [12] R. Dabora, I. Marić, and A. Goldsmith, “Relay strategies for interference-forwarding,” in _Proc. IEEE Inf. Theory Workshop (ITW)_ , May 2008, pp. 46–50.
* [13] I. Marić, R. Dabora, and A. Goldsmith, “On the capacity of the interference channel with a relay,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Jul. 2008, pp. 554–558.
* [14] A. Chaaban and A. Sezgin, “Achievable rates and upper bounds for the interference relay channel,” in _Conf. Record Forty-Fourth Asilomar Conf. Signals, Systems and Computers_ , Nov. 2010, pp. 267–271.
* [15] A. Chaaban and A. Sezgin, “On the generalized degrees of freedom of the Gaussian interference relay channel,” _IEEE Trans. Inf. Theory_ , vol. 58, no. 7, pp. 4432–4461, Jul. 2012.
* [16] J. Lou, P. S. C. Thejaswi, J. Zhang, G. Yue, and T. Luo, “Achievable rates for a relay-aided interference channel,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , May 2010, pp. 1–5.
* [17] O. Sahin and E. Erkip, “Achievable rates for the Gaussian interference relay channel,” in _Proc. Global Telecommun. Conf. (Globecom)_ , Nov. 2007, pp. 1627–1631.
* [18] O. Sahin, O. Simeone, and E. Erkip, “Interference channel with an out-of-band relay,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2746–2764, May 2011.
* [19] Y. Tian and A. Yener, “Symmetric capacity of the Gaussian interference channel with an out-of-band relay to within $1.15$ bits,” _IEEE Trans. Inf. Theory_ , vol. 58, no. 8, pp. 5151–5171, Aug. 2012.
* [20] O. Simeone, E. Erkip, and S. Shamai, “On codebook information for interference relay channels with out-of-band relaying,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2880–2888, May 2011.
* [21] O. Sahin, O. Simeone, and E. Erkip, “Gaussian interference channel aided by a relay with out-of-band reception and in-band transmission,” _IEEE Trans. Commun._ , vol. 59, no. 11, pp. 2976–2981, Nov. 2011.
* [22] P. Razaghi and W. Yu, “Universal relaying for the interference channel,” in _Proc. Inf. Theory and App. (ITA) Workshop_ , Feb. 2010, pp. 1–6.
* [23] P. Razaghi, S. N. Hong, L. Zhou, W. Yu, and G. Caire, “Two birds and one stone: Gaussian interference channel with a shared out-of-band relay,” _Submitted to IEEE Trans. Inf. Theory_ , 2011. [Online]. Available: http://arxiv.org/abs/1104.0430
* [24] L. Zhou and W. Yu, “Incremental relaying for the Gaussian interference channel with a degraded broadcasting relay,” in _Proc. Forty-Ninth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2011, pp. 595–602.
* [25] O. Sahin, E. Erkip, and O. Simeone, “Interference channel with a relay: Models, relaying strategies, bounds,” in _Proc. Inf. Theory and App. (ITA) Workshop_ , Feb. 2009, pp. 90–95.
* [26] H. Chong, M. Motani, H. Garg, and H. El Gamal, “On the Han-Kobayashi region for the interference channel,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 7, pp. 3188–3195, Jul. 2008.
* [27] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 3132–3152, May 2011.
* [28] H. Do, T. Oechtering, and M. Skoglund, “Noisy network coding approach to the interference channel with receiver cooperation,” in _Proc. Forty-Ninth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2011, pp. 839–846.
* [29] H. Do, T. Oechtering, and M. Skoglund, “A new inner bound for the interference relay channel,” in _Proc. IEEE Conf. Inf. Sciences and Systems (CISS)_ , Mar. 2012.
* [30] S. Avestimehr, S. Diggavi, and D. N. C. Tse, “Wireless network information flow: A deterministic approach,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 4, pp. 1872–1905, Apr. 2011.
* [31] S. Mohajer, S. Diggavi, C. Fragouli, and D. N. C. Tse, “Approximate capacity of a class of Gaussian interference-relay networks,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2837–2864, May 2011.
* [32] T. Gou, S. A. Jafar, S.-W. Jeon, and S.-Y. Chung, “Aligned interference neutralization and the degrees of freedom of the 2x2x2 interference channel,” _IEEE Trans. Inf. Theory_ , vol. 58, no. 7, pp. 4381–4395, Jul. 2012.
* [33] I. Shomorony and A. S. Avestimehr, “Two-unicast wireless networks: Characterizing the degrees-of-freedom,” _Submitted to IEEE Trans. Inf. Theory_ , 2011. [Online]. Available: http://arxiv.org/abs/1102.2498
* [34] I.-H. Wang and D. N. C. Tse, “Gaussian interference channels with multiple receive antennas: Capacity and generalized degrees of freedom,” in _Proc. Forty-Sixth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2008, pp. 715–722.
* [35] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interference through structured codes,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 10, pp. 6463–6486, Oct. 2011.
* [36] U. Erez and R. Zamir, “Achieving $\frac{1}{2}\log(1+\mathsf{SNR})$ on the AWGN channel with lattice encoding and decoding,” _IEEE Trans. Inf. Theory_ , vol. 50, no. 10, pp. 2293–2314, Oct. 2004.
* [37] K. Kobayashi and T. S. Han, “A further consideration on the HK and the CMG regions for the interference channel,” in _Proc. Inf. Theory and App. Workshop (ITA)_ , Jan. 2007.
* [38] Y.-H. Kim, “Coding techniques for primitive relay channels,” in _Proc. Forty-Fifth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2007, pp. 129–135.
* [39] R. Dabora and S. Servetto, “On the role of estimate-and-forward with time sharing in cooperative communication,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 10, pp. 4409–4431, Oct. 2008.
* [40] S. Karmakar and M. Varanasi, “The capacity region of the MIMO interference channel and its reciprocity to within a constant gap,” _Submitted to IEEE Trans. Inf. Theory_ , 2011. [Online]. Available: http://arxiv.org/abs/1102.0267
* [41] O. Sahin and E. Erkip, “On achievable rates for interference relay channel with interference cancelation,” in _Conf. Record Forty-First Asilomar Conf. Signals, Systems and Computers_ , Nov. 2007, pp. 805–809.
* [42] X. Shang, G. Kramer, and B. Chen, “A new outer bound and the noisy-interference sum-rate capacity for Gaussian interference channels,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 689–699, Feb. 2009.
* [43] A. S. Motahari and A. K. Khandani, “Capacity bounds for the Gaussian interference channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 620–643, Feb. 2009.
* [44] V. S. Annapureddy and V. Veeravalli, “Gaussian interference networks: Sum capacity in the low interference regime and new outer bounds on the capacity region,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 7, pp. 3032–3050, Jul. 2009.
* [45] Y.-H. Kim, “Capacity of a class of deterministic relay channels,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 3, pp. 1328–1329, Mar. 2008.
* [46] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” _IEEE Trans. Inf. Theory_ , vol. 27, no. 6, pp. 786–788, Nov. 1981.
* [47] A. B. Carleial, “Interference channels,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 1, pp. 60–70, Jan. 1978.
* [48] E. Telatar and D. N. C. Tse, “Bounds on the capacity region of a class of interference channels,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Jun. 2007, pp. 2871–2874.
Lei Zhou (S’05) received the B.E. degree in Electronics Engineering from
Tsinghua University, Beijing, China, in 2003 and M.A.Sc. degree in Electrical
and Computer Engineering from the University of Toronto, ON, Canada, in 2008.
During 2008-2009, he was with Nortel Networks, Ottawa, ON, Canada. He is
currently pursuing the Ph.D. degree with the Department of Electrical and
Computer Engineering, University of Toronto, Canada. His research interests
include multiterminal information theory, wireless communications, and signal
processing. He is a recipient of the Shahid U.H. Qureshi Memorial Scholarship
in 2011, the Alexander Graham Bell Canada Graduate Scholarship in 2011, and
the Chinese government award for outstanding self-financed students abroad in
2012.
---
Wei Yu (S’97-M’02-SM’08) received the B.A.Sc. degree in Computer Engineering
and Mathematics from the University of Waterloo, Waterloo, Ontario, Canada in
1997 and M.S. and Ph.D. degrees in Electrical Engineering from Stanford
University, Stanford, CA, in 1998 and 2002, respectively. Since 2002, he has
been with the Electrical and Computer Engineering Department at the University
of Toronto, Toronto, Ontario, Canada, where he is now Professor and holds a
Canada Research Chair in Information Theory and Wireless Communications. His
main research interests include multiuser information theory, optimization,
wireless communications and broadband access networks. Prof. Wei Yu currently
serves as an Associate Editor for IEEE Transactions on Information Theory. He
was an Editor for IEEE Transactions on Communications (2009-2011), an Editor
for IEEE Transactions on Wireless Communications (2004-2007), and a Guest
Editor for a number of special issues for the IEEE Journal on Selected Areas
in Communications and the EURASIP Journal on Applied Signal Processing. He is
member of the Signal Processing for Communications and Networking Technical
Committee of the IEEE Signal Processing Society. He received the IEEE Signal
Processing Society Best Paper Award in 2008, the McCharles Prize for Early
Career Research Distinction in 2008, the Early Career Teaching Award from the
Faculty of Applied Science and Engineering, University of Toronto in 2007, and
the Early Researcher Award from Ontario in 2006.
---
|
arxiv-papers
| 2011-09-26T00:21:47 |
2024-09-04T02:49:22.476634
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Zhou and Wei Yu",
"submitter": "Lei Zhou",
"url": "https://arxiv.org/abs/1109.5420"
}
|
1109.5456
|
# static flow on complete noncompact manifolds I: short-time existence and
asymptotic expansions at conformal infinity
Xue Hu †, Yuguang Shi† Xue Hu, Key Laboratory of Pure and Applied mathematics,
School of Mathematics Science, Peking University, Beijing, 100871, P.R. China.
huxue@math.pku.edu.cn Yuguang Shi, Key Laboratory of Pure and Applied
mathematics, School of Mathematics Science, Peking University; Beijing
International Center for Mathematical Research, Beijing, 100871, P.R. China.
ygshi@math.pku.edu.cn
(Date: 2011)
###### Abstract.
In this paper, we study short-time existence of static flow on complete
noncompact asymptotically static manifolds from the point of view that the
stationary points of the evolution equations can be interpreted as static
solutions of the Einstein vacuum equations with negative cosmological
constant. For a static vacuum $(M^{n},g,V),$ we also compute the asymptotic
expansions of $g$ and $V$ at conformal infinity.
###### Key words and phrases:
static flow, asymptotical static, asymptotical hyperbolic, complete noncompact
manifolds, short-time existence, asymptotic expansions
###### 2000 Mathematics Subject Classification:
Primary 53C25; Secondary 58J05
† Research partially supported by NSF grant of China 10725101 and 10990013.
## 1\. Introduction
Geometric flow equations play an important role in geometric analysis
nowadays. For example, the Ricci flow has contributed to the thorough
resolution of the Poincar$\acute{e}$ and Thurston conjectures [20], [21], and
the diffeomorphic $\frac{1}{4}$-pinched sphere theorem [5], [6], while the
inverse mean curvature flow has yielded a proof of the Riemannian Penrose
conjecture [13], [14]. From the latter, naturally, the question of whether
these powerful geometric flow equations can be used in physics has become
common concern both in differential geometry and physics.
The class of static spacetimes is the most simple and interesting object in
general relativity. List has proposed a geometric flow from the point of view
that the stationary points of the evolution equations can be interpreted as
static solutions of the Einstein vacuum equations [19].
In this note, we mainly generalize List’s results to the case with negative
cosmological constant. In this case, the spacetime $(X^{n+1},\tilde{h})$
satisfies the equation
(1) $G+\Lambda\tilde{h}=8\pi T$
where $G=Ric(\tilde{h})-\frac{1}{2}R(\tilde{h}),$ is the Einstein tensor of
$(X^{n+1},\tilde{h}),$ $T$ is the energy-momentum tensor, and $\Lambda$ is
negative cosmological constant. To find solutions to (1) is a central issue in
physics but, in general, the equation is hard to solve, thus special cases are
considered. For $T=0,$ we call a solution to (1) a vacuum spacetime. Without
loss of generality, we normalize the constant $\Lambda$ for a vacuum spacetime
such that
(2) $Ric(\tilde{h})=-n\tilde{h}.$
Much like List did, we consider spacetimes which have some symmetry. In the
following, we assume $n\geq 3.$
###### Definition 1.1.
A Lorentzian manifold $(X^{n+1},\tilde{h})$ is said to be stationary, if there
exists a 1-parameter group of isometries with timelike orbits. If in addition,
there exists a hypersurface $M^{n}$ which is orthogonal to these orbits and
therefore spacelike, $(X^{n+1},\tilde{h})$ is said to be static.
###### Remark 1.2.
For a 1-parameter group of isometries with timelike orbits, equivalent is the
existence of a timelike Killing vector field $\xi.$
###### Remark 1.3.
Static spacetime metric splits as a warped product of $\mathbb{R}$ and a
Riemannian manifold, i.e.,
$X^{n+1}=\mathbb{R}\times M^{n},\tilde{h}=-V^{2}dt^{2}+g,$
where $(M^{n},g)$ is a Riemannian manifold and $V:=\sqrt{-\tilde{h}(\xi,\xi)}$
is a positive function on $M^{n}.$
If we apply the Einstein vacuum equation (2) to the static spacetime metric
$\tilde{h}$, we obtain the equations
(3) $Ric(g)+ng=V^{-1}\nabla_{g}^{2}V,$
and
(4) $\Delta_{g}V=nV.$
###### Definition 1.4.
A Riemannian manifold $(M^{n},g)$ with a positive function $V$ on $M^{n}$
satisfying (3) and (4) is called static Einstein vacuum, denoted by
$(M^{n},g,V).$
Examples of static Einstein vacuum are the Anti-de Sitter and Schwarzschild-
AdS metrics. For more details, please refer to [7], [22] and [25] and the
references therein.
We introduce the following static flow for $g$ and $V$ on $M^{n}:$
(5) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}g=-2Ric(g)-2ng+2V^{-1}\nabla_{g}^{2}V\\\ \frac{\partial}{\partial
t}V=\Delta_{g}V-nV.\\\ \end{array}\right.$
In [19], List considered static Einstein vacuum with vanishing cosmological
constant, that is, a triple $(M^{n},g,V)$ satisfies
$Ric(g)=V^{-1}\nabla_{g}^{2}V$
and
$\Delta_{g}V=0.$
He took $u=\ln V$ and took conformal transformation
$\tilde{g}=e^{\frac{2}{n-2}u}\cdot g,$ in this way, the static Einstein vacuum
equation $\Delta_{g}V=0$ became $\Delta_{\tilde{g}}u=0,$ while
$Ric(g)=V^{-1}\nabla_{g}^{2}V$ became
$Ric(\tilde{g})=\frac{n-1}{n-2}\nabla_{\tilde{g}}u\otimes\nabla_{\tilde{g}}u,$
involving second derivatives of $\tilde{g}$ but merely first derivatives of
$u.$ Thus List proposed the extended Ricci flow system
(6) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}g=-2Ric(g)+4du\otimes du,\\\ \frac{\partial}{\partial t}u=\Delta_{g}u.\\\
\end{array}\right.$
Using DeTurck’s trick, this system could be changed to be strictly quasi-
linear parabolic system and then List applied standard parabolic theory as
well as Shi’s ideas [24] to get the short-time existence result.
We point out here that List’s system of flow equations is in fact the pullback
by a certain diffeomorphism of a class of Ricci flows in one higher
dimension(Please see [1]). Comparing with List’s flow, our static flow (5)
comes directly from the one-higher dimensional Ricci flow, more specifically,
the Ricci flow of a warped product. Notice that for static Einstein vacuum
$(M^{n},g,V),$ the Riemannian metric $h=V^{2}d\theta^{2}+g$ can be viewed as
an Einstein metric on $\mathbb{S}^{1}\times M^{n}.$ Due to this fact,
Anderson, Chru$\acute{s}$ciel and Delay could use the result for conformally
compact Einstein manifolds to study non-trivial, static, geodesically
complete, vacuum space-times with a negative cosmological constant. Please see
[2] and [3]. Meanwhile, the Ricci tensor of $h$ is given by
$Ric(h)=\left(\begin{array}[]{cc}-V^{-1}\Delta_{g}V&0\\\
0&Ric(g)-V^{-1}\nabla_{g}^{2}V\\\ \end{array}\right).$
Thus we can see that the normalized Ricci flow equation
$\frac{\partial}{\partial t}h=-2Ric(h)-2nh$
induces the static flow (5).
However, taking into account factor such as the use of the flow, for negative
cosmological constant case, we don’t employ the means List used. List’s flow
(6) preserves two classes of asymptotically flat solutions assuming a uniform
curvature bound (see [19] Theorem 9.5 and Theorem 9.7). Similarly, we hope to
propose a kind of flow which preserves some classes of asymptotically
hyperbolic(see definition1.5) solutions under some circumstances. In the
asymptotically flat category, though List’s flow is an extended Ricci flow, we
can see that the conformal transformation doesn’t bring any trouble in the
preservation of asymptotically flat structure. But even for Anti-de Sitter
spacetime,
$(\mathbb{R}\times\mathbb{H}^{n},-\cosh^{2}rdt^{2}+dr^{2}+\sinh^{2}rdw^{2}),$
that is,
$(M^{n},g,V)=(\mathbb{H}^{n},dr^{2}+\sinh^{2}rdw^{2},\cosh r),$
if we take the same conformal transformation, we find that
$\tilde{g}=e^{\frac{2}{n-2}u}\cdot g$ where $u=\ln V$ induces a new metric
$\tilde{g}=(\cosh r)^{\frac{2}{n-2}}g_{\mathbb{H}^{n}},$ which is obviously
not asymptotically hyperbolic. In other words, List’s conformal transformation
doesn’t preserve asymptotically hyperbolic structure.
Now we give some basic notions in conformally compact geometry that we may use
later.
Suppose that $X^{n+1}$ is a smooth manifold with boundary $\partial
X^{n+1}=Y^{n}$. A defining function $\tau$ of the boundary $Y^{n}$ in
$X^{n+1}$ is a smooth function on $X^{n+1}$ such that
1. (1)
$\tau>0$ in $X^{n+1}$;
2. (2)
$\tau=0$ on $Y^{n}$;
3. (3)
$d\tau\neq 0$ on $Y^{n}$.
A complete Riemannian metric $h$ on $X^{n+1}$ is conformally compact if
$(\bar{X}^{n+1},\bar{h}=\tau^{2}h)$ is a compact Riemannian manifold for a
smooth defining function $\tau$ of the boundary $Y^{n}$ in $X^{n+1}.$ If
$\bar{h}$ is $C^{k,\mu},$ $h$ is said to be conformally compact of regularity
$C^{k,\mu}.$ The restriction of $\bar{h}$ to $TY^{n}$ rescales upon changing
$\tau,$ so defines invariantly a conformal class of metrics on $Y^{n}.$
$(Y^{n},[\bar{h}\mid_{TY^{n}}])$ is called the conformal infinity of the
conformally compact manifold $(X^{n+1},h).$
###### Definition 1.5.
A complete noncompact Riemannian manifold $(X^{n+1},h)$ is called
asymptotically hyperbolic($AH$) of order $a$ if
$||Rm-\mathbf{K}||_{h}\leq Ce^{-a\rho},$
where $Rm$ denotes the Riemann curvature tensor of the metric $h$ and
$\mathbf{K}$ the constant curvature tensor of $-1$, i.e.,
$\mathbf{K}_{ijkl}=-(h_{ik}h_{jl}-h_{il}h_{jk})$; $\rho$ is the distance
function to a fixed point in $X^{n+1}$ with respect to $h$; and $C$ is a
positive constant independent of $\rho$.
###### Remark 1.6.
$AH$ order $a=2$ is important and interesting in mathematics and physics. On
one hand, roughly speaking, there is rigidity when $a>2$ (see [15] and the
references therein); On the other hand, $C^{2}$ conformally compact Einstein
manifolds are $AH$ of order $2.$ In order to avoid the complexity of the end
structure of a hyperbolic manifold, we need the concept of an essential set
here. Please see [15] Definition 1.1 for the definition of an essential set.
In Gicquaud’s PhD thesis, he proved that if a complete noncompact manifold is
$C^{2}$ conformally compact then it contains essential sets([12] Lemma 2.5.11
and Corollary 2.5.12).
The recent work of [4] and [23] investigated the behavior of normalized Ricci
flow on $AH$ manifolds, while it took care of conformal infinities. By virtue
of the ideas in these two papers, we give the following
###### Definition 1.7.
Suppose $(M^{n},g)$ is a smooth complete noncompact Riemannian manifold with
an essential set and $V$ is a positive function on $M^{n}.$ Then a triple
$(M^{n},g,V)$ is called asymptotically static(AS) of order $a\geq 2$ if
$(\mathbb{S}^{1}\times M^{n},h=V^{2}d\theta^{2}+g)$ is $AH$ of order $2,$ and
$||\nabla_{h}Ric(h)||_{h}(\theta,x)\leq Ce^{-a\rho(\theta,x)}.$
where $\rho$ is the distance function to a fixed point in
$\mathbb{S}^{1}\times M^{n}$ with respect to $h$; and $C$ is a positive
constant independent of $\rho.$
If we compute the curvature tensor of the warped product in terms of Gauss-
Codazzi equations, we have the following
###### Proposition 1.8.
If $(M^{n},g,V)$ is AS of order $a\geq 2$, then
1. (1)
$(M^{n},g)$ is AH of order $2;$
2. (2)
$||Ric(g)+ng-V^{-1}\nabla_{g}^{2}V||_{g}\leq Ce^{-2r};$
3. (3)
$||\nabla_{g}(V^{-1}\Delta_{g}V)||_{g}\leq Ce^{-ar}$ and
$||\nabla_{g}(Ric(g)-V^{-1}\nabla_{g}^{2}V)||_{g}\leq Ce^{-ar};$
where $r$ is the distance function to a fixed point in $M^{n}$ with respect to
$g.$
Now, we state the main results:
###### Theorem 1.9.
Suppose $(M^{n},g_{0},V_{0})$ is asymptotically static of order $a\geq 2$ and
$\|e^{-r}V_{0}\|_{C^{2+\alpha}}\leq C$ where $\alpha\in(0,1).$ Then for any
$\epsilon>0,$ there exists $T_{0}=T_{0}(n,C,\epsilon)$ such that the static
flow (5) with initial data $g(0)=g_{0}$ and $V(0)=V_{0}$ has a smooth solution
$(g,V)(x,t)$ on $M\times[0,T_{0}]$ satisfying
$\parallel
g-g_{0}\parallel_{g_{0}}(x)+\parallel\nabla_{g_{0}}g\parallel_{g_{0}}(x)\leq\epsilon
e^{-2r(x)}$
for all $(x,t)\in M\times[0,T_{0}].$
###### Remark 1.10.
Since the static flow for $(M^{n},g,V)$ comes from the normalized Ricci flow
for $(\mathbb{S}^{1}\times M^{n},h=V^{2}d\theta^{2}+g),$ we take advantage of
the results in [8] to obtain the uniqueness of the static flow for curvature
bounded solutions. In Theorem 1.9, if the initial metric is $C^{2+\alpha},$
the solution $(g,V)$ to the static flow we get in Theorem 1.9 is still
$C^{2+\alpha},$ then the curvature for $h$ is bounded. Hence we get the
uniqueness of the static flow.
Given a conformally compact, asymptotically hyerbolic manifold $(X^{n+1},h)$
and a representative $\hat{h}$ in $[\hat{h}]$ on the conformal infinity
$Y^{n},$ there is a unique determined defining function $\tau$ such that, in a
neighborhood of the boundary $[0,\delta)\times Y^{n}\subset\bar{X}^{n+1},$ $h$
has the form
$h=\tau^{-2}(d\tau^{2}+h_{\tau}),$
where $h_{\tau}$ is a $1-$parameter family of metrics on $Y^{n}.$ We call this
$\tau$ the special defining function associated with $\hat{h}.$ In [11],
Graham showed the asymptotic expansion of a conformally compact Einstein
manifold with respect to the special defining function, which is associated
with a conformal infinity. Naturally, we come to the question that whether we
can do the same thing for static Einstein vacuum. But this will arose another
question that how $V$ behaves near the infinity. In [25], X.D. Wang has proved
a uniqueness theorem of AdS spacetime, there he required $(M^{n},g)$ to be
conformally compact and $V^{-1}$ to be a defining function for $(M^{n},g).$
Inspired by this idea, we require that $V$ be with the growth of the inverse
of a special defining function near conformal infinity, then we have
###### Theorem 1.11.
Suppose that $(M^{n},g,V)$ is a static Einstein vacuum, that $(M^{n},g)$ is an
asymptotically hyperbolic manifold with the conformal infinity
$(N^{n-1},[\hat{g}])$ and that $\tau$ is the special defining function
associated with a metric $\hat{g}\in[\hat{g}].$ Assume also that $V$ is with
the growth of $\frac{1}{\tau}$ near conformal infinity. If
$g=\tau^{-2}(d\tau^{2}+g_{\tau}),$ then we have
$g_{\tau}=\hat{g}+g^{(2)}\tau^{2}+(even~{}powers~{}of~{}\tau)+g^{(2l)}\tau^{2l}+\cdots$
$V=\frac{1}{\tau}+V^{(1)}\tau+(odd~{}powers~{}of~{}\tau)+V^{(2l-1)}\tau^{2l-1}+\cdots$
where $g^{(2l)}$ and $V^{(2l-1)}$ are uniquely determined by $\hat{g}$ for
$2l\leq n-1.$
This paper would be the first one among a series of papers, which we are about
to work on. In our sequel paper, we will pay more attention to the conformal
category. We will use Theorem1.11 and the methods in [23] to prove existence
result of the static metric, and discuss the behavior of the conformal
infinity in this flow and other related problems. This paper is organized as
follows: In section 2 we prove the short-time existence of the static flow. In
section 3, we compute the asymptotic expansions at conformal infinity.
Acknowledgements The authors are grateful to Professor Xiaodong Wang,
Professor Jie Qing and Professor Romain Gicquaud for their interests in this
work and many enlightening discussions. The authors are also indebted to
Professor Romain Gicquaud to show us the references [2] and [3], as well as
correct some mistakes. The authors would like to express our gratitude for
Professor Eric Woolgar to point out a fact that we didn’t know and show the
citation [1].
## 2\. short-time existence
In this section, we mainly prove the short-time existence of our static flow.
We follow the ideas of Shi[24] and List[19] to modify the static flow and
derive some formulas from the altered flow equations, and then use a maximum
principle by Ecker and Huisken to get our basic estimates. Next, we construct
a Banach space and an operator, together with standard parabolic theory for
linear equation, we can check the fulfillment of the Schauder fixed point
theorem, which promises the short-time existence of static flow.
We begin with the proof of Proposition 1.8 and then we can use it in the
following estimates.
###### Proof of Proposition 1.8.
For any $x\in M^{n},$ and any 2-plane $\Pi\subset T_{x}M,$ without loss of
generality, we assume that $\Pi$ is spanned by unit orthogonal vectors $Z^{1}$
and $Z^{2}$. Supplementing the basis such that $\\{Z^{i}\\}_{i=1}^{n}$ is
orthonormal basis with respect to $g,$ i.e. $g(Z^{i},Z^{j})=\delta_{ij}.$ Then
$\\{\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i}\\}$ is orthonormal basis
with respect to $h.$
The assumption
$||Rm(h)-\mathbf{K}||_{h}(\theta,x)\leq Ce^{-2\rho(\theta,x)},$
implies
(7)
$\begin{split}&|Rm(h)(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i})-(-1)[h(\frac{1}{V}\frac{\partial}{\partial\theta},\frac{1}{V}\frac{\partial}{\partial\theta})h(Z^{i},Z^{i})-h(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i})^{2}]|\\\
&\leq Ce^{-2\rho(\theta,x)}\end{split}$
and
(8)
$|Rm(h)(Z^{i},Z^{j},Z^{i},Z^{j})-(-1)[h(Z^{i},Z^{i})h(Z^{j},Z^{j})-h(Z^{i},Z^{j})^{2}]|\leq
Ce^{-2\rho(\theta,x)}.$
By direct computation, we have
$\nabla_{Z}\frac{\partial}{\partial\theta}=V^{-1}\cdot
Z(V)\cdot\frac{\partial}{\partial\theta},$
and
$\nabla_{\frac{\partial}{\partial\theta}}\frac{\partial}{\partial\theta}=-\sum_{i=1}^{n}V\cdot
Y^{i}(V)\cdot Y^{i}.$
As $\vartheta=\frac{1}{V}\frac{\partial}{\partial\theta}$ is a unit normal
vector field, for any tangent vector fields $W$ and $Y$ on $M^{n},$ the second
fundamental form of $(M^{n},g)$ in $(\mathbb{S}^{1}\times
M^{n},h=V^{2}d\theta^{2}+g)$ is
$S(W,Y)=h(\nabla_{W}\vartheta,Y)=0.$
In other words, $(M^{n},g)$ is totally geodesic in $(\mathbb{S}^{1}\times
M^{n},h).$
Together with Gauss-Codazzi equations, we know that (8) implies
$\begin{split}&|Rm(g)(Z^{i},Z^{j},Z^{i},Z^{j})-(-1)[g(Z^{i},Z^{i})g(Z^{j},Z^{j})-g(Z^{i},Z^{j})^{2}]|\leq
Ce^{-2\rho(\theta,x)}\\\ &\leq Ce^{-2r(x)},\end{split}$
which means that the sectional curvature of any 2-plane $\Pi\subset T_{x}M,$
for any $x\in M^{n}$ is approaching $-1,$ hence $(M^{n},g)$ is $AH$ of order
$2.$
Now Let us take a deep investigation of (7). We find that
(9)
$\begin{split}Ric(g)(Z^{i},Z^{k})&=\sum_{j}Rm(g)(Z^{i},Z^{j},Z^{k},Z^{j})\\\
&=\sum_{j}Rm(h)(Z^{i},Z^{j},Z^{k},Z^{j})\\\
&=Ric(h)(Z^{i},Z^{k})-Rm(h)(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta},Z^{k})\\\
&=-nh(Z^{i},Z^{k})+O(e^{-2\rho})-Rm(h)(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta},Z^{k})\\\
&=-ng(Z^{i},Z^{k})+O(e^{-2r})-Rm(h)(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta},Z^{k})\\\
\end{split}$
Meanwhile,
(10)
$\begin{split}&Rm(h)(\frac{1}{V}\frac{\partial}{\partial\theta},Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta},Z^{k})\\\
=&h(\nabla_{Z^{i}}\nabla_{(\frac{1}{V}\frac{\partial}{\partial\theta})}(\frac{1}{V}\frac{\partial}{\partial\theta})-\nabla_{(\frac{1}{V}\frac{\partial}{\partial\theta})}\nabla_{Z^{i}}(\frac{1}{V}\frac{\partial}{\partial\theta})-\nabla_{[Z^{i},\frac{1}{V}\frac{\partial}{\partial\theta}]}(\frac{1}{V}\frac{\partial}{\partial\theta}),Z^{k})\\\
=&-V^{-1}\nabla_{g}^{2}V(Z^{i},Z^{k}).\\\ \end{split}$
Substituting (10) to (9), we obtain
$Ric(g)(Z^{i},Z^{k})+ng(Z^{i},Z^{k})-V^{-1}\nabla_{g}^{2}V(Z^{i},Z^{k})=O(e^{-2r}),$
which means
$||Ric(g)+ng-V^{-1}\nabla_{g}^{2}V||_{g}\leq Ce^{-2r}.$
At last, we compute
$\begin{split}&(\nabla_{h}Ric(h))(Z,Y,W)\\\ =&(\nabla_{Z}^{h}Ric(h))(Y,W)\\\
=&\nabla_{Z}^{h}(Ric(h)(Y,W))-Ric(h)(\nabla_{Z}^{h}Y,W)-Ric(h)(Y,\nabla_{Z}^{h}W)\\\
=&\nabla_{Z}^{g}(Ric(h)(Y,W))-Ric(h)(\nabla_{Z}^{g}Y,W)-Ric(h)(Y,\nabla_{Z}^{g}W)\\\
=&\nabla_{Z}^{g}[Ric(g)(Y,W)-V^{-1}\nabla_{g}^{2}V(Y,W)]\\\
-&[Ric(g)(\nabla_{Z}^{g}Y,W)-V^{-1}\nabla_{g}^{2}V(\nabla_{Z}^{g}Y,W)]\\\
-&[Ric(g)(Y,\nabla_{Z}^{g}W)-V^{-1}\nabla_{g}^{2}V(Y,\nabla_{Z}^{g}W)]\\\
=&\nabla_{g}(Ric(g)-V^{-1}\nabla_{g}^{2}V)(Z,Y,W)\\\ \end{split}$
and
$(\nabla_{h}Ric(h))(Z,\frac{1}{V}\frac{\partial}{\partial\theta},\frac{1}{V}\frac{\partial}{\partial\theta})=\nabla_{Z}^{g}(V^{-1}\Delta_{g}V),$
we therefore conclude
$||\nabla_{g}(V^{-1}\Delta_{g}V)||_{g}\leq Ce^{-ar},$
and
$||\nabla_{g}(Ric(g)-V^{-1}\nabla_{g}^{2}V)||_{g}\leq Ce^{-ar}.$
∎
We will now deduce from our static flow some basic evolution equations
involving some quantities we need to estimate in the following. First we
observe that the first equation in the system
(11) $\frac{\partial}{\partial t}g=-2Ric(g)-2ng+2V^{-1}\nabla_{g}^{2}V$ (12)
$\frac{\partial}{\partial t}V=\Delta_{g}V-nV$
is only weakly parabolic due to diffeomorphism invariance of the equation.
Since the last two terms depend on $g$ and $\partial g,$ the principal symbol
of the first equation is the same as for the Ricci flow, we could use
DeTurck’s trick to break the gauge, following the presentation of List [19]
very closely.
###### Lemma 2.1.
Suppose that $(g,V)$ is a solution to
(13) $\frac{\partial}{\partial t}g=-2Ric(g)-2ng+2V^{-1}\nabla_{g}^{2}V+L_{W}g$
(14) $\frac{\partial}{\partial t}V=\Delta_{g}V-nV+dV(W)$
on $M^{n}\times[0,T],$ and that $W$ is a smooth time dependent vector field on
$M^{n},$ and that $\varphi_{t}:M^{n}\longrightarrow M^{n}$ is the 1-parameter
family of diffeomorphisms generated by $W,$ that is,
$\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}\varphi_{t}(x)=W(\varphi_{t}(x),t)\\\ \varphi_{0}=id.\\\ \end{array}\right.$
Then the pullbacks
$\left\\{\begin{array}[]{ll}\bar{g}(t):=(\varphi_{t}^{-1})^{*}g\\\
\bar{V}(t):=(\varphi_{t}^{-1})^{*}V\\\ \end{array}\right.$
satisfy (11) and (12) on $M^{n}\times[0,T].$ Moreover, $(g,V)(t)$ have the
same initial values as $(\bar{g},\bar{V})(t),$ that is,
$(g,V)(t)=(g_{0},V_{0}).$
###### Proof.
Denote by $\\{y^{\alpha}\\}_{\alpha=1...n}$ the coordinates where $\bar{g}$
and $\bar{V}$ are represented by $\bar{g}_{\alpha\beta}$ and $\bar{V}.$ As the
same argument in [24] and [19], we define new coordinates by
$x^{i}:=(y\circ\varphi)^{i}$ for $i=1...n.$ Then
$\begin{split}&\varphi_{t}^{\ast}(\bar{V}^{-1}(y,t)\bar{\nabla}_{\bar{g}}^{2}\bar{V}(y,t))_{ij}\\\
&=V(x)^{-1}\cdot\varphi_{t}^{\ast}(\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}\bar{V}dy^{\alpha}\otimes
dy^{\beta})_{ij}\\\ &=V(x)^{-1}\cdot(\frac{\partial x^{i}}{\partial
y^{\alpha}}\frac{\partial x^{j}}{\partial
y^{\beta}}\nabla_{i}\nabla_{j}V\cdot\frac{\partial y^{\alpha}}{\partial
x^{i}}dx^{i}\frac{\partial y^{\beta}}{\partial x^{j}}dx^{j})\\\
&=V^{-1}\nabla_{g}^{2}V.\\\ \end{split}$
The rest of the proof is unchanged as [19] Lemma 3.1. ∎
An easy computation shows that if $W$ is given by
$W^{k}=g^{ij}(\Gamma_{ij}^{k}-\Gamma_{ij}^{k}(g_{0})),$
the trace of the tensor which is the difference between the Christoffel
symbols of the Levi-Civita connections of $g$ and of $g_{0}$, the equation
(13) is strictly parabolic. The idea is from DeTurck [9]. We pursue this idea
via taking all the derivatives with respect to the initial metric $g_{0}.$
This is characterized as follows. For convenience, we use $\hat{g}$ to
represent $g_{0},$ and then $\hat{\nabla}$ and $\hat{\Delta}$ are covariant
derivative and Laplacian with respect to $g_{0}.$
###### Lemma 2.2.
Let $W^{k}=g^{ij}(\Gamma_{ij}^{k}-\hat{\Gamma}_{ij}^{k}),$ then for given $V,$
the equation (13) is strictly parabolic.
###### Proof.
The calculation goes the same as [19] Lemma 3.2. The equations (13) and (14)
turn to
(15) $\left\\{\begin{array}[]{ll}\begin{split}\frac{\partial}{\partial
t}g_{ij}&=g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}g_{ij}-g^{kl}g_{ip}\hat{g}^{pq}\hat{R}_{jkql}-g^{kl}g_{jp}\hat{g}^{pq}\hat{R}_{ikql}\\\
&+\frac{1}{2}g^{kl}g^{pq}(\hat{\nabla}_{i}g_{pk}\cdot\hat{\nabla}_{j}g_{ql}+2\hat{\nabla}_{k}g_{jp}\cdot\hat{\nabla}_{q}g_{il}-2\hat{\nabla}_{k}g_{jp}\cdot\hat{\nabla}_{l}g_{iq}\\\
&-2\hat{\nabla}_{j}g_{pk}\cdot\hat{\nabla}_{l}g_{iq}-2\hat{\nabla}_{i}g_{pk}\cdot\hat{\nabla}_{l}g_{jq})\\\
&+2V^{-1}\hat{\nabla}_{i}\hat{\nabla}_{j}V-V^{-1}\cdot
g^{kl}(\hat{\nabla}_{i}g_{jl}+\hat{\nabla}_{j}g_{il}-\hat{\nabla}_{l}g_{ij})\cdot\hat{\nabla}_{k}V-2ng_{ij}\\\
\end{split}\\\ \frac{\partial}{\partial
t}V=g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}V-nV.\\\ g(\cdot,0)=g_{0}\\\
V(\cdot,0)=V_{0}.\\\ \end{array}\right.$
It is then obvious to see that these equations are strictly parabolic. ∎
The above two lemmas show that once we obtain the short-time existence of the
initial problem (15), after pulling back the solutions to (15) by a specific
diffeomorphism, then we will also get the short-time existence to our static
flow (5). Hence now we prove the existence of a solution to the initial value
problem (15) on $M^{n}\times[0,T].$ We fix a point $o\in M^{n}$ and $r(x)$ is
the distance function to $o$ with respect to $\hat{g}.$ Since $M^{n}$ is
complete noncompact, it gurantees a family of domains $\\{D_{k}\subset
M^{n}:k=1,2,3...\\}$ such that for each $k:$
1. (1)
the boundary $\partial D_{k}$ is a smooth $(n-1)$-dimensional submanifold of
$M^{n};$
2. (2)
the closure $\bar{D}_{k}$ is compact in $M^{n};$
3. (3)
$\bigcup_{k=1}^{\infty}D_{k}=M^{n};$
4. (4)
$B(o,k)\subset D_{k};$
where $B(o,k)$ is a geodesic ball of center $o$ and radius $k$ with respect to
$\hat{g}.$ The parabolic boundary $\Gamma_{k}$ of $D_{k}\times[0,T]$ is
defined by:
$\Gamma_{k}:=(D_{k}\times\\{0\\})\cup(\partial D_{k}\times[0,T]).$
For the short time existence of the evolution equation, our strategy is to use
Schauder fixed point theorem that a compact mapping of a closed bounded convex
set in a Banach space into itself has a fixed point. Thus we need to construct
a Banach space and its closed bounded convex set as well as an operator, and
meanwhile show that the operator has a fixed point which is just the solution
to the static flow (15). Since our initial data $(M^{n},\hat{g})$ is
asymptotical hyperbolic, which is conformally compactifiable, we can find
special coordinate charts called M$\ddot{o}$bius charts. For a concrete
definition of M$\ddot{o}$bius charts, please see [18] chapter 2. The advantage
is that the geometry of $(M^{n},\hat{g})$ is uniformly bounded in
M$\ddot{o}$bius charts.
Before we introduce our Banach space, we give the definition of general
$H\ddot{o}lder$ space used in standard parabolic theory. Denote
$Q_{T}:=\Omega\times(0,T).$ For $0<\mu<1$ and $l$ a nonnegative integer, we
let
$\begin{split}C^{2l+\mu,l+\frac{\mu}{2}}(\bar{Q}_{T})=&\\{u;\partial^{\beta}\partial_{t}^{r}u\in
C^{\mu,\frac{\mu}{2}}(\bar{Q}_{T}),\\\
&for~{}any~{}\beta,~{}r~{}such~{}that~{}|\beta|+2r\leq 2l\\},\\\ \end{split}$
where $C^{\mu,\frac{\mu}{2}}(\bar{Q}_{T})$ is the set of all functions on
$\bar{Q}_{T}$ such that $[u]_{\mu,\frac{\mu}{2};\bar{Q}_{T}}<+\infty,$ endowed
with the norm
$|u|_{\mu,\frac{\mu}{2};\bar{Q}_{T}}=|u|_{0;Q_{T}}+[u]_{\mu,\frac{\mu}{2};\bar{Q}_{T}},$
here
$|u|_{0;\bar{Q}_{T}}=\sup_{(x,t)\in\bar{Q}_{T}}|u(x,t)|$
and
$[u]_{\mu,\frac{\mu}{2};\bar{Q}_{T}}=\sup_{(x,t),(y,s)\in\bar{Q}_{T},(x,t)\neq(y,s)}\frac{|u(x,t)-u(y,s)|}{(|x-y|^{2}+|t-s|)^{\frac{1}{2}}}.$
To define $H\ddot{o}lder$ norms for tensor bundle $E,$ let
$C^{2l+\mu,l+\frac{\mu}{2}}(M^{n}\times[0,T];E)$ be the space of tensor fields
whose components in each M$\ddot{o}$bius coordinate chart are in
$C^{2l+\mu,l+\frac{\mu}{2}}(M^{n}\times[0,T]).$
We define a space
$\begin{split}\mathbf{B}_{\alpha}=&\\{s(\cdot,t)\in Sym(T^{*}M\otimes
T^{*}M)\mid\\\ &e^{2r}s\in
C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T];T^{*}M\otimes T^{*}M),\\\
&e^{2r}\hat{\nabla}s\in
C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T];T^{*}M\otimes T^{*}M\otimes
T^{*}M),t\in[0,T]\\}\end{split}$
with norms
$\begin{split}\|s\|_{\mathbf{B}_{\alpha}}:&=\|e^{2r}s\|_{C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T];T^{*}M\otimes
T^{*}M)}\\\
&+\|e^{2r}\hat{\nabla}s\|_{C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T];T^{*}M\otimes
T^{*}M\otimes T^{*}M)}\\\ &<+\infty\end{split}$
Obviously, $\mathbf{B}_{\alpha}$ is a Banach space. For any $\epsilon>0,$ we
let
$\mathbf{B}_{\alpha,\epsilon}=\\{s\in\mathbf{B}_{\alpha}\mid\|s\|_{\mathbf{B}_{\alpha}}\leq\epsilon\\}.$
It is a closed bounded convex set in $\mathbf{B}_{\alpha}$.
Next we need to define our operator
$\mathbf{A}:\mathbf{B}_{\alpha,\epsilon}\longrightarrow\mathbf{B}_{\alpha}$
and prove that it is well-defined, the range of $\mathbf{A}$ still lies in
$\mathbf{B}_{\alpha,\epsilon},$ and $\mathbf{A}$ is compact in
$\mathbf{B}_{\alpha,\epsilon}.$ In order to define $\mathbf{A},$ it is
necessary to get the estimates of $V.$ We have the following
###### Lemma 2.3.
Suppose that $(M^{n},\hat{g},\hat{V})$ is $AS$ of order $a\geq 2,$ and
$\|e^{-r}\hat{V}\|_{C^{2+\alpha}}\leq C.$ Consider
$\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}V=g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}V-nV\\\
V(\cdot,0)=\hat{V}>0~{}~{}~{}on~{}~{}~{}M^{n}\times[0,T]\\\
\end{array}\right.$
If $g(x,t)\in\mathbf{B}_{\alpha,\epsilon}+\hat{g},$ then the above initial
problem has a unique positive solution $V$ from the class
$C^{2+\alpha,1+\frac{\alpha}{2}}(M^{n}\times[0,T])$ and it satisfies
$e^{r}(V-\hat{V})\in C^{2+\alpha,1+\frac{\alpha}{2}}(M^{n}\times[0,T]).$
###### Proof.
For any $x\in M^{n},$ there exists $k$ such that $x\in B(x,1)\subset
B(x,2)\subset D_{k}.$ We solve the following initial-boundary value problem
(16) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}V=g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}V-nV,~{}~{}~{}on~{}D_{k}\times[0,T]\\\
V\mid_{\Gamma_{k}}=\hat{V}.\\\ \end{array}\right.$
If we let $v=V-\hat{V},$ (16) is equivalent to the following initial-boundary
value problem
(17) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}v=g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}v-nv+g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}\hat{V}-n\hat{V},~{}~{}~{}on~{}D_{k}\times[0,T]\\\
v\mid_{\Gamma_{k}}=0.\\\ \end{array}\right.$
From the assumption that
$g(x,t)=s(x,t)+\hat{g}\in\mathbf{B}_{\alpha,\epsilon}+\hat{g},$ note that we
choose $\epsilon\ll 1$ such that the above equation is uniformly parabolic,
together with standard parabolic theory(see [17] P320 Theorem 5.2), we know
linear initial-boundary value problem (17) has a unique solution $v_{k}$ from
the class $C^{2+\alpha,1+\frac{\alpha}{2}}(\bar{D}_{k}\times[0,T]),$ and it
satisfies
(18) $\|v_{k}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\leq
C\|g^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}\hat{V}-n\hat{V}\|_{C^{\alpha,\frac{\alpha}{2}}(B(x,2)\times[0,T])}.$
Since the right hand side in (18) is independent of $k$ and we have
$\bigcup_{k=1}^{\infty}D_{k}=M^{n},$ we can take the limit
$k\longrightarrow\infty$ and get the convergence of a subsequence of the
solutions $v_{k}$ in the $C^{2+\beta,1+\frac{\beta}{2}}(\beta<\alpha)$
topology on compact subsets of $M^{n}\times[0,T]$ to a
$C^{2+\alpha,1+\frac{\alpha}{2}}$ solution $v$ on $M^{n}\times[0,T]$ by the
theorem of Arzela-Ascoli for any given
$s(x,t)\in\mathbf{B}_{\alpha,\epsilon}.$ From (18), we obtain
$\begin{split}&\|v\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\\\
\leq&C(\|g-\hat{g}\|_{C^{\alpha,\frac{\alpha}{2}}(B(x,2)\times[0,T])}\cdot\|\hat{V}\|_{C^{2+\alpha}(B(x,2))}+\|\hat{\Delta}\hat{V}-n\hat{V}\|_{C^{\alpha}(B(x,2))}).\\\
\end{split}$
When $y\in B(x,2),$ $r(x)$ and $r(y)$ is equivalent, thus
$\|v\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\leq
Ce^{-r(x)},~{}~{}~{}t\in[0,T].$
Hence $V=v+\hat{V}$ is a solution to (16). In order to get that $V$ is
positive and unique, it is enough to get the same results for the original
equation (12). As $\hat{V}>0$ and $\hat{V}=O(e^{r}),$ though the manifold is
complete noncompact, there still exists a positive constant $d$ such that
$\hat{V}\geq d$ on the whole $M^{n}.$ We define $J(x,t)=d-e^{nt}V(x,t),$ then
$J$ satisfies
$\frac{\partial}{\partial t}J=\Delta_{g}J,~{}~{}on~{}M^{n}\times[0,T]$
and $J(\cdot,0)\leq 0.$ Since
$\begin{split}&\int_{0}^{T}\int_{M^{n}}\exp(-r(x)^{2})J_{+}^{2}(x,t)d\mu_{g(t)}dt\\\
\leq&C\int_{0}^{T}\int_{M^{n}}\exp(-r(x)^{2}+2r(x)+2nt)d\mu_{\hat{g}}dt\\\
<&\infty,\end{split}$
due to the maximum principle of Karp and Li [16], we get $J\leq 0$ on
$M^{n}\times[0,T],$ which means $V$ is always positive on $M^{n}\times[0,T].$
If $V_{1}$ and $V_{2}$ are two solutions to (12) with the same initial data
$\hat{V}.$ Let $I(x,t)=e^{nt}(V_{1}-V_{2}),$ then
$\frac{\partial}{\partial t}I=\Delta_{g}I,~{}~{}on~{}M^{n}\times[0,T]$
and $J(\cdot,0)=0.$ Also by the same maximum principle, we conclude $I=0$ on
$M^{n}\times[0,T],$ that is, $V_{1}=V_{2}$ on $M^{n}\times[0,T].$ ∎
Motivated by the above lemma, we define the operator
$\mathbf{A}:\mathbf{B}_{\alpha,\epsilon}\longrightarrow\mathbf{B}_{\alpha}$ by
$\mathbf{A}(s)=G-\hat{g}$
where
$\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}G_{ij}=g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}G_{ij}+H_{ij}(\hat{R}m,\hat{\nabla}g,g,V^{-1}\nabla_{g}^{2}V),~{}~{}~{}on~{}M^{n}\times[0,T]\\\
G_{ij}(\cdot,0)=\hat{g}_{ij}.\\\ \end{array}\right.$
Here $H_{ij}(\hat{R}m,\hat{\nabla}g,g,V^{-1}\nabla_{g}^{2}V)$ contains all
remaining terms of the evolution equation of $g=s+\hat{g}$ in (15) and $V$ is
the unique positive solution in Lemma 2.3 for given $g=s+\hat{g}$. Since we
first need to get the a priori estimates, we claim the following:
###### Proposition 2.4.
Suppose $(M^{n},\hat{g})$ is a complete noncompact AH Riemannian manifold of
order $\eta$ and it contains an essential set. Let $f,$ which is smooth on
$M^{n}\times(0,T]$ and continuous on $M^{n}\times[0,T]$ satisfy
$(\frac{\partial}{\partial t}-\hat{\Delta})f(x,t)\leq Q(x,t)$
on $M^{n}\times[0,T],$ where $Q(x,t)$ is a function on $M^{n}\times[0,T]$
satisfying $|Q(x,t)|\leq Le^{-\mu r(x)}.$ Here $\mu\in\mathbb{R}$ and $L>0$
are constants and $r(x)$ is the distance function to the essential set with
respect to $\hat{g}.$ Assume that $|f|+|\hat{\nabla}f|\leq C_{1}$ on
$M^{n}\times[0,T]$. If $f(x,0)\leq 0,$ then there exists
$\tilde{T}=\tilde{T}(\mu,n,C)\leq T,$ such that for all $(x,t)\in
M^{n}\times[0,\tilde{T}],$ we have
$f(x,t)\leq 2Lte^{-\mu r(x)}.$
###### Proof.
We will now adapt a maximum principle for heat equations by Ecker and Huisken,
refer to Theorem 4.3 in [10].
Let $S(x,t)=e^{\mu r(x)}f(x,t)-2Lt.$ As $r(x)$ is the distance function to the
essential set, we extend $r(x)$ smoothly to the interior of the essential set,
then $S(x,t)$ is smooth on $M^{n}\times(0,T]$ and continuous on
$M^{n}\times[0,T].$ We have $S(x,0)\leq 0$ and
$\begin{split}\frac{\partial S}{\partial
t}&\leq\hat{\Delta}S-2\mu\hat{\nabla}r\cdot\hat{\nabla}S+(\mu^{2}|\hat{\nabla}r|^{2}-\mu\hat{\Delta}r)S\\\
&+2(\mu^{2}|\hat{\nabla}r|^{2}-\mu\hat{\Delta}r)Lt+e^{\mu r}Q-2L.\\\
\end{split}$
First notice that in view of the assumption $(M^{n},\hat{g})$ is $AH$ of order
$\eta$ and lemma 2.1 in [15], $|\hat{\Delta}r-n|\leq Ce^{-2r}$ for $\eta>2,$
$Cre^{-2r}$ for $\eta=2,$ and $Ce^{-\eta r}$ for $0<\eta<2,$ where the
constant $C$ only depends on the $AH$ constant of $(M^{n},\hat{g}).$
Now observe that $|\hat{\nabla}r|\equiv 1,$ we can therefore choose $T_{1}\leq
T$ such that for $t\leq T_{1},$ we have
$2(\mu^{2}|\hat{\nabla}r|^{2}-\mu\hat{\Delta}r)Lt\leq L.$
Using the assumption $|Q(x,t)|\leq Le^{-\mu r(x)}$ we obtain
$e^{\mu r}Q-L\leq 0.$
We therefore conclude for $t\leq T_{1},$
$\begin{split}\frac{\partial S}{\partial
t}-\hat{\Delta}S\leq\vec{\Lambda}\cdot\hat{\nabla}S+\lambda S,\end{split}$
where the vector $\vec{\Lambda}=-2\mu\hat{\nabla}r,$
$\lambda=\mu^{2}|\hat{\nabla}r|^{2}-\mu\hat{\Delta}r$ and the inner product is
with respect to $\hat{g}.$
We then proceed to show all the conditions in [10] Theorem 4.3 are satisfied
in our situations and we could make the finial conclusion.
Since $(M^{n},\hat{g})$ is $AH$, there exists a constant $k,$ say $-(C+1),$
such that $Ric_{\hat{g}}\geq(n-1)k.$ Applying now the absolute volume
comparison result we arrive at
$vol_{\hat{g}}B(x,r)\leq v(n,k,r),$
where $v(n,k,r)$ denotes the volume of a ball of radius $r$ in the constant-
curvature space form $S_{k}^{n}.$ For $k=-(C+1),$ we have
$\begin{split}v(n,k,r)&=\int_{0}^{r}\int_{S^{n-1}}(\frac{\sinh\sqrt{C+1}\tau}{\sqrt{C+1}})^{n-1}d\tau
d\theta\\\ &\leq\exp(Dr)\\\ &\leq\exp(\frac{D}{2}(1+r^{2}))\\\ \end{split}$
where the constant $D$ only depends on $C$ and $n.$ Therefore
$vol_{\hat{g}}B(x,r)\leq\exp(\frac{D}{2}(1+r^{2})).$
We also estimate
$\sup_{M^{n}\times[0,T_{1}]}|\lambda|\leq\mu^{2}+|\mu|(n+C)<\infty$
and
$\sup_{M^{n}\times[0,T_{1}]}|\vec{\Lambda}|\leq 2|\mu|,$
as well as
$\sup_{M^{n}\times[0,T_{1}]}|\frac{d}{dt}\hat{g}_{ij}|=0.$
Now note that
$\int_{0}^{T_{1}}\int_{M^{n}}\exp(-r(x)^{2})|\hat{\nabla}S|^{2}(x)d\mu_{\hat{g}}dt<\infty,$
since $|\hat{\nabla}S|=|\mu e^{\mu r}f\hat{\nabla}r+e^{\mu
r}\hat{\nabla}f|\leq C(C_{1},\mu)e^{\mu r}.$ From Theorem 4.3 in [10] we
obtain finally $S\leq 0$ on $M^{n}\times[0,T_{1}].$ Therefore, if we let
$\tilde{T}=T_{1},$ we have
$f(x,t)\leq 2Lte^{-\mu r(x)},$
for all $(x,t)\in M\times[0,\tilde{T}].$ ∎
###### Lemma 2.5.
Let $g(x,t)=s(x,t)+\hat{g}\in\mathbf{B}_{\alpha,\epsilon}+\hat{g},$ and $V$ is
the unique positive solution we get in Lemma 2.3. Then the initial problem
(19) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}G_{ij}=g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}G_{ij}+H_{ij}(\hat{R}m,\hat{\nabla}g,g,V^{-1}\nabla_{g}^{2}V),~{}~{}~{}on~{}M^{n}\times[0,T]\\\
G_{ij}(\cdot,0)=\hat{g}_{ij}.\\\ \end{array}\right.$
has a solution and
$e^{2r}(G-\hat{g})\in C^{2+\alpha,1+\frac{\alpha}{2}}(M^{n}\times[0,T]).$
Moreover, there exists $T_{0}=T_{0}(n,C,\epsilon)\in(0,T]$ such that
$G\in\mathbf{B}_{\alpha,\epsilon}+\hat{g}$ for $t\in[0,T_{0}].$
###### Proof.
For any $x\in M^{n},$ there exists $k$ such that $x\in B(x,1)\subset
B(x,2)\subset D_{k}.$ We solve the following linear initial-boundary value
problem
(20) $\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}G_{ij}=g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}G_{ij}+H_{ij}(\hat{R}m,\hat{\nabla}g,g,V^{-1}\nabla_{g}^{2}V),~{}~{}~{}on~{}D_{k}\times[0,T]\\\
G_{ij}\mid_{\Gamma_{k}}=\hat{g}_{ij}\\\ \end{array}\right.$
where
$\begin{split}H_{ij}&=-g^{kl}g_{ip}\hat{g}^{pq}\hat{R}_{jkql}-g^{kl}g_{jp}\hat{g}^{pq}\hat{R}_{ikql}+g^{kl}g^{pq}(\frac{1}{2}\hat{\nabla}_{i}g_{pk}\cdot\hat{\nabla}_{j}g_{ql}+\\\
&\hat{\nabla}_{k}g_{jp}\cdot\hat{\nabla}_{q}g_{il}-\hat{\nabla}_{k}g_{jp}\cdot\hat{\nabla}_{l}g_{iq}-\hat{\nabla}_{j}g_{pk}\cdot\hat{\nabla}_{l}g_{iq}-\hat{\nabla}_{i}g_{pk}\cdot\hat{\nabla}_{l}g_{jq})\\\
&-g^{kl}(\hat{\nabla}_{i}g_{jl}+\hat{\nabla}_{j}g_{il}-\hat{\nabla}_{l}g_{ij})\cdot
V^{-1}\hat{\nabla}_{k}V+2V^{-1}\hat{\nabla}_{i}\hat{\nabla}_{j}V-2ng_{ij}.\\\
\end{split}$
Correspondingly, $q=G-\hat{g}$ satisfies
$\left\\{\begin{array}[]{ll}\frac{\partial}{\partial
t}q_{ij}=g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}q_{ij}+H_{ij}(\hat{R}m,\hat{\nabla}g,g,V^{-1}\nabla_{g}^{2}V),~{}~{}~{}on~{}D_{k}\times[0,T]\\\
q_{ij}\mid_{\Gamma_{k}}=0.\\\ \end{array}\right.$
We rewrite $H$ and find that $H$ is a combination of the following terms:
$\hat{R}ic+n\hat{g}-\hat{V}^{-1}(\hat{\nabla}^{2}\hat{V}),$ $s,$ $s*\hat{R}m,$
$s*s*\hat{R}m,$ $V^{-1}\hat{\nabla}s*\hat{\nabla}V,$
$V^{-1}*s*\hat{\nabla}s*\hat{\nabla}V,$ $s*\hat{\nabla}s*\hat{\nabla}s,$
$s*s*\hat{\nabla}s*\hat{\nabla}s,$
$(V\hat{V})^{-1}(V-\hat{V})\hat{\nabla}^{2}\hat{V},$ and
$V^{-1}\hat{\nabla}^{2}(V-\hat{V}).$
By the assumption that $s\in\mathbf{B}_{\alpha,\epsilon}$ and $V$ comes from
the unique positive solution in Lemma 2.3, we know that $g^{kl}$ and $H_{ij}$
belong to the class $C^{\alpha,\frac{\alpha}{2}}(\bar{D}_{k}\times[0,T]),$
from the standard parabolic theory, we know linear initial-boundary value
problem (20) has a unique solution $G^{k}_{ij}$ from the class
$C^{2+\alpha,1+\frac{\alpha}{2}}(\bar{D}_{k}\times[0,T]),$ and it satisfies
(21) $\|G^{k}_{ij}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(D_{k}\times[0,T])}\leq
C(\|\hat{g}\|_{C^{2+\alpha}(D_{k})}+\|H_{ij}\|_{C^{\alpha,\frac{\alpha}{2}}(D_{k}\times[0,T])})\leq
C$
and
(22)
$\|G^{k}_{ij}-\hat{g}_{ij}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\leq
C(\|H_{ij}\|_{C^{\alpha,\frac{\alpha}{2}}(B(x,2)\times[0,T])}+|G^{k}_{ij}-\hat{g}_{ij}|_{\Gamma_{k}})$
Like what we did above, as the right hand side of (21) is independent of $k,$
we can take the limit $k\longrightarrow\infty$ and get the convergence of a
subsequence of the solutions $G_{ij}^{k}$ in the
$C^{2+\beta,1+\frac{\beta}{2}}(\beta<\alpha)$ topology on compact subsets of
$M^{n}\times[0,T]$ to a $C^{2+\alpha,1+\frac{\alpha}{2}}$ solution $G_{ij}$ on
$M^{n}\times[0,T]$ by the theorem of Arzela-Ascoli for given
$s(x,t)\in\mathbf{B}_{\alpha,\epsilon}.$
From (22), we obtain
$\|G_{ij}-\hat{g}_{ij}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\leq
C\|H_{ij}\|_{C^{\alpha,\frac{\alpha}{2}}(B(x,2)\times[0,T])}$
When $y\in B(x,2),$ $r(x)$ and $r(y)$ is equivalent, thus
(23) $\|G-\hat{g}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times[0,T])}\leq
L_{1}e^{-2r(x)},~{}~{}~{}t\in[0,T].$
Let $f=\|G-\hat{g}\|_{\hat{g}},$ by direct computation, $f$ satisfies
$\begin{split}\frac{\partial f}{\partial t}&\leq
g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}f+F\\\
&\leq\hat{\Delta}f+(g^{kl}-\hat{g}^{kl})\hat{\nabla}_{k}\hat{\nabla}_{l}f+F\\\
&=\hat{\Delta}f+Q\end{split}$
where
$\begin{split}Q(x,t)&\leq
C(\|H\|_{0;B(x,1)\times[0,T]}+\|s\|_{0;B(x,1)\times[0,T]}\cdot\|\nabla^{2}(G-\hat{g})\|_{0;B(x,1)\times[0,T]})\\\
&\leq Ce^{-2r(x)}.\\\ \end{split}$
on $M^{n}\times[0,T].$ By applying Proposition 2.4, there exists
$T_{1}\in(0,T]$ which only depends on $n,$ $a,$ and $AS$ constant such that
(24) $\|G-\hat{g}\|(x,t)\leq L_{2}te^{-2r(x)}.$
According to Proposition 2.4, if we choose $T$ to be small at the beginning,
$T_{1}$ could reach $T.$ Without loss of generality, we may let $T_{1}=T.$
Now observe that once we have (23) and (24), using Interpolation
Inequality(see [17] P80 Lemma3.2), we obtain that for any
$\delta\in(0,\sqrt{T}],$ there exist constants $c_{1},$ $c_{2},$ $d_{1},$ and
$d_{2},$ which only depend on $\alpha$ and $n,$ such that
(25)
$\begin{split}&\|G-\hat{g}\|_{C^{\alpha,\frac{\alpha}{2}}(B(x,1)\times(0,T])}\\\
\leq&c_{1}\delta^{2}\|G-\hat{g}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times(0,T])}+c_{2}\delta^{-\alpha}\|G-\hat{g}\|_{0;B(x,1)\times(0,T]},\\\
\end{split}$
and
(26)
$\begin{split}&\|G-\hat{g}\|_{C^{1+\alpha,\frac{1+\alpha}{2}}(B(x,1)\times(0,T])}\\\
\leq&d_{1}\delta\|G-\hat{g}\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(B(x,1)\times(0,T])}+d_{2}\delta^{-(1+\alpha)}\|G-\hat{g}\|_{0;B(x,1)\times(0,T]}.\\\
\end{split}$
Let $\delta=\sqrt{T},$ then we can always choose $T=T_{0}$ such that both (25)
and (26) are no greater than $\frac{\epsilon}{2}e^{-2r},$ which means
$\begin{split}\|G-\hat{g}\|_{\mathbf{B}_{\alpha}}&=\|e^{2r}(G-\hat{g})\|_{C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T_{0}];T^{*}M\otimes
T^{*}M)}\\\
&+\|e^{2r}\hat{\nabla}(G-\hat{g})\|_{C^{\alpha,\frac{\alpha}{2}}(M^{n}\times[0,T_{0}];T^{*}M\otimes
T^{*}M)}\\\ &\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.\end{split}$
We therefore conclude $G\in\mathbf{B}_{\alpha,\epsilon}+\hat{g}$ when
$t\in[0,T_{0}].$ ∎
###### Lemma 2.6.
Let $G$ be what we obtained in the above lemma. Then $G$ is unique in
$\mathbf{B}_{\alpha,\epsilon}+\hat{g}$ for $t\in[0,T_{0}].$
###### Proof.
Suppose that $G_{1}$ and $G_{2}$ are two solutions to the initial problem
(19), and that both $G_{1}$ and $G_{2}$ satisfy
$e^{2r}(G_{i}-\hat{g})\in C^{2+\alpha,1+\frac{\alpha}{2}}(M^{n}\times[0,T])$
and $G_{i}\in\mathbf{B}_{\alpha,\epsilon}+\hat{g}$ for $t\in[0,T_{0}],$
$i=1,2.$
Let $u(x,t):=\|G_{1}-G_{2}\|_{\hat{g}}(x,t),$ then $u$ satisfies
$\frac{\partial u}{\partial t}\leq g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}u$
with $u(x,0)=0.$ Define $h(x,t):=-\frac{r(x)^{2}}{4(2\sigma-t)},$ where
$\sigma$ is an arbitrary time in $[0,T_{0}]$. Note that $h$ is a locally
Lipschitz function defined on $M^{n}\times[0,2\sigma).$ Since
$|\hat{\nabla}r|_{\hat{g}}=1,$ it follows that
(27) $|\hat{\nabla}h|^{2}_{\hat{g}}+\frac{\partial h}{\partial
t}=0,~{}~{}~{}a.e.~{}~{}$
Let $0\leq\varphi_{s}\leq 1$ be a cutoff function which is $1$ inside $B(o,s)$
and compactly supported in $B(o,s+1)$ with
$|\hat{\nabla}\varphi_{s}|_{\hat{g}}\leq 2.$ Multiplying the inequality
$\frac{\partial u}{\partial t}-g^{kl}\hat{\nabla}_{k}\hat{\nabla}_{l}u\leq 0$
by the compactly supported Lipschitz function $\varphi_{s}^{2}e^{h}u$ and
integrating by parts, we have
$\begin{split}0\geq&\int_{0}^{\sigma}\int_{M^{n}}e^{h}(\varphi_{s}^{2}|\hat{\nabla}u|^{2}_{g}+2\langle\hat{\nabla}\varphi_{s},\hat{\nabla}u\rangle_{g}\varphi_{s}u+\varphi_{s}^{2}u\langle\hat{\nabla}h,\hat{\nabla}u\rangle_{g})d\mu_{\hat{g}}dt\\\
&+\int_{0}^{\sigma}\int_{M^{n}}e^{h}\varphi_{s}^{2}u(\hat{\nabla}_{k}g^{kl}\cdot\hat{\nabla}_{l}u)d\mu_{\hat{g}}dt+\frac{1}{2}\int_{0}^{\sigma}\int_{M^{n}}e^{h}\varphi_{s}^{2}(\frac{\partial}{\partial
t}u^{2})d\mu_{\hat{g}}dt\\\
\geq&\int_{0}^{\sigma}\int_{M^{n}}e^{h}(\varphi_{s}^{2}|\hat{\nabla}u|^{2}_{g}-\frac{1}{\lambda^{2}}u^{2}|\hat{\nabla}\varphi_{s}|_{g}^{2}-\lambda^{2}\varphi_{s}^{2}|\hat{\nabla}u|_{g}^{2})d\mu_{\hat{g}}dt\\\
&+\int_{0}^{\sigma}\int_{M^{n}}e^{h}(-\frac{1}{2}\varphi_{s}^{2}u^{2}|\hat{\nabla}h|_{g}^{2}-\frac{1}{2}\varphi_{s}^{2}|\hat{\nabla}u|_{g}^{2})d\mu_{\hat{g}}dt\\\
&+\int_{0}^{\sigma}\int_{M^{n}}e^{h}\varphi_{s}^{2}u(\hat{\nabla}_{k}g^{kl}\cdot\hat{\nabla}_{l}u)d\mu_{\hat{g}}dt\\\
&+\frac{1}{2}\int_{M^{n}}e^{h}\varphi_{s}^{2}u^{2}d\mu_{\hat{g}}\mid_{0}^{\sigma}+\frac{1}{2}\int_{0}^{\sigma}\int_{M^{n}}e^{h}\varphi_{s}^{2}u^{2}|\hat{\nabla}h|^{2}_{\hat{g}}d\mu_{\hat{g}}dt.\\\
\end{split}$
Here we used the Cauchy-Schwarz inequality and the equality (27). Then
$\begin{split}&(\int_{M^{n}}e^{h}\varphi_{s}^{2}u^{2}d\mu_{\hat{g}})(\sigma)\leq\frac{2}{\lambda^{2}}\int_{0}^{\sigma}\int_{M^{n}}e^{h}u^{2}|\hat{\nabla}\varphi_{s}|_{g}^{2}d\mu_{\hat{g}}dt\\\
&+\int_{0}^{\sigma}\int_{M^{n}}e^{h}\varphi_{s}^{2}\\{|\hat{\nabla}u|^{2}_{g}(2\lambda^{2}-1)-2u(\hat{\nabla}_{k}g^{kl}\cdot\hat{\nabla}_{l}u)+u^{2}(|\hat{\nabla}h|^{2}_{g}-|\hat{\nabla}h|^{2}_{\hat{g}})\\}d\mu_{\hat{g}}dt.\\\
\end{split}$
By the assumption and all the estimates in the above lemma, we know that
$g-\hat{g}\in\mathbf{B}_{\alpha,\epsilon}$ and
$G_{1},G_{2}\in\mathbf{B}_{\alpha,\epsilon}+\hat{g}.$ Then
$u=\|G_{1}-G_{2}\|_{\hat{g}}\leq\|G_{1}-\hat{g}\|_{\hat{g}}+\|G_{2}-\hat{g}\|_{\hat{g}}\leq
2\epsilon e^{-2r},$
Due to Kato’s inequality, for a smooth tensor field $W$ with compact support,
$\mid\nabla|W|\mid\leq\mid\nabla W\mid,$ we get
$\|\hat{\nabla}u\|_{\hat{g}}=\|\hat{\nabla}\|G_{1}-G_{2}\|_{\hat{g}}\|_{\hat{g}}\leq\|\hat{\nabla}(G_{1}-G_{2})\|_{\hat{g}}\leq
2\epsilon e^{-2r}.$
Hence
$|\hat{\nabla}u|_{g}\leq\|g-\hat{g}\|_{\hat{g}}\cdot\|\hat{\nabla}u\|_{\hat{g}}+\|\hat{\nabla}u\|_{\hat{g}}\leq
4\epsilon e^{-2r}.$
As
$|u^{2}(|\hat{\nabla}h|^{2}_{g}-|\hat{\nabla}h|^{2}_{\hat{g}})|\leq
u^{2}\cdot\|g-\hat{g}\|_{\hat{g}}\cdot|\hat{\nabla}h|^{2}_{\hat{g}},$
and the integration domain of time is from $0$ to $\sigma,$ so
$|u^{2}(|\hat{\nabla}h|^{2}_{g}-|\hat{\nabla}h|^{2}_{\hat{g}})|\leq\frac{1}{(2\sigma-t)^{2}}\epsilon^{3}r^{2}e^{-6r}\leq\frac{\epsilon^{3}}{\sigma^{2}}e^{-4r}.$
Since $\epsilon$ is small, we can always choose $\lambda$ such that
$\begin{split}&|\hat{\nabla}u|^{2}_{g}(2\lambda^{2}-1)-2u(\hat{\nabla}_{k}g^{kl}\cdot\hat{\nabla}_{l}u)+u^{2}(|\hat{\nabla}h|^{2}_{g}-|\hat{\nabla}h|^{2}_{\hat{g}})\\}\\\
\leq&\\{16(2\lambda^{2}-1)+8\epsilon+\frac{\epsilon}{\sigma^{2}}\\}\epsilon^{2}e^{-4r}\\\
\leq&0.\end{split}$
Then
$\begin{split}(\int_{M^{n}}e^{h}\varphi_{s}^{2}u^{2}d\mu_{\hat{g}})(\sigma)&\leq\frac{2}{\lambda^{2}}\int_{0}^{\sigma}\int_{M^{n}}e^{h}u^{2}|\hat{\nabla}\varphi_{s}|_{g}^{2}d\mu_{\hat{g}}dt\\\
&\leq\frac{32\epsilon^{3}}{\lambda^{2}}\int_{0}^{\sigma}\int_{B(o,s+1)\setminus
B(o,s)}exp\\{-\frac{1}{8\sigma}r^{2}-6r\\}d\mu_{\hat{g}}dt.\\\ \end{split}$
The right hand side tends to zero as $s\rightarrow\infty,$ and we conclude
that $u\equiv 0$ on $M^{n}\times[0,\sigma].$ Since $\sigma$ is arbitrary in
$[0,T_{0}],$ we conclude that $u\equiv 0$ on $M^{n}\times[0,T_{0}].$ Thus the
uniqueness is obtained. ∎
Finally, we arrive at the proof of our main result.
###### Proof of Theorem 1.9.
For $t\in[0,T_{0}],$ and for any $s\in\mathbf{B}_{\alpha,\epsilon},$ let
$g=\hat{g}+s,$ by Lemma 2.3, we get a unique positive $V$ with good estimates.
Plugging $g$ and $V$ in equation (19), from Lemma 2.5, we get a solution $G.$
Now we find that the operator
$\mathbf{A}:\mathbf{B}_{\alpha,\epsilon}\longrightarrow\mathbf{B}_{\alpha}$
$\mathbf{A}(s)=G-\hat{g}$
has the range in $\mathbf{B}_{\alpha,\epsilon}$ by Lemma 2.5 and is well-
defined in $\mathbf{B}_{\alpha,\epsilon}$ by Lemma 2.6. Lemma 2.5 also shows
that $\mathbf{A}$ is compact in $\mathbf{B}_{\alpha,\epsilon}.$ Then Schauder
fixed point theorem promises us a fixed point
$s_{0}\in\mathbf{B}_{\alpha,\epsilon}$ such that
$\mathbf{A}(s_{0})=s_{0}.$
Let $g=s_{0}+\hat{g},$ and we denote the unique solution from Lemma 2.3 by
$V.$ Obviously, $(g,V)$ are solutions to our static flow (15) in
$C^{2+\alpha,1+\frac{\alpha}{2}}(M^{n}\times[0,T_{0}])$. Then we can improve
the spatial regularity step by step, and by bootstrapping and the equation we
can improve the regularity in time as well, hence we get a smooth solution.
Since $(g,V)$ is a pullback of the solution to the original static flow (5) by
a specific diffeomorphism, we get the short-time existence of the static flow
(5). ∎
## 3\. Asymptotic expansions at conformal infinity
In this section, we mainly prove Theorem 1.11. Suppose the triple
$(M^{n},g,V)$ is as that of Theorem 1.11. We also assume that the metric $g$
is sufficiently regular at infinity, which means that it has asymptotic
expansions to high enough order, in general involving log terms. We follow the
methods used in [11].
###### Lemma 3.1.
Suppose that $(M^{n},g,V),$ $\hat{g}$ and $\tau$ are as that of Theorem 1.11.
Then $\partial_{\tau}^{m}g_{\tau}|_{\tau=0}$ and
$\partial_{\tau}^{(m-1)}V|_{\tau=0}$ can uniquely determine by $\hat{g}$ for
any $1\leq m\leq n-1.$
###### Proof.
By the assumption, and $g_{\tau}|_{\tau=0}=\hat{g}.$ Since $V$ is with the
growth of $\frac{1}{\tau},$ we define a new variable
$u:=\tau V,$
then $u|_{\tau=0}=1.$
We now impose the static Einstein vacuum condition
$Ric(g)+ng=V^{-1}\nabla_{g}^{2}V$ and $\Delta_{g}V=nV$ on a triple
$(M^{n},g,V)$ of the above form.
One can decompose the tensor $Ric(g)+ng-V^{-1}\nabla_{g}^{2}V$ into components
with respect to the product structure $N^{n-1}\times(0,\delta).$ A
straightforward calculation shows that the vanishing of the component with
both indices in $N^{n-1}$ is given by
(28)
$\begin{split}&R_{ij}(g_{\tau})-\frac{1}{2}g_{ij}^{{}^{\prime\prime}}+(\frac{n-2}{2\tau}-\frac{1}{4}g^{kl}g_{kl}^{{}^{\prime}})g_{ij}^{{}^{\prime}}+\frac{1}{2}g^{kl}g_{ik}^{{}^{\prime}}g_{jl}^{{}^{\prime}}+(\frac{1}{2\tau}g^{kl}g_{kl}^{{}^{\prime}}+\frac{1}{\tau^{2}})g_{ij}\\\
&=V^{-1}((\nabla_{g_{\tau}}^{2}V)_{ij}-\frac{V^{{}^{\prime}}}{\tau}g_{ij}+\frac{V^{{}^{\prime}}}{2}g_{ij}^{{}^{\prime}}),\end{split}$
where $g_{ij}$ denotes the tensor $g_{\tau}$ on $N^{n-1},$ ′ denotes
$\partial_{\tau},$ and $R_{ij}(g_{\tau})$ denotes the $Ric$ tensor of
$g_{\tau}$ with $\tau$ fixed.
Meanwhile, we also rewrite $\Delta_{g}V$ in terms of the metric
$g=\tau^{-2}(d\tau^{2}+g_{\tau}),$ then we get
(29)
$\tau^{2}(V^{{}^{\prime\prime}}+\frac{1}{\tau}V^{{}^{\prime}})+\tau^{2}g^{ij}(\nabla_{g}^{2}V)_{ij}=nV,$
where $g_{ij}$ still denotes the tensor $g_{\tau}$ on $N^{n-1}$ and ′ denotes
$\partial_{\tau}.$
Substituting $u$ for $V,$ (28) turns to
(30) $\begin{split}&\tau
ug_{ij}^{{}^{\prime\prime}}+(1-n)ug_{ij}^{{}^{\prime}}-ug^{kl}g_{kl}^{{}^{\prime}}g_{ij}-\tau
ug^{kl}g_{ik}^{{}^{\prime}}g_{jl}^{{}^{\prime}}+\frac{\tau}{2}ug^{kl}g_{kl}^{{}^{\prime}}g_{ij}^{{}^{\prime}}\\\
&-2\tau
uR_{ij}(g_{\tau})+2\tau(\nabla_{g_{\tau}}^{2}u)_{ij}-2u^{{}^{\prime}}g_{ij}+\tau
u^{{}^{\prime}}g_{ij}^{{}^{\prime}}=0.\\\ \end{split}$
And (29) turns to
(31) $\begin{split}\tau\Delta_{g_{\tau}}u+\tau
u^{{}^{\prime\prime}}-nu^{{}^{\prime}}+\frac{\tau}{2}g^{ij}g_{ij}^{{}^{\prime}}u^{{}^{\prime}}-\frac{1}{2}g^{ij}g_{ij}^{{}^{\prime}}u=0.\end{split}$
Differentiating (30) $m-1$ times with respect to $\tau$ and setting $\tau=0$
gives
$(m-n)\partial_{\tau}^{m}g_{ij}-g^{kl}(\partial_{\tau}^{m}g_{kl})g_{ij}-2(\partial_{\tau}^{m}u)g_{ij}=$
(terms involving $\partial_{\tau}^{\mu}g_{ij}$ with $\mu<m$ and
$\partial_{\tau}^{\nu}u$ with $\nu<m$).
Differentiating (31) $m-1$ times with respect to $\tau$ and setting $\tau=0$
gives
$(m-1-n)\partial_{\tau}^{m}u-\frac{1}{2}g^{ij}(\partial_{\tau}^{m}g_{ij})=$
(terms involving $\partial_{\tau}^{p}g_{ij}$ with $p<m$ and
$\partial_{\tau}^{q}u$ with $q<m$).
So long as $m<n,$ we can inductively uniquely determine
$\partial_{\tau}^{m}g_{ij}|_{\tau=0}$ and $\partial_{\tau}^{m}u|_{\tau=0}$ at
each step if only we have $\hat{g}=g_{\tau}|_{\tau=0}.$ We compute that
$u^{{}^{\prime}}|_{\tau=0}=0,~{}~{}~{}g_{ij}^{{}^{\prime}}|_{\tau=0}=0,$
$u^{{}^{\prime\prime}}|_{\tau=0}=\frac{S(\hat{g})}{2(n-1)(n-2)},$
$g_{ij}^{{}^{\prime\prime}}|_{\tau=0}=\frac{1}{2-n}[\frac{S(\hat{g})}{1-n}\hat{g}_{ij}+2Ric_{ij}(\hat{g})].$
Note that $\partial_{\tau}^{m}V|_{\tau=0}$ is determined by
$\partial_{\tau}^{(m+1)}u|_{\tau=0}.$ So we can uniquely determine
$\partial_{\tau}^{m}g_{\tau}|_{\tau=0}$ and
$\partial_{\tau}^{(m-1)}V|_{\tau=0}$ for any $m<n$ if only we have $\hat{g}$.
When $m=n,$ we can only uniquely determine the term
$[2\partial_{\tau}^{n}u-tr_{g_{\tau}}(\partial_{\tau}^{n}g_{\tau})]|_{\tau=0}.$
∎
###### Lemma 3.2.
Suppose that $(M^{n},g,V),$ $\hat{g}$ and $\tau$ are as that of Theorem 1.11.
Let $(X^{n+1},h)=(\mathbb{S}^{1}\times M^{n},h=V^{2}d\theta^{2}+g),$ then
$\tau$ is the special defining function associated with the metric
$d\theta^{2}+\hat{g}.$
###### Proof.
By the assumption, we know that
$g=\tau^{-2}(d\tau^{2}+g_{\tau})$
in a neighborhood of conformal infinity $(N^{n-1},[\hat{g}]).$ We find that
$\tau^{2}h=d\tau^{2}+[(\tau V)^{2}d\theta^{2}+g_{\tau}]$
and
$\hat{h}=\tau^{2}h|_{T(\mathbb{S}^{1}\times N^{n-1})}=[(\tau
V)^{2}d\theta^{2}+g_{\tau}]|_{\tau=0}=d\theta^{2}+\hat{g}.$
∎
Now that we have the above lemma, meanwhile we know that in fact the
Riemannian manifold $(X^{n+1},h)$ is Einstein and satisfies $Ric(h)=-nh.$ By
means of the expansion of an Einstein metric in [11], we arrive at the proof
of the expansion of a static metric.
###### Proof of Theorem 1.11.
For a static Einstein vacuum $(M^{n},g,V)$ satisfying the assumptions of
Theorem 1.11, according to Lemma 3.1, we may write
$g=\hat{g}+g^{(1)}\tau+g^{(2)}\tau^{2}+\cdots+g^{(n-1)}\tau^{n-1}+o(\tau^{n-1}),$
and
$u=\tau
V=1+u^{(1)}\tau+u^{(2)}\tau^{2}+\cdots+u^{(n-1)}\tau^{n-1}+o(\tau^{n-1}).$
Since $h$ is an Einstein metric, by the results in [11], we know that
$h_{\tau}$ satisfies the following:
When $n$ is odd,
$h_{\tau}=\hat{h}+h^{(2)}\tau^{2}+(even~{}powers~{}of~{}\tau)+h^{(n-1)}\tau^{n-1}+h^{(n)}\tau^{n}+\cdots;$
When $n$ is even,
$h_{\tau}=\hat{h}+h^{(2)}\tau^{2}+(even~{}powers~{}of~{}\tau)+h^{(n)}\tau^{n}+s\tau^{n}\log\tau+\cdots;$
where:
1. (1)
$h^{(2i)}$ are determined by $\hat{h}$ for $2i<n;$
2. (2)
$h^{(n)}$ is traceless when $n$ is odd;
3. (3)
the trace part of $h^{(n)}$ is determined by $\hat{h}$ and $s$ is traceless
and determined by $\hat{h};$
4. (4)
the traceless part of $h^{(n)}$ is divergence free.
As we have from Lemma 3.2 that $h_{\tau}=u^{2}d\theta^{2}+g_{\tau},$ we find
that
$h^{(k)}=\sum_{a+b=k,a\geq 0,b\geq 0}u^{(a)}u^{(b)}d\theta^{2}+g^{(k)}.$
When $2i+1<n,$ $h^{(2i+1)}=0,$ which implies
$g^{(2i+1)}=\sum_{a+b=2i+1,a\geq 0,b\geq 0}u^{(a)}u^{(b)}=0.$
Since we have $u^{(0)}=1,$ using
$\begin{split}&\sum_{a+b=2i+1,a\geq 0,b\geq 0}u^{(a)}u^{(b)}\\\
&=\sum_{a+b=2i-1,a\geq 0,b\geq
0}u^{(a)}u^{(b)}+2u^{(0)}u^{(2i+1)}+2u^{(1)}u^{(2i)}\\\ \end{split}$
We conclude that $u^{(2i+1)}=0$ for $2i+1<n,$ which means $V^{(2i)}=0$ for
$2i<n-1.$
Therefore we have
$g_{\tau}=\hat{g}+g^{(2)}\tau^{2}+(even~{}powers~{}of~{}\tau)+g^{(2l)}\tau^{2l}+\cdots$
$V=\frac{1}{\tau}+V^{(1)}\tau+(odd~{}powers~{}of~{}\tau)+V^{(2l-1)}\tau^{2l-1}+\cdots$
where $g^{(2l)}$ and $V^{(2l-1)}$ are uniquely determined by $\hat{g}$ for
$2l\leq n-1.$
∎
## References
* [1] M.M.Akbar, E.Woolgar, Ricci Solitons and Einstein-Scalar Field Theory, Classical Quantum Gravity 26, 055015 (2009)
* [2] M.T.Anderson, P.T.Chruściel and E.Delay, Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant, Jour. High Energy Phys., 10, 063, 1-27 (2002)
* [3] M.T.Anderson, P.T.Chruściel and E.Delay, Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant II, $n\geq 5$, AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, (O. Biquard, Ed.), Euro. Math. Soc., 165-204 (2005)
* [4] E.Bahuaud, Ricci flow of conformally compact metrics arXiv:1011.2999v2 [math.AP] 7 Jun 2011
* [5] S.Brendle and R.M.Schoen, Classification of manifolds with $\frac{1}{4}$-pinched curvature, Acta Math. 200, 1 C13 (2008)
* [6] S.Brendle and R.M.Schoen, Manifolds with $\frac{1}{4}$-pinched curvature are space forms, J. Amer. Math. Soc. 22, 287 C307 (2009)
* [7] P.Chruściel and W.Simon, Towards the classification of static vacuum spacetimes with negative cosmological constant, J. Math. Phys. 42, no.4 1779-1817 (2001)
* [8] B.L.Chen and X.P.Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Diff. Geom. 74, 119-154 (2006)
* [9] D.DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom. 18, 157 C162, (1983)
* [10] K.Ecker and G.Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. math. 105, 547-569 (1991)
* [11] C.R.Graham, Volume and area renormalizations for conformally compact Einstein metrics, in The Proceedings of the 19th Winter School ”Geometry and Physics” (1999, Srn$\grave{i}$), Rend. Circ. Mat. Palermo(2) 63 (2000)
* [12] R.Gicquaud, PhD thesis,
* [13] G.Huisken and T.Ilmanen, The Riemannian Penrose Inequality, Int. Math. Res. Not. 20, 1045 C1058 (1997)
* [14] G.Huisken and T.Ilmanen, The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality, J. Diff. Geom. 59, 353-437 (2001)
* [15] X.Hu, J.Qing and Y.G.Shi, Regularity and rigidity of asymptotically hyperbolic manifolds, arXiv:0910.2060v2 [math.DG] 26 Oct 2009
* [16] L.Karp and P.Li, Unpublished
* [17] O.A.Lady$\check{z}$enskaja, V.A.Solonnikov and N.N.Ural’ceva, Linear and quasi-linear equations of parabolic type, volume 23, Translations of Math. Monographs, AMS, (1968)
* [18] J.M.Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc, 183, (2006)
* [19] B.List, Evolution of an extended Ricci flow system, Dissertation, (2005)
* [20] G.Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 [math.DG] 11 Nov 2002
* [21] G.Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109v1 [math.DG] 10 Mar 2003
* [22] J.Qing, On the uniqueness of AdS space-time in higher dimensions, Annales Henri Poincar$\acute{e}$ 5, 245 C260 (2004)
* [23] J.Qing, Y.G.Shi and J.Wu, Normalized Ricci flows ad conformally compact Einstein metrics, arXiv:1106.0372v1 [math.DG] 2 Jun 2011
* [24] W.X.Shi, Deforming the metric on complete Riemannian Manifolds, J. Diff. Geom. 30, 223-301 (1989)
* [25] X.D.Wang, On the uniqueness of the ADS spacetime, Acta Math. Sinica, Vol.21, No.4, 917-922 (2005)
|
arxiv-papers
| 2011-09-26T07:04:23 |
2024-09-04T02:49:22.494978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue Hu, Yuguang Shi",
"submitter": "Yuguang Shi",
"url": "https://arxiv.org/abs/1109.5456"
}
|
1109.5492
|
# Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems
with Exact Nonreflecting Boundary Conditions
Li-Lian Wang1, Bo Wang2 and Xiaodan Zhao1
###### Abstract.
This paper is concerned with fast and accurate computation of exterior wave
equations truncated via exact circular or spherical nonreflecting boundary
conditions (NRBCs, which are known to be nonlocal in both time and space). We
first derive analytic expressions for the underlying convolution kernels,
which allow for a rapid and accurate evaluation of the convolution with
$O(N_{t})$ operations over $N_{t}$ successive time steps. To handle the
nonlocality in space, we introduce the notion of boundary perturbation, which
enables us to handle general bounded scatters by solving a sequence of wave
equations in a regular domain. We propose an efficient spectral-Galerkin
solver with Newmark’s time integration for the truncated wave equation in the
regular domain. We also provide ample numerical results to show high-order
accuracy of NRBCs and efficiency of the proposed scheme.
###### Key words and phrases:
Time-domain Dirichlet-to-Neumann map, nonreflecting boundary conditions,
inverse Laplace transform, modified Bessel functions, convolution, spectral
methods, Newmark’s time integration
###### 1991 Mathematics Subject Classification:
35J05, 35L05, 65R10, 65N35, 65E05, 65M70
1 Division of Mathematical Sciences, School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371, Singapore. The research of
the authors is partially supported by Singapore AcRF Tier 1 Grant RG58/08.
2 College of Mathematics and Computer Science, Hunan Normal University,
Changsha, Hunan 410081, China. This author would like to thank the Division of
Mathematical Sciences of Nanyang Technological University for the hospitality
during the visit.
## 1\. Introduction
Wave propagation and scattering problems in unbounded media arise from diverse
application areas such as acoustics, aerodynamics, electromagnetics, antenna
design, oceanography and among others (see, e.g., [15, 33, 10]). Various
approaches have been proposed for their numerical studies that include the
boundary element methods (cf. [7]), infinite element methods (cf. [14]),
perfectly matched layers (PML) (cf. [5]), nonreflecting boundary condition
methods (cf. [22, 18]), and among others (cf. [25, 37]). An essential
ingredient for the latter approach is to truncate an unbounded domain to a
bounded domain by imposing an exact or approximate nonreflecting (absorbing or
transparent) boundary condition at the outer artificial boundary, where the
NRBC is designed to prevent spurious wave reflection from the artificial
boundary (cf. the review papers [15, 16] and the references therein). The
frequency-domain approaches for e.g., the time-harmonic Helmholtz problems
have been intensively investigated, while the time-domain simulations, which
are capable of capturing wide-band signals and modeling more general material
inhomogeneities and nonlinearities (cf. [3, 9]), have been relatively less
studied.
Although some types of NRBCs based on different principles have been proposed
(see, e.g., [12, 4, 39, 34, 35, 17, 18, 16]), a longstanding issue of time-
domain computation is the efficient treatment for NRBCs that can scale and
integrate well with the solver for the underlying truncated problem (cf. [38,
19]). In practice, if an accurate NRBC is imposed, the artificial boundary
could be placed as close as possible to the scatter that can significantly
reduce the computational cost. In this paper, we restrict our attention to the
exact NRBC on the circular or spherical artificial boundary. One major
difficulty lies in that such a NRBC is global in space and time in nature, as
it involves the Fourier/spherical harmonic expressions in space, and history
dependence in time induced by a convolution. The convolution kernel, termed as
nonreflecting boundary kernel (NRBK) in [2], is the inverse Laplace transform
of an expression that includes the logarithmic derivative of a modified Bessel
function. The rapid computation of the NRBK and convolution is of independent
interest. Alpert, Greengard and Hagstrom [2] proposed a rational approximation
of the logarithmic derivative with a least square implementation, which allows
for a reduction of the summation of the poles from $O(\nu)$ to
$O(\log\nu\log\frac{1}{\varepsilon})$ (where $\nu\gg 1$ is the order of the
modified Bessel function and $\varepsilon$ is a given tolerance), and a
recursive convolution. Jiang and Greengard [20, 21] further considered some
interesting applications to Schr$\ddot{o}$dinger equations in one and two
dimensions. Li [23] introduced a more accurate low order approximation of the
three-dimensional NRBK at a slightly expensive cost, where the observation
that the Laplace transform of the three-dimensional NRBK is exactly a rational
function lies at the heart of this algorithm. However, in many cases, the
expressions are not rational functions. For instance, the two-dimensional NRBK
also contains the contributions from the brach-cut along the negative real
axis (see Theorem 2.1 below). Lubich and Schädle [24] developed some fast
algorithm for the temporal convolution with $O(N_{t}\log N_{t})$ operations
(over $N_{t}$ successive time steps) arising from NRBCs with non-rational
expressions for other equations (e.g., Schr$\ddot{o}$dinger equations and
damped wave equations).
In this paper, we derive an analytic formula for the NRBK based on a direct
inversion of the Laplace transform by the residue theorem (see Theorem 2.1
below). In fact, Sofronov [35] presented some formulas of similar type by
working on much more complicated expressions of the kernel in terms of
Tricomi’s confluent hypergeometric functions. We show that with these
formulas, we can evaluate the temporal convolution recursively and rapidly
with $O(N_{t})$ operations and almost without extra memory for the history
dependence. Moreover, the analytic expression provides a useful apparatus for
the stability and convergence analysis. It is worthwhile to remark that Chen
[6] reformulated the two-dimensional wave problem into a first-order system in
time and showed the well-posedness of the truncated problem with an
alternative formulation of the NRBC.
It is known that the nonlocality of the NRBC in space can be efficiently
handled by Fourier/spherical harmonic expansions when the scatter is a disk or
a ball. Recently, a systematic approach, based on the boundary perturbation
technique (also called the transformed field expansion (TFE) method (cf.
[28])), has been developed in [29, 13, 30] for time-harmonic Helmholtz
equations in exterior domains with general bounded obstacles, under which the
whole algorithm boils down to solving a sequence of Helmholtz equations in a
2-D annulus or a 3-D spherical shell. In this paper, we highlight that this
notion can be extended to time-domain computation, though it has not been
investigated before as far as we know. In this paper, we propose an efficient
spectral-Galerkin method with Newmark’s time integration for the truncated
wave equations in an annulus or a spherical shell, and provide ample numerical
results to show the efficiency of the solver and high accuracy of NRBC from
several angles.
The rest of the paper is organized as follows. In Section 2, we present the
formulation of NRBCs, and derive the analytic formulas for NRBKs. In Section
3, we present some properties of NRBK and analyze well-posedness of the
truncated wave equation. In Section 4, we outline the notion of the TFE method
and propose an efficient spectral-Galerkin and Newmark’s time integration
scheme for the truncated wave problem in regular domains. We provide ample
numerical results in Section 5.
## 2\. Evaluation of nonreflecting boundary kernels
In this paper, we consider the time-domain acoustic scattering problem with
sound-soft boundary conditions on the bounded obstacle:
$\displaystyle\partial_{t}^{2}U=c^{2}\Delta U+F,\quad{\rm
in}\;\;\Omega_{\infty}:={\mathbb{R}}^{d}\setminus\bar{D},\;\;t>0,\;\;d=2,3;$
(2.1) $\displaystyle U=U_{0},\quad\partial_{t}U=U_{1},\quad{\rm
in}\;\;\Omega_{\infty},\;\;t=0;$ (2.2) $\displaystyle U=G,\quad{\rm
on}\;\;\Gamma_{D},\;\;t>0;\quad\partial_{t}U+c\partial_{\boldsymbol{n}}U=o(|\boldsymbol{x}|^{(1-d)/2}),\;\;|{\boldsymbol{x}}|\to\infty,\;\;t>0.$
(2.3)
Here, $D$ is a bounded obstacle (scatter) with Lipschitz boundary
$\Gamma_{D},$ $c>0$ is a given constant, and the radiation condition (2.3),
where ${\boldsymbol{n}}={\boldsymbol{x}}/|\boldsymbol{x}|$, corresponds to the
well-known Sommerfeld radiation condition in the frequency domain. Assume that
the data $F,U_{0}$ and $U_{1}$ are compactly supported in a 2-D disk or a 3-D
ball $B$ of radius $b.$
A common way is to reduce this exterior problem to the problem in a bounded
domain by imposing an exact or approximate NRBC at the artificial boundary
$\Gamma_{b}:=\partial B.$ In what follows, we shall focus on the wave equation
truncated by the exact circular or spherical NRBC:
$\displaystyle\partial_{t}^{2}U=c^{2}\Delta U+F,\quad{\rm
in}\;\;\Omega:=B\setminus\bar{D},\;\;t>0,\;\;d=2,3;$ (2.4) $\displaystyle
U=U_{0},\quad\partial_{t}U=U_{1},\quad{\rm in}\;\;\Omega,\;\;t=0;\quad
U=G,\quad{\rm on}\;\;\Gamma_{D},\;\;t>0;$ (2.5)
$\displaystyle\partial_{r}U=T_{d}(U),\quad{\rm at}\;\;r=b,\;\;t>0,$ (2.6)
where $T_{d}(U)$ is the so-called time-domain DtN map.
### 2.1. Formulation of $T_{d}(U)$
We first present the expression of $T_{d}(U)$ in (2.6), and refer to e.g.,
[18, 3] (and the original references therein) for the detailed derivation. It
is known that the problem (2.1)-(2.3), exterior to $D=B$ with
$F=U_{0}=U_{1}\equiv 0$ and $G=U|_{r=b}$ (i.e., the Dirichlet data taken from
the interior problem (2.4)-(2.6)), can be solved analytically by using Laplace
transform in time and separation of variables in space in polar coordinate
$(r,\phi)$/spherical coordinate $(r,\theta,\phi)$. By imposing the continuity
of directional derivative with respect to $r$ across the artificial boundary
$r=b,$ we obtain the boundary condition (2.6) with
$T_{d}(U)=\begin{cases}\Big{(}-\dfrac{1}{c}\dfrac{\partial U}{\partial
t}-\dfrac{U}{2r}\Big{)}\Big{|}_{r=b}+\displaystyle\sum_{|n|=0}^{\infty}\sigma_{n}(t)\ast\widehat{U}_{n}(b,t)e^{{\rm
i}n\phi},&d=2,\\\\[7.0pt] \Big{(}-\dfrac{1}{c}\dfrac{\partial U}{\partial
t}-\dfrac{U}{r}\Big{)}\Big{|}_{r=b}+\displaystyle\sum_{n=0}^{\infty}\displaystyle\sum_{|m|=0}^{n}\sigma_{n+1/2}(t)\ast\widehat{U}_{nm}(b,t)Y_{n}^{m}(\theta,\phi),&d=3,\end{cases}$
(2.7)
where
$\sigma_{\nu}(t):=\mathcal{L}^{-1}\left[\frac{s}{c}+\frac{1}{2b}+\frac{s}{c}\frac{K_{\nu}^{\prime}(sb/c)}{K_{\nu}(sb/c)}\right],\quad\nu=n,n+1/2.$
(2.8)
Here, $K_{\nu}$ is the modified Bessel function of the second kind of order
$\nu$ (see, e.g., [1, 40]), and $\mathcal{L}^{-1}[H(s)]$ is the inverse
Laplace transform of a Laplace transformable function $h(t)$ with
$H(s)=\mathcal{L}[h(t)](s)=\int_{0}^{\infty}e^{-st}h(t)\,dt,\quad
s\in{\mathbb{C}},\;\;{\rm Re}(s)>0.$
In (2.7), $\\{Y_{n}^{m}\\}$ are the spherical harmonics, which are orthonormal
as defined in [26], and $\\{\widehat{U}_{n}\\}$/$\\{\widehat{U}_{nm}\\}$ are
the Fourier/spherical harmonic expansion coefficients of $U|_{r=b}.$ Note that
the convolution is defined as usual: $(f\ast
g)(t)=\int_{0}^{t}f(t-\tau)g(\tau)d\tau.$
Alternatively, we can represent $T_{d}(U)$ by the temporal convolution in
terms of the expansion coefficients of $\partial_{t}U|_{r=b}.$ More precisely,
we define
$\omega_{\nu}(t):=\omega_{\nu}(t;d):=-\dfrac{(d-1)c}{2b}+c\displaystyle\int_{0}^{t}\sigma_{\nu}(\tau)d\tau,$
(2.9)
and note that $\omega_{\nu}^{\prime}(t)=c\sigma_{\nu}(t).$ Then, we find from
(2.7) and integration by parts that
$T_{d}(U)=-\dfrac{1}{c}\dfrac{\partial U}{\partial
t}\Big{|}_{r=b}+\frac{1}{c}\begin{cases}\displaystyle\sum_{|n|=0}^{\infty}\omega_{n}(t)\ast\partial_{t}\widehat{U}_{n}(b,t)e^{{\rm
i}n\phi},&d=2,\\\\[7.0pt]
\displaystyle\sum_{n=0}^{\infty}\displaystyle\sum_{|m|=0}^{n}\omega_{n+1/2}(t)\ast\partial_{t}\widehat{U}_{nm}(b,t)Y_{n}^{m}(\theta,\phi),&d=3,\end{cases}$
(2.10)
where for $d=2,3,$
$\omega_{\nu}(t)=\mathcal{L}^{-1}\left[1-\frac{(d-2)c}{2bs}+\frac{K_{\nu}^{\prime}(sb/c)}{K_{\nu}(sb/c)}\right](t),\quad\nu=n,n+1/2.$
(2.11)
Hereafter, we term $\sigma_{\nu}$ and $\omega_{\nu}$ as the nonreflecting
boundary kernels (NRBKs). Since $K_{-n}(z)=K_{n}(z)$ (see Formula 9.6.6 in
[1]), it suffices to consider $\omega_{n}$ and $\sigma_{n}$ with $n\geq 0,$
for $d=2.$
###### Remark 2.1.
In the expressions of $\sigma_{\nu}$ and $\omega_{\nu},$ some terms are added,
e.g., $s/c$ and $1/(2b)$ in (2.8), for the purpose of removing the singular
part from the ratio $K_{\nu}^{\prime}/K_{\nu}$. Indeed, recall the asymptotic
formula for fixed $\nu\geq 0$ and large $|z|$ (see Formula 9.7.2 of [1]):
$K_{\nu}(z)\sim\sqrt{\frac{\pi}{2z}}e^{-z}\Big{\\{}1+\frac{4\nu^{2}-1}{8z}+O(z^{-2})\Big{\\}},$
(2.12)
for $|{\rm arg}z|<{3\pi}/2,$ and the recurrence relation:
$zK_{\nu}^{\prime}(z)=\nu K_{\nu}(z)-zK_{\nu+1}(z).$ (2.13)
One verifies that
$\frac{K_{\nu}^{\prime}(z)}{K_{\nu}(z)}\sim-1-\frac{1}{2z}+O(z^{-2}).\qed$
(2.14)
###### Remark 2.2.
We find that the use of the NRBK $\omega_{\nu}$ is more convenient, if one
reformulates (2.4) into a first-order (with respect to the time variable)
system (cf. [6]), and it is more suitable for analysis as well, while the NRBK
$\sigma_{\nu}$ is more appropriate for computation. ∎
We see that NRBCs are global in both time and space. To solve the truncated
problem (2.4)-(2.6) efficiently, we need to (i) invert Laplace transform to
compute NRBKs; (ii) deal with temporal convolutions efficiently; and (iii)
handle the nonlocality of the NRBC in space effectively. The rest of the paper
will address these issues.
### 2.2. Evaluation of the NRBKs
Our starting point is to invert the Laplace transform via evaluating the
Bromwich’s contour integral:
$\begin{split}\sigma_{\nu}(t)&=\frac{1}{2\pi{\rm i}}\int_{\gamma-\infty{\rm
i}}^{\gamma+\infty{\rm
i}}\Big{(}\frac{s}{c}+\frac{1}{2b}+\frac{s}{c}\frac{K_{\nu}^{{}^{\prime}}(sb/c)}{K_{\nu}(sb/c)}\Big{)}e^{ts}ds=\frac{c}{2b^{2}\pi{\rm
i}}\int_{\gamma-\infty{\rm i}}^{\gamma+\infty{\rm
i}}F_{\nu}(z)e^{czt/b}dz,\end{split}$ (2.15)
for $\nu=n,n+1/2$ with $n\geq 0,$ where
$F_{\nu}(z)=z+\frac{1}{2}+z\frac{K^{\prime}_{\nu}(z)}{K_{\nu}(z)},$ (2.16)
and $\gamma$ is the Laplace convergence abscissa, which is a generic constant
greater than the real part of any singularity of $F_{\nu}(z).$
In order to use the residue theorem to evaluate (2.15), we need to understand
the behavior of the poles of $F_{\nu}(z),$ i.e., the zeros of $K_{\nu}(z).$
###### Lemma 2.1.
Let $\nu\geq 0$ be a real number.
* (i)
If $z$ is a zero of $K_{\nu}(z),$ then its complex conjugate $\bar{z}$ is also
a zero. Moreover, all complex conjugate pairs of zeros lie in the second and
third quadrants with ${\rm Re}(z)<0.$
* (ii)
The total number of zeros of $K_{\nu}(z)$ is the even integer nearest to
$\nu-{1}/{2}$, if $\nu-{1}/{2}$ is not an integer, or exactly $\nu-{1}/{2},$
if $\nu-{1}/{2}$ is an integer.
* (iii)
All zeros of $K_{n}(z)$ and $K_{n+{1}/{2}}(z)$ are simple, and lie
approximately along the left half of the boundary of an eye-shaped domain
around $z=0$ (see Figure 2.1).
###### Proof.
The properties (i) and (ii) can be found from Page 511 of [40]. We now
consider the property (iii). As a consequence of (i), it suffices to consider
the zeros of $K_{\nu}(z)$ in the third quadrant and on the negative real axis
(i.e., with $-\pi\leq{\rm arg}z<-\pi/2$) of the complex plane. According to
Formula 9.6.4 of [1], we have the following relation between $K_{\nu}(z)$ and
the Hankel function of the first kind:
$K_{\nu}(z)=\frac{\pi{\rm i}}{2}e^{\frac{1}{2}\nu\pi{\rm i}}H_{\nu}^{(1)}({\rm
i}z),\quad-\pi<{\rm arg}z\leq\frac{\pi}{2},$ (2.17)
which implies that all zeros of $K_{\nu}(z)$ in the third quadrant (i.e., with
$-\pi<{\rm arg}z<-\pi/2$) are obtained by rotating all zeros of
$H_{\nu}^{(1)}(z)$ in the fourth quadrant (i.e., with $-\pi/2<{\rm arg}z<0$)
by an angle $-\pi/2.$ Recall that the zeros of $H^{1}_{\nu}(z)$ in the fourth
quadrant lie approximately along the boundary of an eye-shaped domain around
$z=0$ (see Figure 9.6 and Page 441 of [1]), whose boundary curve intersects
the real axis at $z=n$ and the imaginary axis at $z=-{\rm i}na$, where
$a=\sqrt{t_{0}^{2}-1}\approx 0.66274$ and $t_{0}\approx 1.19968$ is the
positive root of $\coth t=t$. ∎
For clarity, let $M_{\nu}$ be the total number of zeros of $K_{\nu}(z)$ with
$\nu=n,n+1/2,$ that is,
$M_{\nu}=\begin{cases}\text{the largest even integer nearest to
$n-1/2$},\;&{\rm for}\;K_{n}(z),\\\ n,\quad&{\rm
for}\;K_{n+1/2}(z).\end{cases}$ (2.18)
We plot in Figure 2.1 some samples of zeros of $K_{n}(z)$ (and $K_{n+1/2}(z)$
for various $n,$ and visualize that for a given $n$, the zeros sit on the left
half boundary of an eye-shaped domain that intersects the imaginary axis
approximately at $\pm n{\rm i},$ and the negative real axis at $-na$ with
$a\approx 0.66274$ (see the dashed coordinate grids) as predicted by Lemma 2.1
(iii).
Figure 2.1. Distributions of $\\{z_{j}^{\nu}\\}_{j=1}^{M_{\nu}}$ of $K_{n}$
(left) and $K_{n+1/2}$ (right) for various $n$.
With the above understanding of the poles of the integrand $F_{\nu}(z)$ in
(2.16), we now present the exact formula for the NRBKs $\sigma_{\nu}(t)$ with
$\nu=n,n+1/2.$
###### Theorem 2.1.
Let $\nu=n,n+1/2$ with $n\geq 0,$ and let $\\{z_{j}^{\nu}\\}_{j=1}^{M_{\nu}}$
be the zeros of $K_{\nu}(z).$ Then
* •
for $d=2,$
$\sigma_{n}(t)=\frac{c}{b^{2}}\bigg{\\{}\sum\limits_{j=1}^{M_{n}}z_{j}^{n}e^{ctz_{j}^{n}/b}+(-1)^{n}\int_{0}^{\infty}\frac{e^{-ctr/b}}{K^{2}_{n}(r)+\pi^{2}I^{2}_{n}(r)}dr\bigg{\\}},$
(2.19)
* •
for $d=3,$
$\sigma_{\nu}(t)=\frac{c}{b^{2}}\sum\limits_{j=1}^{M_{\nu}}z_{j}^{\nu}e^{ctz_{j}^{\nu}/b},\quad\nu=n+1/2,$
(2.20)
where $I_{n}(z)$ is the modified Bessel function of the first kind (cf. [1]).
We sketch the proof in Appendix A by applying the residue theorem to the
Bromwich’s contour integral (2.15). We remark that Sofronov [35] derived
formulas of similar type by working on much more complicated expressions in
terms of Tricomi’s confluent hypergeometric functions. However, the formulas
in the above theorem are more compact and informative.
###### Remark 2.3.
Based on a delicate study of the logarithmic derivative of the Hankel function
$H^{(1)}_{\nu}(z)$, Alpert et al. [2] (see Theorem 4.1 and Lemma 4.2 in [2])
obtained the following formula:
$\begin{split}&z\frac{H^{(1)^{\prime}}_{\nu}(z)}{H_{\nu}^{(1)}(z)}={\rm
i}z-\frac{1}{2}+\sum\limits_{j=1}^{N_{\nu}}\frac{h_{\nu,j}}{z-h_{\nu,j}}\\\
&\quad-\frac{1}{\pi{\rm
i}}\int_{0}^{\infty}\frac{\pi\cos(\nu\pi)}{\cos^{2}(\nu\pi)K_{\nu}^{2}(r)+(\pi
I_{\nu}(r)+\sin(\nu\pi)K_{\nu}(r))^{2}}\frac{1}{{\rm i}r+z}dr,\end{split}$
(2.21)
for any $\nu\not=n+1/2,$ where $h_{\nu,1},h_{\nu,2},\cdots,h_{\nu,N_{\nu}}$
are zeros of $H_{\nu}^{(1)}(z),$ which number $N_{\nu}.$ Interestingly, (2.19)
can be derived from (2.21), which is justified in Appendix B. ∎
###### Remark 2.4.
Thanks to (2.9), we obtain from Theorem 2.1 the expression of
$\omega_{\nu}(t):$
* •
for $d=2,$
$\omega_{n}(t)=-\frac{c}{2b}+\frac{c}{b}\bigg{\\{}\sum_{j=1}^{M_{n}}\big{(}e^{ctz_{j}^{n}/b}-1\big{)}+(-1)^{n}\int_{0}^{\infty}\frac{1-e^{-ctr/b}}{r\\{K^{2}_{n}(r)+\pi^{2}I^{2}_{n}(r)\\}}dr\bigg{\\}},$
(2.22)
* •
for $d=3,$
$\omega_{\nu}(t)=-\frac{c}{b}+\frac{c}{b}\sum_{j=1}^{M_{\nu}}\big{(}e^{ctz_{j}^{\nu}/b}-1\big{)},\quad\nu=n+1/2.\qed$
(2.23)
### 2.3. Computation of the improper integral in (2.19)
The computation of the two-dimensional NRBK requires to evaluate the improper
integral involving the kernel function:
$W_{n}(r):=\frac{1}{K_{n}^{2}(r)+\pi^{2}I_{n}^{2}(r)}:=\frac{1}{G_{n}(r)},\quad
n\geq 0,\;\;r>0,$ (2.24)
whose important properties are characterized below.
###### Lemma 2.2.
For any $n\geq 0$ and any real $r>0,$ we have
* (i)
$G_{n}(r)$ is a convex function of $r,$ and $W_{n}(r)$ attains its maximum at
a unique point.
* (ii)
For large $n,$ we have the uniform asymptotic estimate:
$W_{n}(n\kappa)\sim\frac{n\sqrt{1+\kappa^{2}}}{\pi}{\rm
sech}(2n\Theta):=\widetilde{W}_{n}(n\kappa),$ (2.25)
for $\kappa>0,$ where
$\Theta=\Theta(\kappa):=\sqrt{1+\kappa^{2}}+\ln\frac{\kappa}{1+\sqrt{1+\kappa^{2}}}.$
(2.26)
Approximately, the maximum value of $W_{n}(r)$ attains at $r=na$ with
$a\approx 0.66274$ being the root of $\Theta,$ and the maximum value is
approximately $n\sqrt{1+a^{2}}/\pi\approx 0.38187n.$
###### Proof.
(i). We find from Page 374 of [1] that for a given $n,$ $K_{n}(r),I_{n}(r)>0,$
and $K_{n}(r)$ (resp. $I_{n}(r)$) is monotonically descending (resp.
ascending) with respect to $r.$ From the series representation (see Formula
9.6.10 of [1]):
$I_{n}(r)=\frac{r^{n}}{2^{n}}\sum_{k=0}^{\infty}\frac{r^{2k}}{2^{2k}k!(n+k)!},\quad
n\geq 0,$
we conclude that $I_{n}^{\prime\prime}(r)>0.$ Moreover, since $K_{n}(r)$
satisfies (see Formula 9.6.1 of [1])
$r^{2}K_{n}^{\prime\prime}(r)+rK_{n}^{\prime}(r)-(r^{2}+n^{2})K_{n}(r)=0,$
we have $K_{n}^{\prime\prime}(r)>0.$ Therefore, a direct calculation shows
that $G_{n}^{\prime\prime}(r)>0,$ so $G_{n}(r)$ is convex.
One verifies that $G_{n}(0+)=G_{n}(+\infty)=+\infty$ for all $n,$ which
follows from (2.12), and the asymptotic properties (see [1] again):
$\begin{split}K_{\nu}(r)\sim\begin{cases}-\ln r,\;\;&{\rm
if}\;\;\nu=0,\\\\[7.0pt]
\dfrac{\Gamma(\nu)}{2}\Big{(}\dfrac{r}{2}\Big{)}^{-\nu},\;\;&{\rm
if}\;\;\nu>0,\end{cases}\end{split}\qquad
I_{\nu}(r)\sim\frac{1}{\Gamma(\nu+1)}\Big{(}\frac{r}{2}\Big{)}^{\nu},\;\;\nu\geq
0,$ (2.27)
for $0<r\ll 1,$ and
$I_{\nu}(r)\sim\sqrt{\frac{1}{2\pi r}}e^{r},\quad r\gg 1,\;\;\nu\geq 0.$
(2.28)
Since $G_{n}(r)$ is convex, $G_{n}(r)$ attains its minimum at a unique point
$r_{0}.$ Thanks to $G_{n}^{\prime}(r)=-W_{n}^{\prime}(r)/W_{n}^{2}(r),$
$W_{n}(r)$ has a unique maximum at the same point $r_{0}.$
(ii). Recall that for large $n$ (see Formulas (9.7.7)-(9.7.11) of [1]):
$\begin{split}&K_{n}(n\kappa)\sim\sqrt{\frac{\pi}{2n}}\frac{e^{-n\Theta}}{(1+\kappa^{2})^{1/4}},\qquad\qquad
I_{n}(n\kappa)\sim\frac{1}{\sqrt{2\pi
n}}\frac{e^{n\Theta}}{(1+\kappa^{2})^{1/4}},\\\
&K_{n}^{\prime}(n\kappa)\sim-\sqrt{\frac{\pi}{2n}}\frac{(1+\kappa^{2})^{1/4}}{\kappa}{e^{-n\Theta}},\quad
I_{n}^{\prime}(n\kappa)\sim\frac{1}{\sqrt{2\pi
n}}\frac{(1+\kappa^{2})^{1/4}}{\kappa}{e^{n\Theta}},\end{split}$ (2.29)
which, together with (2.24), leads to the asymptotic estimate (2.25). Thus,
the maximum value of $W_{n}(r)$ approximately attains at the unique root of
$\Theta(\kappa),$ which turns out to be $a\approx 0.66274$ as in Figure 2.1,
and the maximum value is about $n\sqrt{1+a^{2}}/\pi\approx 0.38187n.$ ∎
Figure 2.2. Left: graph of $\Theta(\kappa),\kappa\in(0,8]$ defined in (2.26).
Right: $W_{n}$ against the asymptotic estimate $\widetilde{W}_{n}$
(“$\bullet$”) for samples of $n=5,15,30,45.$
We depict in Figure 2.2 (left) the graph of $\Theta(\kappa)$, and highlight
the zero point $(a,0).$ Observe that $\Theta(\kappa)$ grows like $\kappa,$
when $\kappa>a.$ We also plot in Figure 2.2 (right) several sample graphs of
$W_{n}$ (solid lines) and the asymptotic estimate $\widetilde{W}_{n}$
(“$\bullet$”) for $n=5,15,30,45,$ and particularly mark the asymptotic point
$(na,n\sqrt{1+a^{2}}/\pi)$ of the maximum of $W_{n}(r)$ obtained in Lemma 2.2
(ii). Observe that even for small $n,$ the asymptotic estimate provides a very
accurate approximation of $W_{n}.$ These properties greatly facilitate the
computation of the improper integral. Indeed, we can truncate $(0,\infty)$ to
a narrow interval (of length about $20$ for $n\geq 5$ in our numerical
computations in Section 5) around the point $r=na.$
### 2.4. Rapid evaluation of the temporal convolution
Remarkably, the presence of the time variable $t$ in the exponentials in
(2.19) and (2.20) allows us to eliminate the burden of history dependence of
the temporal convolution easily. More precisely, given a function $g(t),$ we
define
$f(t;r):=e^{-ctr/b}\ast g(t)=\int_{0}^{t}e^{-c(t-\tau)r/b}g(\tau)d\tau.$
One verifies readily that
$\begin{split}f(t+\Delta t;r)=e^{-c\Delta tr/b}f(t;r)+\int_{t}^{t+\Delta
t}e^{-c(t+\Delta t-\tau)r/b}g(\tau)d\tau,\end{split}$ (2.30)
so $f(t;r)$ can march in $t$ with step size $\Delta t$ recursively. This
enables us to compute the time convolution rapidly. For example, in the 2-D
case,
$\begin{split}[\sigma_{n}\ast
g](t)&=\int_{0}^{t}\sigma_{n}(t-\tau)g(\tau)d\tau=\frac{c}{b^{2}}\sum_{j=1}^{M_{n}}z_{j}^{n}\int_{0}^{t}e^{c(t-\tau)z_{j}^{n}/b}g(\tau)d\tau\\\
&\quad+\frac{(-1)^{n}c}{b^{2}}\int_{0}^{\infty}\frac{1}{K_{n}^{2}(r)+\pi^{2}I_{n}^{2}(r)}\Big{[}\int_{0}^{t}e^{-c(t-\tau)r/b}g(\tau)d\tau\Big{]}dr\\\
&=\frac{c}{b^{2}}\sum_{j=1}^{M_{n}}z_{j}^{n}f(t;-z_{j}^{n})+\frac{(-1)^{n}c}{b^{2}}\int_{0}^{+\infty}{f(t;r)}W_{n}(r)dr.\end{split}$
(2.31)
Thanks to (2.30), $[\sigma_{n}\ast g](t+\Delta t)$ can be computed recursively
from the previous step and the history dependence is then narrowed down to
$[t,t+\Delta t].$ We also refer to Subsection 4.4 for more detailed
discussions.
## 3\. A priori estimates
In this section, we analyze the well-posedness of the truncated problem
(2.4)-(2.6), and provide a priori estimates for its solution.
We first recall the Plancherel or Parseval results for the Laplace transform.
###### Lemma 3.1.
Let $s=s_{1}+{\rm i}s_{2}$ with $s_{1},s_{2}\in{\mathbb{R}}.$ If $f,g$ are
Laplace transformable, then
$\frac{1}{2\pi}\int_{-\infty}^{+\infty}{\mathcal{L}}[f](s)\,{\mathcal{L}}[\bar{g}](s)\,ds_{2}=\int_{0}^{+\infty}e^{-2s_{1}t}f(t)\bar{g}(t)dt,\quad\forall\,s_{1}>\gamma,$
(3.1)
where $\gamma$ is the absissa of convergence for both $f$ and $g$, and
$\bar{g}$ is the complex conjugate of $g.$
###### Proof.
This identity can be proved by following the same lines as for (2.46) in [8]
(or [11]). ∎
For notational convenience, we introduce the modified spherical Bessel
function (cf. [40]):
$k_{n}(z)=\sqrt{\frac{2}{\pi
z}}K_{n+1/2}(z)\;\;\Rightarrow\;\;\frac{k_{n}^{\prime}(z)}{k_{n}(z)}=-\frac{1}{2z}+\frac{K_{n+1/2}^{\prime}(z)}{K_{n+1/2}(z)},$
(3.2)
and by (2.11),
$\omega_{n+1/2}(t)=\mathcal{L}^{-1}\left[1+\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\right](t),\quad
n\geq 0.$ (3.3)
The following properties are also indispensable for the analysis.
###### Lemma 3.2.
Let $s=s_{1}+{\rm i}s_{2}$ with $s_{1},s_{2}\in{\mathbb{R}}.$ Then for any
$s_{1}>0,$
${\rm(i)}\;{\rm Re}\Big{(}s\frac{Z_{n}^{\prime}(sb/c)}{Z_{n}(sb/c)}\Big{)}\leq
0,\;\;{\rm Re}\Big{(}\frac{Z_{n}^{\prime}(sb/c)}{Z_{n}(sb/c)}\Big{)}\leq
0;\;\;{\rm(ii)}\;{\rm
Im}\Big{(}s\frac{Z_{n}^{\prime}(sb/c)}{Z_{n}(sb/c)}\Big{)}\leq 0,\;\;\forall
s_{2}\geq 0,$ (3.4)
where $Z_{n}(z)=K_{n}(z)$ or $k_{n}(z).$
###### Proof.
The results with $Z_{n}=K_{n}$ were proved in Chen [6]. We next prove (3.4)
with $Z_{n}=k_{n}$ by using a similar argument. By applying Laplace transform
to (2.1)-(2.3), exterior to $D=B$ with $F=U_{0}=U_{1}\equiv 0$ and
$G=U|_{r=b},$ and denoting $u={\mathcal{L}}[U],$ we obtain
$\begin{split}&-c^{2}\Delta u+s^{2}u=0,\quad{\rm in}\;\;\Omega_{\rm
ext}={\mathbb{R}}^{d}\setminus\bar{B},\;\;s\in{\mathbb{C}},\;\;{\rm
Re}(s)>0;\\\ &u|_{r=b}=\psi=\mathcal{L}[U|_{r=b}];\quad
c{\partial_{r}u}+su=o\big{(}r^{(1-d)/2}\big{)},\;\;d=2,3,\end{split}$ (3.5)
which admits the series solution
$u(r,\theta,\phi,s)=\sum_{n=0}^{\infty}\frac{k_{n}(sr/c)}{k_{n}(sb/c)}\sum_{|m|=0}^{n}\hat{\psi}_{nm}(s)Y_{n}^{m}(\theta,\phi),$
(3.6)
where $\\{\hat{\psi}_{nm}\\}$ are the spherical harmonic expansion
coefficients of $\psi$. Multiplying the first equation of (3.5) by $\bar{u}$
and integrating over $\Omega_{b,\rho}:=B_{\rho}\setminus\bar{B},$ where
$B_{\rho}$ is a ball of radius $\rho>b,$ the imaginary part of the resulting
equation reads
$2s_{1}s_{2}\int_{\Omega_{b,\rho}}|u|^{2}d\boldsymbol{x}-c^{2}\,{\rm
Im}\int_{\partial\Omega_{b,\rho}}\frac{\partial
u}{\partial{\boldsymbol{n}}}\bar{u}\,d\gamma=0,$ (3.7)
where $\boldsymbol{n}$ is the unit outer normal of $\partial\Omega_{b,\rho}$.
Since $s_{1}>0,$ multiplying (3.7) by $s_{2}$ yields
$s_{2}\,{\rm Im}\int_{\\{r=b\\}}\frac{\partial u}{\partial
r}\bar{u}\,d\gamma\leq s_{2}\,{\rm Im}\int_{\\{r=\rho\\}}\frac{\partial
u}{\partial r}\bar{u}\,d\gamma.$ (3.8)
It is clear that
$u=k_{n}(sr/c)/k_{n}(sb/c)\hat{\psi}_{nm}(s)Y_{n}^{m}(\theta,\phi)$ is a
solution of (3.5), so using the orthogonality of $\\{Y_{n}^{m}\\},$ we obtain
from (3.8) that
$\frac{b}{c}{\rm
Im}\Big{(}s_{2}s\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}|\hat{\psi}_{nm}|^{2}\leq\frac{\rho}{c}{\rm
Im}\Big{(}s_{2}s\frac{k_{n}^{\prime}(s\rho/c)}{k_{n}(sb/c)}\Big{)}|\hat{\psi}_{nm}|^{2}.$
(3.9)
We find from (2.12) and (3.2) that $k_{n}^{\prime}(s\rho/c)$ decays
exponentially if $s_{1}>0,$ so the right hand side of (3.8) tends to zero as
$\rho\to+\infty.$ Thus, letting $\rho\to+\infty$ in (3.9) leads to
${\rm Im}\Big{(}s_{2}s\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}\leq 0.$
(3.10)
If $s_{2}\geq 0,$ we obtain (ii) in (3.4). Next, we prove the first inequality
of (i) in (3.4). Recall the formula (see Lemma 2.3 of [6]):
$|K_{n+1/2}(sr)|^{2}=\frac{1}{2}\int_{0}^{+\infty}e^{-\frac{|s|^{2}r^{2}}{2r}-\frac{s^{2}+\bar{s}^{2}}{2|s^{2}|}}K_{n+1/2}(\tau)\frac{d\tau}{\tau},\quad
s_{1}>0,$
which implies that $|K_{n+1/2}(sr)|^{2}$ is monotonically descending with
respect to $r.$ The property $\frac{d}{dr}|K_{n+1/2}(sr)|^{2}\leq 0,$ together
with (3.2), implies ${\rm
Re}\Big{(}s\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}\leq 0$. Denoting
$s\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}=\gamma_{1}+{\rm i}\gamma_{2}$ with
$\gamma_{1},\gamma_{2}\in{\mathbb{R}},$ we know from (3.10) that
$\gamma_{1}\leq 0$ and $s_{2}\gamma_{2}\leq 0.$ Therefore,
${\rm Re}\Big{(}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}={\rm
Re}\Big{(}\frac{\gamma_{1}+{\rm
i}\gamma_{2}}{s}\Big{)}=\frac{1}{|s|^{2}}(s_{1}\gamma_{1}+s_{2}\gamma_{2})\leq
0.$
This ends the proof. ∎
With the above preparations, we can derive the following important property.
###### Theorem 3.1.
For any $v\in L^{2}(0,T),$ we have
$\int_{0}^{T}[\omega_{\nu}\ast
v](t)\bar{v}(t)dt\leq\int_{0}^{T}|v(t)|^{2}dt,\quad\forall\;T>0,\;\;n\geq 0,$
(3.11)
where $\omega_{\nu}$ is the NRBK given by (2.11) (or (3.3) for $d=3$).
###### Proof.
Let $\tilde{v}=v\,{\bf 1}_{{}_{[0,T]}},$ where ${\bf 1}_{{}_{[0,T]}}$ is the
characteristic function of $[0,T].$ Then we obtain from (3.1) that for $d=3,$
$\begin{split}\int_{0}^{T}e^{-2s_{1}t}&[\omega_{n+1/2}\ast
v](t)\bar{v}(t)dt=\int_{0}^{+\infty}e^{-2s_{1}t}[\omega_{n+1/2}\ast\tilde{v}](t)\bar{\tilde{v}}(t)dt\\\
&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\Big{[}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}+1\Big{]}\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}\\\
&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}+\frac{1}{2\pi}\int_{-\infty}^{+\infty}\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}.\end{split}$
It is clear that by (3.1) with $f=g=\tilde{v},$
$\frac{1}{2\pi}\int_{-\infty}^{+\infty}\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}=\int_{0}^{+\infty}e^{-2s_{1}t}|\tilde{v}(t)|^{2}dt=\int_{0}^{T}e^{-2s_{1}t}|v(t)|^{2}dt.$
Using (2.13) and the properties $\overline{K_{\nu}(z)}=K_{\nu}(\bar{z})$ and
${\mathcal{L}}[\tilde{v}](\bar{s})=\overline{{\mathcal{L}}[\tilde{v}](s)},$ we
find
$\begin{split}\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}&\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}=\frac{1}{\pi}\int_{0}^{+\infty}{\rm
Re}\Big{(}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}\big{|}{\mathcal{L}}[\tilde{v}](s)\big{|}^{2}ds_{2}\\\
&=\frac{1}{\pi}\int_{0}^{+\infty}{\rm
Re}\Big{(}\frac{k_{n}^{\prime}(sb/c)}{k_{n}(sb/c)}\Big{)}|s|^{2}\Big{|}{\mathcal{L}}\Big{[}\int_{0}^{t}\tilde{v}(\tau)d\tau\Big{]}(s)\Big{|}^{2}ds_{2}.\end{split}$
(3.12)
Thus, we conclude from Lemma 3.2 and the above identities that for $s_{1}>0,$
$\int_{0}^{T}e^{-2s_{1}t}[\omega_{n+1/2}\ast
v](t)\bar{v}(t)dt\leq\int_{0}^{T}e^{-2s_{1}t}|v(t)|^{2}dt.$ (3.13)
Notice that the asymptotic formulas in (2.27) are also valid for complex $r$
(see Formula 9.6.9 of [1]), which, together with Lemma 2.1, implies that
$k_{n}^{\prime}(sb/c)/k_{n}(sb/c)$ is analytic for all ${\rm Re}(s)\geq 0$ but
$|s|\not=0,$ and $\lim_{s_{1}\to
0^{+}}|s|^{2}k_{n}^{\prime}(sb/c)/k_{n}(sb/c)$ exists for all $s_{2}\geq 0.$
Hence, the integral in (3.12) is finite. By letting $s_{1}\to 0^{+}$ in (3.13)
leads to the desired result for $d=3.$
The result (3.11) with $d=2$ can be proved in a similar fashion. ∎
###### Corollary 3.1.
Suppose that $v^{\prime}\in L^{2}(0,T)$ with $v(0)=0.$ Then we have
$\int_{0}^{T}[\sigma_{\nu}\ast
v](t)\bar{v}^{\prime}(t)dt\leq\frac{1}{c}\int_{0}^{T}|v^{\prime}(t)|^{2}dt+\frac{d-1}{4b}|v(T)|^{2},\quad\forall\;T>0,$
(3.14)
for $\nu=n,n+1/2,$ where $\sigma_{\nu}(t)$ is the NRBK defined in (2.8).
###### Proof.
By (2.9), we have
$\omega_{\nu}^{\prime}(t)=c\sigma_{\nu}(t),\omega_{\nu}(0)=-\frac{(d-1)c}{2b}.$
Thus, we obtain from integration by parts and the fact $v(0)=0$ that
$[\omega_{\nu}\ast v^{\prime}](t)=-\frac{(d-1)c}{2b}v(t)+c[\sigma_{\nu}\ast
v](t).$
By Theorem 3.1 with $v^{\prime}$ in place of $v,$
$\begin{split}\int_{0}^{T}&[\omega_{\nu}\ast
v^{\prime}](t)\bar{v}^{\prime}(t)dt=-\frac{(d-1)c}{2b}\int_{0}^{T}v(t)\bar{v}^{\prime}(t)dt+c\int_{0}^{T}[\sigma_{\nu}\ast
v](t)\bar{v}^{\prime}(t)dt\\\
&=-\frac{(d-1)c}{4b}{|v(T)|^{2}}+c\int_{0}^{T}[\sigma_{\nu}\ast
v](t)\bar{v}^{\prime}(t)dt\leq\int_{0}^{T}|v^{\prime}(t)|^{2}dt.\end{split}$
This gives (3.14). ∎
Now, we are ready to analyze the stability of the solution of the truncated
problem (2.4)-(2.6). To this end, we assume that the scatter $D$ is a simply
connected domain with Lipschitz boundary $\Gamma_{D}$, and the Dirichlet data
$G=0$ on $\Gamma_{D}.$ Denote $X:=\big{\\{}U\in
H^{1}(\Omega):U|_{\Gamma_{D}}=0\big{\\}},$ and let
$(\cdot,\cdot)_{L^{2}(\Omega)}$ and $\|\cdot\|_{L^{2}(\Omega)}$ be the inner
product and norm of $L^{2}(\Omega),$ respectively.
###### Theorem 3.2.
Let $U(\in X)$ be the solution of (2.4)-(2.6) with $G=0.$ If $U_{0}\in
H^{1}(\Omega),U_{1}\in L^{2}(\Omega)$ and $F\in L^{1}(0,T;L^{2}(\Omega))$ for
any $T>0,$ then we have $\nabla U\in L^{\infty}(0,T;(H^{1}(\Omega))^{d}),$
$\partial_{t}U\in L^{\infty}(0,T;L^{2}(\Omega)),$ and there holds
$\begin{split}\|\partial_{t}U\|_{L^{\infty}(0,T;L^{2}(\Omega))}&+c\|\nabla
U\|_{L^{\infty}(0,T;L^{2}(\Omega))}\\\ &\leq
C\big{(}\|U_{1}\|_{L^{2}(\Omega)}+{c}\|\nabla
U_{0}\|_{L^{2}(\Omega)}+\|F\|_{L^{1}(0,T;L^{2}(\Omega))}\big{)},\end{split}$
(3.15)
where $C$ is a positive constant independent of any functions and $c,b.$
Moreover, if the source term $F\equiv 0$, we have the conservation of the
energy: $E^{\prime}(t)=0$ for all $t\geq 0,$ where
$E(t)=\int_{\Omega}\big{(}|\partial_{t}U|^{2}+c^{2}|\nabla
U|^{2}\big{)}d\boldsymbol{x}-2c^{2}\int_{0}^{t}\int_{\Gamma_{b}}T_{d}(U)\partial_{\tau}\overline{U}d\gamma
d\tau.$ (3.16)
###### Proof.
Multiplying (2.4) by $2\partial_{t}\overline{U}$ and integrating over
$\Omega,$ we derive from the Green’s formula that for any $t>0,$
$\displaystyle\frac{d}{dt}\Big{(}\|\partial_{t}U\|^{2}_{L^{2}(\Omega)}+{c^{2}}\|\nabla
U\|^{2}_{L^{2}(\Omega)}\Big{)}-2c^{2}\int_{\Gamma_{b}}T_{d}(U)\partial_{t}\overline{U}d\gamma=2(F,\partial_{t}U)_{L^{2}(\Omega)}.$
(3.17)
Integrating the above equation over $(0,t),$ we find that for any $t>0,$
$\displaystyle\begin{split}\|\partial_{t}U\|^{2}_{L^{2}(\Omega)}&+{c^{2}}\|\nabla
U\|^{2}_{L^{2}(\Omega)}-2c^{2}\int_{0}^{t}\int_{\Gamma_{b}}T_{d}(U)\partial_{\tau}\overline{U}d\gamma
d\tau\\\
&=2\int_{0}^{t}(F,\partial_{\tau}U)_{L^{2}(\Omega)}d\tau+\|U_{1}\|^{2}_{L^{2}(\Omega)}+{c^{2}}\|\nabla
U_{0}\|^{2}_{L^{2}(\Omega)}\\\ &\leq
2\|\partial_{t}U\|_{L^{\infty}(0,T;L^{2}(\Omega))}\|F\|_{L^{1}(0,T;L^{2}(\Omega))}+\|U_{1}\|^{2}_{L^{2}(\Omega)}+{c^{2}}\|\nabla
U_{0}\|^{2}_{L^{2}(\Omega)}.\end{split}$ (3.18)
We next show that for any $t>0,$
$\int_{0}^{t}\int_{\Gamma_{b}}T_{d}(U)\partial_{\tau}\overline{U}d\gamma
d\tau\leq 0.$ (3.19)
For $d=3,$ it follows from (2.10), Theorem 3.1 and the orthogonality of
$\\{Y_{n}^{m}\\}$ that
$\begin{split}\int_{0}^{t}\int_{\Gamma_{b}}&T_{d}(U)\partial_{\tau}\overline{U}d\gamma
d\tau=-\frac{1}{c}\int_{0}^{t}\|\partial_{\tau}U\|^{2}_{L^{2}(\Gamma_{b})}d\tau\\\
&\quad+\frac{1}{c}\sum_{n=0}^{\infty}\sum_{|m|=0}^{n}\int_{0}^{t}\big{[}\omega_{n+1/2}\ast\partial_{\tau}\widehat{U}_{nm}(b,\tau)\big{]}{\partial_{\tau}\overline{\widehat{U}}_{nm}(b,\tau)}d\tau\\\
&\overset{(\ref{fgss})}{\leq}-\frac{1}{c}\int_{0}^{t}\|\partial_{\tau}U\|^{2}_{L^{2}(\Gamma_{b})}d\tau+\frac{1}{c}\sum_{n=0}^{\infty}\sum_{|m|=0}^{n}\int_{0}^{t}\big{|}\partial_{\tau}\widehat{U}_{nm}(b,\tau)\big{|}^{2}d\tau=0.\end{split}$
This verifies (3.19) with $d=3.$ Similarly, one can justify (3.19) with $d=2.$
Consequently, the estimate (3.15) follows from (3.18), (3.19) and the Cauchy-
Schwarz inequality.
Letting $F\equiv 0$ in (3.18) leads to the conservation of energy. ∎
###### Remark 3.1.
Note that (i) if $T_{d}(U)$ is given by (2.7), we can use Corollary 3.1 to
verify (3.19), and (ii) a similar estimate for $d=2$ was derived by [6] with a
slightly different argument. ∎
## 4\. Spectral-Galerkin approximation and Newmark’s time integration
This section is devoted to numerical approximation of the truncated problem
(2.4)-(2.6) with a focus on the treatment for the NRBC (2.6). Note that if the
scatter $D$ is a disk/ball (i.e., $\Omega$ in (2.4)-(2.6) is an
annulus/spherical shell), the nonlocality of the NRBC in space becomes local
in the space of Fourier/spherical harmonic coefficients in polar/spherical
coordinates. Correspondingly, the truncated problem can be reduced to a
sequence of one-dimensional problems with mixed boundary conditions at the
outer artificial boundary. In order to deal with a general scatter, one may
resort to the transformed field expansion method (TFE) (cf. [28]), which has
been successfully applied to time-harmonic Helmholtz equations (cf. [29, 13,
30]). The use of this method allows us to solve a sequence of truncated
problems (2.4)-(2.6) in regular domains. Accordingly, it is essential to
construct a fast and accurate solver for (2.4)-(2.6) in an annulus or a
spherical shell, which will be the concern of this section. We shall report
the combination of the solver with the TFE method in a future work as the
implementation is rather involved.
### 4.1. Notion of TFE
We outline the TFE method for (2.4)-(2.6) with $d=2$ and
$\Gamma_{D}=\big{\\{}r=b_{0}+\eta(\phi):0\leq\phi<2\pi\big{\\}},\quad
b>b_{0}+\max_{\phi\in[0,2\pi]}|\eta(\phi)|,$
where $r(\phi)=b_{0}+\eta(\phi)$ is the parametric equation of the boundary of
the scatter $D$.
* •
Make a change of variables
$r^{\prime}=\frac{(b-b_{0})r-b\eta(\phi)}{(b-b_{0})-\eta(\phi)},\quad\phi^{\prime}=\phi,$
which maps $\Omega$ to the annulus
$\Omega_{0}=\big{\\{}(r^{\prime},\phi^{\prime}):b_{0}<r^{\prime}<b,\,0\leq\phi^{\prime}<2\pi\big{\\}}.$
To simplify the notation, we still use $U,F,r,\phi$ etc. to denote the
transformed functions or variables. Then the problem (2.4)-(2.6) becomes
$\begin{split}&\partial_{t}^{2}U=c^{2}\Delta U+F+J(\eta,U),\;\;{\rm
in}\;\;\Omega_{0},\;\;t>0;\\\ &U=U_{0},\;\;\partial_{t}U=U_{1},\;\;{\rm
in}\;\;\Omega_{0},\;\;t=0;\\\
&U|_{r=b_{0}}=G,\;\;\;t>0;\;\;\big{(}\partial_{r}U-T_{d}(U)\big{)}|_{r=b}=L(\eta,U)|_{r=b},\quad
t>0,\end{split}$
where $J(\eta,U)$ and $L(\eta,U)$ contain differential operators with
nonconstant coefficients.
* •
To solve the transformed problem efficiently, we adopt the boundary
perturbation technique by viewing the obstacle as a perturbation of the disk.
More precisely, we write $\eta=\varepsilon\zeta$ and expand the solution $U$
as
$U(r,\phi,t;\varepsilon)=\sum_{l=0}^{\infty}U_{l}(r,\phi,t)\varepsilon^{l},$
and likewise for the data. Formally, we obtain the sequence of equations after
collecting the terms of $\varepsilon^{l}$ (see [29]):
$\begin{split}&\partial_{t}^{2}U_{l}=c^{2}\Delta
U_{l}+F_{l}+\tilde{J}(\eta,U_{l-4},\cdots,U_{l-1}),\quad{\rm
in}\;\;\Omega_{0},\;\;t>0;\\\
&U_{l}=U_{0,l},\quad\partial_{t}U_{l}=U_{1,l},\quad{\rm
in}\;\;\Omega_{0},\;\;t=0;\\\
&U_{l}|_{r=b_{0}}=G_{l},\;\;\;t>0;\quad\big{(}\partial_{r}U_{l}-T_{d}(U_{l})\big{)}|_{r=b}=\tilde{L}(\eta,U_{l-1})|_{r=b},\quad
t>0.\end{split}$
* •
Solve the above equation for $l=0,1,\cdots$, and sum up the series by using a
Pad$\acute{e}$ approximation.
### 4.2. Prototype equation and dimension reduction
Under this notion, the whole algorithm boils down to solving a sequence of
prototype equations. More precisely, we consider
$\begin{split}&\partial_{t}^{2}U=c^{2}\Delta U+F,\quad{\rm
in}\;\;\Omega_{0}=\big{\\{}\boldsymbol{x}\in{\mathbb{R}}^{d}:b_{0}<|\boldsymbol{x}|<b\big{\\}},\;\;t>0;\\\
&U=U_{0},\;\;\partial_{t}U=U_{1},\;\;{\rm in}\;\;\Omega_{0},\;\;t=0;\quad
U|_{r=b_{0}}=0,\;\;\big{(}\partial_{r}U-T_{d}(U)\big{)}\big{|}_{r=b}=0,\;\;t>0,\end{split}$
(4.20)
where $T_{d}(U)$ is the DtN map as before. We expand the solution and given
data in Fourier series/spherical harmonic series. Then the problem (4.20),
after a polar (in 2-D) and spherical (in 3-D) transform, reduces to a sequence
of one-dimensional problems (for brevity, we use $u$ to denote the
Fourier/spherical harmonic expansion coefficients of $U,$ and likewise, we use
$u_{0},u_{1}$ and $f$ to denote the expansion coefficients of $U_{0},U_{1}$
and $F,$ respectively):
$\begin{split}&\frac{\partial^{2}u}{\partial
t^{2}}-\frac{c^{2}}{r^{d-1}}\frac{\partial}{\partial
r}\Big{(}r^{d-1}\frac{\partial u}{\partial
r}\Big{)}+c^{2}\beta_{n}\frac{u}{r^{2}}=f,\quad b_{0}<r<b,\;\;t>0;\\\
&u|_{t=0}=u_{0},\quad\frac{\partial u}{\partial t}\Big{|}_{t=0}=u_{1},\quad
b_{0}<r<b;\quad u|_{r=b_{0}}=0,\;\;t>0;\\\ &\Big{(}\frac{1}{c}\frac{\partial
u}{\partial t}+\frac{\partial u}{\partial
r}+\frac{d-1}{2r}u\Big{)}\Big{|}_{r=b}=\int_{0}^{t}\sigma_{\nu}(t-\tau)u(b,\tau)d\tau,\;\;t>0,\end{split}$
(4.21)
where $\beta_{n}=n^{2},n(n+1)$ and $\nu=n,n+1/2$ for $d=2,3,$ respectively.
Since $\sigma_{\nu}$ is real, the real and imaginary parts of $u$ and the
given data can be decoupled. In what follows, we assume they are real.
### 4.3. Spectral-Galerkin approximation in space
We apply the Legendre spectral Galerkin method to approximate (4.21) in space.
For convenience of implementation, we transform the interval $(b_{0},b)$ to
the reference interval $I=(-1,1)$ by $r=\frac{b-b_{0}}{2}x+\frac{b+b_{0}}{2}$
with $x\in\bar{I},$ and denote the transformed functions by
$v(x,t)=u(r,t),h(x,t)=f(r,t)$ and $v_{i}(x)=u_{i}(r)$ with $i=0,1,$
respectively. Then (4.21) can be reformulated as
$\begin{split}&\frac{\partial^{2}v}{\partial
t^{2}}-\frac{\tilde{c}^{2}}{(x+c_{0})^{d-1}}\frac{\partial}{\partial
x}\Big{(}(x+c_{0})^{d-1}\frac{\partial v}{\partial
x}\Big{)}+\tilde{c}^{2}\beta_{n}\frac{v}{(x+c_{0})^{2}}=h,\quad x\in
I,\;\;t>0;\\\ &v(x,0)=v_{0}(x),\quad\frac{\partial v}{\partial
t}(x,0)=v_{1}(x),\quad x\in I;\quad v(-1,t)=0,\;\;t>0;\\\
&\Big{(}\frac{1}{c}\frac{\partial v}{\partial
t}+\frac{2}{b-b_{0}}\frac{\partial v}{\partial
x}+\frac{d-1}{2b}v\Big{)}(1,t)=\int_{0}^{t}\sigma_{\nu}(t-\tau)v(1,\tau)d\tau,\;\;t>0,\end{split}$
(4.22)
where the constants $\tilde{c}=\frac{2c}{b-b_{0}}$ and
$c_{0}=\frac{b+b_{0}}{b-b_{0}}.$
Let $V_{N}:=\big{\\{}\psi\in P_{N}:\psi(-1)=0\big{\\}},$ where $P_{N}$ is the
set of all algebraic polynomials of degree at most $N.$ The semi-
discretization Legendre spectral-Galerkin approximation of (4.22) is to find
$v_{N}(x,t)\in V_{N}$ for all $t>0$ such that for all $w\in V_{N},$
$\begin{split}&(\ddot{v}_{N},w)_{\varpi}+\tilde{c}(1+c_{0})^{d-1}\dot{v}_{N}(1,t)w(1)+\tilde{c}^{2}(\partial_{x}v_{N},\partial_{x}w)_{\varpi}+\tilde{c}^{2}\beta_{n}\big{(}v_{N}(x+c_{0})^{-2},w\big{)}_{\varpi}\\\
&\qquad+\frac{2c^{2}}{b-b_{0}}(1+c_{0})^{d-1}\Big{(}\frac{d-1}{2b}v_{N}(1,t)-\sigma_{\nu}(t)\ast
v_{N}(1,t)\Big{)}w(1)=(I_{N}h,w)_{\varpi},\\\
&v_{N}(x,0)=v_{0,N}(x),\quad\dot{v}_{N}(x,0)=v_{1,N}(x),\quad x\in
I,\end{split}$ (4.23)
where $(\cdot,\cdot)_{\varpi}$ is the (weighted) inner product of
$L^{2}_{\varpi}(I)$ with the weight function $\varpi=(x+c_{0})^{d-1},$
$\ddot{v}$ denotes $\partial_{t}^{2}v$ or $\frac{d^{2}v}{dt^{2}}$ as usual,
$I_{N}$ is the interpolation operator on $(N+1)$ Legendre-Gauss-Lobatto
points, and $v_{0,N},v_{1,N}\in P_{N}$ are suitable approximations of the
initial values.
Like Theorem 3.2, we have the following a prior estimates.
###### Theorem 4.1.
Let $v_{N}$ be the solution of (4.23). Then for all $t>0,$
$\begin{split}&\|\partial_{t}v_{N}\|_{L^{2}_{\varpi}(I)}^{2}+\tilde{c}^{2}\big{(}\|\partial_{x}v_{N}\|^{2}_{L^{2}_{\varpi}(I)}+\beta_{n}\big{\|}v_{N}/(x+c_{0})\big{\|}^{2}_{L^{2}_{\varpi}(I)}\big{)}\\\
&\quad\leq
C\Big{(}\|v_{1,N}\|_{L^{2}_{\varpi}(I)}^{2}+\tilde{c}^{2}\big{(}\|\partial_{x}v_{0,N}\|^{2}_{L^{2}_{\varpi}(I)}+\beta_{n}\big{\|}v_{0,N}/(x+c_{0})\big{\|}^{2}_{L^{2}_{\varpi}(I)}\big{)}+\|I_{N}h\|^{2}_{L^{2}_{\varpi}(I)}\Big{)},\end{split}$
(4.24)
where $C$ is a positive constant independent of $N,\tilde{c}$ and $b.$
This estimate holds for (4.22) with $v,v_{0},v_{1},h$ in place of
$v_{N},v_{0,N},v_{1,N},I_{N}h,$ respectively.
###### Proof.
Taking $w=\partial_{t}v_{N}$ in (4.23), and integrating the resulted equation
with respect to $t,$ we use Corollary 3.1 and the argument similar to that for
Theorem 3.2 to derive the estimates. ∎
###### Remark 4.1.
Equipped with Theorem 4.1, we can analyze the convergence of the semi-
discretized scheme (4.23) as on Page 341 of [32]. ∎
We next examine the linear system of (4.23). As shown in [31, 32], it is
advantageous to construct basis functions satisfying the underlying
homogeneous Dirichlet boundary conditions. Let $L_{l}(x)$ be the Legendre
polynomial of degree $l$ (see, e.g., [36]), and define
$\phi_{k}(x)=L_{k}(x)+L_{k+1}(x).$ Then $\phi_{k}(-1)=0$ and $V_{N}={\rm
span}\big{\\{}\phi_{k}:0\leq k\leq N-1\big{\\}}.$ Note that $\phi_{k}(1)=2.$
Setting
$\begin{split}&v_{N}(x,t)=\sum_{j=0}^{N-1}\hat{v}_{j}(t)\phi_{j}(x),\;\;{\boldsymbol{v}}(t)=(\hat{v}_{0},\cdots,\hat{v}_{N-1})^{t},\;\;m_{ij}=(\phi_{j},\phi_{i})_{\varpi},\;\;s_{ij}=(\phi_{j}^{\prime},\phi_{i}^{\prime})_{\varpi},\\\
\end{split}$
we obtain the system:
$\begin{split}&M\ddot{\boldsymbol{v}}+\alpha
E\dot{\boldsymbol{v}}+\tilde{c}^{2}\big{(}S+\beta_{n}\widetilde{M}\big{)}{\boldsymbol{v}}+\mu
E{\boldsymbol{v}}-c\alpha\,\Big{(}\sigma_{\nu}\ast\sum_{j=0}^{N-1}\hat{v}_{j}\Big{)}{\boldsymbol{1}}={\boldsymbol{h}},\end{split}$
(4.25)
with $\boldsymbol{v}(0)=\boldsymbol{v}_{0}$ and
$\dot{\boldsymbol{v}}(0)=\boldsymbol{v}_{1},$ where
$M=(m_{ij}),S=(s_{ij}),\widetilde{M}=(\tilde{m}_{ij})$ and
$E={\boldsymbol{1}}{\boldsymbol{1}}^{t}$ is an $N\times N$ matrix of all ones.
In (4.25), $\boldsymbol{v}_{0}$ and $\boldsymbol{v}_{1}$ are column-$N$
vectors of the expansion coefficients of $v_{0,N}$ and $v_{1,N}$ in terms of
$\\{\phi_{k}\\},$ and the constants $\alpha={8c}(1+c_{0})^{d-1}/(b-b_{0})$ and
$\mu={4c^{2}(d-1)}(1+c_{0})^{d-1}/(b(b-b_{0})).$
### 4.4. Newmark’s time integration
To discretize the second-order ordinary differential system (4.25), we resort
to the implicit second-order Newmark’s scheme, which has wide applications in
the field of structural mechanics (see [27, 41]). To this end, let $\Delta t$
be the time-stepping size, and let
$\\{\boldsymbol{v}^{m},\dot{\boldsymbol{v}}^{m},\ddot{\boldsymbol{v}}^{m}\\}$
be the approximation of
$\\{\boldsymbol{v},\dot{\boldsymbol{v}},\ddot{\boldsymbol{v}}\\}$ at
$t=t_{m}=m\Delta t,$ and $\boldsymbol{h}^{m}=\boldsymbol{h}(t_{m}).$
We first take care of the convolution term
$\big{[}\sigma_{\nu}\ast\sum_{j=0}^{N-1}\hat{v}_{j}\big{]}(t_{m+1})$ in
(4.25):
$\begin{split}&\Big{[}\sigma_{\nu}\ast\sum_{j=0}^{N-1}\hat{v}_{j}\Big{]}(t_{m+1})=\sum_{j=0}^{N-1}\int_{t_{m}}^{t_{m+1}}\sigma_{n}(t_{m+1}-\tau)\hat{v}_{j}(\tau)d\tau+\sum_{j=0}^{N-1}\int_{0}^{t_{m}}\sigma_{n}(t_{m+1}-\tau)\hat{v}_{j}(\tau)d\tau\\\
&\qquad\approx\frac{\Delta
t}{2}\Big{(}\sigma_{\nu}(0)\sum_{j=0}^{N-1}\hat{v}_{j}^{m+1}+\sigma_{\nu}(\Delta
t)\sum_{j=0}^{N-1}\hat{v}_{j}^{m}\Big{)}+\sum_{j=0}^{N-1}\int_{0}^{t_{m}}\sigma_{n}(t_{m+1}-\tau)\hat{v}_{j}(\tau)d\tau\\\
&\qquad\approx\frac{\Delta
t}{2}\sigma_{\nu}(0)E{\boldsymbol{v}}^{m+1}+{\boldsymbol{g}}^{m},\end{split}$
where we used the Trapezoidal rule (of order $O(\Delta t^{3})$) to approximate
the integral over $(t_{m},t_{m+1}),$ and denoted by ${\boldsymbol{g}}^{m}$ the
approximation of the remaining terms. Note that ${\boldsymbol{g}}^{m}$ depends
on the history $\boldsymbol{v}^{l}(0\leq l\leq m),$ but fortunately, it can be
evaluated recursively and rapidly as described in Subsection 2.4. Moreover,
only the history data $\sum\hat{v}_{j}^{l}\,(0\leq l\leq m)$ need to be
stored, and no any matrix-vector multiplication is involved. As a result, the
burden of the history dependence can be eliminated.
Denoting
$A=M,\;\;B=\alpha
E,\;\;C=\tilde{c}^{2}\big{(}S+\beta_{n}\widetilde{M}\big{)}+\Big{(}\mu-c\alpha\frac{\Delta
t}{2}\sigma_{\nu}(0)\Big{)}E,$ (4.26)
and $\boldsymbol{f}^{m+1}=\boldsymbol{h}^{m+1}+c\alpha{\boldsymbol{g}}^{m},$
we carry out the full scheme for (4.22) as follows.
* (i)
Set $\boldsymbol{v}^{0}=\boldsymbol{v}_{0}$ and
$\dot{\boldsymbol{v}}^{0}=\boldsymbol{v}_{1},$ and compute
$\ddot{\boldsymbol{v}}^{0}$ from the system (4.25) by
$\ddot{\boldsymbol{v}}^{0}=M^{-1}\big{\\{}\boldsymbol{h}^{0}-\alpha
E\dot{\boldsymbol{v}}^{0}-\tilde{c}^{2}\big{(}S+\beta_{n}\widetilde{M}\big{)}{\boldsymbol{v}}^{0}-\mu
E{\boldsymbol{v}}^{0}\big{\\}},$ (4.27)
where the convolution term vanishes at $t=0.$
* (ii)
For $m\geq 0$, compute ${\boldsymbol{v}}^{m+1}$ from
$\begin{split}(A+\vartheta\Delta tB+\theta\Delta
t^{2}C\big{)}\ddot{\boldsymbol{v}}^{m+1}&={\boldsymbol{f}}^{m+1}-B\big{(}\dot{\boldsymbol{v}}^{m}+(1-\vartheta)\Delta
t\ddot{\boldsymbol{v}}^{m}\big{)}\\\ &\;\;-C\Big{(}\boldsymbol{v}^{m}+\Delta
t\dot{\boldsymbol{v}}^{m}+\frac{1-2\theta}{2}\Delta
t^{2}\ddot{\boldsymbol{v}}^{m}\Big{)},\end{split}$ (4.28)
and update ${\boldsymbol{u}}^{m+1}$ and $\dot{\boldsymbol{u}}^{m+1}$ by
$\displaystyle\boldsymbol{v}^{m+1}=\boldsymbol{v}^{m}+\Delta
t\dot{\boldsymbol{v}}^{m}+\frac{1-2\theta}{2}\Delta
t^{2}\ddot{\boldsymbol{v}}^{m}+\theta\Delta t^{2}\ddot{\boldsymbol{v}}^{m+1},$
(4.29)
$\displaystyle\dot{\boldsymbol{v}}^{m+1}=\dot{\boldsymbol{v}}^{m}+(1-\vartheta)\Delta
t\ddot{\boldsymbol{v}}^{m}+\vartheta\Delta t\ddot{\boldsymbol{v}}^{m+1},$
(4.30)
respectively.
###### Remark 4.2.
In general, the Newmark’s scheme is of second-order and unconditionally
stable, if the parameters satisfy $\vartheta\geq\frac{1}{2}$ and
$\theta\geq\frac{1}{4}\big{(}\frac{1}{2}+\vartheta\big{)}^{2}$ (see, e.g.,
[41]).
The computational cost of the full algorithm lies in solving (4.28) with
(4.26). The matrices $M,\widetilde{M}$ and $S$ are sparse with small bandwidth
under the basis $\\{\phi_{k}\\},$ and their entries can be evaluated exactly
by using the properties of Legendre polynomials. The matrices $B$ and $C$ in
(4.26) appear to be full, but since all entries of $E$ are one, the matrix-
vector multiplication can be carried out in $O(N)$ operations. ∎
## 5\. Numerical results
In this section, we present various numerical results to show the accuracy of
the NRBC and and convergence of the proposed algorithm.
### 5.1. Testing problem and setup
We examine the NRBC and the scheme via the following problem with an exact
solution.
###### Proposition 5.1.
The exterior problem (2.1)-(2.3) with
$d=2,D=\\{\boldsymbol{x}\in{\mathbb{R}^{2}}:|\boldsymbol{x}|<b_{0}\\},F=U_{0}=U_{1}\equiv
0,$ and $U|_{r=b_{0}}=G,$ admits the solution:
$U(r,\phi,t)=\sum_{|n|=0}^{\infty}\widehat{U}_{n}(r,t)e^{{\rm i}n\phi},\quad
r>b_{0},\;\phi\in[0,2\pi),\;t>0,$ (5.1)
where
$\widehat{U}_{n}(r,t)=\begin{cases}0,&\quad t<\beta_{0},\\\
H_{n}(r,t)\ast\widehat{G}_{n}(t-\beta_{0})+\sqrt{b_{0}/r}\widehat{G}_{n}(t-\beta_{0}),&\quad
t\geq\beta_{0},\end{cases}$ (5.2)
with $\beta_{0}=(r-b_{0})/c,$ $\widehat{G}_{n}(t)$ being the Fourier expansion
coefficient of $G(\phi,t),$ and
$\begin{split}H_{n}(r,t)&=\frac{c}{b_{0}}\sum_{j=1}^{M_{n}}\frac{K_{n+1}(rz_{j}^{n}/{b_{0}})}{K^{\prime}_{n}(z_{j}^{n})}e^{tcz_{j}^{n}/b_{0}}\\\
&+(-1)^{n}\frac{c}{b_{0}}\int_{0}^{\infty}\frac{I_{n}(r\rho/b_{0})K_{n}(\rho)-K_{n}(r\rho/b_{0})I_{n}(\rho)}{K^{2}_{n}(\rho)+\pi^{2}I^{2}_{n}(\rho)}e^{-ct\rho/b_{0}}d\rho.\end{split}$
(5.3)
###### Proof.
We sketch its derivation in Appendix C. ∎
###### Remark 5.1.
Notice that $\widehat{U}_{n}$ satisfies the one-dimension problem:
$\displaystyle\frac{\partial^{2}\widehat{U}_{n}}{\partial
t^{2}}-\frac{c^{2}}{r}\frac{\partial}{\partial
r}\Big{(}r\frac{\partial{\widehat{U}_{n}}}{\partial
r}\Big{)}+c^{2}n^{2}\frac{\widehat{U}_{n}}{r^{2}}=0,\quad\;\;b_{0}<r<b,\;t>0;$
(5.4)
$\displaystyle\widehat{U}_{n}|_{t=0}=\frac{\partial\widehat{U}_{n}}{\partial
t}\Big{|}_{t=0}=0,\quad
b_{0}<r<b;\quad\widehat{U}_{n}|_{r=b_{0}}=\widehat{G}_{n},\;\;t>0;$ (5.5)
$\displaystyle\Big{(}\frac{1}{c}\frac{\partial\widehat{U}_{n}}{\partial
t}+\frac{\partial\widehat{U}_{n}}{\partial
r}+\frac{1}{2r}\widehat{U}_{n}\Big{)}\Big{|}_{r=b}=\int_{0}^{t}\sigma_{n}(t-\tau)\widehat{U}_{n}(b,\tau)d\tau,\;\;t>0.\qed$
(5.6)
We take the Dirichlet data:
$G(\phi,t)=A_{1}e^{-\iota((b_{0}\cos\,\phi-x_{s})^{2}+(b_{0}\sin\,\phi-
y_{s})^{2})}\sin^{p}(\omega t),$ (5.7)
where $A_{1}=10,\iota=0.1,x_{s}=y_{s}=2.1,b_{0}=2,$ and $\omega,p$ are to be
specified later. We compute the Fourier coefficient
$\widehat{G}_{n}(t)=\frac{A_{1}}{2\pi}\sin^{p}(\omega
t)\int_{0}^{2\pi}e^{-\iota((b_{0}\cos\,\phi-x_{s})^{2}+(b_{0}\sin\,\phi-
y_{s})^{2})}e^{-{\rm i}n\phi}d\phi,$
via the fast Fourier transform with sufficient Fourier points so that the
error of numerical integration can be neglected.
The first important issue is to compute the exact solution and $\sigma_{n}$
very accurately. We point out that the convolution in (5.2) can be computed
exactly, and the improper integral in (5.3) can be treated by a similar manner
as the improper integral of $\sigma_{n}(t)$ in Subsection 2.3. We adopt a
Newton’s iteration method to compute the zeros of $K_{n}(z)$ accurately for
moderate large $n,$ and use a (composite) Legendre-Gauss quadrature to
evaluate the integral with high precision.
For the readers’ reference, we tabulate in Table 5.1 some samples of
$\sigma_{n}(t)$ with $b=3,c=5.$
Table 5.1. Some samples of $\sigma_{n}(t).$ $n$ | $\sigma_{n}(0.1)$ | $\sigma_{n}(2)$ | $n$ | $\sigma_{n}(0.1)$ | $\sigma_{n}(2)$
---|---|---|---|---|---
0 | 5.922023678764810e-2 | 1.127355852971518e-2 | 5 | -5.394296605508512 | 2.652114252130534e-3
1 | -1.770775252065292e-1 | -1.849758830245009e-2 | 6 | -7.505798337812085 | -1.628640292879232e-3
2 | -8.766785435408012e-1 | -3.516167498630298e-3 | 7 | -9.786660490985828 | 1.032611050422504e-3
3 | -2.012014067208546 | 6.186828109004313e-3 | 8 | -12.14241492386835 | -6.793719076424341e-4
4 | -3.538153424430011 | -4.319820513209325e-3 | 9 | -14.47330651171318 | 4.628600071971645e-4
We plot in Figure 5.1 the exact solution with $p=2,b_{0}=2,b=4,\omega=10\pi$
at various $t.$ We see that the Dirichlet data $G(\phi,t)$ acts as a dynamical
wave-maker.
Figure 5.1. Time evolution of exact solution with $\omega=10\pi$.
### 5.2. Accuracy of the NRBC
Now, we examine accuracy of the NRBC. Notice that
$\max_{|\phi|\leq
2\pi}\big{|}\partial_{r}U_{M}-T_{d}(U_{M})\big{|}\leq\sum_{|n|=0}^{M}e_{n}(t),\;e_{n}(t):=\Big{|}\Big{(}\frac{1}{c}\frac{\partial\widehat{U}_{n}}{\partial
t}+\frac{\partial\widehat{U}_{n}}{\partial
r}+\frac{\widehat{U}_{n}}{2r}\Big{)}\Big{|}_{r=b}-\sigma_{n}(t)\ast\widehat{U}_{n}(b,t)\Big{|},$
where $T_{d}(U_{M})$ is defined in (2.7) with $d=2,$ and
$U_{M}(b,\phi,t)=\sum_{|n|=0}^{M}\widehat{U}_{n}(b,t)e^{{\rm i}n\phi}$ is the
truncation of the exact solution in Proposition 5.1. We choose the same
parameters as those for the exact solution in Figure 5.1, and take $M=32$ so
that $\\{|\widehat{U}_{n}(b,t)|\\}$ are sufficiently small for all modes
$|n|\leq M$ and all $t$ of interest. Note that the differentiations in
$e_{n}(t)$ can be performed exactly on the exact solution.
In Table 5.2, we tabulate the errors:
$E_{M}^{(1)}(t)=\max_{|n|\leq M}e_{n}(t),\quad
E_{M}^{(2)}(t)=\sum_{|n|=0}^{M}e_{n}(t),$
at some samples of $t,$ compute the errors at the outer artificial boundary
$r=b,$ where the exact solution has the magnitude as large as possible.
Table 5.2. The errors $E_{M}^{(1)}(t)$ and $E_{M}^{(2)}(t).$ $t$ | $\omega=10\pi,b=2.22$ | $\omega=10\pi,b=2.75$ | $\omega=20\pi,b=2.38$ | $\omega=20\pi,b=2.87$
---|---|---|---|---
| $E_{M}^{(1)}(t)$ | $E_{M}^{(2)}(t)$ | $E_{M}^{(1)}(t)$ | $E_{M}^{(2)}(t)$ | $E_{M}^{(1)}(t)$ | $E_{M}^{(2)}(t)$ | $E_{M}^{(1)}(t)$ | $E_{M}^{(2)}(t)$
0.5 | 2.637e-16 | 4.902e-16 | 7.026e-17 | 1.148e-16 | 6.245e-17 | 1.603e-16 | 8.327e-17 | 1.752e-16
1 | 4.684e-16 | 8.406e-16 | 2.099e-16 | 2.630e-16 | 9.021e-17 | 2.390e-16 | 2.272e-16 | 4.405e-16
5 | 2.567e-16 | 4.386e-16 | 1.131e-15 | 1.176e-15 | 7.251e-16 | 9.053e-16 | 1.318e-15 | 1.473e-15
10 | 7.772e-16 | 1.020e-15 | 2.207e-15 | 2.227e-15 | 1.457e-15 | 1.960e-15 | 2.387e-15 | 2.575e-15
We see from Table 5.2 that in all tests, the errors are extremely small. This
validates the formula for $\sigma_{n}$ in Theorem 2.1 and high accuracy of the
numerical treatment.
### 5.3. Numerical tests for the spectral-Galerkin-Newmark’s scheme
Hereafter, we provide some numerical results to illustrate the convergence of
the numerical scheme described in Section 4. In the following computations, we
take the Dirichlet data given by (5.7) with
$A_{1}=10,\iota=0.1,x_{s}=y_{s}=2.1$ and $b_{0}=2$ as before.
Given a cut-off number $M>0,$ we compute the numerical solutions
$\\{\widehat{U}_{n}^{N}\\}$ of (5.4)-(5.6) for the modes $|n|\leq M$ by using
the spectral-Galerkin and Newmark’s time integration scheme. The full
numerical solution of the problem in Proposition 5.1 is then defined as
$U_{M}^{N}(r,\phi,t)=\sum_{|n|=0}^{M}\widehat{U}_{n}^{N}(r,t)e^{{\rm
i}n\phi},$ and the exact solution
$U_{M}(r,\phi,t)=\sum_{|n|=0}^{M}\widehat{U}_{n}(r,t)e^{{\rm i}n\phi}$ is
evaluated as before. Once again, we choose $M$ such that the Fourier
coefficient of the exact solution $|\widehat{U}_{n}|\leq 10^{-16}$ for
$|n|>M.$ The numerical errors are measured by
$\widehat{E}_{M}^{N}(t)=\max_{|n|\leq
M}\big{\|}\widehat{U}_{n}^{N}(\cdot,t)-\widehat{U}_{n}(\cdot,t)\big{\|}_{L^{2},N},\quad\widetilde{E}_{M}^{N}(t)=\max_{|n|\leq
M}\big{\|}\widehat{U}_{n}^{N}(\cdot,t)-\widehat{U}_{n}(\cdot,t)\big{\|}_{L^{\infty},N},$
where $\|\cdot\|_{L^{2},N}$ is the discrete $L^{2}$-norm associated with the
Legendre-Gauss-Lobatto interpolation, and $\|\cdot\|_{L^{\infty},N}$ is the
corresponding discrete maximum norm.
To test the second-order convergence of the Newmark’s scheme, we choose $N=50$
so that the error of the spatial discretization is negligible. We provide in
Table 5.3 the numerical errors and the order of convergence for various $t$
with $M=15,b=5,\omega=\pi,$ and $p=6.$ As expected, we observe a second-order
convergence of the time integration.
Table 5.3. The errors $\widehat{E}_{M}^{N}(t)$ and $\widetilde{E}_{M}^{N}(t),$ and order of convergence. $t$ | $\Delta t$ | $\widehat{E}_{M}^{N}(t)$ | order | $\widetilde{E}_{M}^{N}(t)$ | order | $t$ | $\Delta t$ | $\widehat{E}_{M}^{N}(t)$ | order | $\widetilde{E}_{M}^{N}(t)$ | order
---|---|---|---|---|---|---|---|---|---|---|---
| 1e-03 | 1.21e-05 | | 1.93e-05 | | | 1e-03 | 1.23e-05 | | 2.01e-05 |
1 | 5e-04 | 3.02e-06 | 2.00005 | 4.83e-06 | 1.99994 | 3 | 5e-04 | 3.07e-06 | 2.00007 | 5.02e-06 | 1.99997
| 1e-04 | 1.21e-07 | 1.99993 | 1.93e-07 | 1.99986 | | 1e-04 | 1.23e-07 | 1.99997 | 2.01e-07 | 1.99998
| 5e-05 | 3.02e-08 | 2.00012 | 4.83e-08 | 1.99906 | | 5e-05 | 3.07e-08 | 2.00007 | 5.02e-08 | 2.00004
| 1e-03 | 1.23e-05 | | 2.01e-05 | | | 1e-03 | 1.23e-05 | | 2.01e-05 |
2 | 5e-04 | 3.07e-06 | 1.99998 | 5.02e-06 | 1.99997 | 4 | 5e-04 | 3.07e-06 | 2.00007 | 5.02e-06 | 1.99997
| 1e-04 | 1.23e-07 | 2.00001 | 2.01e-07 | 1.99998 | | 1e-04 | 1.23e-07 | 1.99997 | 2.01e-07 | 1.99998
| 5e-05 | 3.07e-08 | 2.00002 | 5.02e-08 | 1.99993 | | 5e-05 | 3.07e-08 | 2.00007 | 5.02e-08 | 2.00004
Next, we fix the time step size $\Delta t=10^{-5}$ and choose different $N$ to
check the accuracy in spatial discretization. The convergence behavior is
illustrated in Table 5.4. With a small number of modes in space, we observe a
fast decay of the errors, which is typical for the spectral approximation.
Table 5.4. The errors $\widehat{E}_{M}^{N}(t)$ and $\widetilde{E}_{M}^{N}(t).$ $t$ | $N=8$ | $N=10$ | $N=16$ | $N=32$
---|---|---|---|---
| $\widehat{E}_{M}^{N}(t)$ | $\widetilde{E}_{M}^{N}(t)$ | $\widehat{E}_{M}^{N}(t)$ | $\widetilde{E}_{M}^{N}(t)$ | $\widehat{E}_{M}^{N}(t)$ | $\widetilde{E}_{M}^{N}(t)$ | $\widehat{E}_{M}^{N}(t)$ | $\widetilde{E}_{M}^{N}(t)$
0.5 | 2.349e-04 | 2.687e-04 | 2.253e-05 | 2.846e-05 | 8.224e-07 | 1.207e-06 | 9.229e-09 | 1.639e-08
1.0 | 3.877e-04 | 4.179e-04 | 1.708e-05 | 2.042e-05 | 1.833e-07 | 2.777e-07 | 1.959e-09 | 2.673e-09
1.5 | 3.276e-04 | 3.596e-04 | 5.965e-06 | 7.146e-06 | 3.072e-08 | 5.238e-08 | 1.508e-09 | 1.762e-09
2.0 | 4.105e-04 | 4.368e-04 | 1.092e-05 | 1.188e-05 | 7.353e-09 | 1.238e-08 | 1.237e-09 | 2.119e-09
2.5 | 3.239e-04 | 3.607e-04 | 5.724e-06 | 6.997e-06 | 2.381e-09 | 3.182e-09 | 1.488e-09 | 1.734e-09
3.0 | 4.113e-04 | 4.380e-04 | 1.095e-05 | 1.189e-05 | 1.404e-09 | 2.893e-09 | 1.227e-09 | 2.009e-09
3.5 | 3.238e-04 | 3.608e-04 | 5.747e-06 | 7.036e-06 | 1.554e-09 | 2.063e-09 | 1.488e-09 | 1.732e-09
4.0 | 4.113e-04 | 4.380e-04 | 1.095e-05 | 1.189e-05 | 1.269e-09 | 2.061e-09 | 1.227e-09 | 2.007e-09
Figure 5.2. Propagation of exact and numerical solutions.
Finally, we plot in Figure 5.2 the numerical solution (red nets) against the
exact solution (blue smooth surface), where the exact solution in (a) and (b)
is in the annulus: $2\leq r\leq 10,$ while the numerical solution within
$2\leq r\leq 5.$ Figure 5.2(c) shows the wave propagation through the
artificial boundary at $b=5,0\leq\phi\leq 2\pi$ for $0\leq t\leq 4$ (numerical
solution) against $0\leq t\leq 4.5$ (exact solution). We see that the red nets
and blue surfaces agree well in these plots. Moreover, we observe the waves
pass the boundary transparently. This shows that the proposed scheme is very
stable and has high-order accuracy.
Concluding remarks
We proposed in this paper analytic and accurate numerical means for the time-
domain wave propagation with exact and global nonreflecting boundary
conditions. We derived the analytic expressions of the convolution kernels and
presented highly accurate methods for their evaluations. We analyzed the
stability of the truncated problem and provided efficient numerical schemes.
Ample numerical results were given to demonstrate the features of the method.
We shall report the combination of the methods with the boundary perturbation
technique in a future work. The techniques and ideas in this paper will be
useful to study the Maxwell equations and elastic wave propagations.
## Appendix A Proof of Theorem 2.1
We first consider $d=2.$ Note that $F_{n}(z)$ is a multi-valued function with
branch points at $z=0$ and infinity. The contour $L$ for (2.15) is depicted in
Figure A.1 (left) with the branch-cut along the negative real axis.
Figure A.1. Contour $L$ for the inverse Laplace transform. Left: $d=2$. Right:
$d=3.$
We know from Lemma 2.1 that $F_{n}(z)$ has a finite number of simple poles in
the second and third quadrants, Therefore, for any $n\geq 0,$ it follows from
the residue theorem that
$\begin{split}&\lim\limits_{\scriptstyle R\to+\infty\hfill\atop\scriptstyle
r\to 0^{+}\hfill}\oint_{L}F_{n}(z)e^{czt/b}dz=2\pi{\rm
i}\sum\limits_{j=1}^{M_{n}}{\rm
Res}\,\big{[}F_{n}(z)e^{czt/b},z_{j}^{n}\big{]}\\\ &\quad=2\pi{\rm
i}\sum\limits_{j=1}^{M_{n}}\lim\limits_{z\rightarrow
z_{j}^{n}}\bigg{\\{}(z-z^{n}_{j})e^{czt/b}\left[z+\frac{1}{2}+z\frac{K^{\prime}_{n}(z)}{K_{n}(z)}\right]\bigg{\\}}=2\pi{\rm
i}\sum\limits_{j=1}^{M_{n}}e^{ctz_{j}^{n}/b}z_{j}^{n}.\end{split}$ (A.1)
Thus, by (2.15) and (A.1),
$\begin{split}\frac{2b^{2}\pi{\rm i}}{c}\sigma_{n}(t)&=2\pi{\rm
i}\sum\limits_{j=1}^{M_{n}}e^{ctz_{j}^{n}/b}z_{j}^{n}-\lim\limits_{\scriptstyle
R\to+\infty\hfill\atop\scriptstyle r\to
0^{+}\hfill}\left[\int_{\wideparen{BC}}F_{n}(z)e^{czt/b}dz+\int_{\overline{CD}}F_{n}(z)e^{czt/b}dz\right.\\\
&\left.+\int_{\wideparen{DEF}}F_{n}(z)e^{czt/b}dz+\int_{\overline{FG}}F_{n}(z)e^{czt/b}dz+\int_{\wideparen{GA}}F_{n}(z)e^{czt/b}dz\right].\end{split}$
(A.2)
In view of (2.14), we find from the Jordan’s lemma (cf. [11, 8]) and a direct
calculation that
$\lim\limits_{\scriptstyle R\to+\infty\hfill\atop\scriptstyle r\to
0^{+}\hfill}\left[\int_{\wideparen{BC}}F_{n}(z)e^{czt/b}dz+\int_{\wideparen{DEF}}F_{n}(z)e^{czt/b}dz+\int_{\wideparen{GA}}F_{n}(z)e^{czt/b}dz\right]=0,$
(A.3)
since each contour integral tends to zero. Thus, we only need to evaluate the
integrals along the line segments $\overline{CD}$ and $\overline{FG}$. We have
$\begin{split}\lim\limits_{\scriptstyle R\to+\infty\hfill\atop\scriptstyle
r\to
0^{+}\hfill}&\left[\int_{\overline{CD}}F_{n}(z)e^{czt/b}dz+\int_{\overline{FG}}F_{n}(z)e^{czt/b}dz\right]=\int_{0}^{\infty}re^{-ctr/b}\left[\frac{K^{\prime}_{n}(re^{-{\rm
i}\pi})}{K_{n}(re^{-{\rm i}\pi})}-\frac{K^{\prime}_{n}(re^{{\rm
i}\pi})}{K_{n}(re^{{\rm i}\pi})}\right]dr.\end{split}$
By Formula 9.6.31 of [1],
$K_{n}(re^{\pi{\rm i}})=e^{-n\pi{\rm i}}K_{n}(r)-\pi{\rm i}I_{n}(r),\quad
K_{n}(re^{-\pi{\rm i}})=e^{n\pi{\rm i}}K_{n}(r)+\pi{\rm i}I_{n}(r),$
which, together with (2.13), implies
$\begin{split}&\frac{K^{\prime}_{n}(re^{{\rm i}\pi})}{K_{n}(re^{{\rm
i}\pi})}-\frac{K^{\prime}_{n}(re^{-{\rm i}\pi})}{K_{n}(re^{-{\rm
i}\pi})}=\frac{K_{n+1}(re^{-{\rm i}\pi})}{K_{n}(re^{-{\rm
i}\pi})}-\frac{K_{n+1}(re^{{\rm i}\pi})}{K_{n}(re^{{\rm i}\pi})}\\\
&\quad=\frac{e^{(n+1)\pi{\rm i}}K_{n+1}(r)+\pi{\rm i}I_{n+1}(r)}{e^{n\pi{\rm
i}}K_{n}(r)+\pi{\rm i}I_{n}(r)}-\frac{e^{-(n+1)\pi{\rm i}}K_{n+1}(r)-\pi{\rm
i}I_{n+1}(r)}{e^{-n\pi{\rm i}}K_{n}(r)-\pi{\rm i}I_{n}(r)}\\\
&\quad=\frac{(-1)^{n}2\pi{\rm
i}[K_{n+1}(r)I_{n}(r)+K_{n}(r)I_{n+1}(r)]}{K^{2}_{n}(r)+\pi^{2}I^{2}_{n}(r)}=\frac{(-1)^{n}2\pi{\rm
i}}{r\big{[}K^{2}_{n}(r)+\pi^{2}I^{2}_{n}(r)\big{]}},\end{split}$ (A.4)
where we used the Wronskian identity (see Formula 9.6.15 of [1]):
$\displaystyle I_{n}(z)K_{n+1}(z)+I_{n+1}(z)K_{n}(z)=z^{-1}.$ (A.5)
A combination of (A.1)-(A.4) leads to
$\sigma_{n}(t)=\frac{c}{b^{2}}\Big{\\{}\sum\limits_{j=1}^{M_{n}}z_{j}^{n}e^{ctz_{j}^{n}/b}+\int_{0}^{\infty}\frac{(-1)^{n}e^{-ctr/b}}{K^{2}_{n}(r)+\pi^{2}I^{2}_{n}(r)}dr\Big{\\}},$
(A.6)
which is the expression (2.19).
Now, we turn to the three dimensional case. It is important to notice that the
kernel function $F_{n+1/2}(z)$ is not a multi-valued complex function, as
opposed to the two dimensional case. Indeed, although $K_{n+{1}/{2}}(z)$ is
multi-valued, in view of the formula (see Page 80 of [40]):
$\displaystyle
K_{n+{1}/{2}}(z)=\sqrt{\frac{\pi}{2z}}\sum\limits_{k=0}^{n}\frac{(n+k)!e^{-z}}{k!(n-k)!(2z)^{k}},\quad\forall
n\geq 0,$ (A.7)
the fact $1/\sqrt{z}$ can be eliminated from the ratio
$K_{n+1/2}^{\prime}/K_{n+1/2}.$ Thanks to this property, we use the contour in
Figure A.1 (right), by the residue theorem and Jordan’s lemma, we have
$\begin{split}\lim_{R\to+\infty}&\oint_{L}F_{n+1/2}(z)e^{czt/b}dz=\int_{\gamma-\infty{\rm
i}}^{\gamma+\infty{\rm i}}F_{n+1/2}(z)e^{czt/b}dz\\\ &=2\pi{\rm
i}\sum\limits_{j=1}^{M_{\nu}}{\rm
Res}\,\big{[}F_{n+1/2}(z)e^{czt/b},z_{j}^{\nu}\big{]}=2\pi{\rm
i}\sum\limits_{j=1}^{M_{\nu}}e^{ctz_{j}^{\nu}/b}z_{j}^{\nu},\end{split}$ (A.8)
for $\nu=n+1/2.$ This leads to the formula (2.20). ∎
## Appendix B Justification for Remark 2.3
Thanks to (2.17), we have
$z\frac{H_{\nu}^{(1)^{\prime}}(z)}{H_{\nu}^{(1)}(z)}=-{\rm
i}z\frac{K_{\nu}^{{}^{\prime}}(-{\rm i}z)}{K_{\nu}(-{\rm i}z)}.$
Thus, by (2.21) with $\nu=n$ and $-{\rm i}z=bs/c,$ we find
$\begin{split}\frac{s}{c}&+\frac{1}{2b}+\frac{s}{c}\frac{K_{n}^{\prime}(bs/c)}{K_{n}(bs/c)}=-\frac{{\rm
i}c}{b^{2}}\sum\limits_{j=1}^{N_{n}}\frac{h_{n,j}}{s+\frac{{\rm
i}ch_{n,j}}{b}}+\frac{c}{b^{2}}\int_{0}^{\infty}\frac{(-1)^{n}}{K_{n}^{2}(r)+\pi^{2}I_{n}^{2}(r)}\frac{1}{\frac{cr}{b}+s}dr.\end{split}$
Applying the inverse Laplace transform to both sides of the above identity
leads to
$\begin{split}\mathcal{L}^{-1}\left[\frac{s}{c}+\frac{1}{2b}+\frac{s}{c}\frac{K_{n}^{\prime}(bs/c)}{K_{n}(bs/c)}\right]=\frac{c}{b^{2}}\sum\limits_{j=1}^{N_{n}}(-{\rm
i}h_{n,j})e^{-{{\rm
i}cth_{n,j}}/{b}}+\frac{c}{b^{2}}\int_{0}^{\infty}\frac{(-1)^{n}e^{-{crt}/{b}}}{K_{n}^{2}(r)+\pi^{2}I_{n}^{2}(r)}dr,\end{split}$
which turns out to be (A.6), since $-{\rm i}h_{n,j}$ are zeros of $K_{n}(z)$
and $N_{n}=M_{n}.$ ∎
## Appendix C Proof of Proposition 5.1
This problem can be solved by Laplace transform and separation of variables.
Indeed, the Fourier coefficients $\\{\widehat{U}_{n}\\}$ in (5.1) can be
expressed as
$\widehat{U}_{n}(r,t)={\mathcal{L}}^{-1}\Big{(}\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}\Big{)}(t)\ast\widehat{G}_{n}(t),\quad
r>b_{0}.$ (C.1)
We next sketch the evaluation of the inverse Laplace transform by using the
residue theorem as in Appendix A for $\sigma_{n}(t).$ Using (2.12), we find
$\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}\sim\sqrt{\frac{b_{0}}{r}}e^{-s\beta_{0}},$
where $\beta_{0}=(r-b_{0})/c.$ In order to use the Jordan’s lemma (cf. [11,
8]), we write
$\begin{split}&{\mathcal{L}}^{-1}\Big{(}\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}\Big{)}(t)={\mathcal{L}}^{-1}\bigg{(}e^{-\beta_{0}s}\Big{\\{}e^{\beta_{0}s}\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}-\sqrt{\frac{b_{0}}{r}}\Big{\\}}\bigg{)}+{\mathcal{L}}^{-1}\Big{(}e^{-\beta_{0}s}\sqrt{\frac{b_{0}}{r}}\Big{)}(t)\\\
&\qquad=\widetilde{H}_{n}(r,t-\beta_{0})U_{\beta_{0}}(t)+\sqrt{\frac{b_{0}}{r}}\delta(t-\beta_{0}),\end{split}$
(C.2)
where
$\widetilde{H}_{n}(r,t)={\mathcal{L}}^{-1}\big{(}e^{\beta_{0}s}\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}-\sqrt{{b_{0}}/{r}}\big{)}(t),$
$U_{\beta_{0}}(t)$ is the unit step function (which takes $1$ for
$t\geq\beta_{0},$ and $0$ for $t<\beta_{0}$), and $\delta(t)$ is the Dirac
delta function. By applying the residue theorem and Jordan’s lemma to the
Bromwich’s contour integral:
$\begin{split}\widetilde{H}_{n}(r,t)=\frac{1}{2\pi{\rm
i}}\int_{\gamma-\infty{\rm i}}^{\gamma+\infty{\rm
i}}\Big{(}e^{\beta_{0}s}\frac{K_{n}(sr/c)}{K_{n}(sb_{0}/c)}-\sqrt{\frac{b_{0}}{r}}\Big{)}e^{ts}ds,\end{split}$
(C.3)
we obtain (5.3) by using the contour in Figure A.1 (left) and the same
argument as in Appendix A. Denoting
$H_{n}(r,t):=\widetilde{H}_{n}(r,t-\beta_{0}),$ the expression (5.2) follows
from (C.2). We leave the details to the interested readers. ∎
## References
* [1] M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York, 1984\. Reprint of the 1972 edition, Selected Government Publications.
* [2] B. Alpert, L. Greengard, and T. Hagstrom. Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal., 37(4):1138–1164 (electronic), 2000.
* [3] B. Alpert, L. Greengard, and T. Hagstrom. Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys., 180(1):270–296, 2002.
* [4] A. Bayliss and E. Turkel. Radiation boundary conditions for wave-like equations. Comm. Pure Appl. Math., 33(6):707–725, 1980.
* [5] J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114(2):185–200, 1994.
* [6] Z.M. Chen. Convergence of the time-domain perfectly matched layer method for acoustic scattering problems. Int. J. Numer. Anal. Model., 6(1):124–146, 2009.
* [7] R.D. Ciskowski and C.A. Brebbia. Boundary Element Methods in Acoutics. Kluwer Academic Publishers, 1991.
* [8] A.M. Cohen. Numerical Methods for Laplace Transform Inversion, volume 5 of Numerical Methods and Algorithms. Springer, New York, 2007.
* [9] G.C. Cohen. Higher-Order Numerical Methods for Transient Wave Equations. Scientific Computation. Springer-Verlag, Berlin, 2002. With a foreword by R. Glowinski.
* [10] M.D. Collins and W.A. Kuperman. Inverse problems in ocean acoustics. Inverse Problems, 10(5):1023–1040, 1994.
* [11] B. Davies. Integral Transforms and Their Applications, volume 41 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition, 2002.
* [12] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139):629–651, 1977.
* [13] Q.R. Fang, D.P. Nicholls, and J. Shen. A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering. J. Comput. Phys., 224(2):1145–1169, 2007.
* [14] K. Gerdes and L. Demkowicz. Solution of $3$D-Laplace and Helmholtz equations in exterior domains using $hp$-infinite elements. Comput. Methods Appl. Mech. Engrg., 137(3-4):239–273, 1996.
* [15] D. Givoli. Numerical Methods for Problems in Infinite Domains, volume 33 of Studies in Applied Mechanics. Elsevier Scientific Publishing Co., Amsterdam, 1992.
* [16] D. Givoli. High-order local non-reflecting boundary conditions: a review. Wave Motion, 39(4):319–326, 2004. New computational methods for wave propagation.
* [17] M.J. Grote and J.B. Keller. Exact nonreflecting boundary conditions for the time-dependent wave equation. SIAM J. Appl. Math, 55(2):280–297, 1995.
* [18] T. Hagstrom. Radiation boundary conditions for the numerical simulation of waves. In Acta numerica, 1999, volume 8 of Acta Numer., pages 47–106. Cambridge Univ. Press, Cambridge, 1999.
* [19] R.N. Huan and L.L. Thompson. Accurate radiation boundary conditions for the time-dependent wave equation on unbounded domains. Internat. J. Numer. Methods Engrg., 47(9):1569–1603, 2000.
* [20] S.D. Jiang and L. Greengard. Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comput. Math. Appl., 47(6-7):955–966, 2004.
* [21] S.D. Jiang and L. Greengard. Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions. Comm. Pure Appl. Math., 61(2):261–288, 2008.
* [22] J.B. Keller and D. Givoli. Exact nonreflecting boundary conditions. J. Comput. Phys., 82(1):172–192, 1989.
* [23] J.R. Li. Low order approximation of the spherical nonreflecting boundary kernel for the wave equation. Linear Algebra Appl., 415(2-3):455–468, 2006.
* [24] C. Lubich and A. Schädle. Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput., 24(1):161–182 (electronic), 2002.
* [25] P.A. Martin. Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles, volume 107. Cambridge University Press, 2006.
* [26] J.C. Nédélec. Acoustic and Electromagnetic Equations, volume 144 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Integral representations for harmonic problems.
* [27] N.M. Newmark. A method of computation for structural dynamics. In Proc. ASCE, volume 85, pages 67–94, 1959.
* [28] D. Nicholls and F. Reitich. Analytic continuation of Dirichlet-Neumann operators. Numer. Math., 94(1):107–146, 2003.
* [29] D. Nicholls and J. Shen. A stable, high–order method for two–dimensional bounded–obstacle scattering. SIAM J. Sci. Comput., 28:1398–1419, 2006.
* [30] D. Nicholls and J. Shen. A rigorous numerical analysis of the transformed field expansion method. SIAM J. Numer. Anal., 47(4):2708–2734, 2009.
* [31] J. Shen. Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput., 15:1489–1505, 1994.
* [32] J. Shen, T. Tang, and L.L. Wang. Spectral Methods: Algorithms, Analysis and Applications, volume 41 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, Heidelberg, 2011.
* [33] K.L. Shlager and J.B. Schneider. A selective survey of the finite-difference time-domain literature. Antennas and Propagation Magazine, IEEE, 37(4):39–57, 1995.
* [34] I.L. Sofronov. Conditions for complete transparency on a sphere for a three-dimensional wave equation. Dokl. Akad. Nauk, 326(6):953–957, 1992.
* [35] I.L. Sofronov. Artificial boundary conditions of absolute transparency for two- and three-dimensional external time-dependent scattering problems. European J. Appl. Math., 9(6):561–588, 1998.
* [36] G. Szegö. Orthogonal Polynomials (Fourth Edition), volume 23. AMS Coll. Publ., 1975.
* [37] A. Taflove and S.C. Hagness. Computational Electrodynamics: the Finite-Difference Time-Domain Method. Artech House Inc., Boston, MA, third edition, 2005. With 1 CD-ROM (Windows).
* [38] L.L. Thompson and R.N. Huan. Computation of far-field solutions based on exact nonreflecting boundary conditions for the time-dependent wave equation. Comput. Methods Appl. Mech. Engrg., 190(11-12):1551–1577, 2000\.
* [39] L. Ting and M.J. Miksis. Exact boundary conditions for scattering problems. The Journal of the Acoustical Society of America, 80:1825, 1986\.
* [40] G.N. Watson. A Treatise of the Theory of Bessel Functions (second edition). Cambridge University Press, Cambridge, UK, 1966.
* [41] W.L. Wood. Practical Time-Stepping Schemes. Oxford Applied Mathematics and Computing Science Series. The Clarendon Press Oxford University Press, New York, 1990.
|
arxiv-papers
| 2011-09-26T09:01:30 |
2024-09-04T02:49:22.506328
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Lian Wang, Bo Wang and Xiaodan Zhao",
"submitter": "Li-Lian Wang Dr.",
"url": "https://arxiv.org/abs/1109.5492"
}
|
1109.5509
|
# On Exponential Convergence of Gegenbauer Interpolation and Spectral
Differentiation
Ziqing Xie1, Li-Lian Wang2 and Xiaodan Zhao2
###### Abstract.
This paper is devoted to a rigorous analysis of exponential convergence of
polynomial interpolation and spectral differentiation based on the Gegenbauer-
Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is
analytic on and within an ellipse. Sharp error estimates in the maximum norm
are derived.
###### Key words and phrases:
Bernstein ellipse, exponential accuracy, Gegenbauer polynomials, Gegenbauer
Gauss-type interpolation and quadrature, spectral differentiation, maximum
error estimates
###### 1991 Mathematics Subject Classification:
65N35, 65E05, 65M70, 41A05, 41A10, 41A25
1 School of Mathematics and Computer Science, Guizhou Normal University,
Guiyang, Guizhou 550001, China; College of Mathematics and Computer Science,
Hunan Normal University, Changsha, Hunan 410081, China. The research of the
author is partially supported by the NSFC (10871066) and the Science and
Technology Grant of Guizhou Province (LKS[2010]05).
2 Division of Mathematical Sciences, School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371, Singapore. The research of
the authors is partially supported by Singapore AcRF Tier 1 Grant RG58/08,
Singapore MOE Grant T207B2202 and Singapore NRF2007IDM-IDM002-010.
## 1\. Introduction
Perhaps the most significant advantage of the spectral method is its high-
order of accuracy. The typical convergence rate of the spectral method is
$O(n^{-m})$ for every $m,$ provided that the underlying function is
sufficiently smooth [22, 4, 6, 28, 8]. If the function is suitably analytic,
the expected rate is $O(q^{n})$ with $0<q<1.$ This is the so-called
exponential convergence, which is well accepted among the community. There has
been much investigation on exponential decay of spectral expansions of
analytic functions. For instance, the justification for Fourier and/or
Chebyshev series can be found in [38, 10, 36, 5, 33, 41]. In the seminal work
of Gottlieb and Shu [24, 23], on the resolution of the Gibbs phenomenon (the
interested readers are referred to Gustafsson [29] for a review of this
significant contribution), the exponential convergence, in the maximum norm
(termed as the so-called regularization error), of Gegenbauer polynomial
expansions was derived, when the index (denoted by $\lambda$ below) grows
linearly with the degree $n$. Boyd [7] provided an insightful study of the
“diagonal limit” (i.e., $\lambda=\beta n$ for some constant $\beta>0$)
convergence of the Gegenbauer reconstruction algorithm in [24]. We remark that
under the assumption of analyticity in [24], the exponential accuracy of
Gegenbauer expansions is actually valid for fixed $\lambda$ (see Appendix A
for the justification).
It is known that the heart of a collocation/pseudospectral method is the
spectral differentiation process. That is, given a set of Gauss-type points
$\\{x_{j}\\}_{j=0}^{n},$ e.g., on $[-1,1],$ the derivative values
$\\{u^{\prime}(x_{j})\\}$ can be approximated by an exact differentiation of
the polynomial interpolant $\\{(I_{n}u)^{\prime}(x_{j})\\}.$ Such a direct
differentiation technique is also called a differencing method in the
literature (see, e.g., [39, 19, 35]). For the first time, Tadmor [39] showed
the exponential accuracy of differencing analytic functions on Chebyshev-Gauss
points, where the main argument was based on analyzing the (continuous)
Chebyshev coefficients and the aliasing error, and where the intimate relation
between Fourier and Chebyshev basis functions played an essential role in the
analysis. Reddy and Weideman [35] took a different approach and improved the
estimate in [39] for Chebyshev differencing of functions analytic on and
within an ellipse with foci $\pm 1.$ As pointed out in [35], although the
exponential convergence of spectral differentiation of analytic functions is
appreciated and mentioned in the literature (see, e.g., [19, 40, 6]), the
rigorous proofs (merely for the Fourier and Chebyshev methods) can only be
found in [39, 35]. Indeed, to the best of our knowledge, the theoretical
justification even for the Legendre method is lacking. It is worthwhile to
point out that under the regularity condition (M):
$\|u^{(k)}\|_{L^{\infty}}\leq cM^{k},$ the super-geometric convergence of
Legendre spectral differentiation was proved by Zhang [46]:
$\max_{0\leq j\leq n}|(u-I_{n}u)^{\prime}(x_{j})|\leq
C\Big{(}\frac{eM}{2(n+1)}\Big{)}^{n+2},$ (1.1)
where $\\{x_{j}\\}_{j=0}^{n}$ are the Legendre-Gauss points. Similar estimate
was nontrivially extended to the Chebyshev collocation method in [47]. The
condition (M) covers a large class of functions, but it is even more
restrictive than analyticity. On the other hand, the regularity index $k$
could be infinite, while the dependence of (1.1) on $k$ is not clear.
The main concern of this paper is to show exponential convergence, in the
maximum norm, of Gegenbauer interpolation and spectral differentiation on
Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, provided that the
underlying function is analytic on and within a sizable ellipse. The essential
argument is based on the classical Hermite’s contour integral (see (2.19)
below), and a delicate estimate of the asymptotics of the Gegenbauer
polynomial on the ellipse of interest. It is important to remark that the
Chebyshev polynomial on the ellipse takes a very simple explicit form (see,
e.g., [10] or (3.5) below), but the Gegenbauer polynomial has a complicated
expression. Accordingly, compared with the Chebyshev case in [35], the
analysis in this paper is much more involved. The Chebyshev and Legendre
methods are commonly used in spectral approximations, but we have also
witnessed renewed applications of the Gegenbauer (or more general Jacobi)
polynomial based methods in, e.g., defeating Gibbs phenomenon (see, e.g., [24,
21]), $hp$-elements (see, e.g., [17, 2, 31, 25]), and numerical solutions of
differential equations (see, e.g., [3, 26, 27, 12, 13, 14, 15, 16, 44, 37])
and integral equations (see, e.g., [9]). The results in this paper might be
useful for a better understanding of the methods and have implications in
other applications.
The rest of the paper is organized as follows. As some preliminaries, we
briefly review basic properties of Gegenbauer polynomials, Gamma functions and
analytic functions in Section 2. We study the asymptotics of the Gegenbauer
polynomials in Section 3, and present the main results on exponential
convergence of interpolation and spectral differentiation, together with some
numerical results and extensions in the last section.
## 2\. Preliminaries
In this section, we collect some relevant properties of Gegenbauer polynomials
and assorted facts to be used throughout the paper.
### 2.1. Gegenbauer polynomials
The analysis heavily relies on the normalization of [38], so we define the
Gegenbauer polynomials111Historically, they are sometimes called
“ultraspherical polynomials” (see, e.g., the footnote on Page 80 of [38] and
Page 302 of [1]). by the three-term recurrence:
$\begin{split}&nC_{n}^{\lambda}(x)=2\left(n+\lambda-1\right)xC_{n-1}^{\lambda}(x)-(n+2\lambda-2)C_{n-2}^{\lambda}(x),\quad
n\geq 2,\\\ &C_{0}^{\lambda}(x)=1,\quad C_{1}^{\lambda}(x)=2\lambda
x,\quad\lambda>-1/2,\;\;x\in[-1,1].\end{split}$ (2.1)
Notice that if $\lambda=0,$ $C_{n}^{\lambda}(x)$ vanishes identically for
$n\geq 1.$ This corresponds to the Chebyshev polynomial, and there holds
$\lim_{\lambda\to
0}\lambda^{-1}C_{n}^{\lambda}(x)=\frac{2}{n}T_{n}(x)=\frac{2}{n}\cos(n\ {\rm
arccos}(x)),\quad n\geq 1.$ (2.2)
Hereafter, if not specified explicitly, we assume $\lambda\not=0,$ and refer
to [35] for the analysis of the Chebyshev case. Notice that for $\lambda=1/2,$
$C^{\lambda}_{n}(x)=L_{n}(x),$ i.e., the usual Legendre polynomial of degree
$n.$
The Gegenbauer polynomials are orthogonal with respect to the weight function
$(1-x^{2})^{\lambda-1/2},$ namely,
$\int_{-1}^{1}C_{n}^{\lambda}(x)C_{m}^{\lambda}(x)(1-x^{2})^{\lambda-1/2}dx=h_{n}^{\lambda}\delta_{mn},$
(2.3)
where $\delta_{mn}$ is the Kronecker symbol, and
$h_{n}^{\lambda}=\frac{2^{1-2\lambda}\pi}{\Gamma^{2}(\lambda)}\frac{\Gamma(n+2\lambda)}{n!(n+\lambda)}.$
(2.4)
Moreover, we have
$C_{n}^{\lambda}(-x)=(-1)^{n}C_{n}^{\lambda}(x),\quad
C_{n}^{\lambda}(1)=\frac{\Gamma{(n+2\lambda)}}{n!\Gamma{(2\lambda)}},$ (2.5)
and
$\frac{d}{dx}C_{n}^{\lambda}(x)=2\lambda C_{n-1}^{\lambda+1}(x).$ (2.6)
By Formula (4.7.1) and Theorems 7.32.1 and 7.33.1 of Szeg$\ddot{o}$ [38], we
have
$|C_{n}^{\lambda}(x)|\leq C_{n}^{\lambda}(1)\;\;\;{\rm
if}\;\lambda>0;\quad|C_{n}^{\lambda}(x)|\leq
D_{\lambda}\,n^{\lambda-1}\;\;\;{\rm if}\;-\frac{1}{2}<\lambda<0\;{\rm
and}\;n\gg 1,$ (2.7)
where $D_{\lambda}$ is a positive constant independent of $n.$ A tight bound
can be found in [34] (also see [32]):
$\max_{|x|\leq
1}\Big{\\{}(1-x^{2})^{\lambda}\big{(}C_{n}^{\lambda}(x)\big{)}^{2}\Big{\\}}\leq\frac{2e(2+\sqrt{2}\lambda)}{\pi}h_{n}^{\lambda},\quad\lambda>0,\;n\geq
0.$ (2.8)
### 2.2. Gamma and incomplete Gamma functions
The following properties of the Gamma and incomplete Gamma functions (cf.
[45]) are found useful. The Gamma function satisfies
$\Gamma(x)\Gamma(x+1/2)=2^{1-2x}\sqrt{\pi}\,\Gamma(2x),\quad\forall x\geq 0,$
(2.9)
and
$\Gamma{(x)}\Gamma{(-x)}=-\frac{\pi}{x\sin(\pi x)},\quad\forall x>0.$ (2.10)
We would like to quote the Stirling’s formula (see, e.g., [23]):
$\sqrt{2\pi}x^{x+1/2}e^{-x}\leq\Gamma(x+1)\leq\sqrt{2\pi}x^{x+1/2}e^{-x}e^{\frac{1}{12x}},\quad\forall
x\geq 1.$ (2.11)
We also need to use the incomplete Gamma function defined by
$\Gamma(\alpha,x)=\int_{x}^{\infty}t^{\alpha-1}e^{-t}dt,\quad\alpha>0,\;\;x\geq
0,$ (2.12)
which satisfies (see P. 899 of [30])
$\Gamma(n+1,x)=n!e^{-x}\sum_{k=0}^{n}\frac{x^{k}}{k!},\quad n=0,1,\cdots.$
(2.13)
### 2.3. Basics of analytic functions
Suppose that $u(x)$ is analytic on $[-1,1].$ Based on the notion of analytic
continuation, there always exists a simple connected region $R$ in the complex
plane containing $[-1,1]$ into which $f(x)$ can be continued analytically. The
analyticity may be characterized by the growth of the derivatives of $u$. More
precisely, let ${\mathcal{C}}$ be a simple positively oriented closed contour
surrounding $[-1,1]$ and lying in $R.$ Then we have (see, e.g., [10]):
$\frac{|u^{(m)}(x)|}{m!}\leq\frac{\max_{z\in{\mathcal{C}}}|u(z)|L({\mathcal{C}})}{2\pi\delta^{m+1}},\quad\forall
x\in[-1,1],$ (2.14)
where $L({\mathcal{C}})$ is the length of ${\mathcal{C}},$ and $\delta$ is the
distance from ${\mathcal{C}}$ to $[-1,1]$ (can be viewed as the distance from
$[-1,1]$ to the nearest singularity of $u$ in the complex plane).
Mathematically, an appropriate contour to characterize the analyticity is the
so-called Bernstein ellipse:
${\mathcal{E}}_{\rho}:=\Big{\\{}z\in{\mathbb{C}}~{}:~{}z=\frac{1}{2}(w+w^{-1})\;\;\text{with}\;\;w=\rho
e^{{\rm i}\theta},\;\theta\in[0,2\pi]\Big{\\}},\quad\rho>1,$ (2.15)
where ${\mathbb{C}}$ is the set of all complex numbers, and ${\rm
i}=\sqrt{-1}$ is the complex unit. The ellipse ${\mathcal{E}}_{\rho}$ has the
foci at $\pm 1$ and the major and minor semi-axes are, respectively
$a=\frac{1}{2}\big{(}\rho+\rho^{-1}\big{)},\quad
b=\frac{1}{2}(\rho-\rho^{-1}),$ (2.16)
so the sum of two axes is $\rho.$ As illustrated in Figure 2.1, $\rho$ is the
radius of the circle $w=\rho e^{{\rm i}\theta}$ that is mapped to the ellipse
${\mathcal{E}}_{\rho}$ under the conformal mapping: $z=\frac{1}{2}(w+w^{-1}).$
Figure 2.1. Circle (left): $|w|=\rho=1.5$, and Bernstein ellipse (right):
${\mathcal{E}}_{\rho}$ with foci $\pm 1$ linked by the conformal mapping:
$z=\frac{1}{2}(w+w^{-1})$.
According to [35], the perimeter of ${\mathcal{E}}_{\rho}$ satisfies
$L({\mathcal{E}}_{\rho})\leq\pi\sqrt{\rho^{2}+\rho^{-2}},$ (2.17)
which overestimates the perimeter by less than 12 percent. The distance from
${\mathcal{E}}_{\rho}$ to the interval $[-1,1]$ is
$\delta_{\rho}=\frac{1}{2}(\rho+\rho^{-1})-1.$ (2.18)
We are concerned with the interpolation and spectral differentiation of
analytic functions on the Gegenbauer-Gauss-type points. Let
$\\{x_{j}:=x_{j}(\lambda,n)\\}_{j=0}^{n}$ be the Gegenbauer-Gauss points
(i.e., the zeros of $C_{n+1}^{\lambda}(x)$) or the Gegenbauer-Gauss-Lobatto
points (i.e., the zeros of $(1-x^{2})C_{n-1}^{\lambda+1}(x)$). The associated
Lagrange interpolation polynomial of $u$ is given by
$I_{n}u\in{\mathbb{P}}_{n}$ (the set of all polynomials of degree $\leq n$)
such that $(I_{n}u)(x_{j})=u(x_{j})$ for $0\leq j\leq n.$ Our starting point
is the Hermite’s contour integral (see, e.g., [11]):
$(u-I_{n}u)(x)=\frac{1}{2\pi{\rm
i}}\oint_{{\mathcal{E}}_{\rho}}\frac{Q_{n+1}(x)}{Q_{n+1}(z)}\frac{u(z)}{z-x}dz,\quad\forall
x\in[-1,1],$ (2.19)
where $Q_{n+1}(x)=C_{n+1}^{\lambda}(x)$ or $(1-x^{2})C_{n-1}^{\lambda+1}(x).$
Consequently, we have
$(u-I_{n}u)^{\prime}(x_{j})=\frac{1}{2\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{Q_{n+1}^{\prime}(x_{j})}{Q_{n+1}(z)}\frac{u(z)}{z-x_{j}}dz,\quad
0\leq j\leq n.$ (2.20)
A crucial component of the error analysis is to obtain a sharp asymptotic
estimate of $Q_{n+1}(z)$ on ${\mathcal{E}}_{\rho}$ with large $n.$ This will
be the main concern of the forthcoming section.
## 3\. Asymptotic estimate of Gegenbauer polynomials on
${\mathcal{E}}_{\rho}$
Much of our analysis relies on the following representation of the Gegenbauer
polynomial.
###### Lemma 3.1.
Let $z=\frac{1}{2}(w+w^{-1}).$ We have
$C_{n}^{\lambda}(z)=\sum_{k=0}^{n}g_{k}^{\lambda}g_{n-k}^{\lambda}w^{n-2k},\quad
n\geq 0,\;\;\lambda>-1/2,$ (3.1)
where
$g_{0}^{\lambda}=1,\quad g_{k}^{\lambda}={{k+\lambda-1}\choose
k}=\frac{\Gamma(k+\lambda)}{k!\Gamma(\lambda)},\quad 1\leq k\leq n.$ (3.2)
This formula is derived from the three-term recurrence formula (2.1) and the
mathematical induction. Its proof is provided in Appendix B.
###### Remark 3.1.
Some consequences of Lemma 3.1 are in order.
* (a)
Comparing the coefficients $w^{n}$ on both sides of (3.1), we find that the
leading coefficient of $C_{n}^{\lambda}$ is $2^{n}g_{n}^{\lambda}.$ This can
be also verified from (2.1) by the mathematical induction.
* (b)
If $\lambda>0,$ then $g_{k}^{\lambda}>0$ for all $0\leq k\leq n.$ On the other
hand, if $\lambda<0,$ we find from (2.10) and (3.2) that
$g_{k}^{\lambda}=\frac{\sin(\pi\lambda)}{\pi}\frac{\Gamma{(k+\lambda)}\Gamma{(1-\lambda)}}{k!}<0,\quad
1\leq k\leq n.$ (3.3)
* (c)
If $\lambda=1/2,$ it follows from (2.9) and (3.2) that
$g_{k}^{\lambda}=\frac{(2k)!}{(k!)^{2}2^{2k}},\quad 0\leq k\leq n.$ (3.4)
Such a representation for the Legendre polynomial can be found in, e.g., [11]
and [38], but the derivation is quite different.
* (d)
If $\lambda=0,$ then by (2.2),
$T_{n}(z)=\frac{1}{2}\big{(}w^{n}+w^{-n}\big{)},\quad n\geq 1.$ (3.5)
* (e)
If $\lambda=1,$ then $g_{k}^{\lambda}\equiv 1$ for $0\leq k\leq n.$ Therefore,
the Chebyshev polynomial of the second kind has the representation
$U_{n}(z)=\frac{w^{n+1}-w^{-(n+1)}}{w-w^{-1}}=w^{n}\sum_{k=0}^{n}w^{-2k}={C_{n}^{1}(z)},\quad
n\geq 0,$ (3.6)
which can be found in [33]. ∎
It is interesting to observe from (3.6) that for $\lambda=1,$
$C_{n}^{\lambda}(z)/w^{n}$ converges to $(1-w^{-2})^{-\lambda}$ uniformly for
all $|w|>1,$ that is,
$\sum_{k=0}^{\infty}w^{-2k}=\frac{1}{1-w^{-2}},\quad|w|>1.$
In what follows, we show a similar property holds for general $\lambda>-1/2$
and $\lambda\not=0.$ More precisely, we estimate the upper bound of remainder:
$\begin{split}\Big{|}\big{(}1-w^{-2}\big{)}^{-\lambda}-\frac{C_{n}^{\lambda}(z)}{g_{n}^{\lambda}w^{n}}\Big{|}&\leq\sum_{k=1}^{n}|d_{n,k}^{\lambda}||g^{\lambda}_{k}|\rho^{-2k}+\sum_{k=n+1}^{\infty}|g_{k}^{\lambda}|\rho^{-2k}:=R_{n}(\rho,\lambda),\end{split}$
(3.7)
where $z\in{\mathcal{E}}_{\rho}$ with $|w|=\rho>1,$ and
$d_{n,k}^{\lambda}=1-\frac{g_{n-k}^{\lambda}}{g_{n}^{\lambda}},\;\;\;1\leq
k\leq n.$ (3.8)
To obtain a sharp estimate of $R_{n}(\rho,\lambda),$ it is necessary to
understand the behavior of the coefficients $\\{g_{k}^{\lambda}\\}_{k=1}^{n}$
and $\\{d_{n,k}^{\lambda}\\}_{k=1}^{n},$ which are summarized in the following
two lemmas.
###### Lemma 3.2.
For $\lambda>-1/2,\,k\geq 1$ and $k+\lambda\geq 1,$
$c_{1}\,\Big{(}1+\frac{\lambda}{k}\Big{)}^{k+1/2}e^{-\lambda}\leq\frac{\Gamma(\lambda)\,g_{k}^{\lambda}}{(k+\lambda)^{\lambda-1}}\leq
c_{2}\,\Big{(}1+\frac{\lambda}{k}\Big{)}^{k+1/2}e^{-\lambda},$ (3.9)
where $c_{1}=e^{-\frac{1}{12k}}$ and $c_{2}=e^{\frac{1}{12(k+\lambda)}}.$
###### Proof.
Applying the Stirling’s formula (2.11) to
$(k+\lambda)\Gamma(\lambda)\,g_{k}^{\lambda}=\frac{\Gamma(k+\lambda+1)}{\Gamma(k+1)}$
leads to (3.9). ∎
###### Lemma 3.3.
Let $\\{g_{k}^{\lambda}\\}_{k=1}^{n}$ and $\\{d_{n,k}^{\lambda}\\}_{k=1}^{n}$
be the sequences as defined in (3.2) and (3.8), respectively.
* (i)
If $\lambda>1$, then there holds
$0<d_{n,1}^{\lambda}<d_{n,2}^{\lambda}<\cdots<d_{n,n}^{\lambda}<1.$ (3.10)
* (ii)
If $-1/2<\lambda<1$ and $\lambda\not=0,$ then
$\cdots<|g_{k+1}^{\lambda}|<|g_{k}^{\lambda}|<\cdots<|g_{1}^{\lambda}|<g_{0}^{\lambda}=1,$
(3.11)
and we have
$0<-d_{n,1}^{\lambda}<-d_{n,2}^{\lambda}<\cdots<-d_{n,n-1}^{\lambda},$ (3.12)
and
$|d_{n,k}^{\lambda}g_{k}^{\lambda}|<1\;\;\;{\rm for}\;\;\;1\leq k\leq
n-1,\;n\geq 3.$ (3.13)
###### Proof.
By (3.2),
$\frac{g_{k+1}^{\lambda}}{g_{k}^{\lambda}}=\frac{k+\lambda}{k+1}\;\;\;{\rm
for}\;\;\lambda\not=0.$ (3.14)
Thus for $\lambda>1,$ $\\{g_{k}^{\lambda}\\}$ is strictly increasing with
respect to $k,$ which, together with the fact $g_{k}^{\lambda}>0,$ implies
$0<d_{n,k}^{\lambda}=1-\frac{g_{n-k}^{\lambda}}{g_{n}^{\lambda}}<1,$
and
$d_{n,k+1}^{\lambda}-d_{n,k}^{\lambda}=\frac{g_{n-k}^{\lambda}-g_{n-k-1}^{\lambda}}{g_{n}^{\lambda}}>0.$
This completes the proof of (i).
The property (3.11) is a direct consequence of (3.14), and (3.12) can be
proved in a fashion similar to (3.10). It remains to verify (3.13). If $k=1,$
a direct calculation by using (3.2) yields
$|d_{n,1}^{\lambda}g_{1}^{\lambda}|=|\lambda|\frac{1-\lambda}{n-(1-\lambda)}<1,\quad\forall
n\geq 3.$
For $2\leq k\leq n-1,$ it follows from (3.2) and (3.12) that
$\begin{split}|d_{n,k}^{\lambda}g_{k}^{\lambda}|&=\Big{(}\frac{g_{n-k}^{\lambda}}{g_{n}^{\lambda}}-1\Big{)}|g_{k}^{\lambda}|<\frac{g_{n-k}^{\lambda}}{g_{n}^{\lambda}}|g_{k}^{\lambda}|=\left(\prod_{j=0}^{k-2}\frac{1-\frac{1-\lambda}{k-j}}{1-\frac{1-\lambda}{n-j}}\right)\frac{|\lambda|}{1-\frac{1-\lambda}{n-k+1}}<1.\end{split}$
This ends the proof. ∎
With the above preparation, we are ready to present the main result on the
upper bound of $R_{n}(\rho,\lambda)$ in (3.7).
###### Theorem 3.1.
Let $R_{n}(\rho,\lambda)$ with $\rho>1$ be the remainder as defined in (3.7).
* (i)
If $\lambda>1,$ then for all $n\geq m\geq 1$ and
$m+2\geq(\lambda-1)\Big{(}\frac{1}{2\ln\rho}-1\Big{)},$ (3.15)
we have
$R_{n}(\rho,\lambda)\leq
d_{n,m}^{\lambda}\Big{(}(1-\rho^{-2})^{-\lambda}-1\Big{)}+A\frac{[\lambda]!}{(2\ln\rho)^{\lambda}}\frac{(m+\lambda)^{[\lambda]}}{\rho^{2(m-1)}},$
(3.16)
where $[\lambda]$ is the largest integer $\leq\lambda,$ and
$A=\frac{1}{\Gamma(\lambda)}{\rm
exp}\Big{(}{\frac{1}{12(m+1+\lambda)}+\frac{\lambda}{2(m+1)}}\Big{)}.$ (3.17)
* (ii)
If $-1/2<\lambda<1$ and $\lambda\not=0,$ then for all $n\geq m\geq 1$ and
$n\geq 3,$
$R_{n}(\rho,\lambda)\leq|d_{n,m}^{\lambda}|\big{|}(1-\rho^{-2})^{-\lambda}-1\big{|}+\frac{\rho^{-2m}}{\rho^{2}-1}+2\rho^{-2n}.$
(3.18)
Here, the factor $d_{n,m}^{\lambda}$ is given by (3.8).
###### Proof.
(i) For $\lambda>1,$ we obtain from (3.10) that
$\begin{split}R_{n}(\rho,\lambda)&=\sum_{k=1}^{m}d_{n,k}^{\lambda}g_{k}^{\lambda}\rho^{-2k}+\sum_{k=m+1}^{n}d_{n,k}^{\lambda}g_{k}^{\lambda}\rho^{-2k}+\sum_{k=n+1}^{\infty}g_{k}^{\lambda}\rho^{-2k}\\\
&\overset{(\ref{dnkcase1})}{\leq}d_{n,m}^{\lambda}\sum_{k=1}^{m}g_{k}^{\lambda}\rho^{-2k}+\sum_{k=m+1}^{n}g_{k}^{\lambda}\rho^{-2k}+\sum_{k=n+1}^{\infty}g_{k}^{\lambda}\rho^{-2k}\\\
&\leq
d_{n,m}^{\lambda}\Big{(}(1-\rho^{-2})^{-\lambda}-1\Big{)}+\sum_{k=m+1}^{\infty}g_{k}^{\lambda}\rho^{-2k}.\end{split}$
(3.19)
By Lemma 3.2,
$\begin{split}\sum_{k=m+1}^{\infty}g_{k}^{\lambda}\rho^{-2k}&\leq{(\Gamma(\lambda))^{-1}}{e^{-\lambda}e^{\frac{1}{12(m+1+\lambda)}}}\sum_{k=m+1}^{\infty}\frac{(k+\lambda)^{\lambda-1}}{\rho^{2k}}\Big{(}1+\frac{\lambda}{k}\Big{)}^{k+1/2}\\\
&\leq{(\Gamma(\lambda))^{-1}}e^{-\lambda}e^{\frac{1}{12(m+1+\lambda)}}\sum_{k=m+1}^{\infty}\frac{(k+\lambda)^{\lambda-1}}{\rho^{2k}}e^{\lambda+\frac{\lambda}{2k}},\end{split}$
where we used the inequality $1+x<e^{x}$ for $x>0.$ Hence,
$\sum_{k=m+1}^{\infty}g_{k}^{\lambda}\rho^{-2k}\leq
A\sum_{k=m+1}^{\infty}\frac{(k+\lambda)^{\lambda-1}}{\rho^{2k}},$ (3.20)
where $A$ is given by (3.17). One verifies that under the condition (3.15),
${(k+\lambda)^{\lambda-1}}/{\rho^{2k}}$ is decreasing with respect to $k.$
Therefore, by (2.12) and (2.13),
$\begin{split}\sum_{k=m+1}^{\infty}\frac{(k+\lambda)^{\lambda-1}}{\rho^{2k}}&\leq\int_{m}^{\infty}{(x+\lambda)^{\lambda-1}}\rho^{-2x}dx=\frac{\rho^{2\lambda}}{(2\ln\rho)^{\lambda}}\int_{2(m+\lambda)\ln\rho}^{\infty}x^{\lambda-1}e^{-x}dx\\\
&=\frac{\rho^{2\lambda}}{(2\ln\rho)^{\lambda}}\Gamma\big{(}\lambda,2(m+\lambda)\ln\rho\big{)}\leq\frac{\rho^{2\lambda}}{(2\ln\rho)^{\lambda}}\Gamma\big{(}[\lambda]+1,2(m+\lambda)\ln\rho\big{)}\\\
&=\frac{[\lambda]!\rho^{-2m}}{(2\ln\rho)^{\lambda}}\sum_{k=0}^{[\lambda]}\frac{(m+\lambda)^{k}(2\ln\rho)^{k}}{k!}\leq\frac{[\lambda]!}{(2\ln\rho)^{\lambda}}\frac{(m+\lambda)^{[\lambda]}}{\rho^{2m}}\sum_{k=0}^{\infty}\frac{(2\ln\rho)^{k}}{k!}\\\
&=\frac{[\lambda]!}{(2\ln\rho)^{\lambda}}\frac{(m+\lambda)^{[\lambda]}}{\rho^{2(m-1)}}.\end{split}$
A combination of the above estimates leads to (3.16).
(ii) Now, we turn to the proof of the second case: $-1/2<\lambda<1$ and
$\lambda\not=0.$ By Lemma 3.3,
$\begin{split}R_{n}(\rho,\lambda)&=\sum_{k=1}^{m}|d_{n,k}^{\lambda}||g_{k}^{\lambda}|\rho^{-2k}+\sum_{k=m+1}^{n}|d_{n,k}^{\lambda}||g_{k}^{\lambda}|\rho^{-2k}+\sum_{k=n+1}^{\infty}|g_{k}^{\lambda}|\rho^{-2k}\\\
&\overset{(\ref{dnkcase2})}{\leq}|d_{n,m}^{\lambda}|\sum_{k=1}^{m}|g_{k}^{\lambda}|\rho^{-2k}\overset{(\ref{dnkcase3})}{+}\sum_{k=m+1}^{n-1}\rho^{-2k}+|d_{n,n}^{\lambda}g_{n}^{\lambda}|\rho^{-2n}\overset{(\ref{dnkcase2aa})}{+}\sum_{k=n+1}^{\infty}\rho^{-2k}\\\
&\quad\leq|d_{n,m}^{\lambda}|\big{|}(1-\rho^{-2})^{-\lambda}-1\big{|}+\frac{\rho^{-2m}}{\rho^{2}-1}+2\rho^{-2n},\end{split}$
where in the last step, we used the following facts:
$\begin{split}\sum_{k=1}^{m}|g_{k}^{\lambda}|\rho^{-2k}&\leq{\rm
sign}(\lambda)\sum_{k=1}^{\infty}g_{k}^{\lambda}\rho^{-2k}={\rm
sign}(\lambda)\big{(}(1-\rho^{-2})^{-\lambda}-1\big{)},\quad\rho>1,\end{split}$
(note: ${\rm sign}(\lambda)$ is the sign of $\lambda$), and
$|d_{n,n}^{\lambda}g_{n}^{\lambda}|=|g_{n}^{\lambda}-1|<2,$ thanks to (3.11).
∎
The estimate in Theorem 3.1 is quite tight and is valid even for small $n.$ By
choosing a suitable $m$ to balance the two error terms in the upper bound, we
are able to derive the anticipated asymptotic estimate.
###### Theorem 3.2.
For any $z\in{\mathcal{E}}_{\rho}$ with $|w|=\rho>1,$ and any $\lambda>-1/2$
and $\lambda\not=0,$ there exists $0<\varepsilon\leq 1/2$ such that
$\Big{|}\big{(}1-w^{-2}\big{)}^{-\lambda}-\frac{C_{n}^{\lambda}(z)}{g_{n}^{\lambda}w^{n}}\Big{|}\leq
A(\rho,\lambda)n^{\varepsilon-1}+O(n^{-1}),$ (3.21)
where
$A(\rho,\lambda)=|1-\lambda|\big{|}(1-\rho^{-2})^{-\lambda}-1\big{|}.$ (3.22)
###### Proof.
We first estimate $|d_{n,m}^{\lambda}|$ in Theorem 3.1, when $n-m$ is large.
Using the Stirling’s formula (2.11) and (3.2) that
$\begin{split}\frac{g_{n-m}^{\lambda}}{g_{n}^{\lambda}}&=\Big{(}1+\frac{1-\lambda}{n+\lambda-1}\Big{)}^{n+\frac{1}{2}}\Big{(}1-\frac{1-\lambda}{n-m}\Big{)}^{n-m+\frac{1}{2}}\Big{(}1-\frac{m}{n+\lambda-1}\Big{)}^{\lambda-1}\Big{\\{}1+O\Big{(}\frac{1}{n-m}\Big{)}\Big{\\}}\\\
&=\Big{(}1-\frac{m}{n+\lambda-1}\Big{)}^{\lambda-1}\Big{\\{}1+O\Big{(}\frac{1}{n-m}\Big{)}\Big{\\}}\\\
&=\Big{\\{}1+\frac{(1-\lambda)m}{n+\lambda-1}+O\Big{(}\frac{m^{2}}{n^{2}}\Big{)}\Big{\\}}\Big{\\{}1+O\Big{(}\frac{1}{n-m}\Big{)}\Big{\\}}.\end{split}$
Hereafter, taking $m=[n^{\varepsilon}]$ with $0<\varepsilon\leq 1/2$ yields
$\begin{split}\frac{g_{n-m}^{\lambda}}{g_{n}^{\lambda}}&=1+(1-\lambda)n^{\varepsilon-1}+O\Big{(}\frac{1}{n-n^{\varepsilon}}\Big{)}\\\
&\Longrightarrow\;\;d_{n,m}^{\lambda}=(\lambda-1)n^{\varepsilon-1}+O(n^{-1}).\end{split}$
(3.23)
One verifies readily that for $\lambda>1$ and any $0<\varepsilon\leq 1/2,$
$\frac{m^{[\lambda]}}{\rho^{2m}}\leq\frac{1}{n}\;\;\Longleftrightarrow\;\;\frac{\ln
n}{n^{\varepsilon}}\leq\frac{2\ln\rho}{1+\varepsilon[\lambda]},$ (3.24)
which, together with (3.16) and (3.23), implies (3.21) with $\lambda>1.$
If $-1/2<\lambda<1$ and $\lambda\not=0,$ it follows from (3.24) that
$\rho^{-2m}\leq n^{-1}$ for any $0<\varepsilon\leq 1/2.$ This validates the
desired estimate. ∎
A direct consequence of Theorem 3.2 is that
$\lim_{n\to\infty}\frac{C_{n}^{\lambda}(z)}{g_{n}^{\lambda}}=\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{g_{n-k}^{\lambda}}{g_{n}^{\lambda}}g_{k}^{\lambda}w^{-2k}=(1-w^{-2})^{-\lambda},$
(3.25)
for all $z\in{\mathcal{E}}_{\rho}$ with $|w|=\rho>1,$ and any $\lambda>-1/2$
and $\lambda\not=0.$
###### Remark 3.2.
Based on a completely different argument, Elliott [18] derived an asymptotic
expansion for large $n$ near $z=1$ (but not near $z=-1$):
$C_{n}^{\lambda}(z)\sim\frac{B(n,\lambda)}{(z^{2}-1)^{\lambda/2}},$ where $B$
is a series involving modified Bessel functions, and some other asymptotic
expansions for $|z|$ large and $n$ fixed. Although they are valid for general
$z$ off the interval $[-1,1],$ our results in Theorems 3.1 and 3.2 provide
tighter and sharper bounds when $z$ is sitting on ${\mathcal{E}}_{\rho}.$ ∎
As the end of this section, we provide some numerical results to illustrate
the tightness of the upper bound in (3.21). Denote by
$E_{n}(\rho;\lambda):=\frac{1}{A(\rho,\lambda)}\max_{z\in{\mathcal{E}}_{\rho}}\Big{|}\big{(}1-w^{-2}\big{)}^{-\lambda}-\frac{C_{n}^{\lambda}(z)}{g_{n}^{\lambda}w^{n}}\Big{|}.$
(3.26)
To approximate the maximum value, we sample a set of points dense on the
ellipse ${\mathcal{E}}_{\rho}$ based on the conformal mapping
$z=\frac{1}{2}(w+w^{-1})$ of the Fourier points on the circle $w=\rho e^{{\rm
i}\theta}.$ We plot in Figure 3.1 in (Matlab) log-log scale of
$E_{n}(\rho;\lambda),n^{-1}$ and $n^{\varepsilon-1}$ (with $\varepsilon=0.1$)
for several sets of parameters $\lambda$ and $\rho,$ and for large $n.$
According to Theorem 3.2, $E_{n}$ should be bounded by $n^{\varepsilon-1}$
from above, and it is anticipated to be bounded below by $n^{-1},$ if the
estimate is tight. Indeed, we observe from Figure 3.1 such a behavior when $n$
is large.
Figure 3.1. $E_{n}$ against $n^{\varepsilon-1}$ (with $\varepsilon=0.1$) and
$n^{-1}$ for large $n$.
## 4\. Error estimates of interpolation and spectral differentiation
After collecting all the necessary results, we are ready to estimate
exponential convergence of interpolation and spectral differentiation of
analytic functions.
Hereafter, the notation $a_{n}\cong b_{n}$ means that $a_{n}/b_{n}\to 1$ as
$n\to\infty,$ for any two sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ (with
$b_{n}\not=0$) of complex numbers.
### 4.1. Gegenbauer-Gauss interpolation and differentiation
We start with the analysis of interpolation and spectral differentiation on
zeros of the Gegenbauer polynomial $C_{n+1}^{\lambda}(x).$
###### Theorem 4.1.
Let $u$ be analytic on and within the ellipse ${\mathcal{E}_{\rho}}$ with foci
$\pm 1$ and $\rho>1$ as defined in (2.15), and let $(I_{n}u)(x)$ be the
interpolant of $u(x)$ at the set of $(n+1)$ Gegenbauer-Gauss points.
* (i)
If $\lambda>0,$ we have
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)(x)\big{|}\leq\frac{c\Gamma(\lambda)M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\Gamma(2\lambda)(\rho-1)^{2}(1+\rho^{-2})^{-\lambda}}\frac{n^{\lambda}}{\rho^{n}}.$
(4.1)
* (ii)
If $-1/2<\lambda<0,$ we have
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)(x)\big{|}\leq\frac{cD_{\lambda}|\Gamma(\lambda)|M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{(\rho-1)^{2}(1-\rho^{-2})^{-\lambda}}\frac{1}{\rho^{n}}.$
(4.2)
Here, $M_{\rho}=\max_{z\in{\mathcal{E}_{\rho}}}|u(z)|$, $D_{\lambda}$ is
defined in (2.7), and $c\cong 1$ is a generic positive constant.
###### Proof.
By the formula (2.19) with $Q_{n+1}=C_{n+1}^{\lambda}$ and (2.17)-(2.18), we
have the bound of the point-wise error:
$\begin{split}\big{|}(u-I_{n}u)(x)\big{|}&\leq\frac{|C_{n+1}^{\lambda}(x)|}{2\pi}\frac{\max_{z\in\mathcal{E}_{\rho}}|u(z)|}{\min_{z\in{\mathcal{E}_{\rho}}}|C_{n+1}^{\lambda}(z)|}\oint_{\mathcal{E}_{\rho}}\frac{|dz|}{|z-x|}\\\
&\leq\frac{M_{\rho}L({\mathcal{E}_{\rho}})}{2\pi\delta_{\rho}}\frac{|C_{n+1}^{\lambda}(x)|}{\min_{z\in{\mathcal{E}_{\rho}}}|C_{n+1}^{\lambda}(z)|}\\\
&\leq\frac{M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\rho+\rho^{-1}-2}\frac{|C_{n+1}^{\lambda}(x)|}{\min_{z\in{\mathcal{E}_{\rho}}}|C_{n+1}^{\lambda}(z)|},\quad
x\in[-1,1],\;n\geq 0.\end{split}$ (4.3)
Therefore, it is essential to obtain the lower bound of
$|C_{n+1}^{\lambda}(z)|.$ Recall that for any two complex numbers $z_{1}$ and
$z_{2},$ we have $\big{|}|z_{1}|-|z_{2}|\big{|}\leq|z_{1}-z_{2}|.$ It follows
from Theorem 3.2 that
$\bigg{|}|1-w^{-2}|^{-\lambda}-\frac{|C_{n+1}^{\lambda}(z)|}{|g_{n+1}^{\lambda}|\rho^{n+1}}\bigg{|}\leq
A(\rho,\lambda)n^{\varepsilon-1}+O(n^{-1}),$
which implies
$\begin{split}\big{|}1-w^{-2}\big{|}^{-\lambda}&-A(\rho,\lambda)n^{\varepsilon-1}-O(n^{-1})\leq\frac{|C_{n+1}^{\lambda}(z)|}{|g_{n+1}^{\lambda}|\rho^{n+1}}\\\
&\leq\big{|}1-w^{-2}\big{|}^{-\lambda}+A(\rho,\lambda)n^{\varepsilon-1}+O(n^{-1}).\end{split}$
(4.4)
Notice that
$1-\rho^{-2}\leq|1-w^{-2}|\leq 1+\rho^{-2}.$ (4.5)
Consequently,
$|C_{n+1}^{\lambda}(z)|\geq
c\frac{n^{\lambda-1}\rho^{n+1}}{|\Gamma(\lambda)|}\begin{cases}(1+\rho^{-2})^{-\lambda},\quad&{\rm
if}\;\;\lambda>0,\\\ (1-\rho^{-2})^{-\lambda},\quad&{\rm
if}\;\;\lambda<0,\end{cases}$ (4.6)
where we used (3.9), and the constant $c\cong 1.$
On the other hand, we derive from (2.5), (2.7) and (2.11) that if $\lambda>0,$
$\max_{|x|\leq
1}|C_{n+1}^{\lambda}(x)|=C_{n+1}^{\lambda}(1)\cong\frac{n^{2\lambda-1}}{\Gamma(2\lambda)}.$
(4.7)
Hence, a combination of (4.3), (4.6) and (4.7) leads to (4.1). Similarly, for
$-1/2<\lambda<0,$ we use (2.7) to derive (4.2). ∎
###### Remark 4.1.
For $\lambda>0,$ we obtain from (2.4), (2.8) and (2.11) that
$|C_{n+1}^{\lambda}(x)|\leq\frac{c2^{1-\lambda}\sqrt{e(2+\sqrt{2}\lambda)}}{\Gamma(\lambda)}n^{\lambda-1}(1-x^{2})^{-\lambda/2},\;\;|x|<1.$
(4.8)
Replacing (4.7) by this bound in the above proof, we can derive the point-wise
estimate for $\lambda>0:$
$\big{|}(u-I_{n}u)(x)\big{|}\leq
D(\rho,\lambda)\frac{(1-x^{2})^{-\lambda/2}}{\rho^{n}},\quad|x|<1,$ (4.9)
where the positive constant $D(\rho,\lambda)$ can be worked out as well. It
appears to be sharper than (4.1) at the points which are not too close to the
endpoints $x=\pm 1.$ A similar remark also applies to the Gegenbauer-Gauss-
Lobatto interpolation to be addressed in a minute. ∎
Now, we turn to the estimate of spectral differentiation.
###### Theorem 4.2.
Let $u$ be analytic on and within the ellipse ${\mathcal{E}_{\rho}}$ with foci
$\pm 1$ and $\rho>1$ as defined in (2.15), and let $(I_{n}u)(x)$ be the
interpolant of $u(x)$ at $(n+1)$ Gegenbauer-Gauss points
$\\{x_{j}\\}_{j=0}^{n}$. Then we have
$\max_{0\leq j\leq
n}\big{|}(u-I_{n}u)^{\prime}(x_{j})\big{|}\leq\Lambda(\rho,\lambda)\frac{n^{\lambda+2}}{\rho^{n}},$
(4.10)
where the constant
$\Lambda(\rho,\lambda)=\frac{2c\Gamma(\lambda+1)M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\Gamma(2\lambda+2)(\rho-1)^{2}}\begin{cases}(1+\rho^{-2})^{\lambda},\quad&{\rm
if}\;\;\lambda>0,\\\ (1-\rho^{-2})^{\lambda},\quad&{\rm
if}\;\;\lambda<0,\end{cases}$ (4.11)
and $c,M_{\rho}$ are the same as in Theorem 4.1.
###### Proof.
In view of (2.19) and (2.20), it is enough to replace $x$ and
$C_{n+1}^{\lambda}(x)$ by $x_{j}$ and $\frac{d}{dx}C_{n+1}^{\lambda}(x),$
respectively, in (4.3). Thus, we have
$\begin{split}\big{|}(u-I_{n}u)^{\prime}(x_{j})\big{|}&\leq\frac{M_{\rho}}{2\pi}\frac{|(C_{n+1}^{\lambda})^{\prime}(x_{j})|}{\min_{z\in\mathcal{E}_{\rho}}|C_{n+1}^{\lambda}(z)|}\oint_{\mathcal{E}_{\rho}}\frac{|dz|}{|z-x_{j}|}\\\
&\leq\frac{M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\rho+\rho^{-1}-2}\frac{|(C_{n+1}^{\lambda})^{\prime}(x_{j})|}{\min_{z\in{\mathcal{E}_{\rho}}}|C_{n+1}^{\lambda}(z)|},\quad
0\leq j\leq n.\end{split}$ (4.12)
By (2.6) and (4.7),
$\max_{|x|\leq
1}\big{|}(C_{n+1}^{\lambda})^{\prime}(x)\big{|}=2|\lambda||C_{n}^{\lambda+1}(1)|\cong\frac{2|\lambda|}{\Gamma(2\lambda+2)}n^{2\lambda+1},$
(4.13)
which, together with (4.6) and (4.12), leads to the desired estimate. ∎
###### Remark 4.2.
Obviously, by (2.19),
$(u-I_{n}u)^{\prime}(x)=\frac{1}{2\pi{\rm
i}}\oint_{{\mathcal{E}}_{\rho}}\Big{(}\frac{(C_{n+1}^{\lambda})^{\prime}(x)}{z-x}+\frac{C_{n+1}^{\lambda}(x)}{(z-x)^{2}}\Big{)}\frac{u(z)}{C_{n+1}^{\lambda}(z)}dz.$
(4.14)
If $x\not=x_{j},$ we need to estimate the second term in the summation, which
can be done in the same fashion in the proof of Theorem 4.1. The first term is
actually estimated above. Consequently, we have
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)^{\prime}(x)\big{|}\leq\Lambda(\rho,\lambda)\frac{n^{\lambda+2}}{\rho^{n}}+\frac{1}{\delta_{\rho}}\max_{|x|\leq
1}\big{|}(u-I_{n}u)(x)\big{|},$
where $\delta_{\rho}$ is given by (2.18). Hence, by Theorem 4.1,
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)^{\prime}(x)\big{|}=O\Big{(}\frac{n^{\lambda+2}}{\rho^{n}}\Big{)}.$
(4.15)
In fact, the results for higher-order derivatives can be derived recursively,
and it is anticipated that
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)^{(k)}(x)\big{|}=O\Big{(}\frac{n^{\lambda+2k}}{\rho^{n}}\Big{)},\quad
k\geq 1.$ (4.16)
A similar remark applies to the Gegenbauer-Gauss-Lobatto case below. ∎
### 4.2. Gegenbauer-Gauss-Lobatto interpolation and differentiation
We are now in a position to estimate the Gegenbauer-Gauss-Lobatto
interpolation and spectral differentiation. In this case,
$Q_{n+1}(x)=(1-x^{2})C_{n-1}^{\lambda+1}(x)$ in (2.19)-(2.20). The main result
is stated as follows.
###### Theorem 4.3.
Let $u$ be analytic on and within the ellipse ${\mathcal{E}_{\rho}}$ with foci
$\pm 1$ and $\rho>1$ as defined in (2.15), and let $(I_{n}u)(x)$ be the
interpolant of $u(x)$ at the set of $(n+1)$ Gegenbauer-Gauss-Lobatto points.
* (a)
We have the interpolation error:
$\max_{|x|\leq
1}\big{|}(u-I_{n}u)(x)\big{|}\leq\frac{4cM_{\rho}\sqrt{\rho^{2}+\rho^{-2}}(1+\rho^{-2})^{\lambda+1}}{(1-\rho^{-1})^{2}(\rho-\rho^{-1})^{2}}\frac{\Gamma(\lambda+1)}{\Gamma(2\lambda+2)}\frac{n^{\lambda+1}}{\rho^{n}}.$
(4.17)
* (b)
We have the estimate:
$\begin{split}&\max_{0\leq j\leq
n}\big{|}(u-I_{n}u)^{\prime}(x_{j})\big{|}\leq\frac{8cM_{\rho}\sqrt{\rho^{2}+\rho^{-2}}(1+\rho^{-2})^{\lambda+1}}{(1-\rho^{-1})^{2}(\rho-\rho^{-1})^{2}}\frac{\Gamma(\lambda+2)}{\Gamma(2\lambda+4)}\frac{n^{\lambda+3}}{\rho^{n}}.\end{split}$
(4.18)
Here, $c\cong 1$ and $M_{\rho}=\max_{z\in{\mathcal{E}}_{\rho}}|u(z)|.$
###### Proof.
For any $z\in{\mathcal{E}}_{\rho},$ one verifies that
$\frac{1}{4}(\rho-\rho^{-1})^{2}\leq|z^{2}-1|\leq\frac{1}{4}(\rho+\rho^{-1})^{2},$
and
$\min_{z\in{\mathcal{E}_{\rho}}}\big{|}(1-z^{2})C_{n-1}^{\lambda+1}(z)\big{|}\geq\frac{1}{4}(\rho-\rho^{-1})^{2}\min_{z\in{\mathcal{E}_{\rho}}}\big{|}C_{n-1}^{\lambda+1}(z)\big{|}.$
(4.19)
(a) By (2.19) with $Q_{n+1}(x)=(1-x^{2})C_{n-1}^{\lambda+1}(x)$ and
(2.17)-(2.18), we have the bound of the point-wise error:
$\begin{split}\big{|}(u-I_{n}u)(x)\big{|}&\leq\frac{\big{|}(1-x^{2})C_{n-1}^{\lambda+1}(x)\big{|}}{2\pi}\frac{\max_{z\in\mathcal{E}_{\rho}}|u(z)|}{\min_{z\in{\mathcal{E}_{\rho}}}\big{|}(1-z^{2})C_{n-1}^{\lambda+1}(z)\big{|}}\oint_{\mathcal{E}_{\rho}}\frac{|dz|}{|z-x|}\\\
&\overset{(\ref{minCestGL})}{\leq}\frac{M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\rho+\rho^{-1}-2}\frac{4(\rho-\rho^{-1})^{-2}|C_{n-1}^{\lambda+1}(x)|}{\min_{z\in{\mathcal{E}_{\rho}}}|C_{n-1}^{\lambda+1}(z)|},\quad
x\in[-1,1],\;n\geq 0.\end{split}$ (4.20)
Thus, the estimate (4.17) follows from (4.6) and (4.7).
(b) Similarly, we have
$\begin{split}\big{|}(u-I_{n}u)^{\prime}(x_{j})\big{|}&\leq\frac{M_{\rho}}{2\pi}\frac{\big{|}[(1-x^{2})C_{n-1}^{\lambda+1}(x)]^{\prime}(x_{j})\big{|}}{\min_{z\in\mathcal{E}_{\rho}}\big{|}(1-z^{2})C_{n-1}^{\lambda+1}(z)\big{|}}\oint_{\mathcal{E}_{\rho}}\frac{|dz|}{|z-x_{j}|}\\\
&\leq\frac{4M_{\rho}\sqrt{\rho^{2}+\rho^{-2}}}{\rho+\rho^{-1}-2}\frac{\big{|}[(1-x^{2})C_{n-1}^{\lambda+1}(x)]^{\prime}(x_{j})\big{|}}{(\rho-\rho^{-1})^{2}\min_{z\in{\mathcal{E}_{\rho}}}\big{|}C_{n-1}^{\lambda+1}(z)\big{|}},\quad
0\leq j\leq n.\end{split}$ (4.21)
A direct calculation leads to
$\big{[}(1-x^{2})C_{n-1}^{\lambda+1}(x)\big{]}^{\prime}=-2xC_{n-1}^{\lambda+1}(x)+(1-x^{2})\big{[}C_{n-1}^{\lambda+1}(x)\big{]}^{\prime},$
(4.22)
which, together with (4.7) and (4.13), gives
$\max\limits_{|x|\leq
1}\big{|}[(1-x^{2})C_{n-1}^{\lambda+1}(x)]^{\prime}\big{|}\cong\frac{2(\lambda+1)}{\Gamma(2\lambda+4)}n^{2\lambda+3}.$
(4.23)
A combination of (4.6), (4.21) and (4.23) yields the desired estimate. ∎
###### Remark 4.3.
Similar to Remarks 4.1 and 4.2, we can derive a sharper point-wise estimate
and analyze higher-order derivatives of interpolation errors. ∎
### 4.3. Analysis of quadrature errors
Recall the interpolatory Gegenbauer-Gauss-type quadrature formula:
$\int_{-1}^{1}u(x)(1-x^{2})^{\lambda-1/2}dx\approx\sum_{j=0}^{n}u(x_{j})\omega_{j}=\int_{-1}^{1}(I_{n}u)(x)(1-x^{2})^{\lambda-1/2}dx,$
(4.24)
where the quadrature weights $\\{\omega_{j}\\}_{j=0}^{n}$ are expressed by the
Lagrange basis polynomials (see, e.g., [38, 20]). Observe that
$\Big{|}\int_{-1}^{1}(u-I_{n}u)(x)(1-x^{2})^{\lambda-1/2}dx\Big{|}\leq
h_{0}^{\lambda}\max_{|x|\leq 1}|(u-I_{n}u)(x)|,$ (4.25)
where $h_{0}^{\lambda}$ is given by (2.4). With the aid of interpolation error
estimates in Theorem 4.1 and Theorem 4.3, we are able to derive the quadrature
errors immediately.
### 4.4. Numerical results
In what follows, we provide two numerical examples to demonstrate the
sharpness of the estimates established in Theorems 4.2 and 4.3.
#### 4.4.1. Example 1
We take
$u(x)=\frac{1}{x^{2}+1},$ (4.26)
which has two simple poles at $\pm{\rm i}.$ By (2.16), we are free to choose
${\mathcal{E}}_{\rho}$ with $\rho$ in the range
$1<\rho<1+\sqrt{2}\approx 2.414,$ (4.27)
such that $u$ is analytic on and within ${\mathcal{E}}_{\rho}.$ To compare the
(discrete) maximum error of spectral differentiation with the upper bound, we
sample about $2000$ values of $\rho$ equally from $(1,1+\sqrt{2}),$ and find a
tighter upper bound (which is usually attained when $\rho$ is close to
$1+\sqrt{2}$ ). In Figure 4.1 (a)-(b), we plot the (discrete) maximum errors
of spectral differentiation against the upper bounds. We visualize the
exponential decay of the errors, and the upper bounds and the errors decay at
almost the same rate. Moreover, it seems that the bounds are slightly sharper
in the Gegenbauer-Gauss case.
#### 4.4.2. Example 2
The estimates indicates that the errors essentially depend on the location of
singularity (although it affects the constant $M_{\rho}$) rather than the
behavior of the singularity. To show this, we test the function with poles at
$\pm{\rm i}$ of order $2:$
$u(x)=\frac{1}{(x^{2}+1)^{2}}.$ (4.28)
We plot in Figure 4.1 (c)-(d) the errors and upper bounds as in (a)-(b).
Indeed, a similar convergence behavior is observed. Indeed, Boyd [6] pointed
out that the type of singularity might change the rate of convergence by a
power of $n,$ but not an exponential function of $n.$
(a) Upper bound vs Max.-error (Example 1)
(b) Upper bound vs Max.-error (Example 1)
(c) Upper bound vs Max.-error (Example 2)
(d) Upper bound vs Max.-error (Example 2)
Figure 4.1. The (discrete) maximum errors of Gegenbauer-Gauss and Gegenbauer-
Gauss-Lobatto spectral differentiation against the upper bounds in Theorems
4.2 and 4.3 with $\lambda=1/2,3/2.$ Example 1 (a)-(b), and Example 2 (c)-(d).
In the legend, GG SP err (resp. GGL SP err) represents the Gegenbauer-Gauss
(resp. Gegenbauer-Gauss-Lobatto) spectral differentiation error.
Concluding remarks In this paper, the exponential convergence of Gegenbauer
interpolation, spectral differentiation and quadrature of functions analytic
on and within a sizable ellipse is analyzed. Sharp estimates in the maximum
norm with explicit dependence on the important parameters are obtained.
Illustrative numerical results are provided to support the analysis. For
clarity of presentation, it is assumed that $\lambda$ is fixed in our
analysis, but the dependence of the error on this parameter can also be
tracked if it is necessary.
Acknowledgement
The first author would like to thank the Division of Mathematical Sciences at
Nanyang Technological University in Singapore for the hospitality during the
visit. The second author is grateful to Prof. Jie Shen at Purdue University
for bringing this topic to attention. The authors would also like to thank the
anonymous referees for their valuable comments and suggestions.
## Appendix A Exponential convergence of Gegenbauer expansions
To this end, we discuss the exponential convergence of Gegenbauer polynomial
(with fixed $\lambda$) expansions of functions satisfying the assumption of
analyticity in [24], that is, (2.14) in the form:
$\max_{|x|\leq 1}\Big{|}\frac{d^{k}u}{dx^{k}}(x)\Big{|}\leq
C(\delta)\frac{k!}{\delta^{k}},\quad\delta\geq 1,\;\;\forall k\geq 0,$ (A.1)
where $C(\delta)$ is a positive constant depending only on $\delta.$ We write
$u(x)=\sum_{l=0}^{\infty}\hat{u}_{l}^{\lambda}C_{l}^{\lambda}(x)\;\;\;{\rm
with}\;\;\;\hat{u}_{l}^{\lambda}=\frac{1}{h_{l}^{\lambda}}\int_{-1}^{1}u(x)C_{l}^{\lambda}(x)(1-x^{2})^{\lambda-1/2}dx,$
where $h_{l}^{\lambda}$ is defined in (2.4).
###### Proposition A.1.
If $u(x)$ is an analytic function on $[-1,1]$ satisfying the assumption (A.1),
then for fixed $\lambda>-1/2,$ we have
$\max_{|x|\leq 1}\big{|}(\pi_{n}^{\lambda}u-u)(x)\big{|}\leq
C(\delta,\lambda)\frac{n^{\lambda}}{(2\delta)^{n}},$ (A.2)
where
$(\pi_{n}^{\lambda}u)(x)=\sum_{l=0}^{n}\hat{u}_{l}^{\lambda}C_{l}^{\lambda}(x),$
and $C(\delta,\lambda)$ is a positive constant depending on $\delta$ and
$\lambda.$
###### Proof.
The estimate follows directly from Theorem 4.3 in [24]. Indeed, by Theorem 4.3
in [24], we have
$\begin{split}\max_{|x|\leq 1}\big{|}(\pi_{n}^{\lambda}u-u)(x)\big{|}\leq
A\frac{C(\delta)\Gamma(\lambda+1/2)\Gamma(n+2\lambda+1)}{n\sqrt{\lambda}(2\delta)^{n}\Gamma(2\lambda)\Gamma(n+\lambda)},\quad
n\geq 1,\;\lambda>-1/2,\end{split}$
where $A$ is a generic positive constant independent of $\lambda,n$ and $u.$
For fixed $\lambda>-1/2,$ using (2.11) leads to
$\begin{split}\frac{\Gamma(n+2\lambda+1)}{\Gamma(n+\lambda+1)}&=\frac{n+\lambda+1}{n+2\lambda+1}\frac{\Gamma(n+2\lambda+2)}{\Gamma(n+\lambda+2)}\\\
&\leq
A(n+2\lambda+1)^{\lambda}\Big{(}1+\frac{\lambda}{n+\lambda+1}\Big{)}^{n+\lambda+3/2}e^{-\lambda}\\\
&\leq A(n+2\lambda+1)^{\lambda}e^{\frac{\lambda}{2(n+\lambda+1)}},\end{split}$
where in the last step, we used the inequality $1+x\leq e^{x}$ for all real
$x.$ Therefore, we obtain the estimate
$\begin{split}\max_{|x|\leq 1}\big{|}(\pi_{n}^{\lambda}u-u)(x)\big{|}\leq
C(\delta,\lambda)\frac{n+\lambda}{n}\frac{(n+2\lambda+1)^{\lambda}}{(2\delta)^{n}}.\end{split}$
This implies (A.2). ∎
###### Remark A.1.
The error $\max_{|x|\leq 1}\big{|}(\pi_{n}^{\lambda}u-u)(x)\big{|}$ is termed
as the regularization error in [24, 23]. This, together with the so-called
truncation error, contributes to the error of the Gegenbauer reconstruction.
As shown in [24], an exponential convergence could be recovered from Fourier
partial sum of a nonperiodic analytic function, when $\lambda$ grows linearly
with $n.$ However, the estimate (A.2) shows that the regularization error
alone decays exponentially for fixed $\lambda$. ∎
###### Remark A.2.
It is also interesting to study the exponential convergence of Gegenbauer
expansions of analytic functions, which are analytic on and within the
Bernstein ellipse ${\mathcal{E}}_{\rho}$ in (2.15). We refer to [10, 36, 33]
for the results on Chebyshev expansions. However, the analysis for the
Legendre expansions is much more involved. Davis [10] stated an estimate due
to K. Neumann (see Page 312 of [10]), that is, if $u$ is analytic on and
inside ${\mathcal{E}}_{\rho}$ with $\rho>1,$ then the Legendre expansion
coefficient satisfies
$\underset{{n\to\infty}}{\lim\sup}\,|\hat{u}_{n}|^{1/n}=\rho^{-1}.$ (A.3)
A more informative estimate:
$|\hat{u}_{n}|\leq C(\rho)\frac{n}{\rho^{n+1}},\quad\forall n\geq 0,$ (A.4)
was presented in the very recent paper [42]. Some discussions on the
Gegenbauer expansions can be found in [23]. We shall report the analysis for
the general Jacobi expansions in [43]. ∎
## Appendix B Proof of Lemma 3.1
We carry out the proof by induction.
Apparently, by (2.1), $C_{0}^{\lambda}(z)=1,$ so (3.1)-(3.2) holds for $n=0.$
Similarly, we can verify the case with $n=1.$
Assume that the formula holds for $C_{n-2}^{\lambda}(z)$ and
$C_{n-1}^{\lambda}(z)$ with $n\geq 2$. It follows from the three-term
recurrence relation (2.1) that
$\begin{split}nC_{n}^{\lambda}(z)&=2(n+\lambda-1)z\sum_{k=0}^{n-1}g_{k}^{\lambda}g_{n-1-k}^{\lambda}w^{n-2k-1}\\\
&\quad-(n+2\lambda-2)\sum_{k=0}^{n-2}g_{k}^{\lambda}g_{n-2-k}^{\lambda}w^{n-2k-2}\\\
&=(n+\lambda-1)\sum_{k=0}^{n-1}g_{k}^{\lambda}g_{n-1-k}^{\lambda}w^{n-2k}+(n+\lambda-1)\sum_{k=0}^{n-1}g_{k}^{\lambda}g_{n-1-k}^{\lambda}w^{n-2k-2}\\\
&\quad-(n+2\lambda-2)\sum_{k=0}^{n-2}g_{k}^{\lambda}g_{n-2-k}^{\lambda}w^{n-2k-2}\\\
&=(n+\lambda-1)g_{n-1}^{\lambda}(w^{n}+w^{-n})+\sum_{k=1}^{n-1}D_{n,k}^{\lambda}\,g_{k}^{\lambda}g_{n-k}^{\lambda}w^{n-2k},\end{split}$
where
$D_{n,k}^{\lambda}=(n+\lambda-1)\left(\frac{g_{n-k-1}^{\lambda}}{g_{n-k}^{\lambda}}+\frac{g_{k-1}^{\lambda}}{g_{k}^{\lambda}}\right)-(n+2\lambda-2)\frac{g_{n-k-1}^{\lambda}}{g_{n-k}^{\lambda}}\frac{g_{k-1}^{\lambda}}{g_{k}^{\lambda}}.$
One verifies from (3.2) that
${g_{n}^{\lambda}}=\frac{n+\lambda-1}{n}{g_{n-1}^{\lambda}},\quad
D_{n,k}^{\lambda}=n.$ (B.1)
This completes the induction. ∎
## References
* [1] G.E. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
* [2] I. Babuška and B.Q. Guo. Direct and inverse approximation theorems for the $p$-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces. SIAM J. Numer. Anal., 39(5):1512–1538 (electronic), 2001/02.
* [3] C. Bernardi, M. Dauge, and Y. Maday. Spectral Methods for Axisymmetric Domains, volume 3 of Series in Applied Mathematics (Paris). Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaïez.
* [4] C. Bernardi and Y. Maday. Spectral Methods. In Handbook of Numerical Analysis, Vol. V, Handb. Numer. Anal., V, pages 209–485. North-Holland, Amsterdam, 1997.
* [5] J.P. Boyd. The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval. J. Comput. Phys., 110(2):360–372, 1994.
* [6] J.P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Publications Inc., 2001.
* [7] J.P. Boyd. Trouble with Gegenbauer reconstruction for defeating Gibbs’ phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations. J. Comput. Phys., 204(1):253–264, 2005.
* [8] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods: Fundamentals in Single Domains. Springer, Berlin, 2006.
* [9] Y.P. Chen and T. Tang. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comp., 79(269):147–167, 2010.
* [10] P.J. Davis. Interpolation and Approximation. Dover Publications, Inc, New Year, 1975.
* [11] P.J. Davis and P. Rabinowitz. Methods of Numerical Integration. Computer Science and Applied Mathematics. Academic Press Inc., Orlando, FL, second edition, 1984.
* [12] E.H. Doha. The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. Comput. Math. Appl., 21(2-3):115–122, 1991.
* [13] E.H. Doha. The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function. J. Comput. Appl. Math., 89(1):53–72, 1998.
* [14] E.H. Doha and A.H. Bhrawy. Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials. Numer. Algorithms, 42(2):137–164, 2006.
* [15] E.H. Doha and A.H. Bhrawy. A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differential Equations, 25(3):712–739, 2009\.
* [16] E.H. Doha, A.H. Bhrawy, and W.M. Abd-Elhameed. Jacobi spectral Galerkin method for elliptic Neumann problems. Numer. Algorithms, 50(1):67–91, 2009.
* [17] M. Dubiner. Spectral methods on triangles and other domains. J. Sci. Comput., 6(4):345–390, 1991.
* [18] David Elliott. Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp., 25:309–315, 1971.
* [19] B. Fornberg. A Practical Guide to Pseudospectral Methods. Cambridge Univ Pr, 1998.
* [20] D. Funaro. Polynomial Approxiamtions of Differential Equations. Springer-Verlag, 1992.
* [21] A. Gelb and Z. Jackiewicz. Determining analyticity for parameter optimization of the Gegenbauer reconstruction method. SIAM J. Sci. Comput., 27(3):1014–1031, 2006.
* [22] D. Gottlieb and S.A. Orszag. Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial Mathematics, 1977.
* [23] D. Gottlieb and C.W. Shu. On the Gibbs phenomenon and its resolution. SIAM Rev., 39(4):644–668, 1997.
* [24] D. Gottlieb, C.W. Shu, A. Solomonoff, and H. Vandeven. On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math., 43(1-2):81–98, 1992. Orthogonal polynomials and numerical methods.
* [25] B.Q. Guo. Mathematical Foundation and Applications of the $p$ and $h$-$p$ Finite Element Methods, volume 4 of Series in Analysis. World Scientific, 2011.
* [26] B.Y. Guo. Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity. Appl. Numer. Math., 38(4):403–425, 2001.
* [27] B.Y. Guo and L.L. Wang. Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math., 14(3):227–276, 2001.
* [28] B.Y. Guo and L.L. Wang. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory, 128(1):1–41, 2004.
* [29] B. Gustafsson. The work of David Gottlieb: A success story. Commun. Comput. Phys., 9(3):481–496, 2011.
* [30] A. Jeffrey. Table of Integrals, Series, and Products (Seventh Edition). Elsevier, San Diego, CA, 2007.
* [31] G.E. Karniadakis and S.J. Sherwin. Spectral/$hp$ Element Methods for Computational Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, second edition, 2005.
* [32] I. Krasikov. An upper bound on Jacobi polynomials. J. Approx. Theory, 149(2):116–130, 2007.
* [33] J.C. Mason and D.C. Handscomb. Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL, 2003.
* [34] P. Nevai, T. Erdélyi, and A.P. Magnus. Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal., 25(2):602–614, 1994.
* [35] S.C. Reddy and J.A.C. Weideman. The accuracy of the Chebyshev differencing method for analytic functions. SIAM J. Numer. Anal., 42(5):2176–2187 (electronic), 2005.
* [36] T.J. Rivlin and V. Kalashnikov. Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory. Wiley New York, 1990.
* [37] J. Shen, T. Tang, and L.L. Wang. Spectral Methods : Algorithms, Analysis and Applications, volume 41 of Series in Computational Mathematics. Springer, 2011.
* [38] G. Szegö. Orthogonal Polynomials (Fourth Edition). AMS Coll. Publ., 1975.
* [39] E. Tadmor. The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal., 23(1):1–10, 1986.
* [40] L.N. Trefethen. Spectral Methods in Matlab. Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
* [41] L.N. Trefethen. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev., 50(1):67–87, 2008.
* [42] H.Y. Wang and S.H. Xiang. On the convergence rates of Legendre approximation. Math. Comput., In press, 2011.
* [43] L.L. Wang, Z.Q. Xie, and X.D. Zhao. A note on exponential convergence of Jacobi polynomial expansions. In preparation, 2011.
* [44] Z.Q. Wang and B.Y. Guo. Jacobi rational approximation and spectral method for differential equations of degenerate type. Math. Comp., 77(262):883–907, 2008.
* [45] G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1995.
* [46] Z. Zhang. Superconvergence of spectral collocation and $p$-version methods in one dimensional problems. Math. Comp., 74(252):1621–1636 (electronic), 2005.
* [47] Z. Zhang. Superconvergence of a Chebyshev spectral collocation method. J. Sci. Comput., 34(3):237–246, 2008.
|
arxiv-papers
| 2011-09-26T09:56:45 |
2024-09-04T02:49:22.518760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ziqing Xie, Li-Lian Wang and Xiaodan Zhao",
"submitter": "Li-Lian Wang Dr.",
"url": "https://arxiv.org/abs/1109.5509"
}
|
1109.5842
|
# A proposal for the measurement of Rashba and Dresselhaus spin-orbit
interaction strengths in a single sample
Santanu K. Maiti santanu@post.tau.ac.il School of Chemistry, Tel Aviv
University, Ramat-Aviv, Tel Aviv-69978, Israel Shreekantha Sil Department of
Physics, Visva-Bharati, Santiniketan, West Bengal-731 235, India Arunava
Chakrabarti Department of Physics, University of Kalyani, Kalyani, West
Bengal-741 235, India
###### Abstract
We establish an exact analytical treatment for the determination of the
strengths of the Rashba and Dresselhaus spin-orbit interactions in a single
sample by measuring persistent spin current. A hidden symmetry is exploited in
the Hamiltonian to show that the spin current vanishes when the strength of
the Dresselhaus interaction becomes equal to the strength of the Rashba term.
The results are sustained even in the presence of disorder and thus an
experiment in this regard will be challenging.
###### pacs:
73.23.-b, 71.70.Ej, 73.23.Ra
Spintronics is a rapidly growing field that aims at the achievement of
efficient quantum devices, such as the magnetic memory circuits and computers,
in which one needs to manipulate the spin of the electron rather than its
charge zutic ; datta ; ando ; ding ; bellucci ; aharony1 ; aharony2 . Magnetic
nano-structures and quasi-one dimensional semiconductor rings have been
acknowledged as ideal candidates for testing the effects of quantum coherence
in low dimensions, and have been extensively investigated to test their
potential as new generation quantum devices mentioned above moldo ; engels .
Spin-orbit interaction (SOI) in semiconductors is a central mechanism that
determines the essential physics in the meso- and nano-scales, and is largely
responsible for the prospect of semiconducting structures as potential quantum
devices. Rashba spin-orbit interaction (RSOI) and the Dresselhaus spin-orbit
interaction (DSOI) are the two typical spin-orbit interactions that one
encounters in a conventional semiconductor.
The Rashba spin-orbit field in a solid is attributed to an electric field that
originates from a structural inversion asymmetry whereas, the Dresselhaus
interaction comes from bulk inversion asymmetry meier . Quantum rings formed
at the interface of two semiconducting materials are ideal candidates to
unravel the interplay of the two kinds of SOI. A quantum ring in a
heterojunction, formed by trapping a two-dimensional gas of electrons in a
quantum well, generates a band offset at the interface of two different
semiconducting materials. This creates an electric field. This electric field
is described by a potential gradient normal to the interface premper . The
potential at the interface is asymmetric, leading to the presence of a RSOI.
On the other hand, at such interfaces, the bulk inversion symmetry is
naturally broken, and DSOI plays its part.
Needless to say, an accurate estimation of the SOI’s is crucial in the field
of spintronics. The RSOI can be controlled by a gate voltage placed in the
vicinity of the sample engels ; meier ; premper ; cmhu ; grundler . Thus, in
principle, all possible values of the RSOI can be achieved. Measurement of
RSOI has already been reported in the literature nitta . Comparatively
speaking, reports on the techniques of measurement of the DSOI are relatively
few, and are mainly based on an optical monitoring of the spin precession of
the electrons meier ; studer , measurement of electrical conductance of nano-
wires scheid ; jaya , or photo-galvanic methods yin .
In the present communication we propose a method to measure the strengths of
both the RSOI and DSOI by measuring the spin current flowing through a single
sample. An important issue, while the measurement of a spin current
Figure 1: (Color online). Schematic view of a mesoscopic ring where a section
of the ring (shaded region measured by the angle $\gamma$) is subjected to
SOI, while the other section is free from the SOI.
in a mesoscopic system is concerned, is the non-conservation of spin caused by
the ubiquitous presence of the spin-orbit coupling. A proper definition of
spin current is therefore urgent wang and, its measurement in a sample is
challenging. It has only recently been possible, using a novel version of the
Doppler effect, to quantify the spin current in a ferromagnetic wire vlaminck
, and by using spin relaxation modulation induced by spin injection ando1 ;
ando2 .
Inspired by the success in the measurement of spin current we provide a
completely analytical scheme that immediately suggests an experiment to
measure the strengths of either RSOI or the DSOI in the same sample. Our
system is a mesoscopic ring grafted at a heterojunction, but in such a manner
that the spin-orbit interaction is effective only in a fraction of the
perimeter of the ring (see Fig. 1). The spin current should be measured in the
SO interaction-free region where the spin flip scattering is absent. One thus
need not worry about the non-conservation of spin and the spin current in any
‘bond’ in the non-shaded region in Fig. 1 is the same, and becomes a well
defined quantity.
In this letter we prove that, in such a mesoscopic ring the spin current will
be zero whenever the strengths of the RSOI and the DSOI are equal. As we have
already mentioned above, the RSOI strength can in principle, be smoothly
varied with applied gate voltage. This leads to the idea of estimating the
strength of the DSOI if we happen to know a measured value of the RSOI. The
idea is as follows. We devise a unitary transformation that acts on the full
Hamiltonian of the system to extract a subtle symmetry that is finally
exploited to achieve the result. Incidentally, similar kind of symmetry has
previously been reported in literature schliemann ; andrei ; brataas . In this
work we have focused on the effect of the symmetry on the spin current for a
system in the presence of both the Rashba and the Dresselhaus interactions.
Our result is analytically exact, and is true for any value of the spin-orbit
interactions. It should therefore be observable in a suitably designed
experiment. In addition, we numerically calculate the persistent spin current
in the system, and determine its dependence on the length of the sample
involving the SOI (determined by the angle $\gamma$ in Fig. 1). The numerical
results corroborate our analytical work, and remain robust even in the
presence of disorder.
$\bullet$ Analytical treatment for the determination of SOI strengths: Let us
refer to Fig. 1. Only the bold portion of the arc contains the spin-orbit
interactions. Within a tight-binding framework the Hamiltonian for such a ring
reads,
$\mbox{\boldmath$H$}=\mbox{\boldmath$H_{0}$}-i\alpha\mbox{\boldmath$H_{RSO}$}+i\beta\mbox{\boldmath$H_{DSO}$}$
(1)
where,
$\mbox{\boldmath$H$}_{0}=\sum_{n}\mbox{\boldmath$c_{n}^{\dagger}\epsilon_{n}c_{n}$}+\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}t$}\mbox{\boldmath$c_{n}$}+h.c.\right).$
(2)
The Rashba and the Dresselhaus spin-orbit parts of the Hamiltonian, viz,
$H_{RSO}$ and $H_{DSO}$, are given by,
$H_{RSO}$ $\displaystyle=$
$\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\cos\varphi_{n,n+1}\mbox{\boldmath$c_{n+1}$}\right.$
$\displaystyle+$
$\displaystyle\left.\mbox{\boldmath$c_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\sin\varphi_{n,n+1}\mbox{\boldmath$c_{n+1}$}+h.c.\right)$
$H_{DSO}$ $\displaystyle=$
$\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\cos\varphi_{n,n+1}\mbox{\boldmath$c_{n+1}$}\right.$
(3) $\displaystyle+$
$\displaystyle\left.\mbox{\boldmath$c_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\sin\varphi_{n,n+1}\mbox{\boldmath$c_{n+1}$}+h.c.\right)$
where, $n=1$, $2$, $\dots$, $N$ is the site index along the azimuthal
direction $\varphi$ of the ring. The other factors in Eq. 3 are as follows.
$c_{n}$=$\left(\begin{array}[]{c}c_{n\uparrow}\\\
c_{n\downarrow}\end{array}\right);$
$\epsilon_{n}$=$\left(\begin{array}[]{cc}\epsilon_{n\uparrow}&0\\\
0&\epsilon_{n\downarrow}\end{array}\right);$
$t$=$t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$.
Here $\epsilon_{n\sigma}$ is the site energy of an electron at the $n$-th site
of the ring with spin $\sigma$ ($\uparrow,\downarrow$). $t$ is the nearest-
neighbor hopping integral. $\alpha$ and $\beta$ are the isotropic nearest-
neighbor transfer integrals which measure the strengths of Rashba and
Dresselhaus SOI, respectively, and
$\varphi_{n,n+1}=\left(\varphi_{n}+\varphi_{n+1}\right)/2$, where
$\varphi_{n}=2\pi(n-1)/N$. $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the
Pauli spin matrices. $c_{n\sigma}^{\dagger}$ ($c_{n\sigma}$) is the creation
(annihilation) operator of an electron at the site $n$ with spin $\sigma$
($\uparrow,\downarrow$). The form of the Hamiltonian written in the case of a
ring is discussed elsewhere santanu .
We now define a transformation,
$\mbox{\boldmath$U$}=\mbox{\boldmath$\sigma_{z}\left(\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}}\right)$}$
(4)
which, by definition is unitary. It is simple to verify that,
$\mbox{\boldmath$U\sigma_{x}U^{\dagger}$}=-\mbox{\boldmath$\sigma_{y}$};\,\mbox{\boldmath$U\sigma_{y}U^{\dagger}$}=-\mbox{\boldmath$\sigma_{x}$};\,\mbox{\boldmath$U\sigma_{z}U^{\dagger}$}=-\mbox{\boldmath$\sigma_{z}$}.$
(5)
Using the operator defined in Eq. 4 we now make a change of basis and describe
the full Hamiltonian in terms of the operators
$\mbox{\boldmath$\tilde{c}_{n}$}=\mbox{\boldmath$Uc_{n}$}$, and
$\mbox{\boldmath$\tilde{c}_{n}^{\dagger}$}=\mbox{\boldmath$c_{n}^{\dagger}U^{\dagger}$}$.
The Hamiltonian in the new basis reads,
$\mbox{\boldmath$\tilde{H}$}=\mbox{\boldmath$\tilde{H}_{0}$}-i\beta\mbox{\boldmath$\tilde{H}_{RSO}$}+i\alpha\mbox{\boldmath$\tilde{H}_{DSO}$}$
(6)
where,
$\mbox{\boldmath$\tilde{H}_{0}$}=\sum_{n}\mbox{\boldmath$\tilde{c}_{n}^{\dagger}\epsilon_{n}\tilde{c}_{n}$}+\sum_{n}\left(\mbox{\boldmath$\tilde{c}_{n+1}^{\dagger}t$}\mbox{\boldmath$\tilde{c}_{n}$}+h.c.\right)$
(7)
and, the spin-orbit parts of the Hamiltonian in this new basis are given by,
$\tilde{H}_{RSO}$ $\displaystyle=$
$\displaystyle\sum_{n}\left(\mbox{\boldmath$\tilde{c}_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\cos\varphi_{n,n+1}\mbox{\boldmath$\tilde{c}_{n+1}$}\right.$
$\displaystyle+$
$\displaystyle\left.\mbox{\boldmath$\tilde{c}_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\sin\varphi_{n,n+1}\mbox{\boldmath$\tilde{c}_{n+1}$}+h.c.\right)$
$\tilde{H}_{DSO}$ $\displaystyle=$
$\displaystyle\sum_{n}\left(\mbox{\boldmath$\tilde{c}_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\cos\varphi_{n,n+1}\mbox{\boldmath$\tilde{c}_{n+1}$}\right.$
(8) $\displaystyle+$
$\displaystyle\left.\mbox{\boldmath$\tilde{c}_{n}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\sin\varphi_{n,n+1}\mbox{\boldmath$\tilde{c}_{n+1}$}+h.c.\right).$
A simultaneous look at the Eqs. 6 and 8 reveal that in the new basis, the
strengths of the Rashba and Dresselhaus spin-orbit interactions viz, $\alpha$
and $\beta$ have been interchanged. As a consequence, the polarized spin
current operator in the quantized direction ($+Z$), for the free region,
defined as
$\mbox{\boldmath$J_{s}^{z}$}=\frac{1}{2}\left(\mbox{\boldmath${\sigma_{z}}$}\mbox{\boldmath${\dot{x}}$}+\mbox{\boldmath${\dot{x}}$}\mbox{\boldmath${\sigma_{z}}$}\right)$
flips its sense of circulation, and becomes $-\mbox{\boldmath$J_{s}^{z}$}$.
Here, $x$$=\sum n\mbox{\boldmath$c_{n}^{\dagger}c_{n}$}$. This immediately
leads to the most important observation that, as soon as $\alpha=\beta$, $H$
and $\tilde{H}$ become identical. That is, the full Hamiltonian remains
invariant under the unitary transformation defined by Eq. 4 whenever the
strengths of the Rashba and the Dresselhaus spin-orbit interactions become
equal. The energy eigenvalues obviously remain unchanged in this case.
At the same time, from the last equation in Eq. 5 we see that
$\mbox{\boldmath$U\sigma_{z}U^{\dagger}$}=-\mbox{\boldmath$\sigma_{z}$}$ which
means that, the spin current $J_{s}^{z}$ calculated in the original basis
becomes exactly equal to the spin current $-J_{s}^{z}$ calculated in the new
basis. This can happen only when $J_{s}^{z}=0$ as $\alpha=\beta$. The RSOI can
be controlled by a gate voltage, and hence its strength is determined. So, an
experiment on the measurement of spin current in a mesoscopic ring, where the
RSOI is continuously varied, will show a complete disappearance of the spin
current as soon as the strength of the DSOI becomes equal to that of the RSOI.
We thus have a means of determining DSOI from a measurement of the spin
current. The spin current, however, as stated earlier, is to be measured in
that region of the ring which is free from the spin-orbit coupling.
Needless to say, the strength of the Rashba interaction can be obtained by the
same method, if we know the Dresselhaus interaction beforehand.
$\bullet$ Calculation of persistent spin current: In the second quantized form
the spin current operator $J_{s}^{z}$ for the free region can be written as,
$J_{s}^{z}$ $\displaystyle=$ $\displaystyle 2i\pi
t\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}\sigma_{z}$}\mbox{\boldmath$c_{n+1}$}-\mbox{\boldmath$c_{n+1}^{\dagger}\sigma_{z}$}\mbox{\boldmath$c_{n}$}\right).$
(9)
Therefore, for a particular eigenstate $|\psi_{k}\rangle$ the persistent spin
current becomes,
$J_{s}^{z,k}=\langle\psi_{k}|\mbox{\boldmath$J_{s}^{z}$}|\psi_{k}\rangle$,
where
$|\psi_{k}\rangle=\sum\limits_{p}a_{p,\uparrow}^{k}|p\uparrow\rangle+a_{p,\downarrow}^{k}|p\downarrow\rangle$.
Here $|p\uparrow\rangle$’s and $|p\downarrow\rangle$’s are the Wannier states
and $a_{p,\uparrow}^{k}$’s
Figure 2: (Color online). Persistent spin current as a function of SO coupling
strength for an ordered $40$-site half-filled ring, when $\gamma$ is set at
$\pi$. (a) $\beta=0$ and (b) $\alpha=0$.
and $a_{p,\downarrow}^{k}$’s are the corresponding coefficients. In terms of
these coefficients, the final expression of persistent spin current for $k$-th
eigenstate reads,
$\displaystyle J_{s}^{z,k}$ $\displaystyle=$ $\displaystyle 2\pi
it\sum_{n}\left\\{a_{n,\uparrow}^{k\,*}\,a_{n+1,\uparrow}^{k}-a_{n+1,\uparrow}^{k\,*}\,a_{n,\uparrow}^{k}\right\\}$
(10) $\displaystyle-$ $\displaystyle 2\pi
it\sum_{n}\left\\{a_{n,\downarrow}^{k\,*}\,a_{n+1,\downarrow}^{k}-a_{n+1,\downarrow}^{k\,*}\,a_{n,\downarrow}^{k}\right\\}.$
Let us rename the polarized spin current $J_{s}^{z,k}$ as $J_{s}^{k}$ for the
sake of simplicity. At absolute zero temperature ($T=0$ k), net persistent
spin current in a mesoscopic ring for a particular filling can be obtained by
taking the sum of individual contributions from the energy levels with
energies less than or equal to Fermi energy $E_{F}$. Therefore, for $N_{e}$
electron system total spin current becomes $J_{s}=\sum_{k}^{N_{e}}J_{s}^{k}.$
We measure spin current in the region which is free from the SOI, and, since
the spin currents between any two neighboring sites in this interacting free
region are identical to each other we compute the current only in a single
bond.
In the present work we compute all the essential features of persistent spin
current and related issues at absolute zero temperature and choose the units
where $c=h=e=1$. Throughout our numerical work we fix $t=1$ and measure the
energy scale in unit of $t$.
$\bullet$ Numerical results: In Fig. 2, we present the variation of the spin
current as a function of the RSOI and the DSOI for a $40$-site ordered half-
filled ring. The angle $\gamma=\pi$. Figs. 2(a) and (b) refer to
Figure 3: (Color online). Persistent spin current as a function of Rashba SO
interaction strength for a $40$-site ordered ring in the half-filled case
considering different values of $\beta$, when $\gamma$ is fixed at $\pi$.
Figure 4: (Color online). Persistent spin current as a function of $\gamma$
for a $40$-site ordered ring in the half-filled case considering different
values of $\alpha$, when $\beta$ is fixed at $0$.
the cases where only the RSOI and the DSOI are present respectively. The RSOI
and the DSOI drive the spin current in opposite directions (with equal
magnitudes when $\alpha=\beta$), as is evident from the figure.
Figure 3 shows the variation of the spin current as a function of the strength
of the RSOI, plotted for various values of the strength of the DSOI. This
figure is a clear demonstration of the fact that, whenever the strength of the
DSOI becomes equal to that of the RSOI, the spin current becomes zero. We
again emphasize that, the vanishing of the persistent spin current will be
observed for any strength of the RSOI, provided it equals the DSOI strength.
Hence, it is expected to be observed in experiments as well. The region of the
mesoscopic ring in which the SOI is ‘on’, plays an important role in
determining the strength of the spin current in the system. To get an idea, we
systematically study the variation of $J_{s}$ as a function of $\gamma$ with
the DSOI $\beta=0$. For a given value of the angle $\gamma$, the current
increases with increasing values of the RSOI strength $\alpha$ (see Fig. 4).
Finally, in view of a possible experiment, we test the robustness of our
results by considering a $40$-site disordered ring. Disorder is introduced via
a random distribution (width $W=1$) of the values of the on-site
Figure 5: (Color online). Persistent spin current as a function of Rashba SO
interaction strength for a $40$-site disordered ($W=1$) ring in the half-
filled case considering different values of $\beta$, when $\gamma$ is fixed at
$\pi$.
potentials (diagonal disorder), and results averaged over $40$ disorder
configurations have been presented (see Fig. 5). Since $H_{0}$ remains
invariant under the unitary transformation defined by Eq. 4, all the
qualitative results should remain unchanged even in the presence of disorder.
However, the disorder will reduce the amplitude of the spin current. This is
precisely what we find in our numerical analysis. The current still becomes
zero as soon as $\alpha=\beta$. This observation strengthens our claim that a
suitably designed experiment on the measurement of the spin current will lead
to an exciting method of measurement of the Dresselhaus spin-orbit interaction
from a knowledge of the gate controlled Rashba spin-orbit interaction in a
mesoscopic ring.
A relevant question in view of the above discussion is the possibilities of
the experimental detection of persistent spin current. The spin current, as
obtained from the theoretical calculations already existing in the literature
as well as from the present calculation, is a robust phenomenon undisturbed by
the presence of disorder. Determination of spin current is also very much
possible, and already a few experiments have been carried out in this
direction expt1 ; expt2 ; expt3 ; expt4 . For example, by studying the Kerr
effect expt1 ; expt2 ; expt3 associated with spin accumulations induced by
the spin current or by investigating the reciprocal spin Hall effect expt4
persistent spin current can be detected. It is also well known that a spin
current may produce a spin torque which can be measured experimentally expt5 ;
expt6 ; expt7 . This definitely provides a way of estimating the spin current.
Probably a more convenient way of detecting persistent spin current is the
measurement of the electric field and electric potential induced by it expt8 ;
expt9 ; expt10 ; expt11 ; expt12 . This is quite analogous to the detection of
persistent charge current in a mesoscopic ring by measuring its induced
magnetic field expt13 ; expt14 . We can also measure the strengths of spin-
orbit interactions by attaching two electrodes in a mesoscopic ring. In that
case also the persistent spin current as well as the transport spin current
have well defined expressions and simply by measuring the transport spin
current we can have an estimate of the SO interaction strengths. These latter
observations are the major issues of our forthcoming paper.
In conclusion, we present an exact analytical method which shows that the
strengths of the Rashba or the Dresselhaus spin-orbit interactions can be
determined in a single mesoscopic ring by noting the vanishing of the spin
current in the sample. A unitary transformation is prescribed, which when
applied to the spin-orbit Hamiltonian brings out a hidden symmetry. The
symmetry is exploited to prove that, by making the strengths of the two
interactions equal, one achieves a zero spin current in the system. We provide
numerical results which support all our analytical findings, and show that the
vanishing of the spin current is a robust effect even in the presence of
disorder. This last observation gives us confidence to propose an experiment
in this line.
## References
* (1) I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
* (2) S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
* (3) Mesoscopic Physics and Electronics, edited by T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, and H. Nakashima (Springer, Berlin, 1998).
* (4) G.-H. Ding and B. Dong, Phys. Rev. B 76, 125301 (2007).
* (5) S. Bellucci and P. Onorato, Phys. Rev. B 78, 235312 (2008).
* (6) A. Aharony, O. E.-Wohlman, Y. Tokura, and S. Katsumoto, Phys. Rev. B 78, 125328 (2008).
* (7) A. Aharony, O. E.-Wohlman, Y. Tokura, and S. Katsumoto, Physica E 42, 629 (2010).
* (8) V. Moldoveanu and B. Tanatar, Phys. Rev. B 81, 035326 (2010).
* (9) G. Engels, J. Lange, Th. Schäpers, and H. Lüth, Phys. Rev. B 55, R1958 (1997).
* (10) L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Schön, and K. Ensslin, Nature Physics 3, 650 (2007).
* (11) J. Premper, M. Trautmann, J. Henk, and P. Bruno, Phys. Rev. B 76, 073310 (2007).
* (12) C.-M. Hu, J. Nitta, T. Akazaki, H. Takayanagi, J. Osaka, P. Pfeffer, and W. Zawadzki, Phys. Rev. B 60, 7736 (1999).
* (13) D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).
* (14) J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997).
* (15) M. Studer, M. P. Walser, S. Baer, H. Rusterholz, S. Schön, D. Schuh, W. Wegscheider, K. Ensslin, and G. Salis, Phys. Rev. B 82, 235320 (2010).
* (16) M. Scheid, M. Kohda, Y. Kunihashi, K. Richter, and J. Nitta, Phys. Rev. Lett. 101, 266401 (2008).
* (17) A. R. Dedigama, D. Jayathilaka, S. H. Gunawardana, S. Q. Murphy, M. Edirisoorya, N. Goel, T. D. Mishima, and M. B. Santos, Springer Proceedings in Physics 119, 35 (2008).
* (18) C. Yin, B. Shen, Q. Zhang, F. Xu, N. Tang, L. Cen, X. Wang, Y. Chen, and J. Yu, Appl. Phys. Lett. 97, 181904 (2010).
* (19) Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. Lett. 98, 196801 (2007).
* (20) V. Vlaminck and M. Bailleul, Science 322, 410 (2008).
* (21) K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008).
* (22) K. Ando, H. Nakayama, Y. Kajiwara, D. Kikuchi, K. Sasage, K. Uchida, K. Ikeda, and E. Saitoh, J. Appl. Phys. 105, 07C913 (2009).
* (23) J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003).
* (24) B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. Lett. 97, 236601 (2006).
* (25) Y. Tserkovnyak and A. Brataas, Phys. Rev. B 76, 155326 (2007).
* (26) S. K. Maiti, M. Dey, S. Sil, A. Chakrabarti, and S. N. Karmakar, Europhys. Lett. 95, 57008 (2011).
* (27) Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).
* (28) V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and D. D. Awschalom, Nat. Phys. 1, 31 (2005).
* (29) V. Sih, W. H. Lau, R. C. Myers, V. R. Horowitz, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 97, 096605 (2005).
* (30) S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176 (2006).
* (31) T.-W. Chen, C.-M. Huang, and G. Y. Guo, Phys. Rev. B 73, 235309 (2006).
* (32) E. B. Sonin, Phys. Rev. Lett. 99, 266602 (2007).
* (33) E. B. Sonin, Phys. Rev. B 76, 033306 (2007).
* (34) Q.-F. Sun and X. C. Xie, Phys. Rev. B 72, 245305 (2005).
* (35) F. Schütz, M. Kollar, and P. Kopietz, Phys. Rev. Lett. 91, 017205 (2003).
* (36) Q.-F. Sun, H. Guo, and J. Wang, Phys. Rev. B 69, 054409 (2004).
* (37) J. E. Hirsch, Phys. Rev. B 60, 14787 (1999).
* (38) Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. B 77, 035327 (2008).
* (39) L. P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074 (1990).
* (40) D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993).
|
arxiv-papers
| 2011-09-27T11:38:44 |
2024-09-04T02:49:22.536714
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti, Shreekantha Sil and Arunava Chakrabarti",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1109.5842"
}
|
1109.5953
|
# On the Nonlinear Theory of Viscoelasticity of Differential Type
Edvige Pucci, Giuseppe Saccomandi,
Dipartimento di Ingegneria Industriale,
Università degli Studi di Perugia, 06125 Perugia, Italy.
###### Abstract
We consider nonlinear viscoelastic materials of differential type and for some
special models we derive exact solutions of initial boundary value problems.
These exact solutions are used to investigate the reasons of non-existence of
global solutions for such equations.
Keywords: Viscoelasticity of differential type; nonlinear viscosity functions,
exact solutions, existence and uniqueness of boundary value problems.
## 1 Introduction and basic equations
In empirical rheology the standard sketch of the one dimensional Kelvin Voigt
model of linear viscoelasticity is of a purely viscous damper and a purely
elastic spring connected in parallel. In continuum mechanics, this model may
be recast in the class of the simple materials denoted as viscoelastic solids
of differential type. For such materials, in the isotropic case, the Cauchy
stress tensor, $\boldsymbol{T}$, is defined by the constitutive equation
$\boldsymbol{T}=\boldsymbol{T}(\boldsymbol{B};\boldsymbol{D}),$ (1.1)
where $\boldsymbol{B}=\boldsymbol{F}\boldsymbol{F}^{T}$ and
$\boldsymbol{D}=(\dot{\boldsymbol{F}}\boldsymbol{F}^{-1}+\boldsymbol{F}^{-T}\dot{\boldsymbol{F}}^{T})/2$
are the left Cauchy-Green strain tensor and the stretching tensor respectively
(Truesdell and Noll, 1965). (Here $\boldsymbol{F}$ is the gradient of
deformation). Therefore $\boldsymbol{B}$ stands for the elastic spring and
$\boldsymbol{D}$ stands for the viscous damper.
Nowadays there is an increasing interest in the study of nonlinear models of
viscoelasticity and this because in the experimental studies of rubber-like
materials and soft tissues the viscous and nonlinear effects are prominent and
clearly measurable (Lakes, 2009). For these reasons the possibility to
elaborate feasible and reliable mathematical models of nonlinear
viscoelasticity is a major problem of contemporary continuum mechanics
(Wineman, 2009). In this framework the class of constitutive equations (1.1)
seems to have a central role in view of its direct and clear connection with
the linear Kelvin-Voigt model.
The aim of the present note is to show that many of the nonlinear models
contained in the constitutive equation (1.1) are not _acceptable_ because they
are clearly unphysical.
To make the algebra simple as possible we shall restrict our attention to the
following family of shearing motions
$x=X+u(Z,t),\qquad y=Y,\qquad z=Z.$ (1.2)
Moreover, it is convenient to consider only incompressible materials. In the
case of compressible materials some complications arise, because the family of
motions (1.2) is in general not admissible (Destrade and Saccomandi, 2005) and
because the constitutive issues are more complex.
In this framework, it is straightforward to check that the balance equations
(in the absence of body forces) reduce to the scalar equations
$\rho u_{tt}=\partial T_{13}/\partial Z+p_{x},\quad p_{y}=0,\quad p_{z}=0,$
(1.3)
where $\rho$ is the (constant) density of the material and $p$ is the Lagrange
multiplier associated with the constraint of incompressibility
($det\boldsymbol{F}=1$). For an isotropic material the classical
representation formula (Truesdell and Noll, 1965) gives
$T_{13}=Q(K^{2})+\nu(K^{2},K_{t}^{2},KK_{t})K_{t}$ (1.4)
where $K(Z,t)=u_{Z}$ is the amount of shear, $Q=Q(K^{2})$ is the generalized
shear modulus and $\nu=\nu(K^{2},K_{t}^{2},KK_{t})$ is the viscosity
_function_. The unknowns of our problem are the pressure field $p=p(x,y,z,t)$
and the shearing motion $u=u(Z,t)$ (or equivalently the amount of shear $K$).
From the three scalar equations (1.3) we first deduce that
$p(x,t)=f(t)x+g(t),$
where $g(t)$ is an unessential function. Then by derivation, with respect of
$Z$, of the first equation in (1.3) we obtain
$\rho K_{tt}=\frac{\partial}{\partial
Z^{2}}\left[Q(K^{2})K+\nu(K^{2},K_{t}^{2},KK_{t})K_{t}\right].$ (1.5)
Once that (1.5) has been solved for $K(Z,t)$ and we integrate to obtain
$u(Z,t)$ from (1.3) it is possible to recover the pressure field, i.e. the
function $f(t)$.
Assuming that there is a characteristic time, $\omega^{-1}$, such that
$\tau=\omega t$, a characteristic length, $L$ ,such that $\zeta=Z/L$ and
assuming that the limit
$\mu_{0}=lim_{K\rightarrow 0}Q$
is a positive number, we define the quantities
$\epsilon=\frac{\omega^{2}L^{2}\rho}{\mu_{0}},\quad\hat{Q}=\frac{Q}{\mu_{0}},\quad\hat{\nu}=\frac{\omega\nu}{\mu_{0}},$
such that the dimensionless version of (1.5) reads
$\epsilon
K_{\tau\tau}=\frac{\partial}{\partial\zeta^{2}}\left[\hat{Q}(K^{2})K+\hat{\nu}(K^{2},\omega^{2}K_{\tau}^{2},\omega
KK_{\tau})K_{\tau}\right].$ (1.6)
When $\hat{Q}=1$, $\nu=\nu_{0}$ choosing $\omega=\mu_{0}/\nu_{0}$, (i.e.
$\hat{\nu}=1$) from (1.6) we recover the classical _linear_ Kelvin-Voigt
material
$\epsilon K_{\tau\tau}=K_{\zeta\zeta}+K_{\tau\zeta\zeta}..$ (1.7)
If in (1.7) we let $\epsilon\rightarrow 0$ we obtain the _quasistatic_
approximation. This approximation is fundamental to model the usual creep and
recovery experimental tests (Lakes, 2009). In Pucci and Saccomandi (2010) it
has been shown that when:
* •
the empirical inequalities are in force111In this case the generalized shear
modulus, $Q$, is positive for any value of the amount of shear and the strain-
energy density function associated with the elastic part of equation (1.6)
must be convex. (Beatty, 1987);
* •
the infinitesimal shear modulus $\mu_{0}$ is positive and finite;
* •
the viscosity function is such that
$\lim_{K\rightarrow 0,K_{t}\rightarrow 0}\nu=\nu_{0},$
where $\nu_{0}$ is a positive constant;
then the constitutive equations (1.1) are able to describe, at least
qualitatively, the mechanical behavior observed in standard quasi-static
experimental tests.
The quasi-static approximation is a _singular perturbation_ of the full
dynamical equation (1.6) and therefore is an approximation valid for large
times. For this reason the global existence of solutions of (1.6) is a main
issue. The existence, uniqueness and regularity of solutions for initial-
boundary value problems (IBVPs) associated to (1.5) (or its dimensionless
version (1.6)) have been studied by several authors. A state of art of the
various results both for classical and weak solutions is provided in (Tvedt,
2008).
The main problem with (1.5) is that the existence of smooth solutions global
in time is not guaranteed when the viscosity function is nonlinear. Indeed
Tvedt points out that:
> To my knowledge, all existence results for weak solutions of any kind to
> (1.1) in multiple space dimensions, including the Young-measure valued
> constructed in [6], require at least
>
> $\left|S(p,q)\right|\leq C(1+\left|p\right|+\left|q\right|)$
> for some constant $C$. _This condition is difficult to justify on physical
> grounds_.
(Using the present notation $S=T_{13}$, $p=K$, $q=K_{t}$.).
It seems that the consequences of this negative result quoted by Tvedt are not
full appreciated by workers in nonlinear viscoelasticity. For example, if we
have not a global existence result it is clear that we can consider a quasi-
static approximation. Our show note is centered on the impact of this result
in very simple problems of viscoelasticity.
The plan of the paper is the following. In the next Section we derive some
simple exact solutions for simple IBVPs with very smooth initial and boundary
conditions of special cases of (1.6) . For all these solutions there is a
finite time $t^{*}$ associated with the blow-up or the extinction of the
solution. In Section 3, we suggest that the _catastrophic_ problem associated
with $t^{*}$ is located in the initial layer of the solution (i.e. for short
times). By considering the case where the viscosity function depends only on
the amount of shear we prove that the dynamics in the initial layer is
governed by an equation equivalent to a nonlinear diffusion equation. The
blow-up characteristics of such an equation have been studied into details in
(Galaktionov and Vazquez, 1995) and the conditions derived there show the non-
existence of global solutions. The last Section is devoted to concluding
remarks.
## 2 Some exact solutions
We start by considering the problem associated with some viscous function that
does not have an upper bound. To this end we consider the following three
nonlinear polynomial viscous functions in turn
$\hat{\nu}_{e}(K^{2})=1+\hat{\nu}_{e1}K^{2},\quad\hat{\nu}_{v}(K_{\tau}^{2})=1+\hat{\nu}_{v1}K_{\tau}^{2},\quad\hat{\nu}_{m}(KK_{\tau})=1+\hat{\nu}_{m1}KK_{\tau}.$
For all these viscosity functions we have that $\lim_{K\rightarrow
0,K_{\tau}\rightarrow 0}\hat{\nu}=1>0$ but none of these functions are bounded
from above.
We start considering, for $\zeta\in[0,1]$, the equation
$\epsilon
K_{\tau\tau}=K_{\zeta\zeta}+K_{\tau\zeta\zeta}+\hat{\nu}_{e1}\left(K^{2}K_{\tau}\right)_{\zeta\zeta},$
(2.1)
i.e. an equation corresponding to a material whose elastic part is neo-Hookean
and with a super linear viscosity function in the amount of shear defined as
$\hat{\nu}_{e}$.
This equation has an interesting and simple exact solution that may be
computed as
$K(\zeta,\tau)=\alpha(\tau)\zeta+\beta(\tau),$ (2.2)
where
$\epsilon\frac{d\alpha}{d\tau}=2\hat{\nu}_{e1}\alpha^{3}+c_{1},\quad\epsilon\frac{d\beta}{d\tau}=2\hat{\nu}_{e1}\alpha^{2}\beta+c_{2},$
(here $c_{1}$ and $c_{2}$ are two constants of integration). If we consider
$c_{1}=c_{2}=0$ and
$\beta(\tau)\equiv-\frac{1}{2}\alpha(\tau)\equiv\frac{1}{2}\sqrt{\frac{\epsilon}{\epsilon-\hat{\nu}_{e1}\tau}}$
we obtain
$K(\zeta,\tau)=\frac{1}{2}\sqrt{\frac{\epsilon}{\epsilon-\hat{\nu}_{e1}\tau}}(\zeta-\frac{1}{2})\rightarrow\hat{u}(\zeta,\tau)=\frac{1}{4}\sqrt{\frac{\epsilon}{\epsilon-\hat{\nu}_{e1}\tau}}(\zeta^{2}-\zeta).$
(2.3)
(Here $\hat{u}=u/L$).
The shearing motion in (2.3) satisfies the following set of smooth initial and
boundary conditions
$\hat{u}(0,\tau)=\hat{u}(1,\tau)=0,\,\hat{u}(\zeta,0)=(\zeta^{2}-\zeta)/4,\,\hat{u}_{\tau}(\zeta,0)=\hat{\nu}_{1}(\zeta^{2}-\zeta)/(8\epsilon).$
In this way we have provided a simple example of non-existence (because of
finite time blow-up) of the solution for a very regular IVBPs for equation
(2.1) in $[0,1]$.
If now we consider the equation
$\epsilon
K_{\tau\tau}=K_{\zeta\zeta}+K_{\tau\zeta\zeta}+\hat{\nu}_{v1}\left(K_{\tau}^{3}\right)_{\zeta\zeta},$
(2.4)
a similar solution may be obtained using the ansatz in (2.2) for the amount of
shear. The exact solution is now
$\hat{u}(\zeta,\tau)=\frac{1}{2}\sqrt{\frac{12\hat{\nu}_{v1}-\epsilon\tau}{3\hat{\nu}_{v1}}}(\zeta^{2}-\zeta).$
(2.5)
This solution may be used to solve the boundary-initial value problem
$\hat{u}(0,\tau)=\hat{u}(1,\tau)=0,\,\hat{u}(\zeta,0)=\zeta^{2}-\zeta,\,\hat{u}_{\tau}(\zeta,0)=-\frac{\epsilon}{24\hat{\nu}_{{v1}}}(\zeta^{2}-\zeta).$
Here we record that the reason why the existence of the solution fails in
finite time is the extinction phenomena.
The last equation we consider is
$\epsilon
K_{\tau\tau}=K_{\zeta\zeta}+K_{\tau\zeta\zeta}+\hat{\nu}_{m1}\left(KK_{\tau}^{2}\right)_{\zeta\zeta}.$
(2.6)
Here the mathematics is more involved. Let us consider that the shearing
motion is in the form
$\hat{u}(\zeta,\tau)=\alpha(\tau)(\zeta^{2}-\zeta).$
Then the time dependent function $\alpha$ is determined by the ordinary
differential equation
$\frac{\epsilon}{\hat{\nu}_{m1}}\alpha^{\prime\prime}=24\alpha\alpha^{\prime
2}.$
The solution of this second order ordinary differential equation is given by
$\text{Erf}\left(\tilde{\alpha}\right)=c_{1}\tau+c_{2},$
where
$\tilde{\alpha}=2\sqrt{\frac{3\hat{\nu}_{m1}}{\epsilon}}\alpha,$
Erf is the standard error function and $c_{1},\,c_{2}$ two integration
constants. This exact solution solves the following IBVP (in the normalized
variables)
$\tilde{u}(0,\tau)=\tilde{u}(1,\tau)=0,\,\tilde{u}(\zeta,0)=\delta(\zeta^{2}-\zeta),$
with $0<\delta<1$ and $\tilde{u}_{\tau}(\zeta,0)=\mu(\zeta^{2}-\zeta)$ where
$\mu$ is an ad hoc constant determined by $\alpha^{\prime}$. From
$\text{Erf}\left(\tilde{\alpha}\right)-\text{Erf}\left(\delta\right)=c_{1}\tau,$
(2.7)
it is clear that this solution blows up in a finite time.
To summarize, the following remarks apply to all the three examples we have
proposed. First of all in the case $\epsilon\rightarrow 0$ all the exact
solutions we propose degenerate and are of no interest. Second, for all three
solutions there exists a critical time $\tau^{*}$ such that for
$\tau>\tau^{*}$ the solution does not exists anymore. For the solution in
(2.3) and (2.7) we record a blow-up phenomenon and for the solution (2.5) we
have an extinction phenomenon.
Therefore we have provided explicit examples of the lack of global existence
for equations with nonlinear viscous function and a neo-Hookean elastic term.
For all these solutions the inertia term (i.e. $\epsilon\neq 0$) is crucial in
delineating why the global existence fails.
## 3 The initial layer
We point out once again that in the quasi-static limit all the models we have
considered in the previous Section are ”good” models. This fact suggests that
because the quasi-static limit is the result of a singular perturbation
procedure, the failure of well-posedness must occur for short times or, using
the terminology of singular perturbations in the _initial layer_.
When we consider a general viscosity function defined as
$\hat{\nu}=\hat{\nu}(K^{2})$ and we introduce the _stretched_ time
$\tilde{\tau}=\tau/\epsilon$, it is possible to obtain easily, for any regular
viscosity function, the equation governing the dynamics in the initial layer.
Restricting the attention to this class of materials it is then possible to
rewrite (1.6) as
$K_{\tilde{\tau}\tilde{\tau}}=\frac{\partial}{\partial\zeta^{2}}\left[\epsilon\hat{Q}(K^{2})K+\hat{\nu}(K^{2})K_{\tilde{\tau}}\right],$
(3.1)
and when $\epsilon\rightarrow 0$ we derive the equation
$K_{\tilde{\tau}\tilde{\tau}}=\frac{\partial^{3}\Pi}{\partial\zeta^{2}\partial\tilde{\tau}}.$
(3.2)
where $\Pi=\int\hat{\nu}(K^{2})dK$.
By a simple integration the above equation is rewritten as
$K_{\tilde{\tau}}=\frac{\partial^{2}\Pi}{\partial\zeta^{2}}+\varphi(\zeta).$
(3.3)
Equation (3.3) is a nonlinear diffusion equation. This equation has been
considered by many authors and the necessary and sufficient conditions for the
complete blow-up and extinction of this equation have been considered in
(Galaktionov and Vazquez, 1995). It is interesting to note that for equation
(3.3) we have blow-up of the solution in finite time if the integral
$\int_{1}^{\infty}\frac{\Pi^{\prime}(s)}{s}ds,$
does not converge. (Here a dash denotes the derivative with respect the amount
of shear). This is exactly the case of the examples in the previous Section.
It is clear that there are several choices of the viscosity functions for
which the above integral may converge, but the usual higher order polynomial
models that are used in the standard fitting procedures do blow up. We remark
that a model with a constant viscosity, in the initial layer, is associated
with a linear heat equation this explain why it is possible to find a unique
global solution for such a model in the class (1.1).
We remark that via a multiple scale methods is possible to deduce the equation
in the initial layer for more general viscosity functions, but the case we
have just examined is sufficient to point out the problems inherent with the
constitutive equation (1.1) when the sub linear growth condition pointed out
by Tvedt (2009) is not satisfied.
## 4 Concluding remarks
If we believe that materials of differential type have a ”good” quasi-static
behavior, the fact that a global solution for the full dynamical equations
does not exist is a major problem. Indeed, if a solution blows up, or is
associated with an extinction phenomena, in finite time it is not possible to
consider the asymptotic limit for $t\rightarrow\infty$ and therefore it is not
possible to consider the quasi-static approximation.
Clearly such situation may be arranged considering a sort of ”cut-off” for the
usual polynomial constitutive equations. For example we may be consider a
constitutive equation such as
$\hat{\nu}(K^{2})=\left\\{\begin{array}[]{ll}1+\nu_{e1}K^{2}&\mbox{if
$K^{2}\leq K^{*}$};\\\ \nu_{e}&\mbox{if $K^{2}>K^{*}$}.\end{array}\right.$
We cannot be satisfied with this approach for two reasons. First of all we
introduce some oddities in the quasi-static regime as it is possible to check
by analyzing the corresponding creep problem that loose regularity. Moreover,
we do not solve the ”unphysical” situation generated by the fact: greater
viscous effects lead to greater problems with the regularity of the solution.
This situation seems to suggest that there is something missing in the
”dynamics” of the nonlinear viscoelastic models we are considering. It seems
that the model (1.1) must be refined taking into account phenomena that at
first sight seemed to be irrelevant but indeed are fundamental. These
phenomena seems to be of the same order of the inertia terms and therefore
disappear when $\epsilon\rightarrow 0$.
A possibility is to consider the presence of objective derivatives of the
stretching tensor $\boldsymbol{D}$ i.e. higher order models of differential
type where the constitutive equations not only depends on the stretching
tensor $\boldsymbol{D}$ but for example also on the subsequent Rivlin-Ericksen
tensors (Truesdell and Noll, 1965). These tensors may introduce some
additional terms that regularize the equations and allows to avoid the
problems presented here. This direction is outlined in the context of non
dissipative materials in (Destrade and Saccomandi, 2006).
## Acknowledgements
The research is supported by PRIN-2009 project _Matematica e meccanica dei
sistemi biologici e dei tessuti molli_.
GS thanks Michel Destrade and Jeremiah Murphy for their remarks on a previous
draft of this paper.
This paper is dedicated with great admiration to Professor Robin Knops for his
80-th birthday.
## References
* [1] Beatty, M.F. : Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Appl. Mech. Rev. 40 (1987) 1699–1734.
* [2] Destrade, M. and Saccomandi, G.: On finite amplitude elastic waves propagating in compressible solids, Physical Review E, 72 (2005) 016620.
* [3] Destrade, M. and Saccomandi, G.: Solitary and compact-like shear waves in the bulk of solids, Physical Review E, 73 (2006) 065604.
* [4] Galaktionov, V.A. and Vazquez, J. L.: Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations. Arch. Rational Mech. Anal. 129 (1995) 225–244.
* [5] Lakes, R. S.: Viscoelastic Materials, Cambridge University Press, 2009
* [6] Pucci, E. and Saccomandi, G.: On a Special Class of Nonlinear Viscoelastic Solids. Mathematics and Mechanics of Solids 15 (2010) 803–811.
* [7] C. Truesdell and W. Noll.: The Non-linear Field Theories of Mechanics. Volume III/3, Encyclopedia of Physics (S. Flugge Ed.). Springer, Berlin, 1965.
* [8] B. Tvedt.: Quasilinear Equations for Viscoelasticity of Strain-Rate Type. Archive for Rational Mechanics and Analysis 189 (2008) 237–281.
* [9] Wineman, A.: Nonlinear viscoelastic solids-a review, Mathematics and Mechanics of Solids 14 (2009) 300-366.
|
arxiv-papers
| 2011-09-27T16:12:32 |
2024-09-04T02:49:22.548244
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edvige Pucci, Giuseppe Saccomandi",
"submitter": "Giuseppe Saccomandi",
"url": "https://arxiv.org/abs/1109.5953"
}
|
1109.6353
|
# Thin films flowing down inverted substrates: Three dimensional flow
T.-S. Lin and L. Kondic
Department of Mathematical Sciences
and
Center for Applied Mathematics and Statistics
New Jersey Institute of Technology, Newark, NJ 07102
A. Filippov
2920 Shadowbriar Dr., Apt. 134, Houston, TX 77082
###### Abstract
We study contact line induced instabilities for a thin film of fluid under
destabilizing gravitational force in three dimensional setting. In the
previous work (Phys. Fluids, 22, 052105 (2010)), we considered two dimensional
flow, finding formation of surface waves whose properties within the
implemented long wave model depend on a single parameter,
$D=(3Ca)^{1/3}\cot\alpha$, where $Ca$ is the capillary number and $\alpha$ is
the inclination angle. In the present work we consider fully 3D setting and
discuss the influence of the additional dimension on stability properties of
the flow. In particular, we concentrate on the coupling between the surface
instability and the transverse (fingering) instabilities of the film front. We
furthermore consider these instabilities in the setting where fluid viscosity
varies in the transverse direction. It is found that the flow pattern strongly
depends on the inclination angle and the viscosity gradient.
## 1 Introduction
The problem of spreading of thin films on a solid surface is of interest in a
variety of applications, many of which were discussed and elaborated upon in
excellent review articles [1, 2, 3, 4, 5]. Perhaps the largest amount of work
has been done in the direction of analyzing properties of the flow of a
uniform film spreading down an incline. Starting from a pioneering work by
Kapitsa and Kapitsa [6] and progressing to more contemporary contributions [7,
2, 8], a rich mathematical structure of the solutions of governing evolution
equations, usually obtained under the framework of lubrication (long wave)
approach has been uncovered. The developed models have lead to evolution
equations nowadays known as Kuramoto-Sivashinksy [9, 10], Benney [3] and
Kapitsa-Shkadov [11, 12]. A variety of nonlinear waves which have been found,
as well as conditions for their formation were briefly discussed in the
Introduction to our earlier work [13]. For the purpose of the present work, it
is worth emphasizing that the linear and nonlinear waves which were discussed
in the cited literature resulted from either natural or forced perturbations
of the film surface, in the setup where inertial effects were relevant: flow
of uniform film down an inclined plane (with inclination angle
$\alpha\leq{\pi/2}$) is stable in the limit of zero Reynolds number.
In another direction, there has also been a significant amount of work
analyzing different type of instabilities caused by the presence of fluid
fronts, bounded by contact lines where the three phases (gas, liquid, solid)
meet. The fluid fronts are unstable, leading to formation of finger-like
structures [14] whose properties depend on the relative balance of the in-
plane and out-of-plane components of gravity [15, 16], as well as on wetting
properties of the fluid [17, 18, 19]. The analysis of the contact-line induced
instabilities has so far concentrated on the films flowing down an incline, so
with $\alpha\leq{\pi/2}$. In these configurations, fluid surface itself is
stable - typically, the only structure visible on the main body of the film
involves capillary ridge which forms just behind the contact line.
It is also of interest to consider situations where body forces (such as
gravity) are destabilizing, as it is the case during spreading down an
inverted surface, with $\alpha>{\pi/2}$. Such a flow is expected to be
unstable, even if inertial effects are neglected. An example of related
instabilities are wave and drop structures seen in experimental studies of a
pendant rivulet [20, 21]. Furthermore, if contact lines are present, one may
expect coupling of different types of instabilities discussed above. In this
context of front/contact line induced instabilities, two incompressible
viscous fluids in an inclined channel were considered [22], while more
recently the configuration where the top layer is denser than the bottom one
was studied [23]. Such configurations were found to be unstable, and give rise
to large amplitude interfacial waves.
In addition to mathematical complexity, this setting is of significant
technological relevance, in particular in the problems where there is also a
temperature gradient present, which may lead to a significant variation of
viscosity of the film. Despite significant progress in study of isoviscous
thin film flows, a surprisingly few studies have been devoted to analyses of
fluid dynamics of thin films with variable viscosity. Meanwhile, such flows
are rather common in various industrial applications.
Well-known examples include layers of liquid plastics and paints used for
coatings, as well as other materials, whose viscosities are strong functions
of temperature. Very often, the temperature variations are difficult to detect
and prevent. Corresponding variations of viscosity affect the processes and,
as we will discuss in this work, can be included in the model in the
relatively straightforward manner.
In our previous work that concentrated on isoviscous flow [13] we found that
for a film flowing down an inverted plane in 2D, fluid front bounded by
contact line (contact point in 2D) played a role of local disturbance that
induce instabilities. As the inclination angle approaches $180^{\circ}$ (that
is, the parameter $D=(3Ca)^{1/3}\cot\alpha$ becomes smaller), the fluid front
influences strongly the flow behind it and induces waves. In particular, the
governing equation, obtained under lubrication approximation, is found to
allow for three types of solutions. These types can be distinguished by the
value of the parameter $D$ as follows:
$\bullet$ Type 1: $-1.1\leq D$, traveling waves;
$\bullet$ Type 2: $-1.9\leq D<-1.1$, mixed waves;
$\bullet$ Type 3: $-3.0\leq D<-1.9$, waves resembling solitary ones.
Solutions could not be found for flows characterized by even smaller values of
$D$. We presume that this is due to the fact that gravitational force is so
strongly destabilizing that detachment is to be expected. It is of interest to
discuss how the discussion of different wave types extends to three spatial
dimensions (3D).
The present paper consists of two related parts with slightly different focus.
In the first part, we consider in general terms the 3D flow of a film
spreading on an inverted surface. The problem is formulated in Sec. 2. In Sec.
3 we consider instabilities occurring in the flow of a single rivulet, as a
simplest example of a 3D flow. Fully 3D flow is considered in Sec. 4, where we
discuss in particular the interaction between the surface instabilities
considered previously in the 2D setting [13], and the transverse fingering
instabilities at the front. We also briefly comment on the connection of the
instability considered here and Rayleigh-Taylor instability mechanism. In the
second part of the paper, Sec. 5, we discuss a setting which is perhaps more
closely related to applications: dynamics of a finite width fluid film
characterized by a nonuniform viscosity which varies in the transverse
direction. Such a setting may be relevant, for example, to the fluid exposed
to a temperature gradient, leading to nonuniform viscosity.
## 2 Problem formulation
We consider completely wetting fluid flowing down a planar surface enclosing
an angle $\alpha$ with horizontal. The flow is considered within the framework
of lubrication approximation, see e.g. [13]. In particular, the spatial and
velocity scalings, denoted by $x_{c}$ and $U$, respectively, are chosen as
$x_{c}=\left(\frac{a^{2}\,h_{c}}{\sin\alpha}\right)^{1/3},\quad
U=\frac{\gamma\,h^{2}_{c}}{3\mu\,a^{2}}\,\sin\alpha,$ (1)
where $a=\sqrt{\gamma/\rho g}$ is the capillary length, $\gamma$ is the
surface tension, $\rho$ is the fluid density, $g$ is the gravity, $\mu$ is the
viscosity and $h_{c}\ll x_{c}$ is the scaling on film thickness. Within this
approach, one obtains the depth averaged velocity $v$
$3\mu\mbox{\boldmath$v$}=\gamma h^{2}\nabla\nabla^{2}h-\rho gh^{2}\nabla
h\cos\alpha+\rho gh^{2}\sin\alpha\bf{i},$ (2)
where $h=h(x,y,t)$ is the fluid thickness, $\nabla=(\partial x,\partial y)$,
$x,y$ are two spatial variables, $t$ is time and $\bf{i}=(1,0)$ is the unit
vector pointing in the $x$ direction. Using this expression and the mass
conservation $h_{t}+\nabla\cdot(h\mbox{\boldmath$v$})=0$, we obtain the
following dimensionless PDE discussed extensively in the literature, see e.g.,
[24, 25, 26]
$\frac{\partial h}{\partial
t}+\nabla\cdot\left[\frac{h^{3}}{\bar{\mu}}\left(\nabla\nabla^{2}h-D\nabla
h+\bf{i}\right)\right]=0.$ (3)
The parameter $D=(3Ca)^{1/3}\cot\alpha$ measures the size of the normal
gravity, where $Ca=\mu U/\gamma$ is the capillary number and $\bar{\mu}$ is
the normalized viscosity scaled by the viscosity scale $\mu_{0}$. To avoid
well known contact line singularity, we implement precursor film approach,
therefore assuming that the surface is prewetted by a thin film of thickness
$b$, see [13] for further discussion regarding this point.
## 3 Inverted rivulet
In this section we consider a single rivulet with front on the underside of a
solid surface. The related problem of an infinite rivulet was studied in a
number of works. For example, the exact solution of Navier-Stokes equation for
steady infinite rivulet has been obtained [27], and under the lubrication
assumption the stability of such setting was studied for completely and
partially wetting fluids [28, 29]. However, the influence of fluid front on
the stability has not been studied. In the following, we will first extend the
steady infinite rivulet solution [30] to include presence of a precursor film.
Then we will examine the effect of fluid front on the stability of an inverted
rivulet.
### 3.1 Inverted infinite rivulet
Consider infinite length rivulet flowing down an inverted planar surface. A
steady state shape of the rivulet, independent of the downstream coordinate
$x$ can be found by solving Eq. (3), which in this special case reduces to
$\left(h^{3}h_{yyy}\right)_{y}-D\left(h^{3}h_{y}\right)_{y}=0.$ (4)
Integrating this equation and applying the boundary conditions
$h_{y}=h_{yyy}=0$ on $\partial\Omega$ and integrating again yields
$h_{yy}-Dh=a,$ (5)
where $a$ is a constant. Concentrating on inverted case, ie., $D<0$, the
general solution can be written as
$h_{r}(y)=A\cos(\sqrt{-D}\,y)+B\sin(\sqrt{-D}\,y)-\frac{a}{D}.$ (6)
Without loss of generality, we impose the symmetry condition at $y=0$ to find
$B=0$, and the complete wetting assumption further determines the rivulet’s
width as $\Omega=\left[-\pi/\sqrt{-D},\pi/\sqrt{-D}\right]$. At the boundaries
$h\left(\pm\pi/\sqrt{-D}\right)=b$, so we obtain
$h_{r}(y)=\frac{A-b}{2}\cos\left(\sqrt{-D}\,y\right)+\frac{A+b}{2},$ (7)
where $A=h_{r}(0)$ is a constant.
According to the above analysis, we find a family of exact rivulet solutions
for a given $D$. The unique solution can be obtained by specifying flux or
average thickness at the inlet.
### 3.2 Inverted rivulet with a front
#### 3.2.1 Initial and boundary conditions
To analyze the effect of the front bounded by contact line (regularized by the
precursor) on the rivulet flow, we perform numerical simulations of the 3D
thin film equation via ADI method. The Appendix gives the complete details of
the implemented method. The boundary conditions are such that constant flux at
the inlet is maintained with additional assumption that the shape is taken as
a steady rivulet profile. The choice implemented here is
$h(0,y,t)=h_{r}(y),\quad h_{xxx}(0,y,t)-Dh_{x}(0,y,t)=0.$ (8)
In addition, we choose $A=2$ in the steady rivulet solution so that the
average thickness at the inlet is $1$. At the outlet, $x=L$, as well as at the
$y$ boundaries, we assume zero-slope and a precursor film
$\displaystyle h(L,y,t)=h(x,\pm M,t)=b,$ (9) $\displaystyle
h_{x}(L,y,t)=h_{y}(x,\pm M,t)=0,$ (10)
where $[0,L]$ is the domain size in the $x$ direction, $[-M,M]$ is the domain
in the $y$ direction and $b$ is typically set to $0.1$. The initial shape of
the rivulet is chosen as a hyperbolic tangent to connect smoothly the steady
solution and the precursor film at $x=x_{f}$ \- here we choose $x_{f}=5$. It
has been verified that the results are independent of the details of this
procedure.
#### 3.2.2 Results
The presence of the contact line modifies the steady solution discussed in
Sec. 3.1. Without going into the details of this modification, for the present
purposes it is sufficient to realize that the speed of the traveling rivulet,
$V_{r}$, can be easily computed by comparing the net flux with the average
film thickness as
$V_{r}=\frac{\int(h_{r}^{3}-b^{3})\,dy}{\int(h_{r}-b)\,dy}=\frac{5}{8}\,A^{2}+A\,b+O(b^{2}).$
(11)
As we will see in what follows, $V_{r}$ is important for the purpose of
understanding the computational results in the context of 2D instabilities
discussed previously [13].
Figure 1 shows the computational results for different $D$’s at $t=25$. In
order to compare with 2D simulations, we have renormalized $D$ with $V_{r}$
(recall that the 2D traveling wave velocity, $V$, equals $1+b+b^{2}$ while
$V_{r}=2.7$ for $A=2$ in our 3D rivulet simulations). We call the renormalized
$D$ by $Dn=D/(V_{r})^{1/3}$.
Figure 1: (Color online) Rivulet flow for different $Dn$’s at $t=25$. The
domain size is specified by $L=90$ and $M=5$. Left hand side shows the contour
plot and the right hand side shows the cross section ($y=\mbox{const}$.) at
the middle of a rivulet. We will use similar way of presenting results, and
the same color map in the other figures given in this paper.
For $Dn=-1.0$, we still observe traveling type solution. We have examined
speed of the capillary ridge and found that it equals $V_{r}$, as predicted.
For $Dn$ larger in absolute value, the traveling type solution becomes
unstable and waves keep forming right behind the capillary ridge. For
$Dn>-1.5$, simulations suggest that the instability is convective, since it is
carried by the flow and moves downstream from the initial contact line
position, $x_{f}$. For $Dn<-1.5$, the instability is absolute. At the time
shown, the whole rivulet is covered by waves which in the cross section
resemble solitary ones.
The rivulet simulations show a qualitative similarity to our 2D simulations
(vis. the right hand side of Fig. 1 and Fig. 4 in [13]). Therefore, on one
hand this result validates the accuracy of our 3D simulations. On the other
hand, it also suggests that the instability regimes, (type 1, 2 and 3) can be
extended to 3D rivulet geometry. Clearly, it was necessary to use renormalized
value of $D$, $Dn$, to be able to carry out this comparison.
It is also of interest to relate the present results to stability properties
of an inverted infinite length rivulet without a front. In this case, it is
known that there exists a critical angle between $\pi/2$ and $\pi$ such that
the inverted infinite rivulet is unstable if the inclination angle is larger
than the critical one [29]. In order to be able to directly compare the two
problems (with and without a front), we have carried out simulations of an
inverted infinite rivulet. The steady state is fixed by choosing $A=2$. We
perturb the rivulet at $t=0$ by a single perturbation defined by
$h(x,y,0)=h_{r}(y)(1+A_{r}sech(x-x_{p}))$ with $A_{r}=0.2$ and $x_{p}=5.0$. In
this work we consider only this type of perturbation and do not discuss in
more detail the influence of its properties on the results. For this case, we
find that an inverted infinite rivulet is unstable for $Dn\leq-0.74$, which is
consistent with the fact that there exists a critical angle for stability
[29]. An obvious question to ask is which instability is dominant for
sufficiently small $Dn$’s, such that both front-induced and surface-
perturbation induced instabilities are present. This question will also appear
later in the context of thin film flow - to avoid repetition we consider it
for that problem, in Sec. 4.3.
## 4 Inverted film with a front
We proceed with analyzing stability of an inverted film with a front flowing
down a plane. In Sec. 4.1 we extend the results of the linear stability
analysis in the transverse direction to the inverted case. Then, we proceed
with fully nonlinear time dependent simulations in Sec. 4.2. We start with
addressing a simple case where we perturb the fluid front by a single
wavelength only – this case allows us to correlate the results with the 2D
surface instabilities discussed previously [13], with the instabilities of a
single inverted rivulet discussed in Sec. 3, and also with the well known
results for transverse instability of a film front flowing down an inclined
plane, see e.g. [15]. We proceed with more realistic simulations of a front
perturbed by a number of modes with random amplitudes, where all discussed
instability mechanisms come into play. To illustrate complex instability
evolution, we also include animations of the flow dynamics for few selected
cases [31]. We conclude the section by discussing in Sec. 4.3 the connection
between the instabilities considered here with Rayleigh-Taylor type of
instability of an infinite film flowing down an inverted plane.
### 4.1 Linear stability analysis in the transverse direction
In order to analyze the stability of the flow in the transverse, $y$,
direction, we perform a linear stability analysis (LSA). The results of
similar analysis were reported in previous works, see, e.g., [24], but they
typically concentrated on the downhill flows, with $D\geq 0$. Here we extend
the analysis to also consider films on an inverted surface, with $D<0$.
Consider a moving frame defined by $s=x-Vt$, and assume a solution of the form
$h(s,y,t)=H(s)+\epsilon h_{1}(s,y,t),$ (12)
where $\epsilon\ll 1$, $H(s)$ is the traveling wave solution with the speed
$V$. Then, plug this ansatz into Eq. (3). The leading order term
($O(\epsilon^{0})$) gives the 2D equation
$-VH^{\prime}+[H^{3}(H^{\prime\prime\prime}-DH^{\prime}+1)]^{\prime}=0$ (13)
while the first order term ($O(\epsilon^{1})$) yields
$\displaystyle\frac{\partial h_{1}}{\partial t}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\left[H^{3}\nabla\nabla^{2}h_{1}+3H^{2}h_{1}\nabla\nabla^{2}H\right]$
(14) $\displaystyle+D\nabla\cdot\left[H^{3}\nabla h_{1}+3H^{2}h_{1}\nabla
H\right]$ $\displaystyle-\left(3H^{2}h_{1}\right)_{s}+Vh_{1s},$
where $\nabla=(\partial_{s},\partial_{y})$. The next step is to express the
solution, $h_{1}$, as a continuous superposition of Fourier modes,
$h_{1}(s,y,t)=\int^{0}_{-\infty}\phi(s,q)e^{\sigma t}e^{iqy}dq,$ (15)
where $q$ is the wavenumber and $\sigma$ is the growth rate that determines
the temporal evolution of $h_{1}$. For a given $q$, there is an associated
eigenvalue problem, see e..g, [24]. The largest eigenvalue corresponds to the
growth rate, which is the quantity of interest.
Figure 2 shows the LSA results. Each curve represents the corresponding
largest eigenvalue for a given wavenumber and for fixed $D$. One can see that
sufficiently long wavelengths are unstable. Consequently, there is a critical
wavenumber, $q_{c}(D)$, which determines the range of unstable wavenumbers to
be $[0,q_{c}]$. Concentrating now on negative $D$’s, we find that $q_{c}$
increases with $|D|$, suggesting shorter unstable wavelength for larger
$|D|$’s, which are furthermore expected to grow faster. Therefore, for $D<0$,
as $|D|$ is increased (for example by increasing the angle $\alpha$, or by
making the film thicker), LSA predicts formation of more unstable fingers
spaced more densely.
Figure 2: (Color online) Wavenumber, $q$, and corresponding growth rate
$\sigma$ for different $D$’s.
One may note that Fig. 2 only shows the results for $D$ down to $-1.0$. The
reason is that a base state could be found only for $D\geq-1.1$, and therefore
LSA could not be carried out for smaller $D$’s. This issue was discussed in
[13], where we were able to find traveling wave solutions only for $D$’s in
type 1 regime, and could not find such solutions in type 2 and 3 regimes.
### 4.2 Fully 3D simulations
In this section we discuss the results of fully 3D simulations of the thin
film equation, Eq. (3). Throughout this section as our initial condition we
chose the same profile as in 2D simulations [13] \- that is, two flat regions
of thickness $h=1$ and $h=b$ connected at $x=x_{f}$ by a smooth transition
zone described by a hyperbolic tangent, perturbed as follows
$x_{f}(y)=x_{f0}-A_{0}\cos(2\pi y/\lambda),$
where $\lambda$ is the wavelength of the perturbation and $x_{f0}$ is the
unperturbed position. Here we choose $x_{f0}=5$. The boundary conditions in
the flow direction are such that constant flux at the inlet is maintained,
while at $x=L$, we assume that the film thickness is equal to the precursor.
The boundary conditions implemented here are
$\displaystyle h(0,y,t)=1,\quad h_{xxx}(0,y,t)-D\,h_{x}(0,y,t)=0,$
$\displaystyle h(L,y,t)=b,\quad h_{x}(L,y,t)=0.$
For the $y$ boundaries, we use the no-flow boundary conditions as
$\displaystyle h_{y}(x,0,t)=h_{y}(x,M,t)=0,$ (16) $\displaystyle
h_{yyy}(x,0,t)=h_{yyy}(x,M,t)=0,$
where $M$ is the width of the domain in the $y$ direction.
Figure 3 shows the results of simulations for $D=-1.0$ perturbed by specified
single mode perturbations. For $\lambda=8$ and $\lambda=10$, the perturbation
evolves into a single finger, whereas for $\lambda=20$, which corresponds to
the wavelength larger than the most unstable one (vis. Fig. 2), we observe a
secondary instability: in addition to the finger that corresponds to the
initial perturbation, there is another one (which appears as half fingers at
$y=0,M$), developing at later time (in this case after $t=10$). This
phenomenon can be explained by the fact that the domain is large enough to
allow for two fingers to coexist.
Figure 3: (Color online) Time evolution for perturbations of different
wavelengths ($\lambda$) and $D=-1.0$. Here the domain is specified by $L=50$
and $M=\lambda$.
Next, we compare the growth rate of a finger from 3D simulation with the LSA
results. This comparison is shown in Fig. 4 for $D=-0.5$ and $D=-1.0$. The
initial condition for these simulations is chosen as a single mode
perturbation, $\lambda=M=10$. The growth rate is extracted by considering the
finger’s length, $A$, defined as the distance between tip and root. As shown
in Fig. 4, for early times, the finger grows exponentially with the same
growth rate as predicted by the LSA. For later times, the finger length
exhibits a linear growth. This late time behavior can be simply explained by
the fact that the finger evolves into the rivulet solution discussed in Sec.
3.
Figure 4: (Color online) Length of a finger, $A$, divided by the initial
length, $A_{0}$. Here $\lambda=M=10$. Red (curved) lines show the
computational results while the black (straight) lines show the LSA prediction
for two different $D$’s.
Figure 5 shows the numerical results for $D=-1.5$ with three single mode
perturbations in a fixed domain, $[0,90]\times[0,20]$. Both contour plots and
the cross sections at middle of the domain ($y=10$) are presented. In this
case, in addition to the secondary instability in the case of the perturbation
by $\lambda=20$, we also see secondary instability developing for
$\lambda=10$, suggesting that as the absolute value of (negative) $D$ grows,
shorter and shorter wavelengths become unstable. For this $D$, even
$\lambda=5$ is unstable. This result is consistent with the trend of the LSA
results shown in Fig. 2; note however that LSA cannot be carried out for such
small $D$’s due to the fact that traveling wave solution could not be found.
Instead of traveling waves, we find surface instabilities on the film. As
shown in the contour plot at $t=50$, several red(dark) dots, which represent
waves, appear on the fingers, as it can be seen in the cross section plots as
well. The red(dark) dots keep forming, moving forward and interact with the
capillary ridge in the fronts. This interaction can be seen much more clearly
in the animations available as supplementary materials [31].
Figure 5: (Color online) Time evolution for perturbations of different
wavelengths ($\lambda$) at $D=-1.5$. The domain is chosen as $L=90$ and
$M=20$. In each sub-block, the upper figure shows the contour plot, and the
lower figure shows the cross section at $y=10$.
The cross section plots in Fig. 5 show formation of solitary-like waves.
Behind these solitary-like waves, there exists a second region where waves
appear as ‘stripes’ (vis. the straight stripes in the middle part of the
contour plot at $t=50$). These stripe-waves move forward for a short distance
and then break into several waves localized on the surface of the finger-like
rivulets. In the cross-section plots, these strip-like waves appear as
sinusoidal waves. Finally, flat film is observed in the region far behind the
contact line. The appearance of such a flat film indicates that the flow
instability is of convective type. Therefore, for this $D$, the contact line
induced waves are carried by the fluid and they eventually move away from any
fixed position.
Figure 6 shows the numerical results for $D=-2.0$ with the same set of initial
perturbation as in Fig. 5. Both contour plot and the cross sections at the
middle of the $y$ domain are presented. As mentioned in [13], $D=-2.0$
corresponds to the Type 2 regime and is of absolute instability type. This is
exactly what we see in the contour plot. Localized waves shown as red (dark)
dots appear all over the surface and we do not see neither strip waves nor
flat film appearing. In the cross section plots, we only see solitary-like
waves.
Figure 6: (Color online) Time evolution for perturbations of different
wavelengths ($\lambda$) at $D=-2.0$. $t_{end}=50$ for $\lambda=6.67$,
$t_{end}=40$ for $\lambda=10$, $t_{end}=30$ for $\lambda=20$. The domain is
chosen as $L=90$ and $M=20$. In each sub-block, the upper figure shows the
contour plot, and the lower figure shows the cross section at $y=10$.
Next we proceed to analyze the behavior in the case where initially multiple
perturbations are present. The imposed perturbation consists of $50$
sinusoidal modes with amplitudes randomly selected from $[-0.2,0.2]$
$x_{f}(y)=x_{f0}-\sum_{i=1}^{50}\,A_{i}\,\cos((i-1)\,\pi\,y),\quad-0.2\leq
A_{i}\leq 0.2\,.$
Figure 7 shows the simulations for different $D$’s with the same random
initial perturbations. The initial profile is shown in Fig. 7 ($t=0$). The
initial condition is set to be the same for all $D$’s so that the non-
dimensional parameter $D$ is the only difference between the 4 panels in Fig.
7.
Figure 7: (Color online) Time evolution of perturbations for different $D$’s
with the same random initial perturbations. Here $t_{end}=50$ for $D=-1.0$,
$t_{end}=40$ for $D=-1.5$, $t_{end}=30$ for $D=-2.0$, $t_{end}=20$ for
$D=-3.0$. The domain is chosen as $L=90$ and $M=50$ See also the animations
available [31].
The first obvious observation is that the number of fingers increases as the
absolute value of (negative) $D$ becomes larger. This is due to the fact that
the most unstable wavelength decreases as $|D|$ of (negative) $D$ becomes
larger. Therefore more fingers can fit into the flow domain. Secondly, the
fingers become more narrow for these $D$’s, consistently with the predictions
for rivulet flow. Thirdly, the absolute/convective instability argument in
[13] is a good explanation for these contact line induced instabilities: there
is no surface instability seen for $D=-1.0$; we see convective instability,
shown as localized waves/stripes/flat film for $D=-1.5$; and absolute
instability for $D=-2.0$ and $D=-3.0$.
Remark
Here we comment on two additional sets of simulations - corresponding figures
are omitted for brevity. One set involves the case when the initial condition
is chosen as $y$-independent. Mathematically such initial condition reduces
the problem to 2D, and the solution should remain $y$-independent for all
times. However, this is not the case in numerical simulations. Numerical
errors are present and grow with time. To estimate this effect, one can
calculate the largest grows rate in the $x$ and $y$ direction based on the LSA
results, and estimate the time for which the numerical noise becomes
significant. For example, for $D=-1.0$, the largest growth rate for
instability of a flat film in the $x$ direction is $0.25$ and the largest
growth rate in the $y$ direction is $0.46$. That is, it takes $70$ time units
for noise of initial amplitude $10^{-16}$ (typical for double precision
computer arithmetic) to grow to $10^{-2}$. So as long as the final time is
less than $70$, the numerical noise is still not visible. By carrying out such
an analysis, we are able to distinguish between the numerical noise induced
instability and the contact line induced one, and further separate the effect
of numerical noise.
The other set or simulations has to do with the case when the initial single
mode perturbation is chosen as a stable one. In such a case, the amplitude of
perturbation decays exponentially and the surface profile soon becomes
$y$-independent. Again, after sufficiently long time, numerical noise will
become relevant and break the $y$-independence.
#### 4.2.1 The width of a finger
It is of interest to discuss how fingers’ width depend on $D$. As a reminder,
the LSA shows that there exists a most unstable wavenumber, $q_{m}(D)$, and
the distance between two neighboring fingers in physical experiments, in the
presence of natural or other noise, is expected to center around the most
unstable wavelength, $\lambda_{m}=2\pi/q_{m}(D)$. On the other hand, the width
of the rivulet part of a single finger is not known to the best of our
knowledge. In the following we define this width and discuss how it relates to
the LSA results.
First we check whether there is a difference between the fingers for different
single mode perturbations. Figure 8 shows the $y$-orientation cross section of
the fingers’ rivulet part for $D=-0.5$ and $D=-1.0$. As shown in the figure,
for a given $D$, the fingers are very similar in the cross section, with their
shape almost independent of the initial perturbation. Note that this result
still holds even for the $\lambda$’s that are very close to the critical one,
$\lambda_{c}={2\pi/q_{c}(D)}$ \- eg., see $\lambda=6.3$ in Fig. 8(b) (here
$\lambda_{c}\approx 6.25$). By direct comparison of the parts (a) and (b) of
this figure, we immediately observe that the fingers are more narrow for more
negative $D$’s.
(a) $D=-0.5$.
(b) $D=-1.0$.
Figure 8: (Color online) Cross sections of film thickness as a function of the
transverse coordinate, $y$, for different wavelengths of initial perturbation,
$\lambda$. The cross sections are taken from the rivulet part of a finger, at
$x=35$, $t=30$; some of results from which the cross sections are extracted
can be seen in Fig. 3. The centers of the cross sections are shifted to $y=10$
for the purpose of comparison. For $D=-0.5$, $\lambda=6.3$ is stable (not
shown).
To make this discussion more precise, we define the width of a rivulet, $w$,
as the distance between two dips on each side of a finger (the two dips are
shown at $y\approx 7$ and $y\approx 13$ in Fig. 8). The main finding is that
$w\approx\lambda_{c}$. This finding applies for all $D$’s and all perturbation
wavelengths that we considered. In particular note that Fig. 8 shows that $w$
becomes smaller as absolute value of (negative) $D$ increases, consistently
with the decrease of $\lambda_{c}$, see Fig 2.
While it is clear that $w$ can not be larger than $\lambda_{c}$ (since $w$ is
independent of the initial perturbation, and for $\lambda>\lambda_{c}$ it must
be that $\lambda>w$), at this point we do not have a precise argument why $w$
is so close to $\lambda_{c}$ for all considered perturbations and the values
of $D$. Of course, one could argue that $w\approx\lambda_{c}$ is also
consistent with stability of any perturbation characterized by
$\lambda<\lambda_{c}$, since such a perturbation cannot support a finger of
$w\approx\lambda_{c}$.
### 4.3 Rayleigh-Taylor instability of inverted film
The instabilities we have discussed so far are discussed in the fluid
configurations where contact line is present, and an obvious question is what
happens if there is no contact line, that is, if we have an infinite film
spreading down an (inverted) surface. This configuration is expected to be
susceptible to Rayleigh-Taylor (R-T) type of instability since we effectively
have a heavier fluid (liquid) above the lighter one (air). The question is
what are the properties of this instability for a film flowing down an
inverted surface, and how this instability relates to the contact line induced
one, discussed so far.
Figure 9 shows the results of 3D simulation of an infinite film (no contact
line). The initial condition at $t=0$ is chosen as a flat film with localized
half-sphere-like perturbation of amplitude $0.1$ (marked by the black circle
in Fig. 9). As an example, we use $D=-1.0$. At time $t=20$, the perturbation
had been amplified, as expected. The properties of this instability are,
however, very different from the one observed in thin film flow where contact
line is present. For this $D$, if a contact line is present, we see only a
capillary wave behind the front (vis. Fig. 3 and Fig. 7), and we do not find
upstream propagating waves as for the infinite film shown in Fig. 9.
Therefore, the instability discussed so far is not of R-T type - instead, it
is induced by the presence of a contact line.
Figure 9: (Color online) Simulation of Rayleigh-Taylor instability for hanging
film on inclined plane at $D=-1.0$. The black circle indicates the initial
profile. The surface shown in the downstream is the surface profile at $t=20$.
The computational domain is $[0,90]\times[0,50]$.
One obvious question is why we do not observe (additional) R-T instability in
the flow with fronts. The answer is that the speed with which a perturbation
on a main body of a film (such as infinite film shown in Fig. 9) propagates
downstream is faster than the speed of the contact line itself. Therefore, in
the simulations of films with fronts presented so far, flat film instabilities
do not have time to develop since they reach the contact line before having a
chance to grow. We note that in Fig. 9 we used large scale perturbation to
illustrate the point; in a physical problem, one would expect surface
perturbations to be characterized by much smaller amplitudes and would
therefore require much longer time to grow to the scale comparable to the film
thickness. As a consequence, R-T instability could be expected to become
relevant only for the films characterized by the spatial extend which is much
larger than the one considered here. Similar conclusion extends to stability
of infinite rivulets, discussed briefly in Sec. 3.
## 5 Inverted film of variable viscosity with a front
Consider now a situation, when the film is of finite width and the fluid
viscosity is variable in the transverse direction, ie.,
$\bar{\mu}=\bar{\mu}(y)$ in Eq. (3). We have to substitute the no-flow
boundary conditions, Eq. (16), at the borders of the computational domain with
the following ones
$\displaystyle h(x,0,t)=h(0,M,t)=b,$ $\displaystyle
h_{x}(x,0,t)=h_{x}(0,M,t)=0,$
The flow starts at the top of the domain according to the condition
$h(0,y,t)=b+F_{0}\left(\frac{y}{M}\right)\frac{(at)^{2}}{1+(at)^{2}},$ (17)
where the value of parameter $a$ is $0.775$ and the bell shape of the entrance
thickness profile determined by the function $F_{0}$ is shown in Fig. 10.
Equation (17) mimics the growth of the flow rate and the film thickness at the
entrance boundary as the liquid starts being delivered to the substrate during
the ramp-up in industrial processes or experiments. Characteristic patterns of
flow depend on the value of parameter $D$, width of the film and distribution
of viscosity, and establish when the product $at$ in Eq. (17) reaches values
of several units.
Figure 10: Entrance profile of film thickness.
In many practical situations, the variation of viscosity is a result of its
dependence on temperature, which is usually non-linear. For example, assume
that the film temperature $T$ drops linearly with the lateral coordinate $y$
and the viscosity (in $Pa\cdot s$) is given by a generic equation:
$\displaystyle T$ $\displaystyle=$ $\displaystyle T_{0}-ay,\quad
a=\mbox{const.};$ (18) $\displaystyle\log_{10}(\mu)$ $\displaystyle=$
$\displaystyle-A+\frac{B}{T-C},$ (19)
where $A=5.5$, $B=700$ K and $C=52$ K are the material constants, and $T$ is
the liquid temperature. The viscosity function is normalized by its value at
$y=0$. The constant value $T_{0}$ is assumed to be equal $153$ K, while the
lateral temperature drop across the computational domain is assumed to be
equal $11$ K in all cases discussed hereafter, resulting in $7.5$ fold
viscosity growth from bottom to top of the domain.
Similarly to isoviscous cases discussed earlier, the film front is always
unstable, producing finger-like rivulets. The morphology of the non-isoviscous
film (as compared to an isoviscous case) stems from the fact that similar
structures such as fingers at different parts of the film move with different
speeds. As an example, Fig. 11 (a) and (b) show distribution of film thickness
at $t=9.4$, $16.5$, with parameter $D=-0.88$ and the dimensionless domain
width $M$ equal to $133$. Morphology of individual fingers is similar to the
one of fingers in isoviscous cases shown in Fig. 7, but there is obvious mass
redistribution along the front line because of its inclination, resulting in
coalescence of some of the fingers and variation of the finger-to-finger
distance. A finger produced by coalescence of two parent fingers, such as
finger $5$ in Fig. 11(b), has a higher flow rate and moves faster than its
neighbors.
(a) $t=9.4$.
(b) $t=16.5$.
Figure 11: (Color online) Film thickness for $D=-0.88$. The circle in the part
(a) indicates two coalescing fingers.
These specific properties of non-isoviscous film are common for flows in the
whole spectrum of parameter $D$, but morphology of individual fingers strongly
depends on the type of the flow, similar to already considered isoviscous
cases. Figure 12 shows distribution of film thickness for parameter $D=-1.5$,
which corresponding to type 2, with film width $M=72.2$, at $t=25.4$. The
leading head capillary ridges are followed by a rivulet with followup smaller
waves moving faster than the leader. The pattern is common for all fingers,
but the speed of propagation increases with temperature. Each of the fingers
has a faster-moving neighbor on the higher temperature side, causing slight
increase of the background film thickness on that side compared to the colder
side. As a result, the fingers may have a tendency of being diverted and
coalesce with warmer neighbors resulting in mass transport from colder areas
to warmer areas of the flow, despite the fact that the film velocity does not
explicitly depend on the temperature gradient.
Figure 12: (Color online) Film thickness for $D=-1.5$ at $t=25.4$.
In type 3 film flow, the height of follow-up drops is already close to that of
finger head drops, as shown in Fig. 13, showing thickness distribution in film
with parameter $D=-2.54$, film width $M=52.2$ at $t=44.4$. In type 3 flows,
there is no propagation front and the flow itself consists of a series of
propagating fingers. Another peculiarity of this type of flows is existence of
small drops, separating from the leading drops to be immediately consumed by
the following drop in the train, as indicated by circles in Fig. 13. These
features are also observed in isoviscous flows for similar values of $D$; see,
e.g., cross sectional profile in Fig. 6. The viscosity effect in the developed
type 3 flows shows mainly as the difference in speeds for droplets in low and
high viscosity regions.
Figure 13: (Color online) Film thickness for $D=-2.54$ at $t=44.4$.
## 6 Conclusions
In the previous work [13] we carried out extensive computational and
asymptotic analysis of the two dimensional flow of a completely wetting fluid
down an inverted surface. Complex behavior was uncovered with different
families of waves evolving in the configurations characterizing by different
values of the governing parameter $D$. In the present work, we have considered
fully three dimensional problem of spreading down an inverted surface. We find
that there is an elaborate interaction of surface instabilities and contact
line instabilities. For the values of $D$ which are not too small
(approximately $D\geq-1.1$) we find similar behavior as already known for the
flow down an inclined surface, with the main difference that the finger-like
patterns that form are spaced more closely and the fingers themselves are more
narrow for negative $D$’s. As $D$ is further decreased, we still find
instabilities of the contact line leading to formation of fingers, but in
addition we observe formation of surface waves, which propagate down the
fingers with the speed larger than the speed of the fingers themselves:
therefore, these propagation waves (which may appear as drops on top of the
base film) travel down a finger, reach the front and merge with the leading
capillary ridge. Behind the fingers, in this regime we find strip-like waves
(whose fronts are independent of the transverse direction). These waves are
convective in nature and leave behind a portion of a flat film whose length
increases with time. For even smaller $D$’s (smaller than approximately
$-2.0$), these transverse strip-like waves disappear, and the whole film is
covered by localized waves. These localized waves travel faster than the film
itself, and converge towards the fingers which form at the front.
It is worth emphasizing that the properties of the surface waves which form
due to the presence of fronts are different from the ones which would be
expected if the fronts were not present. To illustrate this effect, we
consider an infinite film with a localized perturbation which is expected to
be unstable by a Rayleigh-Taylor type of instability. We find that this
instability leads to a different type of surface waves, which may or may not
be observable in physical experiments, depending on the size of the fluid
domain.
In the second part of the paper we consider flow where fluid viscosity is not
constant, but varies in the transverse direction. The most important
difference is the loss of flow periodicity in the lateral direction. The
fingers in the warmer parts of the flow move faster than those in the colder
areas, yielding slight increase of the background film thickness on warmer
side of each finger compared to the colder side. This results in mass
redistribution from colder areas to warmer areas of the flow, more pronounced
for lower values of $|D|$, despite explicit independence of the film velocity
on the temperature gradient.
Acknowledgments. This work was partially supported by NSF grant No.
DMS-0908158.
## Appendix A Numerical method for 3D thin film equation
### A.1 Solving nonlinear time dependent PDE
In general, time dependent PDE, Eq. (3), can be expressed as $h_{t}+f(h)=0$,
where $h=h(\mbox{\boldmath$x$},t)$ is the unknown function with time variable
$t$, spatial variables $x$, and $f$ is a nonlinear discretization operator for
spatial variables. From time $t^{n}=n\Delta t$ to $t^{n+1}$, where $\Delta t$
is the time step, the PDE can be integrated numerically by the so-called
$\theta$ method leading to a nonlinear system:
$h^{n+1}+(1-\theta)\Delta tf^{n+1}=h^{n}-\theta\Delta tf^{n},$ (20)
where $f^{n}=f(h^{n})$ and $h^{n}=h(\mbox{\boldmath$x$},t^{n})$. To solve the
nonlinear system (20), we apply the Newton’s method. Firstly, we linearize
$h^{n+1}$ about a guess for the solution by assuming $h^{n+1}=h^{*}+c$, where
$h^{*}$ is a guess and $c$ is the correction. Then we express the nonlinear
part using Taylor’s expansion
$f^{n+1}=f(h^{*}+c)\approx f(h^{*})+J_{f}(h^{*})\cdot c=f^{*}+J_{f}^{*}\cdot
c,$
where $J_{f}$ is the Jacobian matrix for function $f$ and
$J_{f}^{*}=J_{f}(h^{*})$. After substituting the above quantities into Eq.
(20), we obtain a linear system for the correction term, $c$:
$\left(I+(1-\theta)\Delta tJ_{f}^{*}\right)c=-h^{*}-(1-\theta)\Delta
tf^{*}+h^{n}-\theta\Delta tf^{n},$ (21)
$I$ is the identity matrix that has the same size as the Jacobian matrix,
$J^{*}_{f}$. The solution at $t=t^{n+1}$ is obtained by correcting the guess
iteratively until the process converges, ie., the new correction is small
enough.
### A.2 Spatial discretization
We discretize the spatial derivatives of thin film equation, Eq. (3), through
finite difference method. The grid points in the computational domain,
$[0,L]\times[0,M]$, is defined as
$\displaystyle x_{i}$ $\displaystyle=$
$\displaystyle\left(i-\frac{1}{2}\right)\Delta x,\quad i=1,\cdots,n_{x},$
$\displaystyle y_{j}$ $\displaystyle=$
$\displaystyle\left(j-\frac{1}{2}\right)\Delta y,\quad j=1,\cdots,n_{y},$
where $\Delta x=L/n_{x}$, $\Delta y=M/n_{y}$ are the step size in the $x$, $y$
domain; $n_{x}$, $n_{y}$ are number of grid points in the $x$, $y$ domain,
respectively.
The scheme presented here is 2nd order central difference scheme. In the
following, the subscripts, $i,j$, denote that the value been taken at
$(x_{i},y_{j})$. The notation $h_{i+1/2,j}$ denotes an average at the point
$(x_{i+1/2},y_{j})$ as
$h_{i+1/2,j}=\frac{h_{i+1,j}+h_{i,j}}{2}.$
Also we use the standard difference notation as
$\displaystyle\delta_{x}h_{i+1/2,j}$ $\displaystyle=$ $\displaystyle
h_{i+1,j}-h_{i,j},$ $\displaystyle\delta_{y}h_{i,j+1/2}$ $\displaystyle=$
$\displaystyle h_{i,j+1}-h_{i,j},$ $\displaystyle\delta^{3}_{x}h_{i+1/2,j}$
$\displaystyle=$ $\displaystyle h_{i+2,j}-3h_{i+1,j}+3h_{i,j}-h_{i-1,j},$
$\displaystyle\delta^{3}_{y}h_{i,j+1/2}$ $\displaystyle=$ $\displaystyle
h_{i,j+2}-3h_{i,j+1}+3h_{i,j}-h_{i,j-1},$
$\displaystyle\delta_{x}\delta^{2}_{y}h_{i+1/2,j}$ $\displaystyle=$
$\displaystyle h_{i+1,j+1}-2h_{i+1,j}+h_{i+1,j-1}$ $\displaystyle-
h_{i,j+1}+2h_{i,j}-h_{i,j-1},$
$\displaystyle\delta_{y}\delta^{2}_{x}h_{i,j+1/2}$ $\displaystyle=$
$\displaystyle h_{i+1,j+1}-2h_{i,j+1}+h_{i-1,j+1}$ $\displaystyle-
h_{i+1,j}+2h_{i,j}-h_{i-1,j}.$
The discretization of each term in Eq. (3) involving spatial derivatives is as
follows:
The surface tension term
$\displaystyle\nabla\cdot[h^{3}\,\nabla\nabla^{2}h]_{i,j}=$
$\displaystyle\left(h^{3}_{i+1/2,j}\delta^{3}_{x}h_{i+1/2,j}-h^{3}_{i-1/2,j}\delta^{3}_{x}h_{i-1/2,j}\right)/\Delta
x^{4}$
$\displaystyle\left(h^{3}_{i+1/2,j}\delta_{x}\delta^{2}_{y}h_{i+1/2,j}-h^{3}_{i-1/2,j}\delta_{x}\delta^{2}_{y}h_{i-1/2,j}\right)/\Delta
x^{2}\Delta y^{2}$
$\displaystyle\left(h^{3}_{i,j+1/2}\delta_{y}\delta^{2}_{x}h_{i,j+1/2}-h^{3}_{i,j-1/2}\delta_{y}\delta^{2}_{x}h_{i,j-1/2}\right)/\Delta
x^{2}\Delta y^{2}$
$\displaystyle\left(h^{3}_{i,j+1/2}\delta^{3}_{y}h_{i,j+1/2}-h^{3}_{i,j-1/2}\delta^{3}_{y}h_{i,j-1/2}\right)/\Delta
y^{4}.$
The normal gravity term
$\displaystyle\nabla\cdot[h^{3}\,\nabla h]_{i,j}=$
$\displaystyle\left(h^{3}_{i+1/2,j}\delta_{x}h_{i+1/2,j}-h^{3}_{i-1/2,j}\delta_{x}h_{i-1/2,j}\right)/\Delta
x^{2}$
$\displaystyle+\left(h^{3}_{i,j+1/2}(h_{i,j+1}-h_{i,j})-h^{3}_{i,j-1/2}(h_{i,j}-h_{i,j-1})\right)/\Delta
y^{2}.$
The tangential gravity term
$\frac{\partial}{\partial
x}\left(h^{3}\right)_{i,j}=\left(h^{3}_{i+1/2,j}-h^{3}_{i-1/2,j}\right)/\Delta
x.$ (22)
### A.3 Fully implicit algorithm
Applying algorithm Eq. (21) to 3D thin film Eq. (3), we have [32]
$\displaystyle\left(I+(1-\theta)\Delta
t(J_{fx}^{*}+J_{fy}^{*}+J_{fm}^{*})\right)\cdot c=$
$\displaystyle-h^{*}-(1-\theta)\Delta tf^{*}+h^{n}-\theta\Delta tf^{n},$ (23)
where $J_{fx}$, $J_{fy}$ and $J_{fm}$ are the Jacobian matrices for $x$, $y$
and mixed derivative terms of function $f$, respectively. Equation (23) is a
non-symmetric sparse linear system that has $n_{x}n_{y}$ unknowns. For large
$n_{x}$ and $n_{y}$, as it is the case for the problems discussed in the
present work, solving such a system carries a significant computational cost.
In general, the operation count is proportional to $O(n_{x}^{3}n_{y}^{3})$ or
$O(n_{x}^{2}n_{y}^{2})$, depending on the matrix solver.
### A.4 ADI method
To decrease the computational cost, Witelski and Bowen suggested to use the
approximate-Newton approach [33]. The idea is to replace the Jacobian matrix
by an approximated one
$\displaystyle\left[I+(1-\theta)\Delta
t\left(J^{*}_{fx}+J_{fy}^{*}+J_{fm}^{*}\right)\right]$ (24)
$\displaystyle\sim\left[I+(1-\theta)\Delta
tJ^{*}_{fy}\right]\left[I+(1-\theta)\Delta tJ^{*}_{fx}\right]$ .
Therefore we get a new system of equations
$\left[I+(1-\theta)\Delta tJ^{*}_{fy}\right]\left[I+(1-\theta)\Delta
tJ^{*}_{fx}\right]\cdot c=R,$ (25)
where $R=-h^{*}-(1-\theta)\Delta tf^{*}+h^{n}-\theta\Delta tf^{n}$ is the
right hand side of Eq. (23). One should note that as long as $c$ decreases
after each iteration and approaches $0$ in some norm, we have $R=0$, leading
to Eq. (20). That is, such an approach does not affect the stability and
accuracy of the original space-time discretization.
Under the same spirit of alternating direction implicit method (ADI), equation
(25) can be easily split into two steps:
$\displaystyle\left(I+(1-\theta)\Delta tJ_{fx}^{*}\right)\cdot w$
$\displaystyle=$ $\displaystyle R,$ $\displaystyle\left(I+(1-\theta)\Delta
tJ_{fy}^{*}\right)\cdot c$ $\displaystyle=$ $\displaystyle w.$ (26)
The main advantage of such splitting is that the operations in the $x$ and $y$
directions are decoupled and therefore the computational cost reduced
significantly. Specifically for our discretization, the Jacobian matrices in
the $x$ and $y$ direction are penta-diagonal matrices, leading to system that
can be solved in $O(n_{x})$ and $O(n_{y})$ arithmetic, and the overall
computational cost for solving Eq. (26) is proportional to $O(n_{x}n_{y})$.
The approach presented here deals with a matrix that is an approximation to
the original Jacobian one. Therefore we should not expect the convergent rate
of the ADI method to be quadratic. However, since the approximation error is
proportional to $O(\Delta t)$, for small enough time step, we expect the rate
of convergence to be close to quadratic; see [33] for further discussion of
this issue. Furthermore, the ratio in the operation count between fully
implicit discretization and the ADI method is $O(n_{x}n_{y})$. That is, even
if we need to decrease the time step or increase the number of iterations to
achieve convergence, the ADI method is still more efficient as long as the
additional effort is of $o(n_{x}n_{y})$. In our experience, under the same
conditions, ADI method is significantly more efficient compared to fully
implicit discretization.
## References
* [1] K.J. Ruschak. Coating flows. Ann. Rev. Fluid Mech., 17:65, 1999.
* [2] H.-C. Chang and E. A. Demekhin. Complex wave dynamics on thin films. Elsevier, New York, 2002.
* [3] A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69:931, 1997.
* [4] H.A. Stone, A.D. Stroock, and A. Ajdari. Engineering flows in small devices. Ann. Rev. Fluid Mech., 36:381, 2004.
* [5] R. V. Craster and O. K. Matar. Dynamics and stability of thin liquid films. Rev. Mod. Phys., 81:1131, 2009.
* [6] P. L. Kapitsa and S. P. Kapitsa. Wave flow of thin fluid layers of liquid. Zh. Eksp. Teor. Fiz., 19:105, 1949.
* [7] J. Liu and J. P. Gollub. Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett., 70:2289, 1993.
* [8] S. V. Alekseenko, V. E. Nakoryakov, and B. G. Pokusaev. Wave flow of liquid films. Begell House, New York, 1994.
* [9] H.-C. Chang. Wave evolution on a falling film. Annu. Rev. Fluid Mech., 26:103, 1994.
* [10] S. Saprykin, E. A. Demekhin, and S. Kalliadasis. Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses. Phys. Fluids, 17:117105, 2005.
* [11] Y. Y. Trifonov and O. Y. Tsvelodub. Nonlinear waves on the surface of a falling liquid film. part 1. waves of the first family and their stability. J. Fluid Mech., 229:531, 1991.
* [12] H.-C. Chang, E. A. Demekhin, and S. S. Saprikin. Noise-driven wave transitions on a vertically falling film. J. Fluid Mech., 462:255, 2002.
* [13] T.-S. Lin and L. Kondic. Thin film flowing down inverted substrates: two dimensional flow. Phys. Fluids, 22:052105, 2010.
* [14] S. M. Troian, E. Herbolzheimer, S. A. Safran, and J. F. Joanny. Fingering instabilities of driven spreading films. Europhys. Lett., 10:25, 1989.
* [15] J. A. Diez and L. Kondic. Contact line instabilities of thin liquid films. Phys. Rev. Lett., 86:632, 2001.
* [16] L. Kondic and J. A. Diez. Contact line instabilities of thin film flows: Constant flux configuration. Phys. Fluids, 13:3168, 2001.
* [17] H. E. Huppert. Flow and instability of a viscous current down a slope. Nature, 300:427, 1982.
* [18] N. Silvi and E. B. Dussan V. On the rewetting of an inclined solid surface by a liquid. Phys. Fluids, 28:5, 1985.
* [19] J. R. de Bruyn. Growth of fingers at a driven three–phase contact line. Phys. Rev. A, 46:4500, 1992.
* [20] S. V. Alekseenko, D. M. Markovich, and S. I. Shtork. Wave flow of rivulets on the outer surface of an inclined cylinder. Phys. Fluids, 8:3288, 1996.
* [21] A. Indeikina, I. Veretennikov, and H.-C. Chang. Drop fall-off from pendent rivulets. J. Fluid Mech., 338:173, 1997.
* [22] T.M. Segin, B. S. Tilley, and L. Kondic. On undercompresive shocks and flooding in countercurrent two-layer flows. J. Fluid Mech., 532:217, 2005.
* [23] M. Mavromoustaki, O.K. Matar, and R.V. Craster. Shock-wave solutions in two-layer channel flow. I. One-dimensional flows. Phys. Fluids, 22:112102, 2010.
* [24] L. Kondic. Instabilities in gravity driven flow of thin liquid films. SIAM Review, 45:95, 2003.
* [25] L. W. Schwartz. Viscous flows down an inclined plane: instability and finger formation. Phys. Fluids A, 1:443, 1989.
* [26] A. Oron and P. Rosenau. Formation of patterns induced by thermocapillarity and gravity. J. Phys. (France), 2:131, 1992.
* [27] A. J. Tanasijczuk, C. A. Perazzo, and J. Gratton. Navier-Stokes solutions for steady parallel-sided pendent rivulets. Euro. J. Mech. B/Fluids, 29:465, 2010.
* [28] J.S. Sullivan, S.K. Wilson, and B.R. Duffy. A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress. Quart. J. Mech. Appl. Math., 61:25, 2008.
* [29] E.S. Benilov. On the stability of shallow rivulets. J. Fluid Mech., 636:455, 2009.
* [30] B.R. Duffy and S.K. Wilson. A rivulet of perfectly wetting fluid with temperature-dependent viscosity draining down a uniformly heated or cooled slowly varying substrate. Phys. Fluids, 15:3236, 2003.
* [31] See supplementary material at [url will be inserted by aip].
* [32] J. A. Diez and L. Kondic. Computing three-dimensional thin film flows including contact lines. J. Comp. Phys., 183:274, 2002.
* [33] T. P. Witelski and M. Bowen. ADI schemes for higher-order nonlinear diffusion equations. Applied Numer. Math., 45:331, 2003.
|
arxiv-papers
| 2011-09-28T20:50:49 |
2024-09-04T02:49:22.565477
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T.-S. Lin, L. Kondic, A. Filippov",
"submitter": "Te-Sheng Lin",
"url": "https://arxiv.org/abs/1109.6353"
}
|
1109.6394
|
# The geometry of Grassmannian manifolds and Bernstein type theorems for
higher codimension
J. Jost, Y. L. Xin and Ling Yang Max Planck Institute for Mathematics in the
Sciences, Inselstr. 22, 04103 Leipzig, Germany Department of Mathematics and
Computer Science, University of Leipzig, 04109 Leipzig, Germany.
jost@mis.mpg.de Institute of Mathematics, Fudan University, Shanghai 200433,
China. ylxin@fudan.edu.cn Institute of Mathematics, Fudan University,
Shanghai 200433, China. lingyang@fudan.edu.cn
###### Abstract.
We identify a region ${\tenmsb W}_{\frac{1}{3}}$ in a Grassmann manifold
$\mathbb{G}_{n,m}$, not covered by a usual matrix coordinate chart, with the
following important property. For a complete $n-$submanifold in
$\mathbb{R}^{n+m}\,(n\geq 3,m\geq 2)$ with parallel mean curvature whose image
under the Gauss map is contained in a compact subset $K\subset{\tenmsb
W}_{\frac{1}{3}}\subset\mathbb{G}_{n,m}$, we can construct strongly
subharmonic functions and derive a priori estimates for the harmonic Gauss
map. While we do not know yet how close our region is to being optimal in this
respect, it is substantially larger than what could be achieved previously
with other methods. Consequently, this enables us to obtain substantially
stronger Bernstein type theorems in higher codimension than previously known.
###### 1991 Mathematics Subject Classification:
58E20,53A10.
The first author is supported by the ERC Advanced Grant FP7-267087, the second
and third authors are partially supported by NSFC. They are grateful to the
Max Planck Institute for Mathematics in the Sciences in Leipzig for its
hospitality and continuous support.
## 1\. Introduction
The classical Bernstein theorem states that any complete minimal graph $M^{2}$
in $\mathbb{R}^{3}$ has to be an affine plane. Equivalently, all tangent
planes are parallel to each other, and the Gauss map $\gamma:M\to S^{2}$ is
constant. This scheme extends to higher dimensions, that is, minimal graphs in
$\mathbb{R}^{n+1}$, although the result becomes somewhat weaker for $n\geq 7$.
In this work, we are interested in such a scheme for higher codimensional
minimal submanifolds of Euclidean space.
We now make this more precise. Let $M^{n}$ be a complete $n$-dimensional
submanifold in Euclidean space $\mathbb{R}^{n+m}$. The Grassmann manifold
$\mathbb{G}_{n,m}$ is the target manifold of the Gauss map
$\gamma:M\to\mathbb{G}_{n,m}$ that assigns to each point of $M$ the direction
of its tangent space in $\mathbb{R}^{n+m}$. By the Ruh-Vilms theorem [20] the
Gauss map is a harmonic map into the Grassmann manifold iff $M$ has parallel
mean curvature. In particular, the Gauss map is harmonic for minimal
submanifolds of Euclidean space. When $m=1$, the Grassmann manifold reduces to
the sphere $S^{n}$, and investigating the Gauss map has identified many
conditions under which a minimal submanifold has to be affine. In this scheme,
the key is to show that the Gauss map is constant. The aim of the present
paper then is to obtain Bernstein type result for higher codimension by using
the geometric properties of the Grassmann manifolds $\mathbb{G}_{n,m}$ and
studying the corresponding harmonic Gauss maps into $\mathbb{G}_{n,m}.$
This method was introduced by Hildebrandt-Jost-Widman [12]. The distance
function from a fixed point $P_{0}\in\mathbb{G}_{n,m}$ in a geodesic ball of
radius $\frac{\sqrt{2}}{4}\pi$ and centered at $P_{0}$ is convex. Using this
fact one obtains a strongly subharmonic function on $M$ by composing the
distance function with the Gauss map $\gamma$ under the assumption that the
image of the Gauss map is contained in a closed subset of the geodesic convex
ball. Suppressing some basic technical difficulties in this introduction, an
application of the maximum principle should then yield that this subharmonic
function is constant, and hence so then is the Gauss map. Using this local
convex geometry of Grassmann manifolds and advanced harmonic map regularity
theory to overcome the indicated technical difficulties, Hildebrandt-Jost-
Widman obtained Bernstein type results for higher dimension and codimension.
Somehow, however, while a geodesic ball of radius $\frac{\sqrt{2}}{4}\pi$ is
the largest convex ball for $m\geq 2$ (for $m=1$, we can take the ball of the
radius $\frac{1}{2}\pi$, that is, a hemisphere), it is not the largest convex
set, and therefore the result of [12] is not yet the best possible, and there
seems opportunity for improvement. In [15], then, the largest such geodesic
convex set was found, and stronger results were obtained. But, as we shall
explore here, there is still further opportunity for improvement, via a deeper
understanding of the convex geometry of Grassmann manifolds.
Since the geometry of general Grassmann manifolds is not as easy to visualize
as the one for the special case $m=1$ where the Grassmannian reduces to the
sphere $S^{n}$, let us briefly discuss the situation for the latter. In
general, the domain of a strictly convex function cannot contain any closed
geodesic, and therefore, a closed hemipshere in $S^{n}$ cannot support a
strictly convex function. Thus, no ball with a radius $\geq\frac{\pi}{2}$ can
support a strictly convex function. Nevertheless, there exist larger open sets
that contain an open hemisphere, but still do not contain any closed geodesic.
In fact, on $S^{2}$, we can take the complement $U$ of a semicircle. In [16],
we have shown that any compact subset of $U$ supports a strictly convex
function. (It is, however, not true that this convex supporting property holds
for any open subset of $S^{n}$ containing no closed geodesic.) First of all,
this inspires us to look for such convex supporting sets in general Grassmann
manifolds. Secondly, the reason why we need convex functions is that the
postcomposition of a harmonic map (in our case, the Gauss map of a minimal
submanifold) with a convex function is subharmonic so that we can apply the
maximum principle. When suitable technical conditions are met (their
verification is part of our technical achievements), this tells us that the
composition, being subharmonic, is constant. When the function is nontrivial,
we can then conclude that the Gauss map itself is constant. Bingo – the
minimal submanifold is affine! Now, however, it might happen that the
postcomposition of the Gauss map with a function $f$ turns out to be
subharmonic without $f$ being convex. Here, the more subtle geometry of the
Grassmann manifolds for $m\geq 2$ enters. An intuition about the geometry
comes from the following observation. We can rotate the plane $e_{1}\wedge
e_{2}$ in $\mathbb{R}^{4}$ either into the plane $e_{1}\wedge e_{3}$ or into
the plane $e_{3}\wedge e_{4}$. Each of these rotations yields a geodesic arc
in $\mathbb{G}_{2,2}$. However, these two geodesics stop diverging from their
neighboring geodesics at different distances ($\frac{\sqrt{2}}{4}\pi$ vs.
$\frac{\pi}{2}$) (equivalently, the first conjugate point is reached after a
distance of $\frac{\sqrt{2}}{2}\pi$ vs. $\pi$), and therefore, a maximal
geodesically convex set is larger than a ball of radius
$\frac{\sqrt{2}}{4}\pi$. While this geometric intuition will be important for
the present paper, nevertheless, due to more subtle features of Grassmannian
geometry, we can construct suitable functions that are no longer convex, but
still give the required subharmonicity on even larger sets than such maximal
geodesically convex sets. The function that we use is defined in a geometric
manner from the Plücker embedding, or equivalently, in terms of Jordan angles.
It is designed to take into account the phenomenon just described, namely that
rotations of subspaces of Euclidean space are geometrically different
depending on how many independent normal directions are involved. Again, this
is a feature particular for $m\geq 2$, not yet apparent on a sphere. In fact,
and this is a main point of this paper, we shall find such functions $f$ here,
defined on larger subsets of $\mathbb{G}_{n,m}$ than in previous work (as we
shall explain in more detail below, these subsets are larger and more natural
than the ones identified in our previous work [17]). Therefore, we can obtain
a larger possible range of the Gauss map of a minimal submanifold that still
implies that it is constant. Thus, we obtain stronger Bernstein type results
than previously known. Actually, our approach, being quantitative, also yields
estimates for the second fundamental form of minimal submanifolds under
suitable geometric conditions.
Of course, there are limits how far one can push Bernstein theorems in higher
codimension. For codimension 1, we have Moser’s theorem [19] that any entire
minimal graph with bounded slope is affine linear (for $n\leq 7$, the bounded
slope condition is not needed – this is Simons’ theorem [21]). In higher
codimensions, the situation is not so good. After all, there is the important
example of Lawson-Osserman [18] of a nontrivial minimal graph (to be analyzed
in detail in the Appendix of the present paper) of bounded slope that sets a
limit for how far one can go. The ultimate aim then is to narrow the gap
between the range of Bernstein theorems and the counterexamples as far as
possible. Our paper is a step in this direction. We do not know yet whether
one can still go further, or whether there exist even more striking examples
than the one of [18].
One more remark: Most of what we do in this paper holds for submanifolds of
parallel mean curvature, and not only for those of vanishing mean curvature,
the minimal ones. Since, however, according to mathematical tradition, the
minimal submanifolds are the most interesting ones, in this introduction, we
mostly restrict ourselves to discuss those minimal ones in place of the more
general ones of parallel mean curvature.
Let us now describe the results in more precise terms. The Grassmann manifold
$\mathbb{G}_{n,m}$ can be imbedded into Euclidean space by the Plücker
imbedding. This simply means that we consider an oriented $n$-plane in
$\mathbb{R}^{n+m}$ as an element of $\Lambda^{n}(\mathbb{R}^{n+m})$. In this
space, we also have a scalar product. We then can introduce suitable
functions, called $w$-functions and $v$-functions, on $\mathbb{G}_{n,m}$. The
$w$-function is simply given by the above scalar product with some fixed
reference $n$-plane, and $v$ is the inverse of $w$. By precomposition with the
Gauss map, we then get corresponding functions on $M$. The region of convexity
for the $v$-function is the same as for the corresponding distance function
[26], but in our previous work [17] we have already found that there is a
larger region in $\mathbb{G}_{n,m}$ on which the $v$-function is no longer
convex, but where its composition with the Gauss map is still strongly
subharmonic by rather delicate estimates. We then obtain the following theorem
###### Theorem 1.1.
Let $z^{\alpha}=f^{\alpha}(x^{1},\cdots,x^{n}),\ \alpha=1,\cdots,m$, be smooth
functions defined everywhere in ${\tenmsb R}^{n}$ ($n\geq 3,m\geq 2$). Suppose
their graph $M=(x,f(x))$ is a submanifold with parallel mean curvature in
${\tenmsb R}^{n+m}$. Suppose that there exists a number $\beta_{0}<3$ such
that
(1.1) $\Delta_{f}=\Big{[}\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\Big{]}^{\frac{1}{2}}\leq\beta_{0}.$
Then $f^{1},\cdots,f^{m}$ has to be affine linear, i.e., it represents an
affine $n$-plane.
In our previous work (see [15, 17, 25, 26]) the image under the Gauss map for
a submanifold in Euclidean space is contained in an $(n\times m)-$matrix chart
for a Grassmann manifold. There is, however, still further room for
improvement, and here, we consider a more general situation. Recall that a
Grassmann manifold can be viewed as a minimal submanifold in the Euclidean
sphere via the Plücker imbedding. This leads us to employ a technique from our
previous work [16] for the case of codimension $m=1$, where the convex
geometry of the Euclidean sphere has been thoroughly investigated.
We shall introduce new notions of S-orthogonality and of S-maps on a Grassmann
manifold. Let $P,Q\in\mathbb{G}_{n,m}$ be $S-$orthogonal to each other. This
means that their intersection is of dimension $n-1$ and one is obtained from
the other by rotating a single tangent vector into a normal direction by an
angle of $\frac{\pi}{2}$. Using the $w$-function we define an $S$-map
$\mathscr{S}:\mathbb{G}_{n,m}\to\overline{{\tenmsb D}}$ relative to $P$ and
$Q$. This is a map onto the closed unit disk.
Define
(1.2) ${\tenmsb W}_{c}:=\mathscr{S}^{-1}\Big{(}\overline{{\tenmsb
D}}\backslash\big{(}\overline{{\tenmsb D}}_{c}\cup\big{\\{}(a,0):a\leq
0\big{\\}}\big{)}\Big{)},$
with $\overline{{\tenmsb D}}_{c}=\big{\\{}(x_{1},x_{2})\in{\tenmsb
R}^{2}:x_{1}^{2}+x_{2}^{2}\leq c^{2}\big{\\}}$.
Now, for our method to apply, the image of our Gauss map can be any compact
subset of ${\tenmsb W}_{\frac{1}{3}}.$ This is a global region in
$\mathbb{G}_{n,m}$ that is not contained in any matrix coordinate chart. More
precisely, even when the Gauss image is somewhat larger, we still can find
subharmonic functions on our submanifold by combining some tricks in our
previous work in [16] and [17]. In this way, we obtain the following Bernstein
type theorem.
###### Theorem 1.2.
Let $z^{\alpha}=f^{\alpha}(x^{1},\cdots,x^{n}),\ \alpha=1,\cdots,m$, be smooth
functions defined everywhere in ${\tenmsb R}^{n}$ ($n\geq 3,m\geq 2$), such
that their graph $M=(x,f(x))$ is a submanifold with parallel mean curvature in
${\tenmsb R}^{n+m}$. Suppose that there exist $\beta_{0}<+\infty$ and
$\beta_{1}<3$, such that
(1.3) $\Delta_{f}:=\Big{[}\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\Big{]}^{\frac{1}{2}}\leq\beta_{0}.$
and for certain $\alpha$ and $i$
(1.4) $\Delta_{f}\leq\beta_{1}\Big{(}1+\big{(}\frac{\partial
f^{\alpha}}{\partial x^{i}}\big{)}^{2}\Big{)}^{\frac{1}{2}}.$
Then $f^{1},\cdots,f^{m}$ has to be affine linear (representing an affine
$n$-plane).
In fact, we can prove more general results, see Theorem 6.1 in §6, but the
preceding statement perhaps best highlights the main achievement of this
paper.
In the Appendix we see that for the Lawson-Osserman cone the $v-$function is
the constant $9$. The Jordan angles between the Gauss image of this cone and
the coordinate $n-$ plane are constants. The largest value for a Jordan angle
permitted by the refined term $(1+\frac{\partial f^{2}}{\partial
x^{1}})^{\frac{1}{2}}$ in the above Theorem is $\sqrt{6}$. Theorem 1.2 is a
substantial improvement of the previous results which only could reach smaller
values.
The paper is organized as follows. We will describe in §2 that the basic
geometry of Grassmann manifolds and define S-orthogonality and S-maps and show
their properties. §3 will be devoted to the computation of $\Delta\log w$. For
the general case the results come from [26]. We also compute it for minimal
submanifolds with rank of the Gauss map less than or equal $2$. We obtain a
formula more general than that in [6]. In §4 we will construct subharmonic
functions $F$ on our submanifolds by using the techniques in [16]. The level
sets of $F$ coincide with those of $\log w$. We first prove an important
transition lemma (Lemma 4.1). Then, we can estimate $\mbox{Hess}\,F$ in terms
of $\mbox{Hess}\,\log w$, which is already computed in the previous section.
Once we have subharmonic functions, the extrinsic rigidity results for compact
minimal submanifolds in the sphere (Theorem 4.2) follows immediately. Using
strongly subharmonic functions we can also study complete submanifolds in
Euclidean space by using harmonic map regularity theory, as Hildebrandt-Jost-
Widman in [12]. In §5 we will obtain curvature estimates (Lemma 5.3 ). In §6
the iteration method will be used and the quantitatively controlled Gauss
image shrinking Lemmas 6.2 and 6.3 will be obtained. Finally, in this section
we prove our main results, Theorem 6.1 and Theorem 6.2 of which Theorem 1.2 is
a direct corollary. Finally, in an Appendix, we provide computations for the
Lawson-Osserman cone and related coassociated $4-$manifolds in
$\mathbb{R}^{7}.$
## 2\. Geometry of Grassmann manifolds
Let $\mathbb{G}_{n,m}$ be the Grassmann manifold consisting of the oriented
linear $n-$subspaces in $(n+m)-$Euclidean space ${\tenmsb R}^{n+m}.$ The
canonical Riemannian structure on $\mathbb{G}_{n,m}$ makes it a natural
generalization of the Euclidean sphere. $\mathbb{G}_{n,m}=SO(n+m)/SO(n)\times
SO(m)$ is an irreducible symmetric space of compact type.
For every $P\in\mathbb{G}_{n,m}$, we choose an oriented basis
$\\{u_{1},\cdots,u_{n}\\}$ of $P$, and let
(2.1) $\psi(P):=u_{1}\wedge\cdots\wedge u_{n}\in\Lambda^{n}({\tenmsb
R}^{n+m}).$
A different basis for $P$ shall give a different exterior product, but the two
products differ only by a positive scalar; $\psi(P)$ is called the Plücker
coordinate of $P$, which is a homogenous coordinate.
Via the Plücker embedding, $\mathbb{G}_{n,m}$ can be viewed as a submanifold
of some Euclidean space $\mathbb{R}^{N}(N=C_{n+m}^{n})$. The restriction of
the Euclidean inner product is denoted by
$w:\mathbb{G}_{n,m}\times\mathbb{G}_{n,m}\rightarrow{\tenmsb R}$
(2.2)
$w(P,Q)=\frac{\langle\psi(P),\psi(Q)\rangle}{\langle\psi(P),\psi(P)\rangle^{\frac{1}{2}}\langle\psi(Q),\psi(Q)\rangle^{\frac{1}{2}}}.$
If $\\{e_{1},\cdots,e_{n}\\}$ is an oriented orthonormal basis of $P$ and
$\\{f_{1},\cdots,f_{n}\\}$ is an oriented orthonormal basis of $Q$, then
$w(P,Q)=\langle e_{1}\wedge\cdots\wedge e_{n},f_{1}\wedge\cdots\wedge
f_{n}\rangle=\det W$
with the W-matrix $W=\big{(}\langle e_{i},f_{j}\rangle\big{)}$. It is well-
known that
$W^{T}W=O^{T}\Lambda O$
with an orthogonal matrix $O$ and
$\Lambda=\left(\begin{array}[]{ccc}\mu_{1}^{2}&&\\\ &\ddots&\\\
&&\mu_{n}^{2}\end{array}\right).$
Here each $0\leq\mu_{i}^{2}\leq 1$. Putting $p:=\min\\{m,n\\}$, then at most
$p$ elements in $\\{\mu_{1}^{2},\cdots,\mu_{n}^{2}\\}$ are not equal to $1$.
Without loss of generality, we can assume $\mu_{i}^{2}=1$ whenever $i>p$. We
also note that the $\mu_{i}^{2}$ can be expressed as
(2.3) $\mu_{i}^{2}=\frac{1}{1+\lambda_{i}^{2}}$
with $\lambda_{i}\in[0,+\infty)$.
The Jordan angles between $P$ and $Q$ are critical values of the angle
$\theta$ between a nonzero vector $u$ in $P$ and its orthogonal projection
$u^{*}$ in $Q$ as $u$ runs through $P$. Let $\theta_{i}$ be a Jordan angle
between $P$ and $Q$ determined by a unit vector $e_{i}$ and its projection
$e_{i}^{*}$ in $Q$, we call $e_{i}$ an angle direction of $P$ relative to $Q$,
and the 2-plane spanned by $e_{i}$ and $e_{i}^{*}$ an angle 2-plane between
$P$ and $Q$ (see [22]). A direct calculation shows there are $n$ Jordan angles
$\theta_{1},\cdots,\theta_{n}$, with $\theta_{p+1}=\cdots=\theta_{n}=0$ and
$\theta_{i}=\arccos(\mu_{i})\qquad 1\leq i\leq p.$
Thus
(2.4)
$|w|=\big{(}\det(W^{T}W)\big{)}^{\frac{1}{2}}=\det(\Lambda)^{\frac{1}{2}}=\prod_{i=1}^{n}\cos\theta_{i}$
and (2.3) becomes
(2.5) $\lambda_{i}=\tan\theta_{i}.$
If $w(P,Q)>0$, arrange all the Jordan angles between $P$ and $Q$ as
$\frac{\pi}{2}>\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{r}>\theta_{r+1}=\cdots=\theta_{n}=0$
with $0\leq r\leq p$. (If $\theta_{i}=\frac{\pi}{2}$ for some $i$, then (2.4)
implies $w(P,Q)=0$, which contradicts $w(P,Q)>0$.) Then one can find an
orthonormal basis $\\{e_{1},\cdots,e_{n+m}\\}$ of ${\tenmsb R}^{n+m}$, such
that $P$ is spanned by $\\{e_{1},\cdots,e_{n}\\}$, which are angle directions
of $P$ relative to $Q$, and $\\{e_{i}\wedge e_{n+i}:1\leq i\leq r\\}$ are
angle 2-planes between $P$ and $Q$. Denote
(2.6)
$f_{i}:=\left\\{\begin{array}[]{cc}\cos\theta_{i}e_{i}+\sin\theta_{i}e_{n+i}&1\leq
i\leq r,\\\ e_{i}&r+1\leq i\leq n.\end{array}\right.$
and
(2.7)
$f_{n+\alpha}:=\left\\{\begin{array}[]{cc}-\sin\theta_{\alpha}e_{\alpha}+cos\theta_{\alpha}e_{n+\alpha}&1\leq\alpha\leq
r,\\\ e_{n+\alpha}&r+1\leq\alpha\leq m.\end{array}\right.$
then $\\{f_{1},\cdots,f_{n+m}\\}$ is also an orthonormal basis of ${\tenmsb
R}^{n+m}$, and $f_{1}\wedge\cdots\wedge f_{n}$ is a Plücker coordinate of $Q$.
The distance between $P$ and $Q$ is defined by
(2.8) $d(P,Q)=\sqrt{\sum\theta_{i}^{2}}.$
It is a natural generalization of the canonical distance of Euclidean spheres.
Now we fix $P_{0}\in\mathbb{G}_{n,m}.$ We represent it by $n$ vectors
$\epsilon_{i}$, which are complemented by $m$ vectors $\epsilon_{n+\alpha}$.
Denote
(2.9) ${\tenmsb U}:=\\{P\in\mathbb{G}_{n,m}:w(P,P_{0})>0\\}.$
We can span an arbitrary $P\in{\tenmsb U}$ by $n$ vectors $e_{i}$:
(2.10) $e_{i}=\epsilon_{i}+Z_{i\alpha}\epsilon_{n+\alpha}.$
Here $Z=(Z_{i\alpha})$ could be regarded as the $(n\times m)$-matrix
coordinate of $P$. The canonical Riemannian metric in ${\tenmsb U}$ can be
described as
(2.11) $ds^{2}=tr((I_{n}+ZZ^{T})^{-1}dZ(I_{m}+Z^{T}Z)^{-1}dZ^{T}),$
where $I_{m}$ (res. $I_{n}$) denotes the $(m\times m)$-identity (res. $n\times
n$) matrix. It is shown that (2.11) can be derived from (2.8) in [23].
Here and in the sequel, we use the summation convention and agree on the
ranges of indices:
$1\leq\alpha,\beta\leq m,\;1\leq i,j,k\leq n.$
We shall now introduce the new concepts of S-orthogonal and S-map, which will
play a crucial role for our investigations.
###### Proposition 2.1.
For any $P,Q\in\mathbb{G}_{n,m}$, the following statements are equivalent:
(a) $\dim(P\cap Q)=n-1$ and $w(P,Q)=0$;
(b) $\dim(P+Q)=n+1$ and $w(P,Q)=0$, where $P+Q=\\{u+v:u\in P,v\in Q\\}$;
(c) There exists a single Jordan angle between $P$ and $Q$ being
$\frac{\pi}{2}$, and the other $(n-1)$ Jordan angles are all $0$;
(d) $Q$ is the nearest point to $P$ in $\partial{\tenmsb U}$, where ${\tenmsb
U}$ is the $(n\times m)$-matrix coordinate chart centered at $P$;
(e) There exists an orthonormal basis $\\{e_{i},e_{n+\alpha}\\}$ of ${\tenmsb
R}^{n+m}$, such that the Plücker coordinates of $P$ and $Q$ are $e_{1}\wedge
e_{2}\wedge\cdots\wedge e_{n}$ and $e_{n+1}\wedge e_{2}\wedge\cdots\wedge
e_{n}$, respectively.
###### Proof.
(a)$\Longleftrightarrow$(b) follows from $\dim(P+Q)=\dim P+\dim Q-\dim(P\cap
Q)$.
(a)$\Longleftrightarrow$(c) is an immediate corollary of the definition of
Jordan angles and (2.4).
$\partial{\tenmsb U}$ consists of all $S\in\mathbb{G}_{n,m}$ satisfying
$w(P,S)=0$; it follows from $(\ref{w1})$ that at least 1 Jordan angle between
$S$ and $P$ is $\frac{\pi}{2}$; hence one can obtain
$d(P,S)\geq\frac{\pi}{2}\qquad\text{for all }S\in\partial{\tenmsb U}$
from (2.8), and the equality holds if and only if the other $(n-1)$ Jordan
angles all vanish. Hence (c)$\Longleftrightarrow$(d).
(e)$\Longrightarrow$(a) is trivial. To prove (a)$\Longrightarrow$(e), it
suffices to choose $\\{e_{2},\cdots,e_{n}\\}$ as an orthonormal basis of
$P\cap Q$, and put $e_{1}$ (or $e_{n+1}$) to be the unit vector in $P$ (or
$Q$) that is orthogonal to $P\cap Q$.
∎
###### Definition 2.1.
Two points $P$ and $Q$ in a Grassmann manifold are called S-orthogonal, if
they satisfy one of the properties in Proposition 2.1.
###### Remark 2.1.
There is only one Jordan angle between any two points in the sphere $S^{n}$.
If it equals $\frac{\pi}{2}$, they are orthogonal each other. The notion of
S-orthogonal is a natural generalization of orthogonality in $S^{n}$.
Let $P,Q\in\mathbb{G}_{n,m}$ be S-orthogonal. By Proposition 2.1, there exists
an orthonormal basis $\\{e_{i},e_{n+\alpha}\\}$ of ${\tenmsb R}^{n+m}$, such
that $\psi(P)=e_{1}\wedge e_{2}\wedge\cdots\wedge e_{n}$ and
$\psi(Q)=e_{n+1}\wedge e_{2}\wedge\cdots\wedge e_{n}$. Let $\gamma:{\tenmsb
R}/(2\pi{\tenmsb Z})\rightarrow\mathbb{G}_{n,m}$ be a closed curve, such that
$\gamma(t)$ is spanned by $\\{\cos t\ e_{1}+\sin t\
e_{n+1},e_{2},\cdots,e_{n}\\}$. Then whenever $|t-s|<\frac{\pi}{2}$, there
exists one and only one Jordan angle between $\gamma(t)$ and $\gamma(s)$ being
$|t-s|$, and the other Jordan angles are all $0$; hence by (2.8),
$d(\gamma(t),\gamma(s))=|t-s|\qquad\text{whenever }|t-s|<\frac{\pi}{2}.$
It implies $\gamma$ is the closed geodesic that is extended from the minimal
geodesic between $P$ and $Q$. In the following, we write $P_{t}=\gamma(t)$; in
particular, $P_{0}=P$ and $P_{\frac{\pi}{2}}=Q$.
The S-map $\mathscr{S}:\mathbb{G}_{n,m}\rightarrow{\tenmsb R}^{2}$ is defined
by
(2.12) $S\mapsto\big{(}w(S,P_{0}),w(S,P_{\frac{\pi}{2}})\big{)}.$
If there exists a nonzero vector in $S$ orthogonal to $P_{0}\cap
P_{\frac{\pi}{2}}$, then $w(S,P_{0})=w(S,P_{\frac{\pi}{2}})=0$, and hence
$\mathscr{S}(S)=0$. Otherwise, the orthogonal projection $p$ from $P_{0}\cap
P_{\frac{\pi}{2}}$ to $S$ has rank $n-1$, which enables us to find a unit
vector $f_{1}$ in $S$ which is orthogonal to the $(n-1)$-dimensional
projective image $p(P_{0}\cap P_{\frac{\pi}{2}})$. Certainly, one can get
(2.13) $\displaystyle w(S,P_{0})$ $\displaystyle=\langle f_{1},e_{1}\rangle
w\big{(}p(P_{0}\cap P_{\frac{\pi}{2}}),P_{0}\cap P_{\frac{\pi}{2}}\big{)},$
$\displaystyle w(S,P_{\frac{\pi}{2}})$ $\displaystyle=\langle
f_{1},e_{n+1}\rangle w\big{(}p(P_{0}\cap P_{\frac{\pi}{2}}),P_{0}\cap
P_{\frac{\pi}{2}}\big{)}.$
It follows that
$w^{2}(S,P_{0})+w^{2}(S,P_{\frac{\pi}{2}})\leq\langle
f_{1},e_{1}\rangle^{2}+\langle f_{1},e_{n+1}\rangle^{2}\leq 1$
and the equality holds if and only if $p(P_{0}\cap
P_{\frac{\pi}{2}})=P_{0}\cap P_{\frac{\pi}{2}}$ and
$f_{1}\in\text{span}\\{e_{1},e_{n+1}\\}$; i.e. $S=P_{t}$ for some
$t\in[-\pi,\pi)$. Hence $\mathscr{S}$ is a smooth map onto the closed unit
2-disk $\overline{{\tenmsb D}}$. Although $\mathscr{S}$ is not one-to-one, any
point on $\partial\overline{{\tenmsb D}}$ has one and only one preimage. For
simplicity we write
(2.14) $x_{1}=w(\cdot,P_{0}),\qquad x_{2}=w(\cdot,P_{\frac{\pi}{2}}).$
Then from (2.13) one can obtain
(2.15) $\displaystyle(\cos t\ x_{1}+\sin t\ x_{2})(S)$ $\displaystyle=\cos t\
w(S,P_{0})+\sin t\ w(S,P_{\frac{\pi}{2}})$ $\displaystyle=\langle f_{1},\cos
t\ e_{1}+\sin t\ e_{n+1}\rangle w\big{(}p(P_{0}\cap
P_{\frac{\pi}{2}}),P_{0}\cap P_{\frac{\pi}{2}}\big{)}$
$\displaystyle=w(S,P_{t}).$
## 3\. Laplacian of $\log w$
Let $M^{n}\to\bar{M}^{n+m}$ be an isometric immersion with second fundamental
form $B,$ which can be viewed as a cross-section of the vector bundle
Hom($\odot^{2}TM,NM$) over $M,$ where $TM$ and $NM$ denote the tangent bundle
and the normal bundle along $M$, respectively. The connection on $TM$ and $NM$
(denoted by $\nabla$) can be induced naturally from the Levi-Civita connection
on $\bar{M}$ (denoted by $\bar{\nabla}$). We define the mean curvature $H$ as
the trace of the second fundamental form. It is a normal vector field on $M$
in $\bar{M}$. If $\nabla H=0$, we say $M$ has parallel mean curvature;
moreover if $H$ vanishes on $M$ everywhere, it is called a minimal
submanifold.
The second fundamental form, the curvature tensor of the submanifold (denoted
by $R$), the curvature tensor of the normal bundle (denoted by $R^{N}$) and
that of the ambient manifold (denoted by $\bar{R}$) satisfy the Gauss
equations, the Codazzi equations and the Ricci equations (see [23] for
details, for example). Here and in the sequel, $\\{e_{i}\\}$ denotes a local
orthonormal tangent frame field and $\\{\nu_{\alpha}\\}$ is a local
orthonormal normal frame field of $M$;
$h_{\alpha,ij}:=\langle\langle B(e_{i},e_{j}),\nu_{\alpha}\rangle$
are the coefficients of the second fundamental form $B$ of $M$ in $\bar{M}$.
Now we consider a submanifold $M^{n}$ in Euclidean space $\mathbb{R}^{n+m}$
with parallel mean curvature.
Let $0$ be the origin of ${\tenmsb R}^{n+m}$, $SO(n+m)$ be the Lie group
consisting of all orthonormal frames $(0;e_{i},\nu_{\alpha})$.
$TF=\big{\\{}(x;e_{1},\cdots,e_{n}):x\in M,e_{i}\in T_{p}M,\langle
e_{i},e_{j}\rangle=\delta_{ij}\big{\\}}$ be the principle bundle of
orthonormal tangent frames over $M$, and
$NF=\big{\\{}(x;\nu_{1},\cdots,\nu_{m}):x\in M,\nu_{\alpha}\in
N_{x}M\big{\\}}$ be the principle bundle of orthonormal normal frames over
$M$. Then $\bar{\pi}:TF\oplus NF\rightarrow M$ is the projection with fiber
$SO(n)\times SO(m)$.
The Gauss map $\gamma:M\rightarrow\mathbb{G}_{n,m}$ is defined by
$\gamma(x)=T_{x}M\in\mathbb{G}_{n,m}$
via the parallel translation in ${\tenmsb R}^{n+m}$ for every $p\in M$. Then
the following diagram commutes
$\begin{CD}TF\oplus NF@>{i}>{}>SO(n+m)\\\ @V{\bar{\pi}}V{}V@V{}V{\pi}V\\\
M@>{\gamma}>{}>\mathbb{G}_{n,m}\end{CD}$
where $i$ denotes the inclusion map and
$\pi:SO(n+m)\rightarrow\mathbb{G}_{n,m}$ is defined by
$(0;e_{i},\nu_{\alpha})\mapsto e_{1}\wedge\cdots\wedge e_{n}.$
From the above diagram we know that the energy density of the Gauss map (see
[23] Chap.3, §3.1)
$e(\gamma)=\frac{1}{2}\left<\gamma_{*}e_{i},\gamma_{*}e_{i}\right>=\frac{1}{2}|B|^{2}.$
Ruh-Vilms proved that the mean curvature vector of $M$ is parallel if and only
if its Gauss map is a harmonic map [20].
We define
(3.1) $w:=w(\cdot,P_{0})\circ\gamma.$
Let $\\{\varepsilon_{i}\\}$ be an orthonormal basis of $P_{0}$, then
$w=\langle e_{1}\wedge\cdots\wedge
e_{n},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}\rangle$ at every point
$x\in M$. From the Codazzi equations, we shall now get basic formulas for the
function $w$.
###### Lemma 3.1.
[6][24] If $M$ is a submanifold in ${\tenmsb R}^{n+m}$, then
(3.2) $\nabla_{e_{i}}w=h_{\alpha,ij}\langle
e_{j\alpha},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}\rangle$
with
(3.3) $e_{j\alpha}=e_{1}\wedge\cdots\wedge\nu_{\alpha}\wedge\cdots\wedge
e_{n}$
that is obtained by replacing $e_{j}$ by $\nu_{\alpha}$ in
$e_{1}\wedge\cdots\wedge e_{n}$. Moreover if $M$ has parallel mean curvature,
then
(3.4) $\Delta w=-|B|^{2}w+\sum_{i}\sum_{\alpha\neq\beta,j\neq
k}h_{\alpha,ij}h_{\beta,ik}\langle
e_{j\alpha,k\beta},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}\rangle.$
with
(3.5)
$e_{j\alpha,k\beta}=e_{1}\wedge\cdots\wedge\nu_{\alpha}\wedge\cdots\wedge\nu_{\beta}\wedge\cdots\wedge
e_{n}$
that is obtained by replacing $e_{j}$ by $\nu_{\alpha}$ and $e_{k}$ by
$\nu_{\beta}$ in $e_{1}\wedge\cdots\wedge e_{n}$, respectively.
Now we compute $\Delta\log w$ under the additional assumption that $H=0$ and
the rank of the Gauss map $\gamma$ is at most 2. Without loss of generality,
one can assume $h_{ij}^{\alpha}=0$ whenever $i\geq 3$ or $j\geq 3$. Then, we
need to calculate $\langle
e_{1\alpha,2\beta},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}\rangle$.
The following identity shall play an important role.
###### Lemma 3.2.
Fix $A=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$, then for any
distinct indices $\alpha,\beta$,
(3.6) $\langle e_{1}\wedge\cdots\wedge e_{n},A\rangle\langle
e_{1\alpha,2\beta},A\rangle-\langle e_{1\alpha},A\rangle\langle
e_{2\beta},A\rangle+\langle e_{1\beta},A\rangle\langle
e_{2\alpha},A\rangle=0.$
###### Proof.
Let $Q\in\mathbb{G}_{n-2,m+2}$ spanned by $\\{e_{i}:3\leq i\leq n\\}$. If
there is a nonzero vector in $Q$ which is orthogonal to the $n$-dimensional
space $P_{0}$ spanned by $\\{\varepsilon_{i}\\}$, then by the definition of
the inner product on $\Lambda^{n}({\tenmsb R}^{n+m})$, all the terms on the
left hand side of (3.6) equal $0$ and (3.6) trivially holds true.
Otherwise the orthogonal projection $p:Q\rightarrow P_{0}$ has rank $n-2$.
Without loss of generality we assume $p(Q)$ is spanned by
$\\{\varepsilon_{i}:3\leq i\leq n\\}$; the Jordan angles between $Q$ and
$p(Q)$ are denoted by $\theta_{3},\cdots,\theta_{n}$; and $\theta_{i}$ is
determined by $e_{i}$ and $\varepsilon_{i}$. Hence $\langle
e_{i},\varepsilon_{j}\rangle=\cos\theta_{i}\delta_{ij}$ whenever $i\geq 3$,
and moreover
(3.7) $\displaystyle\langle e_{1}\wedge\cdots\wedge e_{n},A\rangle$
$\displaystyle=\left|\begin{array}[]{ccccc}(e_{1},\varepsilon_{1})&(e_{1},\varepsilon_{2})&&&\\\
(e_{2},\varepsilon_{1})&(e_{2},\varepsilon_{2})&&\lx@intercol\hfil\raisebox{-2.15277pt}[0.0pt]{\Huge*}\hfil\lx@intercol&\\\
&&\cos\theta_{3}&&\\\ &\lx@intercol\raisebox{-2.15277pt}[0.0pt]{\Huge
0}\hfil\lx@intercol&&\ddots&\\\ &&&&\cos\theta_{n}\end{array}\right|$
$\displaystyle=\langle e_{1}\wedge
e_{2},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\prod_{i=3}^{n}\cos\theta_{i}$
Similarly,
(3.8) $\langle
e_{1\alpha,2\beta},A\rangle=\langle\nu_{\alpha}\wedge\nu_{\beta},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\prod_{i=3}^{n}\cos\theta_{i}$
and
(3.9) $\displaystyle\langle e_{1\gamma},A\rangle$ $\displaystyle=$
$\displaystyle\langle\nu_{\gamma}\wedge
e_{2},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\prod_{i=3}^{n}\cos\theta_{i}$
(3.10) $\displaystyle\langle e_{2\gamma},A\rangle$ $\displaystyle=$
$\displaystyle\langle
e_{1}\wedge\nu_{\gamma},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\prod_{i=3}^{n}\cos\theta_{i}$
for $\gamma=\alpha$ or $\beta$. A direct calculation shows
(3.11) $\displaystyle\langle e_{1}\wedge
e_{2},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\langle\nu_{\alpha}\wedge\nu_{\beta},\varepsilon_{1}\wedge\varepsilon_{2}\rangle-\langle\nu_{\alpha}\wedge
e_{2},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\langle
e_{1}\wedge\nu_{\beta},\varepsilon_{1}\wedge\varepsilon_{2}\rangle$
$\displaystyle+\langle\nu_{\beta}\wedge
e_{2},\varepsilon_{1}\wedge\varepsilon_{2}\rangle\langle
e_{1}\wedge\nu_{\alpha},\varepsilon_{1}\wedge\varepsilon_{2}\rangle$
$\displaystyle=$
$\displaystyle+(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{2})+(e_{2},\varepsilon_{1})(e_{1},\varepsilon_{2})(\nu_{\beta},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})$
$\displaystyle-(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{2})(\nu_{\beta},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})-(e_{2},\varepsilon_{1})(e_{1},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{2})$
$\displaystyle-(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{2})-(e_{2},\varepsilon_{1})(e_{1},\varepsilon_{2})(\nu_{\beta},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})$
$\displaystyle+(e_{1},\varepsilon_{2})(e_{2},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{1})+(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})(\nu_{\beta},\varepsilon_{2})$
$\displaystyle+(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{2})(\nu_{\beta},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})+(e_{2},\varepsilon_{1})(e_{1},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{2})$
$\displaystyle-(e_{1},\varepsilon_{2})(e_{2},\varepsilon_{2})(\nu_{\alpha},\varepsilon_{1})(\nu_{\beta},\varepsilon_{1})-(e_{1},\varepsilon_{1})(e_{2},\varepsilon_{1})(\nu_{\alpha},\varepsilon_{2})(\nu_{\beta},\varepsilon_{2})$
$\displaystyle=$ $\displaystyle 0.$
From (3.7)-(3.11), (3.6) immediately follows.
∎
$H=0$ implies $0=h_{\alpha,ii}=h_{\alpha,11}+h_{\alpha,22}$ for every
$1\leq\alpha\leq m$, thus
(3.12) $\displaystyle h_{\alpha,i1}h_{\beta,i2}=$ $\displaystyle
h_{\alpha,11}h_{\beta,12}+h_{\alpha,21}h_{\beta,22}$ $\displaystyle=$
$\displaystyle-
h_{\alpha,22}h_{\beta,12}-h_{\alpha,21}h_{\beta,11}=-h_{\alpha,i2}h_{\beta,i1}$
and in particular $h_{\alpha,i1}h_{\alpha,i2}=0$. Therefore (3.4) can be
rewritten as
(3.13) $\displaystyle\Delta w$
$\displaystyle=-|B|^{2}w+\sum_{\alpha\neq\beta}h_{\alpha,i1}h_{\beta,i2}\langle
e_{1\alpha,2\beta},A\rangle+\sum_{\alpha\neq\beta}h_{\alpha,i2}h_{\beta,i1}\langle
e_{1\beta,2\alpha},A\rangle$
$\displaystyle=-|B|^{2}w+2\sum_{\alpha,\beta}h_{\alpha,i1}h_{\beta,i2}\langle
e_{1\alpha,2\beta},A\rangle.$
By (3.2),
(3.14) $|\nabla
w|^{2}=\sum_{i}|\nabla_{e_{i}}w|^{2}=h_{\alpha,ij}h_{\beta,ik}\langle
e_{j\alpha},A\rangle\langle e_{k\beta},A\rangle.$
With the aid of (3.6) and (3.12) one can get
(3.15) $\displaystyle w\Delta w-|\nabla w|^{2}=$
$\displaystyle-|B|^{2}w^{2}+2h_{\alpha,i1}h_{\beta,i2}\langle
e_{1}\wedge\cdots\wedge e_{n},A\rangle\langle e_{1\alpha,2\beta},A\rangle-
h_{\alpha,ij}h_{\beta,ik}\langle e_{j\alpha},A\rangle\langle
e_{k\beta},A\rangle$ $\displaystyle=$
$\displaystyle-|B|^{2}w^{2}+2h_{\alpha,i1}h_{\beta,i2}\Big{(}\langle
e_{1\alpha},A\rangle\langle e_{2\beta},A\rangle-\langle
e_{1\beta},A\rangle\langle e_{2\alpha},A\rangle\Big{)}$ $\displaystyle-
h_{\alpha,ij}h_{\beta,ik}\langle e_{j\alpha},A\rangle\langle
e_{k\beta},A\rangle$ $\displaystyle=$
$\displaystyle-|B|^{2}w^{2}+2h_{\alpha,i1}h_{\beta,i2}\langle
e_{1\alpha},A\rangle\langle
e_{2\beta},A\rangle+2h_{\alpha,i2}h_{\beta,i1}\langle
e_{2\alpha},A\rangle\langle e_{1\beta},A\rangle$ $\displaystyle-
h_{\alpha,i1}h_{\beta,i1}\langle e_{1\alpha},A\rangle\langle
e_{1\beta},A\rangle-h_{\alpha,i2}h_{\beta,i2}\langle
e_{2\alpha},A\rangle\langle e_{2\beta},A\rangle$ $\displaystyle-
h_{\alpha,i1}h_{\beta,i2}\langle e_{1\alpha},A\rangle\langle
e_{2\beta},A\rangle-h_{\alpha,i2}h_{\beta,i1}\langle
e_{2\alpha},A\rangle\langle e_{1\beta},A\rangle$ $\displaystyle=$
$\displaystyle-|B|^{2}w^{2}-\sum_{i}\Big{(}\sum_{\alpha}\big{(}h_{\alpha,i1}\langle
e_{1\alpha},A\rangle-h_{\alpha,i2}\langle
e_{2\alpha},A\rangle\big{)}\Big{)}^{2}.$
Noting that $\Delta\log w=w^{-2}(w\Delta w-|\nabla w|^{2})$, we arrive at
###### Proposition 3.1.
Let $M$ be a minimal submanifold of ${\tenmsb R}^{n+m}$, with the rank of
Gauss map at most $2$. If $w>0$ at a point, then locally
(3.16) $\Delta\log w\leq-|B|^{2}.$
###### Remark 3.1.
If $N$ is a 2-dimensional minimal submanifold in $S^{2+m}$, then the cone $CM$
over $M$ is a 3-dimensional minimal submanifold in ${\tenmsb R}^{3+m}$. The
rank of the Gauss map from $CM$ is no more than 2, as will be shown in the
next section. Hence Proposition 3.1 is a generalization of Proposition 2.2 in
[6]. We note that Lemma 3.2 is also a generalization of Lemma 2.1 in [6].
Finally, we consider the general case. For any $x\in M$ satisfying $w(x)>0$,
then $w(P,P_{0})>0$ with $P=\gamma(x)$. Let $\theta_{1},\cdots,\theta_{n}$ be
the Jordan angles between $P$ and $P_{0}$ which are arranged as
(3.17)
$\frac{\pi}{2}>\theta_{1}\geq\cdots\geq\theta_{r}>\theta_{r+1}=\cdots=\theta_{n}=0$
with $0\leq r\leq\min\\{n,m\\}$. As shown in Section 2, one can find an
orthonormal basis $\\{\varepsilon_{i},\varepsilon_{n+\alpha}\\}$ of ${\tenmsb
R}^{n+m}$, such that
$\psi(P_{0})=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$; if we denote
(3.18)
$e_{i}:=\left\\{\begin{array}[]{cc}\cos\theta_{i}\varepsilon_{i}+\sin\theta_{i}\varepsilon_{n+i}&1\leq
i\leq r\\\ \varepsilon_{i}&r+1\leq i\leq n\end{array}\right.$
and
(3.19)
$\nu_{\alpha}:=\left\\{\begin{array}[]{cc}-\sin\theta_{\alpha}\varepsilon_{\alpha}+\cos\theta_{\alpha}\varepsilon_{n+\alpha}&1\leq\alpha\leq
r\\\ \varepsilon_{n+\alpha}&r+1\leq\alpha\leq m\end{array}\right.$
then $\\{e_{i},\nu_{\alpha}\\}$ is also an orthonormal basis of ${\tenmsb
R}^{n+m}$ and $\psi(P)=e_{1}\wedge\cdots\wedge e_{n}$.
Now we use the notation
(3.20) $v:=w^{-1}$
as in [26] and [17]; the function $v$ is well-defined in a neighborhood of
$x$. By a direct calculation based on metric forms in terms of matrix
coordinates (2.11), one can derive
(3.21) $\Delta
v=v|B|^{2}+v\sum_{i,j}2\lambda_{j}^{2}h_{j,ij}^{2}+v\sum_{i}\sum_{j\neq
k}\lambda_{j}\lambda_{k}(h_{j,ij}h_{k,ik}+h_{k,ij}h_{j,ik})$
as in [17]; where $\lambda_{j}=\tan\theta_{j}$ and $h_{\alpha,ij}=\langle
B(e_{i},e_{j}),\nu_{\alpha}\rangle$. Grouping the terms according to different
types of the indices of the coefficients of the second fundamental form, one
can proceed as in the proof of [17] Proposition 3.1 to get
(3.22) $v^{-1}\Delta v\geq C_{1}|B|^{2}\qquad\text{whenever }v\leq\beta<3.$
Noting that $v=\exp(-\log w)$, one can immediately get
###### Proposition 3.2.
If $M$ is a submanifold in ${\tenmsb R}^{n+m}$ with parallel mean curvature,
once $w\geq\frac{1}{3}+\delta$ with a positive number $\delta$, then
(3.23) $\Delta\log w-|\nabla\log w|^{2}\leq-C_{1}|B|^{2}\qquad\text{at }x$
with a positive constant $C_{1}$ depending only on $\delta$.
## 4\. Subharmonic functions and extrinsic rigidity problem
We already defined the S-map
$\mathscr{S}:\mathbb{G}_{n,m}\to\overline{{\tenmsb D}}$ in §2. For every
$(x_{1},x_{2})\in\overline{{\tenmsb D}}\backslash\big{\\{}(a,0):a\leq
0\big{\\}}$ that is obtained by deleting a radius from the disk, there exist
unique $r\in(0,1]$ and $\theta\in(-\pi,\pi)$, such that
$(x_{1},x_{2})=(r\cos\theta,r\sin\theta).$
Here $r$ and $\theta$ can be seen as smooth functions on
$\mathscr{S}^{-1}\Big{(}\overline{{\tenmsb D}}\backslash\big{\\{}(a,0):a\leq
0\big{\\}}\Big{)}$. For any $c\in[0,1)$, we put
(4.1) ${\tenmsb W}_{c}:=\mathscr{S}^{-1}\Big{(}\overline{{\tenmsb
D}}\backslash\big{(}\overline{{\tenmsb D}}_{c}\cup\big{\\{}(a,0):a\leq
0\big{\\}}\big{)}\Big{)},$
with $\overline{{\tenmsb D}}_{c}=\big{\\{}(x_{1},x_{2})\in{\tenmsb
R}^{2}:x_{1}^{2}+x_{2}^{2}\leq c^{2}\big{\\}}$, then the function $r$ takes
values between $c$ and $1$ on ${\tenmsb W}_{c}$.
###### Remark 4.1.
If $m=1$, then $\mathbb{G}_{n,m}=S^{n}$ and ${\tenmsb W}_{0}$ is the
complement of a half-equator $\overline{S}_{+}^{n-1}$; in particular, if
$n=2$, then ${\tenmsb W}_{0}$ can be obtained by deleting half of a great
circle between 2 antipodal points from $S^{2}$. It has been proved in [16]
that ${\tenmsb W}_{0}$ is a maximal convex-supporting set in $S^{n}$.
###### Remark 4.2.
Obviously, $P_{t},P_{t+\pi}\in{\tenmsb W}_{c}$ consist of an antipodal pair in
$\mathbb{G}_{n,m}$ for each $t\in(-\frac{\pi}{2},0)$, which implies
$w(P_{t},S)=-w(P_{t+\pi},S)$ for arbitrary $S\in\mathbb{G}_{n,m}$. Hence
${\tenmsb W}_{c}$ cannot lie in an $(n\times m)$-matrix coordinate chart
centered at any point of $\mathbb{G}_{n,m}$. Furthermore, they are conjugate
to each other. Since the Grasmann manifolds are simply connected symmetric
spaces of compact type, the cut point of $P_{t}$ along a geodesic coincides
with the first conjugate point along the geodesic. Therefore, ${\tenmsb
W}_{c}$ contains a cut point of $P_{t}$ at least.
On ${\tenmsb W}_{c}$ there are smooth functions $r$ and $\theta$. By the
previous work [16] our concerned function $F$ was constructed from $r$ and
$\theta$ and we need to estimate $\mbox{Hess}\ F$. Hopefully, the level set of
$F=t$ coincides with level set of $w(\cdot,P_{t})$. For estimating
$\mbox{Hess}\ F$ in terms of $\mbox{Hess}\ w(\cdot,P_{t})$ we need the
following transition Lemma, which is crucial in our construction, but may
also be useful in other contexts.
###### Lemma 4.1.
Let $(N,h)$ be a Riemannian manifold, $F$ be a smooth function on $N$, and
$\\{H(\cdot,t):t\in{\tenmsb R}\\}$ be a smooth family of smooth functions on
$N$. Assume that there are smooth real functions $t(s)$ and $l(s)$; for all
$s\in{\tenmsb R}$, the level set $F_{s}:=\\{p\in N:F(p)=s\\}$ coincides with
$\\{p\in N:H\big{(}p,t(s)\big{)}=l(s)\\}$; $\nabla F$ and $\nabla
H\big{(}\cdot,t(s)\big{)}$ are nonzero normal vector fields on $F_{s}$,
pointing in the same direction. Then for arbitrary $\varepsilon>0$, there
exists a smooth nonpositive function $\lambda$ on $N$, depending on
$\varepsilon$, such that
(4.2) $\mbox{Hess}\ F\geq\frac{|\nabla F|}{\big{|}\nabla
H\big{(}\cdot,t(s)\big{)}\big{|}}\mbox{Hess}\
H\big{(}\cdot,t(s)\big{)}-\varepsilon h+\lambda dF\otimes dF$
on $N$.
###### Proof.
For any $s\in{\tenmsb R}$, $F_{s}$ can be regarded as a hypersurface of $N$.
Denote by $\nabla$ and $\nabla^{s}$ the Levi-Civita connection on $N$ and
$F_{s}$, respectively. We put $\nu=\frac{\nabla F}{|\nabla F|}$, then $\nu$ is
the unit normal vector field on $F_{s}$; and denote by $B$ the second
fundamental form of $F_{s}$ in $N$.
For every $X\in T(F_{s})$,
(4.3) $\displaystyle\mbox{Hess}\ F(X,X)$
$\displaystyle=\nabla_{X}\nabla_{X}F-(\nabla_{X}X)F$
$\displaystyle=-(\nabla^{s}_{X}X)F-\langle B(X,X),\nu\rangle\langle\nu,\nabla
F\rangle$ $\displaystyle=-|\nabla F|\langle B(X,X),\nu\rangle.$
Similarly,
(4.4) $\mbox{Hess}\ H\big{(}\cdot,t(s)\big{)}(X,X)=-\big{|}\nabla
H\big{(}\cdot,t(s)\big{)}\big{|}\langle B(X,X),\nu\rangle.$
Hence
(4.5) $\Big{(}\mbox{Hess}\ F-\frac{|\nabla F|}{\big{|}\nabla
H\big{(}\cdot,t(s)\big{)}\big{|}}\mbox{Hess}\
H\big{(}\cdot,t(s)\big{)}\Big{)}(X,X)=0\qquad\text{for all }X\in T(F_{s}).$
It implies the existence of a smooth 1-form $\omega$ on $N$, such that
(4.6) $\mbox{Hess}\ F-\frac{|\nabla F|}{\big{|}\nabla
H\big{(}\cdot,t(s)\big{)}\big{|}}\mbox{Hess}\
H\big{(}\cdot,t(s)\big{)}=\omega\otimes dF+dF\otimes\omega$
on $F_{s}$. Denote by $\omega^{*}$ the associated vector field of $\omega$
with respect to $h$, i.e.
$\langle Y,\omega^{*}\rangle=\omega(Y)\qquad\text{for all }Y\in TN.$
Then
(4.7) $\displaystyle(\omega\otimes dF+dF\otimes\omega)(Y,Y)$ $\displaystyle=$
$\displaystyle 2\omega(Y)dF(Y)=2\langle\omega^{*},Y\rangle dF(Y)$
$\displaystyle\geq$
$\displaystyle-2|\omega^{*}||Y|dF(Y)\geq-\varepsilon|Y|^{2}-\varepsilon^{-1}|\omega^{*}|^{2}dF(Y)^{2}.$
Let $\lambda=-\varepsilon^{-1}|\omega^{*}|^{2}$, then substituting (4.7) into
(4.6) yields (4.2).
∎
Given a compact subset $K$ of ${\tenmsb W}_{c}$, there exists a positive
number $\delta$, such that $r\geq c+2\delta$ on $K$. Hence the following
function
(4.8) $F=\theta-\arccos\left(\frac{c+\delta}{r}\right)$
is well-defined on $K$. Once $F(S)=t$, one can deduce that
$\theta-t=\arccos(\frac{c+\delta}{r})\in(0,\frac{\pi}{2})$, and
(4.9) $\displaystyle c+\delta$ $\displaystyle=r(\cos(\theta-t))=r(\cos
t\cos\theta+\sin t\sin\theta)$ $\displaystyle=(\cos t\ x_{1}+\sin t\
x_{2})(S)=w(S,P_{t}).$
In other words, the level set $F_{t}$ of $F$ overlaps with the level set
$\\{S\in\mathbb{G}_{n,m}:-\log w(S,P_{t})=-\log(c+\delta)\\}$.
To apply Lemma 4.1, we need gradient estimates for the functions $-\log
w(\cdot,P_{t})$ and $F$ along the common level set.
Let ${\tenmsb U}$ be the matrix coordinate chart centered at $P_{t}$. Recall
that $w(\cdot,P_{t})$ has an expression
(4.10) $w=\Big{[}\mbox{tr}(I_{n}+ZZ^{T})\Big{]}^{-1}$
in terms of matrix coordinates (see [26]). It is easily seen that both $w$ and
the metric on ${\tenmsb U}$ are invariant under $SO(n)\times SO(m)$-actions.
Hence for any $S\in{\tenmsb U}$, without loss of generality one can assume
$Z(S)=(\lambda_{i}\delta_{i\alpha})=(\tan\theta_{i}\delta_{i\alpha})$
with $\\{\theta_{i}\\}$ being the Jordan angles between $S$ and $P_{t}$. As in
[26], one can obtain
$dw(\cdot,P_{t})=-\sum_{1\leq i\leq
p}\lambda_{i}w(\cdot,P_{t})\omega_{ii}\qquad\text{at }R$
with $p=\min\\{n,m\\}$ and $\\{\omega_{i\alpha}\\}$ is the dual basis of
$\big{\\{}(1+\lambda_{i}^{2})^{\frac{1}{2}}(1+\lambda_{\alpha}^{2})^{\frac{1}{2}}E_{i\alpha}\big{\\}}$,
which is an orthogonal basis of $T_{S}\mathbb{G}_{n,m}$. (Here $E_{i\alpha}$
is the matrix with 1 in the intersection of row $i$ and column $\alpha$ and 0
otherwise.) Therefore
(4.11) $\big{|}\nabla\log w(\cdot,P_{t})\big{|}^{2}=\sum_{1\leq i\leq
p}\lambda_{i}^{2}\geq p\big{(}w(\cdot,P_{t})^{-\frac{2}{p}}-1\big{)}$
and the equality holds if and only if
$\lambda_{1}^{2}=\cdots=\lambda_{p}^{2}=w(\cdot,P_{t})^{-\frac{2}{p}}-1$. In
particular, on $F_{t}$,
(4.12) $\big{|}\nabla\log w(\cdot,P_{t})\big{|}^{2}\geq
p\big{(}(c+\delta)^{-\frac{2}{p}}-1\big{)}:=C_{2}(p,\delta).$
By (2.15),
$\displaystyle\nabla w(\cdot,P_{t})$ $\displaystyle=\cos t\nabla x_{1}+\sin
t\nabla x_{2}=\cos t\nabla(r\cos\theta)+\sin t\nabla(r\sin\theta)$
$\displaystyle=\cos(t-\theta)\nabla r+r\sin(t-\theta)\nabla\theta$
and moreover by (4.9),
(4.13) $\displaystyle-\nabla\log w(\cdot,P_{t})$
$\displaystyle=-w^{-1}(\cdot,P_{t})\nabla
w(\cdot,P_{t})=-(c+\delta)^{-1}\nabla w(\cdot,P_{t})$
$\displaystyle=-r^{-1}\nabla r+\tan(\theta-t)\nabla\theta$
on $F_{t}$. On the other hand, from (4.8) we have
(4.14) $\displaystyle\nabla F$
$\displaystyle=\nabla\theta-\left(1-\left(\frac{c+\delta}{r}\right)^{2}\right)^{-\frac{1}{2}}\frac{c+\delta}{r^{2}}\nabla
r$ $\displaystyle=\nabla\theta-r^{-1}\cot(\theta-t)\nabla r.$
Comparing (4.13) with (4.14) gives
(4.15) $-\nabla\log w(\cdot,P_{t})=\tan(\theta-t)\nabla F.$
$c+2\delta\leq r\leq 1$ gives
$\theta-t=\arccos\big{(}\frac{c+\delta}{r}\big{)}\in\big{[}\arccos\big{(}\frac{c+\delta}{c+2\delta}\big{)},\arccos(c+\delta)\big{]}$;
hence $-\nabla\log w(\cdot,P_{t})$ and $\nabla F$ point in the same direction,
and
(4.16) $\frac{|\nabla\log w(\cdot,P_{t})|}{|\nabla
F|}=\tan(\theta-t)\leq\frac{\sqrt{1-(c+\delta)^{2}}}{c+\delta}:=C_{3}(\delta).$
Now we put $H(\cdot,t)=-\log w(\cdot,P_{t})$, then by (4.12) and (4.16), one
can apply Lemma 4.1 to get the following conclusion
###### Proposition 4.1.
For a compact set $K\subset{\tenmsb W}_{c}$, put
$\delta:=\frac{1}{2}\inf_{K}(r-c)$ and
$F:=\theta-\arccos\big{(}\frac{c+\delta}{r}\big{)}$, then for arbitrary
$\varepsilon>0$, there exists a smooth nonpositive function $\lambda$ on $K$,
depending on $\varepsilon$, such that
(4.17) $\mbox{Hess}\ F\geq-C_{3}^{-1}\mbox{Hess}\ \log
w(\cdot,P_{t})-\varepsilon g+\lambda dF\otimes dF$
on the level set $F_{t}$. Here $g$ is the canonical metric on
$\mathbb{G}_{n,m}$ and $C_{3}$ is a positive constant depending only on
$\delta$.
###### Remark 4.3.
Put
$\displaystyle\phi(t)(a_{i}e_{i}+b_{\alpha}e_{n+\alpha})=$
$\displaystyle(a_{1}\cos t+b_{1}\sin t)e_{1}+(-a_{1}\sin t+b_{1}\cos
t)e_{n+1}$ $\displaystyle+\sum_{i\geq 2}a_{i}e_{i}+\sum_{\alpha\geq
2}b_{\alpha}e_{n+\alpha},$
then $t\in{\tenmsb R}\mapsto\phi(t)$ is a 1-parameter subgroup of $SO(n+m)$,
which induces a 1-parameter isometry group of $\mathbb{G}_{n,m}$. Since
$w(\cdot,\cdot)$ is invariant under the action of $SO(n+m)$, it is easily-seen
that $F_{t}$ is the orbit of $F_{0}$ under the action of $\phi(t)$. This
observation can explain why the level sets of $F$ share similar properties in
terms of exterior geometry.
Combining Propositions 3.1-3.2 and Proposition 4.1, we can find a strongly
subharmonic function on a submanifold with parallel mean curvature under the
assumptions on the Gauss image.
###### Proposition 4.2.
Let $M$ be an oriented submanifold in ${\tenmsb R}^{n+m}$ with parallel mean
curvature, if its Gauss image is contained in a compact subset $K$ of
${\tenmsb W}_{1/3}$, then there exists a positive bounded function $f$ on $M$,
such that
(4.18) $\Delta f\geq K_{0}|B|^{2}$
with a positive constant $K_{0}$ depending only on $K$. Moreover, if $M$ is
minimal and the rank of the Gauss map is not larger than $2$, then the above
conclusion still holds when ${\tenmsb W}_{1/3}$ is replaced by ${\tenmsb
W}_{0}$.
###### Proof.
We only prove the general case, since the proof for the case with the
additional assumption on the rank of the Gauss map is similar.
Using the definition of $F$ in (4.8) with $c=\frac{1}{3}$ and combining (4.17)
and (4.16) gives
(4.19) $\displaystyle\mu_{0}^{-1}\exp(\mu_{0}F)\mbox{Hess}\exp(\mu_{0}F)$
$\displaystyle\geq$ $\displaystyle-C_{3}^{-1}\mbox{Hess}\log
w(\cdot,P_{t})-\varepsilon g+(\lambda+\mu_{0})dF\otimes dF$
$\displaystyle\geq$ $\displaystyle-C_{3}^{-1}\mbox{Hess}\log
w(\cdot,P_{t})-\varepsilon g+C_{3}^{-2}|\lambda+\mu_{0}|d\log
w(\cdot,P_{t})\otimes d\log w(\cdot,P_{t})$
where $\mu_{0}$ and $\varepsilon$ are positive constants to be chosen. Define
(4.20) $f=\exp(\mu_{0}F)\circ\gamma$
with $\gamma$ the Gauss map into $K\subset{\tenmsb W}_{1/3}$, then using the
composition formula implies
(4.21) $\displaystyle\mu_{0}^{-1}f^{-1}\Delta f=$
$\displaystyle\mu_{0}^{-1}\exp(\mu_{0}F)\Big{(}\mbox{Hess}\
\exp(\mu_{0}F)(\gamma_{*}e_{i},\gamma_{*}e_{i})+d\exp(\mu_{0}F)(\tau(\gamma)\Big{)}$
$\displaystyle\geq$ $\displaystyle-C_{3}^{-1}\mbox{Hess}\ \log
w(\cdot,P_{t})(\gamma_{*}e_{i},\gamma_{*}e_{i})-\varepsilon\langle\gamma_{*}e_{i},\gamma_{*}e_{i}\rangle$
$\displaystyle+C_{3}^{-2}|\lambda+\mu_{0}|\sum_{i}\Big{(}\nabla_{e_{i}}\big{|}\log
w(\cdot,P_{t})\circ\gamma\big{|}\Big{)}^{2}$ $\displaystyle=$ $\displaystyle-
C_{3}^{-1}\Delta\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}-\varepsilon|B|^{2}+C_{3}^{-2}|\lambda+\mu_{0}|\Big{|}\nabla\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}\Big{|}^{2}$
on the level set $\\{x\in M:f(x)=\exp(\mu_{0}t)\\}$ with the tension field
$\tau(\gamma)$ of the Gauss map $\gamma$. Here $\\{e_{i}\\}$ is an orthonormal
basis of the tangent space at the considered point of $M$, and we have used
the harmonicity of the Gauss map. Since $w(\cdot,P_{t})$ equals
$\frac{1}{3}+\delta$ everywhere on the level set $F_{t}$ (see (4.9)), one can
apply Proposition 3.2 to obtain
(4.22) $\displaystyle\Delta\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}-\Big{|}\nabla\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}\Big{|}^{2}\leq-C_{1}|B|^{2}$
with a positive constant $C_{1}$ depending only on $\delta$. Substituting this
into (4.21) yields
$\mu_{0}^{-1}f^{-1}\Delta
f\geq(C_{3}^{-1}C_{1}-\varepsilon)|B|^{2}+\big{(}C_{3}^{-2}|\lambda+\mu_{0}|-C_{3}^{-1}\big{)}\Big{|}\nabla\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}\Big{|}^{2}.$
By letting $\varepsilon:=\frac{1}{2}C_{3}^{-1}C_{1}$ and
$\mu_{0}:=\sup_{K}|\lambda|+C_{3}$ we arrive at (4.18) with
$K_{0}=\frac{1}{2}C_{3}^{-1}C_{1}\mu_{0}\exp(\mu_{0}\inf_{K}F)$.
∎
We can immediately get the following Bernstein type theorem with the aid of
the strongly subharmonic functions.
###### Theorem 4.1.
If $M$ is a parabolic, oriented minimal surface in ${\tenmsb R}^{2+m}$, and if
its Gauss image is contained in a compact subset of ${\tenmsb W}_{0}$, then
$M$ has to be an affine linear subspace.
###### Proof.
One can find a strongly subharmonic function $f$ on $M$ satisfying (4.18) by
applying Proposition 4.2, since the mean curvature vector field vanishes and
the rank of the Gauss map is $\leq 2$. Since a parabolic surface cannot admit
any nonconstant bounded subharmonic function, $f$ has to be constant; hence
$\Delta f\equiv 0$ and moreover $|B|^{2}\equiv 0$, which means $M$ is totally
geodesic.
∎
We now study the extrinsic rigidity problem, initiated by J. Simons [21],
finding better conditions on the Gauss image for a compact minimal submanifold
$M$ in $S^{n+m}$ to be an equator.
As pointed out and utilized by J. Simons [21], the properties of the (minimal)
submanifolds $M$ in the sphere are closed related to those of the cone $CM$
generated by $M$, which is the image under the map of
$M\times[0,+\infty)\rightarrow{\tenmsb R}^{n+m+1}$ defined by $(x,t)\mapsto
tx$. To avoid the singularity $t=0$, we also consider the truncated cone
$CM_{\varepsilon}$ ($\varepsilon>0$) that is the image of
$M\times(\varepsilon,+\infty)$ under the same map.
We choose a local orthonormal frame field $\\{e_{i},\nu_{\alpha}\\}$ of
$S^{n}$ along $M$, then by parallel translating along rays issuing from the
origin we obtain local vector fields $\\{E_{i}\\}$ and
$\\{\mathcal{N}_{\alpha}\\}$; obviously $E_{i}=\frac{1}{t}e_{i}$ and
$\mathcal{N}_{\alpha}=\frac{1}{t}\nu_{\alpha}$. Let $\tau$ be the unit tangent
vector along the rays, i.e. $\tau=\frac{\partial missing}{\partial t}$, then
$\\{E_{i},\mathcal{N}_{\alpha},\tau\\}$ is a local orthonormal frame field in
${\tenmsb R}^{n+m+1}$ and $\\{E_{i},\tau\\}$ is a frame field in
$CM_{\varepsilon}$.
Let $B$ and $B^{c}$ denote the second fundamental form of $M\subset S^{n+m}$
and $CM_{\varepsilon}\subset{\tenmsb R}^{n+m+1}$, respectively. Then a
straightforward calculation shows (see [23] p.18)
(4.23) $\langle
B^{c}(E_{i},E_{j}),\mathcal{N}_{\alpha}\rangle=\frac{1}{t}\langle
B(e_{i},e_{j}),\nu_{\alpha}\rangle$
and
(4.24) $B^{c}(E_{i},\tau)=B^{c}(\tau,\tau)=0.$
Hence $CM_{\varepsilon}$ is a minimal submanifold in ${\tenmsb R}^{n+m+1}$ if
and only if $M$ is a minimal submanifold in $S^{n+m}$.
The normal Gauss map $\gamma^{N}:M\rightarrow\mathbb{G}_{m,n+1}$ is defined by
$\gamma^{N}(x)=N_{x}M$
via parallel translation in ${\tenmsb R}^{n+m+1}$. Let $\eta$ be the natural
isometry between $\mathbb{G}_{n+1,m}$ and $\mathbb{G}_{m,n+1}$ which maps each
$(n+1)$-dimensional oriented linear subspace to its orthogonal complementary
$m$-space. It is easily seen that the Gauss map
$\gamma:CM_{\varepsilon}\rightarrow\mathbb{G}_{n+1,m}$ is a cone-like map;
more precisely, for every $t\in(\varepsilon,+\infty)$ and $x\in M$,
$(\eta\circ\gamma)(tx)=\gamma^{N}(x).$
Thus $\gamma_{*}\tau=0$ and the rank of $\gamma$ at $tx$ equals the rank of
$\gamma^{N}$ at $x$ for all $x\in M$ and any $t\in(\varepsilon,+\infty)$.
One can assume $\nabla e_{i}=0$ at the considered point without loss of
generality, then by computing (see [23] p.18),
(4.25) $\nabla^{c}_{E_{i}}E_{j}=-\frac{1}{t}\delta_{ij}\tau$
with $\nabla^{c}$ the Levi-Civita connection on $CM_{\varepsilon}$. Let $f$ be
a cone-like real function on $CM_{\varepsilon}$, then
(4.26)
$\Delta^{c}f=\nabla^{c}_{E_{i}}\nabla^{c}_{E_{i}}f-(\nabla_{E_{i}}^{c}E_{i})f=\frac{1}{t^{2}}\nabla_{e_{i}}\nabla_{e_{i}}f_{1}+\frac{1}{t}\nabla_{\tau}^{c}f=\frac{1}{t^{2}}\Delta
f_{1}.$
Here $\Delta^{c}$ is the Laplace-Beltrami operator on $CM_{\varepsilon}$ and
$f_{1}$ is a function on $M$ satisfying $f(tx)=f_{1}(x)$ for every $x\in M$
and $t\in(\varepsilon,+\infty)$.
Based on the strongly subharmonic functions constructed in Proposition 4.2, we
can derive the following extrinsic rigidity result.
###### Theorem 4.2.
Let $M^{n}$ be a compact, oriented minimal submanifold in $S^{n+m}$,
$P,Q\in\mathbb{G}_{m,n+1}$ that are S-orthogonal to each other. Assume
(4.27) $w(N,P)^{2}+w(N,Q)^{2}>\left\\{\begin{array}[]{cc}0&\text{if
rank}(\gamma^{N})\leq 2\\\ \frac{1}{9}&\text{otherwise}\end{array}\right.$
holds for all normal $m$-vectors $N$ of $M$, and there exists no point $x\in
M$, such that $w(N,Q)=0$ and $w(N,P)<0$ at $x$. Then $M$ has to be an equator.
###### Proof.
As above, we only give the proof for the general case without the assumptions
on the rank of the Gauss map.
By the definition of $\eta$, we have
$\psi\big{(}\eta(S)\big{)}=*\big{(}\psi(S)\big{)}$
with $\psi$ denoting the Plücker embedding and $*$ being the Hodge star
operator. $*$ keeps the inner product invariant, i.e.
$\langle*A,*B\rangle=\langle A,B\rangle\qquad\text{for
}A,B\in\Lambda^{m}({\tenmsb R}^{n+m+1}).$
Hence
$w\big{(}\eta(S_{1}),\eta(S_{2})\big{)}=w(S_{1},S_{2})\qquad\text{for
}S_{1},S_{2}\in\mathbb{G}_{m,n+1}.$
If $S_{1},S_{2}$ are S-orthogonal to each other, then $w(S_{1},S_{2})=0$ and
$\dim(S_{1}\cap S_{2})=m-1$, which implies
$w\big{(}\eta(S_{1}),\eta(S_{2})\big{)}=0$ and
$\dim\big{(}\eta(S_{1})+\eta(S_{2})\big{)}=n+m+1-\dim(S_{1}\cap S_{2})=n+2$,
hence $\eta(S_{1})$ and $\eta(S_{2})$ are S-orthogonal to each other (see
Proposition 2.1).
Denote
${\tenmsb
W}_{\frac{1}{3}}=\Big{\\{}S\in\mathbb{G}_{n+1,m}:\big{(}w(S,\eta(P)),w(S,\eta(Q))\big{)}\in{\tenmsb
D}\backslash\big{(}\overline{{\tenmsb D}}_{\frac{1}{3}}\cup\\{(a,0):a\leq
0\\}\big{)}\Big{\\}},$
then the assumption on the normal Gauss map $\gamma^{N}$ implies that the
image under the Gauss map of $CM_{\varepsilon}$ is contained in a compact
subset of ${\tenmsb W}_{1/3}$. This enables us to find a strongly subharmonic
function $f$ satisfying
(4.28) $\Delta^{c}f\geq K_{0}|B^{c}|^{2}.$
Noting that $f$ is the composition of the Gauss map $\gamma$ and a function on
$K$, it should be a cone-like function. Denote $f_{1}=f|_{M}$, then combining
(4.28), (4.26) and (4.23) gives
(4.29) $\Delta f_{1}=t^{2}\Delta^{c}f\geq K_{0}t^{2}|B^{c}|^{2}=K_{0}|B|^{2}.$
Integrating both sides along $M$ implies
$0=\int_{M}\Delta f_{1}*1\geq K_{0}\int_{M}|B|^{2}*1.$
Thus $|B|^{2}\equiv 0$ and $M$ has to be totally geodesic.
∎
###### Remark 4.4.
Obviously $w(N,P)>0$ implies $w(N,P)^{2}+w(N,Q)^{2}>0$. Hence for the case of
dimension 2, our result is an improvement of extrinsic rigidity theorems of
Barbosa [2] and Fischer-Colbrie [6].
## 5\. Curvature estimates
In the previous section, we have found a strongly subharmonic function $f$ on
a submanifold $M^{n}\subset{\tenmsb R}^{n+m}$ with parallel mean curvature
under the assumption on the Gauss image. To make Stokes’ theorem applicable,
we take a $H^{1,2}$-function $\eta$ with compact support; multiplying $\eta$
with both sides of (4.18) and then integrating it along $M$ gives
(5.1) $\displaystyle K_{0}\int_{M}|B|^{2}\eta*1\leq$
$\displaystyle\int_{M}\eta\Delta f*1=\int_{M}\big{(}\text{div}(\eta\nabla
f)-\langle\nabla\eta,\nabla f\rangle\big{)}*1$ $\displaystyle=$
$\displaystyle-\int_{M}\langle\nabla\eta,\nabla f\rangle*1.$
In order to obtain a-priori estimates for $|B|^{2}$, we have to choose a
’good’ test function $\eta$.
We shall use the Green test function technique employed in [11] and [12].
Unfortunately, this method cannot be applied for arbitrary noncompact
Riemannian manifolds. We have to impose a so-called DVP-condition as in [16].
###### Definition 5.1.
Let $(M,g)$ be a Riemannian manifold (not necessarily complete) satisfying:
(D) There is a distance function $d$ on $M$, such that the metric topology
induced by $d$ is equivalent to the Riemannian topology of $M$, and
$d(x,y)\leq r(x,y)$ for any $x,y\in M$, where $r$ is the distance function
induced by the metric tensor of $M$;
(V) Let $B_{R}(x_{0})=\\{x\in M:d(x,x_{0})<R\\}$,
$V(x_{0},R)=\text{Vol}\big{(}B_{R}(x_{0})\big{)}$, then there exists a
positive constant $K_{1}$ not depending on $R$ and $x_{0}$, such that
$V(x_{0},2R)\leq K_{1}\ V(x_{0},R)\qquad\text{whenever
}B_{2R}(x_{0})\subset\subset M;$
(P) For any $x_{0}\in M$ and $R>0$ satisfying $B_{R}(x_{0})\subset\subset M$,
the following Neumann-Poincaré inequality
$\int_{B_{R}(x_{0})}|v-\bar{v}_{R}|^{2}*1\leq
K_{2}R^{2}\int_{B_{R}(x_{0})}|\nabla v|^{2}*1$
holds with $\bar{v}_{R}$ denoting the average value of $v$ in $B_{R}(x_{0})$
and $K_{2}$ being a positive constant not depending on $x_{0}$ and $R$.
Then we say that $(M,g)$ satisfies a DVP-condition.
###### Remark 5.1.
If $M={\tenmsb D}^{n}(r_{0})\subset{\tenmsb R}^{n}$ with metric
$g=g_{ij}dx^{i}dx^{j}$, and the eigenvalues of $(g_{ij})$ are uniformly
bounded from below by a positive constant $\lambda$ and from above by a
positive constant $\mu$, then by denoting $d(x,y)=\lambda|x-y|$ with $|\cdot|$
the standard Euclidean norm, we can show $(M,g)$ satisfies the DVP-condition
with $K_{1}=\big{(}\frac{4\mu}{\lambda}\big{)}^{n}$ and
$K_{2}=4\pi^{-2}\big{(}\frac{\mu}{\lambda}\big{)}^{n+2}$ (see [16]). Thus
every Riemannian manifold has a coordiate chart centered at any point that
satisfies the DVP-condition. If $r_{0}=+\infty$, we call $M$ a simple
manifold. Other complete Riemannian manifolds satisfying a DVP-condition
include $(M,g)$ with nonnegative Ricci curvature (see [4] for the proof of the
Neumann-Poincaré inequality) and area-minimizing embedded hypersurfaces in
${\tenmsb R}^{n+1}$. (Here $d$ is taken to be the extrinsic distance function,
see [3] for the proof of the Neumann-Poincaré inequality.)
Based on the DVP-condition, one can apply Moser’s iteration [19] to derive the
Harnack inequality for superharmonic functions on $M$ (see [16] Lemma 4.1),
which implies the following estimates:
###### Lemma 5.1.
([16], Corollary 4.1) Let $M$ be a Riemannian manifold satisfying a DVP-
condition, then there exists a constant $C_{4}\in(0,1)$, only depending on
$K_{1}$ and $K_{2}$, such that
(5.2) $v_{+,\frac{R}{2}}\leq(1-C_{4})v_{+,R}+C_{4}\bar{v}_{\frac{R}{2}}$
for any subharmonic function $v$ on $B_{R}(x_{0})$
($B_{2R}(x_{0})\subset\subset M$). Here $v_{+,R}$ denotes the supremum of $v$
on $B_{R}(x_{0})$.
Let $\Omega$ be an open domain of $M$ and $\rho>0$. Using the Lax-Milgram
Theorem, it is easy to prove that there is a unique
$H_{0}^{1,2}(\Omega)$-function $G^{\rho}(\cdot,x)$, such that
(5.3) $\int_{\Omega}\langle\nabla
G^{\rho}(\cdot,x),\nabla\phi\rangle*1=-\hskip-12.80373pt\int_{B_{\rho}(x)}\phi\qquad\text{for
any }\phi\in H_{0}^{1,2}(\Omega)$
whenever $B_{\rho}(x)\subset\Omega$. Here
$-\hskip-12.80373pt\int_{B_{\rho}(x)}\phi=\frac{\int_{B_{\rho}(x)}\phi}{V(x,\rho)}.$
$G^{\rho}$ is called the mollified Green function. As in [9] and [16], we can
obtain a-priori estimates for mollified Green functions:
###### Lemma 5.2.
([16], Lemma 4.3) Let $(M,g)$ be a Riemannian manifold (dimension $\geq 3$)
satisfying a DVP-condition. Let $x_{0}\in M$ and $R>0$ satisfying
$B_{2R}(x_{0})\subset\subset M$. Denote
(5.4)
$\omega^{R}=\frac{V(x_{0},\frac{R}{2})}{R^{2}}G^{\frac{R}{2}}(\cdot,x_{0})$
with $G^{\frac{R}{2}}$ being the mollified Green function on $B_{R}(x_{0})$,
then
(5.5) $\omega^{R}\leq c_{1}\qquad\text{on }B_{R}(x_{0})$
and
(5.6) $\omega^{R}\geq c_{2}\qquad\text{on }B_{\frac{R}{2}}(x_{0})$
with $c_{1}$ and $c_{2}$ denoting positive constants depending only on $K_{1}$
and $K_{2}$, not depending on $x_{0}$ and $R$.
Choosing $\eta=(\omega^{R})^{2}$ as a test function in (5.1), we can derive
the following curvature estimates with the aid of Lemmas 5.1 and 5.2. See [16]
and [17] for details.
###### Lemma 5.3.
(Curvature estimates) Let $M^{n}$ be a submanifold in ${\tenmsb R}^{n+m}$
($n\geq 3$) with parallel mean curvature. Assume there is a distance function
$d$ on $M$, the metric ball $B_{4R_{0}}(x_{0})$ defined by $d$ satisfies the
DVP-condition for some $x_{0}\in M$ and $R_{0}\in(0,+\infty]$, and the Gauss
image of $B_{4R_{0}}(x_{0})$ is contained in a compact set $K\subset{\tenmsb
W}_{1/3}$. Then there exists a positive constant $C_{5}$, depending only on
$K,K_{1},K_{2}$, such that for arbitrary $R<R_{0}$,
(5.7) $\frac{R^{2}}{V(x_{0},R)}\int_{B_{R}(x_{0})}|B|^{2}*1\leq
C_{5}(f_{+,R}-f_{+,\frac{R}{2}})$
with $f$ being the strongly subharmonic function constructed in Proposition
4.2. Moreover, there exists a positive constant $C_{6}$ depending only on
$K,K_{1},K_{2}$, such that for arbitrary $\varepsilon>0$, we can find
$R\in\big{[}\exp(-C_{6}\varepsilon^{-1})R_{0},R_{0}\big{]}$, such that
(5.8) $\frac{R^{2}}{V(x_{0},R)}\int_{B_{R}(x_{0})}|B|^{2}*1\leq\varepsilon.$
If $M$ is a minimal submanifold with $rank(\gamma)\leq 2$, then the condition
on the Gauss image can be relaxed to $\gamma(B_{4R_{0}}(x_{0}))\subset
K\subset{\tenmsb W}_{0}$.
## 6\. Gauss image shrinking lemmas and Bernstein type theorems
Let $u$ be a harmonic map from $M$ and $H$ be a smooth function on the target
manifold, then $H\circ u$ defines a smooth function on $M$. Take a cut-off
function $\eta$ supported in the interior of $B_{R}(x_{0})$, $0\leq\eta\leq
1$, $\eta\equiv 1$ on $B_{\frac{R}{2}}(x_{0})$ and $|\nabla\eta|\leq
c_{0}R^{-1}$; and denote by $G^{\rho}$ the mollified Green function in
$B_{R}(x_{0})$. Then by Stokes’ theorem
(6.1) $\int_{B_{R}(x_{0})}\text{div}\big{(}\eta G^{\rho}(\cdot,x)\nabla(H\circ
u)\big{)}*1=0\qquad\text{whenever }B_{\rho}(x)\subset B_{R}(x_{0}).$
After careful calculation as in [16][17], we can arrive at a pointwise
estimate for $H\circ u$:
###### Proposition 6.1.
Let $M$ be as in Lemma 5.3. Let $u$ be a harmonic map of
$B_{4R_{0}}(x_{0})\subset M$ into a convex domain $V\subset{\tenmsb R}^{l}$
with metric $h_{ij}dx^{i}dx^{j}$, such that
$K_{3}|\xi|^{2}\leq\xi^{i}h_{ij}\xi^{j}\leq K_{4}|\xi|^{2}$
everywhere on $V$. Then for any smooth function $H$ on $V$,
(6.2) $\displaystyle H\circ u(x)\leq$ $\displaystyle
H(\bar{u}_{R})+C_{7}\sup_{V}|\nabla
H|\Big{(}\frac{R^{2}}{V(x_{0},R)}\int_{B_{R}(x_{0})}|du|^{2}*1\Big{)}^{\frac{1}{2}}$
$\displaystyle-\liminf_{\rho\rightarrow
0^{+}}\int_{B_{R}(x_{0})}G^{\rho}(\cdot,x)\eta\Delta(H\circ u)*1$
for all $x\in B_{\frac{R}{4}}(x_{0})$ with $R<R_{0}$. Here
$\bar{u}_{R}\in\overline{V}$ (the closure of $V$) denotes the average value of
$u$ in $B_{R}(x_{0})$ and $C_{7}$ is a positive constant depending only on
$K,K_{1},K_{2},K_{3},K_{4}$, but independent of $H$, $x_{0}$ and $R$.
If we take $u$ to be the harmonic Gauss map, then
$|du|^{2}=|d\gamma|^{2}=|B|^{2}$, and Lemma 5.3 implies that the second term
on the right hand side of (6.2) becomes arbitrarily small when $R$ is
sufficiently small. Hence, for proving the Gauss image shrinking property as
in [16][17] it remains to show:
* •
${\tenmsb W}_{0}$ is diffeomorphic to a convex domain in Euclidean space;
* •
For any given $S\in K\subset{\tenmsb W}_{1/3}$, one can find a smooth function
$H$, such that $H\circ\gamma$ is a subharmonic function on $B_{R}(x_{0})$, and
the sublevel set $\big{(}H\leq H(S)\big{)}$ is contained in a matrix
coordinate chart of some point in $\mathbb{G}_{n,m}$.
In the sequel, $x_{1},x_{2},r,\theta,e_{i},e_{n+\alpha}$ and $P_{t}$ will be
as in Section 4.
###### Lemma 6.1.
There is a diffeomorphism $\phi=(\phi_{1},\phi_{2}):{\tenmsb
W}_{0}\rightarrow(-\pi,\pi)\times{\tenmsb R}^{nm-1}$ defined by
$S\mapsto\big{(}\phi_{1}(S),\phi_{2}(S)\big{)},$
such that
(6.3) $\displaystyle\phi_{1}=\theta,$
$\displaystyle\big{|}\phi_{2}(S)\big{|}=\frac{1}{r}-1.$
###### Proof.
As shown in Section 4, $\theta$ is a smooth $(-\pi,\pi)$-valued function on
${\tenmsb W}_{0}$, and we shall investigate the level sets of $\theta$.
For any $t\in(-\pi,\pi)$, when $\theta(S)=t$ we have
$w(S,P_{t+\frac{\pi}{2}})=0$ and $w(S,P_{t})>0$ by (2.15). Now we write
$y_{1}=w(\cdot,P_{t}),\qquad y_{2}=w(\cdot,P_{t+\frac{\pi}{2}}),$
then
(6.4) $y_{1}=\cos t\ x_{1}+\sin t\ x_{2}\qquad y_{2}=-\sin t\ x_{1}+\cos t\
x_{2}.$
Denote ${\tenmsb U}_{t}=\big{\\{}S\in\mathbb{G}_{n,m}:w(S,P_{t})>0\big{\\}}$.
Let $\chi_{t}:{\tenmsb U}_{t}\rightarrow M_{n\times m}$ be the diffeomorphism
mapping $S\in{\tenmsb U}_{t}$ to its $(n\times m)$-matrix coordinate, then the
Plücker coordinate of $S$ is
(6.5)
$\psi(S)=(f_{1}+Z_{1\alpha}f_{n+\alpha})\wedge\cdots\wedge(f_{n}+Z_{n\alpha}f_{n+\alpha})\qquad\text{with
}Z=\chi_{t}(S).$
Here $f_{1}=\cos t\ e_{1}+\sin t\ e_{n+1}$, $f_{n+1}=-\sin t\ e_{1}+\cos t\
e_{n+1}$, $f_{i}=e_{i}$ for all $2\leq i\leq n$ and
$f_{n+\alpha}=e_{n+\alpha}$ for every $2\leq\alpha\leq m$. Noting that
$\psi(P_{t})=f_{1}\wedge\cdots\wedge f_{n}$ and
$\psi(P_{t+\frac{\pi}{2}})=f_{n+1}\wedge f_{2}\wedge\cdots\wedge f_{n}$. A
direct calculation shows that for $S\in U_{t}$
(6.6)
$y_{1}(S)=w(S,P_{t})=\frac{\langle\psi(S),\psi(P_{t})\rangle}{\langle\psi(S),\psi(S)\rangle^{\frac{1}{2}}\langle\psi(P_{t}),\psi(P_{t})\rangle^{\frac{1}{2}}}=(I_{n}+ZZ^{T})^{-\frac{1}{2}},$
and
(6.7)
$y_{2}(S)=w(S,P_{t+\frac{\pi}{2}})=\frac{\langle\psi(S),\psi(P_{t+\frac{\pi}{2}})\rangle}{\langle\psi(S),\psi(S)\rangle^{\frac{1}{2}}\langle\psi(P_{t+\frac{\pi}{2}}),\psi(P_{t+\frac{\pi}{2}})\rangle^{\frac{1}{2}}}=Z_{11}(I_{n}+ZZ^{T})^{-\frac{1}{2}}.$
Hence, $y_{2}(R)=0$ if and only if $Z_{11}=0$, i.e.
(6.8) $\\{S\in{\tenmsb W}_{0}:\theta(S)=t\\}=\chi_{t}^{-1}\\{Z\in M_{n\times
m}:Z_{11}=0\\}.$
A straightforward calculation gives
$\displaystyle\nabla y_{1}$ $\displaystyle=\cos(t-\theta)\nabla
r+r\sin(t-\theta)\nabla\theta,$ $\displaystyle\nabla y_{2}$
$\displaystyle=-\sin(t-\theta)\nabla r+r\cos(t-\theta)\nabla\theta.$
Hence, along the level set $(\theta=t)$
$r=y_{1},\quad\nabla y_{1}=\nabla r,\quad\nabla y_{2}=r\nabla\theta.$
Differentiating both sides of (6.7) and noting (6.8), we have
(6.9) $\nabla\theta=r^{-1}\nabla
y_{2}=r^{-1}\nabla(Z_{11}y_{1})=r^{-1}(y_{1}\nabla Z_{11}+Z_{11}\nabla
y_{1})=\nabla Z_{11},$
whenever $\theta(S)=t.$ In $U_{t}$ let $E_{i\alpha}$ be the matrix with 1 in
the intersection of row $i$ and column $\alpha$ and 0 otherwise. Denote
$g_{i\alpha,j\beta}=\langle E_{i\alpha},E_{j\beta}\rangle$ and let
$(g^{i\alpha,j\beta})$ be the inverse matrix of $(g_{i\alpha,j\beta})$. A
standard computation shows $|\nabla Z_{11}|^{2}=g^{11,11}$. It has been shown
in [26] that the eigenvalues of $(g^{i\alpha,j\beta})$ are
$\big{\\{}(1+\lambda_{i}^{2})(1+\lambda_{\alpha}^{2}):1\leq i\leq
n,1\leq\alpha\leq m\big{\\}}$
with $\lambda_{i}=\tan\theta_{i}$ and $\\{\theta_{i}\\}$ being the Jordan
angles between $S$ and $P_{t}$. Hence
$1\leq
g^{11,11}\leq\max_{(i,\alpha)}\big{(}(1+\lambda_{i}^{2})(1+\lambda_{\alpha}^{2})\big{)}\leq
w^{-4}(\cdot,P_{t})=r^{-4};$
i.e.
(6.10) $1\leq|\nabla\theta|^{2}\leq r^{-4}.$
From (6.8) and (6.10), we see that $\theta$ is a non-degenerate function on
${\tenmsb W}_{0}$, and each level set of $\theta$ is diffeomorphic to
${\tenmsb R}^{nm-1}$. $T:M_{n\times m}\rightarrow M_{n\times m}$ defined by
$Z\mapsto\Big{(}\det(I_{n}+ZZ^{T})^{\frac{1}{2}}-1\Big{)}\frac{Z}{\big{(}\text{tr}(ZZ^{T})\big{)}^{\frac{1}{2}}}$
obviously is a diffeomorphism. Note that here
$\big{(}\text{tr}(ZZ^{T})\big{)}^{\frac{1}{2}}=\big{(}\sum_{i,\alpha}Z_{i\alpha}^{2}\big{)}^{\frac{1}{2}}$
equals the Euclidean norm of $Z$ when $Z$ is treated as a vector in ${\tenmsb
R}^{nm}$. Thus, $T\circ\chi_{t}$ is a diffeomorphism between the level set
$(\theta=t)$ and ${\tenmsb R}^{nm-1}$, moreover
$|T\circ\chi_{t}|=w^{-1}(\cdot,P_{t})-1=r^{-1}-1.$
Therefore, $\phi_{1}=\theta$ and $\phi_{2}=T\circ\chi_{t}$ are the required
mappings.
∎
###### Proposition 6.2.
For any $c\in(\frac{1}{3},1)$ and any compact subset $\Theta$ of $(-\pi,\pi)$,
there exists a smooth family of nonnegative, smooth functions
$H(\cdot,t)(t\in\Theta)$ on
$\overline{{\tenmsb W}}_{c,\Theta}:=\\{S\in{\tenmsb W}_{0}:r(S)\geq
c,\theta(S)\in\Theta\\},$
such that
(i) $H(S,t)=0$ if and only if $S=P_{t}$;
(ii) $H(S,t)\leq 1$ (or $H(S,t)<1$) if and only if
$w(S,P_{t})\geq\frac{3}{4}c+\frac{1}{12}$ (or
$w(S,P_{t})>\frac{3}{4}c+\frac{1}{12}$);
(iii) For any submanifold $M$ in ${\tenmsb R}^{n+m}$ with parallel mean
curvature, if the Gauss image of $M$ is contained in $\overline{{\tenmsb
W}}_{c,\Theta}$, then $H(\cdot,t)\circ\gamma$ is a subharmonic function on $M$
for all $t\in\Theta$.
###### Proposition 6.3.
For any $c\in(0,1)$ and any compact subset $\Theta$ of $(-\pi,\pi)$, there
exists a smooth family of nonnegative, smooth functions
$H(\cdot,t)(t\in\Theta)$ on $\overline{{\tenmsb W}}_{c,\Theta},$ such that
(i) $H(S,t)=0$ if and only if $S=P_{t}$;
(ii) $H(S,t)\leq 1$ (or $H(S,t)<1$) if and only if
$w(S,P_{t})\geq\frac{3}{4}c$ (or $w(S,P_{t})>\frac{3}{4}c$);
(iii) If $M$ is a minimal submanifold in ${\tenmsb R}^{n+m}$ with
$rank(\gamma)\leq 2$, then $H(\cdot,t)\circ\gamma$ is a subharmonic function
on $M$ for all $t\in\Theta$ if the Gauss image of $M$ is contained in
$\overline{{\tenmsb W}}_{c,\Theta}$.
###### Proof.
We only prove Proposition 6.2, since the proof of Proposition 6.3 is similar.
Let $\varphi$ be a smooth function on $[0,+\infty)$ satisfying
$\left\\{\begin{array}[]{ll}\varphi(u)=0,&u\in[0,\frac{11}{12}-\frac{3}{4}c];\\\
\varphi(u)=u-\frac{5}{6}+\frac{1}{2}c,&u\in[\frac{3}{4}-\frac{1}{4}c,+\infty);\\\
0\leq\varphi^{\prime}\leq 1.&\end{array}\right.$
Then one can define $\Psi$ on $\overline{{\tenmsb
W}}_{c,\Theta}\times(-\pi,\pi)\times[0,+\infty)$ by
(6.11)
$(S,t,u)\mapsto\left\\{\begin{array}[]{cc}r\cos(\theta-t-\varphi(u))+u-\varphi(u)-1&\text{if
}\theta(S)\geq t;\\\ r\cos(\theta-t+\varphi(u))+u-\varphi(u)-1&\text{if
}\theta(S)<t.\end{array}\right.$
Now we fix $t$ and denote $\Psi_{t}(S,u)=\Psi(S,t,u)$. For arbitrary
$S\in\overline{{\tenmsb W}}_{c,\Theta}$, we put
$I_{S}:=\Big{\\{}u\in[0,+\infty):\max\\{0,|\theta-t|-\pi\\}\leq\varphi(u)\leq|\theta-t|\Big{\\}},$
then obviously $I_{S}$ is a closed interval, $I_{S}:=[m_{S},M_{S}]$. If
$\theta(S)\geq t$, then
$\partial_{2}\Psi_{t}=1-\big{(}1+r\sin(\theta-t-\varphi(u))\big{)}\varphi^{\prime}(u)\geq
0$
and the equality holds if and only if $\varphi(u)=|\theta-t|$ or
$|\theta-t|-\pi$ and $\varphi^{\prime}(u)=1$; which implies $\varphi(u)=m_{S}$
or $M_{S}$. Thus
(6.12) $\partial_{2}\Psi_{t}(S,\cdot)>0\qquad\text{on }(m_{S},M_{S}).$
Similarly (6.12) holds when $\theta(S)<t$.
When $|\theta(S)-t|\leq\pi$, we have $m_{S}=0$ and hence
(6.13) $\Psi_{t}(S,m_{S})=r\cos(\theta-t)-1\leq 0.$
Otherwise $|\theta(S)-t|>\pi$ and $m_{S}$ satisfies
$\varphi(m_{S})=|\theta-t|-\pi$, thus
(6.14) $\displaystyle\Psi_{t}(S,m_{S})$
$\displaystyle=-r+m_{S}-\varphi(m_{S})-1\leq-r-1+\lim_{u\rightarrow+\infty}\big{(}u-\varphi(u)\big{)}$
$\displaystyle=-r-1+\frac{5}{6}-\frac{1}{2}c\leq-\frac{3}{2}c-\frac{1}{6}<0.$
Here the first inequality follows from
$\big{(}u-\varphi(u)\big{)}^{\prime}=1-\varphi^{\prime}(u)\geq 0$.
Obviously $\varphi(M_{S})=|\theta-t|$. By the definition of $\varphi$,
$\varphi$ cannot be identically zero on a neighborhood of $M_{S}$. Thus
$M_{S}\geq\frac{11}{12}-\frac{3}{4}c$, and
(6.15) $\displaystyle\Psi_{t}(S,M_{S})$
$\displaystyle=r+M_{S}-\varphi(M_{S})-1\geq r-1+\frac{11}{12}-\frac{3}{4}c$
$\displaystyle\geq\frac{1}{4}c-\frac{1}{12}>0.$
By (6.12)-(6.15), for each $S\in\overline{{\tenmsb W}}_{c,\Theta}$, there
exists a unique $\tilde{H}=\tilde{H}(S,t)\in[m_{S},M_{S})$, such that
(6.16)
$\Psi\big{(}S,t,\tilde{H}(S,t)\big{)}=\Psi_{t}\big{(}S,\tilde{H}(S,t)\big{)}=0.$
Denote
(6.17) $\Omega:=\big{\\{}(S,t)\in\overline{{\tenmsb
W}}_{c,\Theta}\times(-\pi,\pi):\theta(S)\neq t\big{\\}},$
then $\Psi$ is obviously smooth on $\Omega\times[0,+\infty)$. Hence the
implicit theorem implies $\tilde{H}$ is smooth on $\Omega$. To show smoothness
of $\tilde{H}$, it remains to prove that $\tilde{H}$ is smooth on a
neighborhood of $\Omega^{c}=\big{\\{}(S,t)\in\overline{{\tenmsb
W}}_{c,\Theta}\times(-\pi,\pi):\theta(S)=t\big{\\}}$. Denote
(6.18) $\Omega_{0}:=\big{\\{}(S,t)\in\overline{{\tenmsb
W}}_{c,\Theta}:w(S,P_{t})\geq\frac{1}{12}+\frac{3}{4}c\big{\\}},$
then $\Omega^{c}\subset\subset\Omega_{0}$ since $\theta(S)=t$ implies
$w(S,P_{t})=r\geq c>\frac{1}{12}+\frac{3}{4}c$. For all $(S,t)\in\Omega_{0}$,
$1-w(S,P_{t})\leq 1-(\frac{1}{12}+\frac{3}{4}c)=\frac{11}{12}-\frac{3}{4}c$
and hence $\varphi\big{(}1-w(S,P_{t})\big{)}=0$. Noting that $w(S,P_{t})>0$
implies $|\theta-t|<\pi$, we have $1-w(S,P_{t})\in[m_{S},M_{S})$, and moreover
$\Psi_{t}\big{(}S,1-w(S,P_{t})\big{)}=r\cos(\theta-t)+\big{(}1-w(S,P_{t})\big{)}-1=0.$
Therefore
(6.19) $\tilde{H}(S,t)=1-w(S,P_{t})\qquad\text{for all }(S,t)\in\Omega_{0}$
and the smoothness of $\tilde{H}$ follows.
Now we put
(6.20) $U_{t}:=\Big{\\{}S\in\overline{{\tenmsb
W}}_{c,\Theta}:w(S,P_{t})\geq\frac{1}{12}+\frac{3}{4}c\Big{\\}},$
then obviously $0\leq\tilde{H}(S,t)\leq\frac{11}{12}-\frac{3}{4}c$ whenever
$S\in U_{t}$. On the other hand, for arbitrary $S\in\overline{{\tenmsb
W}}_{c,\Theta}\backslash U_{t}$, one of the following two cases has to occur:
(I) $|\theta(S)-t|\geq\pi$; (II) $|\theta(S)-t|\leq\pi$ and
$w(S,P_{t})<\frac{1}{12}+\frac{3}{4}c$. If Case (I) holds, then
$\tilde{H}(S,t)\geq m_{S}>\frac{11}{12}-\frac{3}{4}c$. For the second case,
since
$\Psi_{t}\big{(}s,\frac{11}{12}-\frac{3}{4}c\big{)}=r\cos(\theta-t)+\frac{11}{12}-\frac{3}{4}c-1<0,$
one can deduce that $\tilde{H}(S,t)>\frac{11}{12}-\frac{3}{4}c$ due to the
monotonicity of $\Psi_{t}$ with respect to the $u$ variable. Therefore
(6.21) $U_{t}=\Big{\\{}S\in\overline{{\tenmsb
W}}_{c,\Theta}:\tilde{H}(S,t)\leq\frac{11}{12}-\frac{3}{4}c\Big{\\}}.$
Similarly
(6.22) $\text{int}(U_{t})=\Big{\\{}S\in\overline{{\tenmsb
W}}_{c,\Theta}:\tilde{H}(S,t)<\frac{11}{12}-\frac{3}{4}c\Big{\\}}.$
It is easily seen from (6.19) and (6.21) that
(6.23)
$\mbox{Hess}\tilde{H}(\cdot,t)\geq\big{(}\frac{1}{12}+\frac{3}{4}c\big{)}\big{(}-\mbox{Hess}\log
w(\cdot,P_{t})+d\log w(\cdot,P_{t})\otimes d\log
w(\cdot,P_{t})\big{)}\qquad\text{on }U_{t}.$
For each $a\geq\frac{11}{12}-\frac{3}{4}c$ and $S\in\overline{{\tenmsb
W}}_{c,\Theta}$ satisfying $\theta(S)>t$, $\tilde{H}(S,t)=a$ if and only if
$0=\Psi_{t}(S,a)=r\cos(\theta-t-\varphi(a))+a-\varphi(a)-1;$
i.e.
(6.24) $w(S,P_{t+\varphi(a)})=1+\varphi(a)-a$
with
$\frac{1}{12}+\frac{3}{4}c\geq 1+\varphi(a)-a\geq
1+\lim_{u\rightarrow+\infty}(\varphi(u)-u)=\frac{1}{6}+\frac{1}{2}c>\frac{1}{3}.$
Hence
(6.25) $N_{t,a}^{+}:=\\{S\in\overline{{\tenmsb
W}}_{c,\Theta}:\theta(S)>t,\tilde{H}(S,t)=a\\}$
overlaps the level set $\big{\\{}S\in\mathbb{G}_{n,m}:-\log
w(S,P_{t+\varphi(a)})=-\log(1+\varphi(a)-a)\big{\\}}$.
From (6.16) we have
$\tilde{\Psi}_{t}:=\Psi_{t}\big{(}S,\tilde{H}(S,t)\big{)}\equiv 0$.
Differentiating both sides implies
(6.26)
$0=\nabla_{\nu}\tilde{\Psi}_{t}=\nabla_{\nu}\Psi_{t}(\cdot,a)+(\partial_{2}\Psi_{t})\nabla_{\nu}\tilde{H}(\cdot,t)\qquad\text{on
}N_{t,a}^{+}$
with $\nu$ the unit normal vector field on $N_{t,a}^{+}$. Noting that
$\Psi_{t}(\cdot,a)=w(\cdot,P_{t+\varphi(a)})-(1+\varphi(a)-a)$ and
$\partial_{2}\Psi_{t}>0$, (6.26) tells us $\nabla\tilde{H}(\cdot,t)$ and
$-\nabla\log w(\cdot,P_{t+\varphi(a)})$ are both nonzero normal vector fields
on $N_{t,a}^{+}$, pointing in the same direction. The compactness of
$\bigcup_{a\geq\frac{11}{12}-\frac{3}{4}c}N_{t,a}^{+}$ and $\Theta$ implies
the existence of a positive constant $C_{8}$ not depending on $a$ and $t$,
such that
(6.27) $\frac{|\nabla\tilde{H}(\cdot,t)|}{\big{|}\nabla\log
w(\cdot,P_{t+\varphi(a)})\big{|}}\geq C_{8}\qquad\text{on }N_{t,a}^{+}.$
Hence applying Lemma 4.1 gives
(6.28) $\mbox{Hess}\ \tilde{H}(\cdot,t)\geq-C_{8}\log
w(\cdot,P_{t+\varphi(a)})-\varepsilon g+\lambda d\tilde{H}(\cdot,t)\otimes
d\tilde{H}(\cdot,t)\qquad\text{on }N_{t,a}^{+}$
with $g$ being the canonical metric on $\mathbb{G}_{n,m}$, $\varepsilon$ being
a positive constant to be chosen and $\lambda$ denoting a continuous
nonpositive function depending on $\varepsilon$.
Similarly,
(6.29) $N_{t,a}^{-}:=\\{S\in\overline{{\tenmsb
W}}_{c,\Theta}:\theta(S)<t,\tilde{H}(S,t)=a\\}$
overlaps the level set $\big{\\{}S\in\mathbb{G}_{n,m}:\log
w(S,P_{t-\varphi(a)})=-\log(1+\varphi(a)-a)\big{\\}}$ for each
$a\geq\frac{11}{12}-\frac{3}{4}c$. On it $\nabla\tilde{H}(\cdot,t)$ and
$-\nabla\log w(\cdot,P_{t-\varphi(a)})$ are both nonzero normal vector fields
pointing in the same direction. Again using Lemma 4.2 yields
(6.30) $\mbox{Hess}\tilde{H}(\cdot,t)\geq-C_{8}\log
w(\cdot,P_{t-\varphi(a)})-\varepsilon g+\lambda d\tilde{H}(\cdot,t)\otimes
d\tilde{H}(\cdot,t)\qquad\text{on }N_{t,a}^{-}.$
Let $M$ be a submanifold in ${\tenmsb R}^{n+m}$ with parallel mean curvature
whose Gauss image is contained in $\overline{{\tenmsb W}}_{c,\Theta}$, denote
$\tilde{h}(\cdot,t)=\exp(\mu_{0}\tilde{H}(\cdot,t))\circ\gamma$
where $\mu_{0}$ is a positive constant to be chosen. Once $\gamma(x)\in
U_{t}$, combining (6.23) and (3.23) gives
(6.31) $\displaystyle\mu_{0}^{-1}\tilde{h}(\cdot,t)^{-1}\tilde{h}(\cdot,t)$
$\displaystyle\geq$
$\displaystyle-\big{(}\frac{1}{12}+\frac{3}{4}c\big{)}\Delta\Big{(}\log
w(\cdot,P_{t})\circ\gamma\Big{)}+\big{(}\frac{1}{12}+\frac{3}{4}c+\mu_{0}\big{)}\Big{|}\nabla\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}\Big{|}^{2}$ $\displaystyle\geq$
$\displaystyle\big{(}\frac{1}{12}+\frac{3}{4}c\big{)}C_{1}|B|^{2}+\mu_{0}\Big{|}\nabla\big{(}\log
w(\cdot,P_{t})\circ\gamma\big{)}\Big{|}^{2}$
at $x\in M$, where $C_{1}$ is a positive constant depending only on $c$. If
$\gamma(x)\in N_{t,a}^{+}$, based on (6.28) and (3.23), one can proceed as in
the proof of Proposition 4.2 to obtain
(6.32)
$\mu_{0}^{-1}\tilde{h}(\cdot,t)^{-1}\Delta\tilde{h}(\cdot,t)\geq(C_{8}C_{1}-\varepsilon)|B|^{2}+(C_{8}^{2}|\lambda+\mu_{0}|-C_{8})\big{|}\nabla(\log
w(\cdot,P_{t+\varphi(a)})\circ\gamma)\big{|}^{2}$
at $x$. Similarly, once $\gamma(x)\in N_{t,a}^{-}$,
(6.33)
$\mu_{0}^{-1}\tilde{h}(\cdot,t)^{-1}\Delta\tilde{h}(\cdot,t)\geq(C_{8}C_{1}-\varepsilon)|B|^{2}+(C_{8}^{2}|\lambda+\mu_{0}|-C_{8})\big{|}\nabla(\log
w(\cdot,P_{t-\varphi(a)})\circ\gamma)\big{|}^{2}.$
Now we put
(6.34) $\varepsilon:=C_{8}C_{1},\qquad\mu_{0}:=\sup|\lambda|+C_{8}^{-1}$
(where $\sup|\lambda|<+\infty$, since $\lambda$ is a continuous function on a
compact set $(\overline{{\tenmsb
W}}_{c,\Theta}\times\Theta)\backslash\text{int}(\Omega_{0})$), then from
(6.31)-(6.33) we see $\tilde{h}$ is a subharmonic function on $M$. Therefore
(6.35)
$H(\cdot,t):=\frac{\exp\big{(}\mu_{0}\tilde{H}(\cdot,t)\big{)}-1}{\exp\big{(}(\frac{11}{12}-\frac{3}{4}c)\mu_{0}\big{)}-1}$
are required functions satisfying (i)-(iii). ∎
###### Remark 6.1.
The auxiliary function $\varphi$ in the above proof can be easily obtained
from the standard bump function. Choose $\xi_{1}$ to be a nonnegative smooth
function on ${\tenmsb R}$, whose support is
$(\frac{11}{12}-\frac{3}{4}c,\frac{3}{4}-\frac{1}{4}c)$, then
$\xi_{2}(u):=\frac{\int_{\frac{11}{12}-\frac{3}{4}c}^{u}\xi_{1}}{\int_{\frac{11}{12}-\frac{3}{4}c}^{\frac{3}{4}-\frac{1}{4}c}\xi_{1}}$
is a smooth function on ${\tenmsb R}$ satisfying $0\leq\xi_{2}\leq 1$,
$\xi_{2}(u)=0$ whenever $u\leq\frac{11}{12}-\frac{3}{4}c$ and $\xi_{2}(u)=1$
whenever $u\geq\frac{3}{4}-\frac{1}{4}c$.
$\alpha\in(0,+\infty)\mapsto\int_{\frac{11}{12}-\frac{3}{4}c}^{\frac{3}{4}-\frac{1}{4}c}\xi_{2}^{\alpha}$
is a strictly decreasing function, which converges to $0$ as
$u\rightarrow+\infty$ and converges to $-\frac{1}{6}+\frac{1}{2}c$ as
$u\rightarrow 0$. There exists a unique $\beta\in(0,+\infty)$, such that
$\int_{\frac{11}{12}-\frac{3}{4}c}^{\frac{3}{4}-\frac{1}{4}c}\xi_{2}^{\beta}=-\frac{1}{12}+\frac{1}{4}c.$
Then
$\varphi(u):=\int_{0}^{u}\xi_{2}^{\beta}$
is the required auxiliary function.
Based on Lemma 6.1 and Proposition 6.2-6.3, one can derive a Gauss image
shrinking property for submanifolds with parallel mean curvature as in
[16][17]:
###### Lemma 6.2.
Let $M^{n}$ be a submanifold in ${\tenmsb R}^{n+m}$ ($n\geq 3$) with parallel
mean curvature. Assume there is a distance function $d$ on $M$, the metric
ball $B_{4R_{0}}(x_{0})$ given by $d$ satisfies DVP-condition for some
$x_{0}\in M$ and $R_{0}\in(0,+\infty]$, and the Gauss image of
$B_{4R_{0}}(x_{0})$ is contained in a compact set $K\subset{\tenmsb W}_{1/3}$.
Then there exists a constant $\delta_{1}\in(0,1)$, depending only on
$K,K_{1},K_{2}$, not depending on $x_{0}$ and $R_{0}$, such that the image of
$B_{\delta_{1}R_{0}}(x_{0})$ under the Gauss map is contained in
$\\{S\in\mathbb{G}_{n,m}:w(S,P)\geq\frac{1}{12}+\frac{3}{4}c\\}$ for some
$P\in\mathbb{G}_{n,m}$, where $c:=\inf_{K}r$.
###### Lemma 6.3.
Let $M^{n}$ be a minimal submanifold in ${\tenmsb R}^{n+m}$ ($n\geq 3$) with
$rank(\gamma)\leq 2$ ($\gamma$ denotes the Gauss map). Assume there is a
distance function $d$ on $M$, the metric ball $B_{4R_{0}}(x_{0})$ given by $d$
satisfies the DVP-condition for some $x_{0}\in M$ and $R_{0}\in(0,+\infty]$,
and the Gauss image of $B_{4R_{0}}(x_{0})$ is contained in a compact set
$K\subset{\tenmsb W}_{0}$. Then there exists a constant $\delta_{1}\in(0,1)$,
depending only on $K,K_{1},K_{2}$, not depending on $x_{0}$ and $R_{0}$, such
that the image of $B_{\delta_{1}R_{0}}(x_{0})$ under the Gauss map is
contained in $\\{S\in\mathbb{G}_{n,m}:w(S,P)\geq\frac{3}{4}c\\}$ for some
$P\in\mathbb{G}_{n,m}$, where $c:=\inf_{K}r$.
###### Proof.
As above, we only prove Lemma 6.2.
Since $K\subset{\tenmsb W}_{1/3}$ is compact, $c:=\inf_{K}r>\frac{1}{3}$ and
$\Theta:=\big{\\{}t\in(-\pi,\pi):t=\theta(S)\text{ for some }S\in K\big{\\}}$
is a compact subset of $(-\pi,\pi)$. Obviously $K\subset\overline{{\tenmsb
W}}_{c,\Theta}$.
By Lemma 6.1, $\overline{{\tenmsb W}}_{c,\Theta}$ is diffeomorphic to
$\Theta\times\overline{{\tenmsb D}}_{c^{-1}-1}^{nm-1}$. Thus
$\overline{{\tenmsb W}}_{c,\Theta}$ can be seen as a bounded convex domain in
${\tenmsb R}^{nm}$ equipped with the induced metric. The eigenvalues of the
metric matrices are bounded from below by $K_{3}$ and from above by $K_{4}$,
where $K_{3},K_{4}$ are positive constants depending only on $K$.
The Ruh-Vilms Theorem [20] implies that $\gamma$ is a harmonic function.
Putting $u=\gamma$ in (6.2) gives $|du|^{2}=|B|^{2}$. Let
$\\{H(\cdot,t):t\in\Theta\\}$ be a family of smooth functions on
$\overline{{\tenmsb W}}_{c,\Theta}$ as constructed in Proposition 6.2.
Choosing one of the functions as a test function in (6.2) yields
(6.36) $H(\gamma(x),t)\leq
H(\bar{\gamma}_{R},t)+C_{7}C_{9}\Big{(}\frac{R^{2}}{V(x_{0},R)}\int_{B_{R}(x_{0})}|B|^{2}*1\Big{)}^{\frac{1}{2}}.$
for arbitrary $R\leq R_{0}$ and all $x\in B_{\frac{R}{4}}$. Here
(6.37) $C_{9}:=\sup_{\Theta}\sup_{\overline{{\tenmsb
W}}_{c,\Theta}}\big{|}\nabla H(\cdot,t)\big{|}$
and the last term in (6.2) has been thrown out, since $H(\cdot,t)\circ\gamma$
is a subharmonic function.
For arbitrary $S\in\overline{{\tenmsb W}}_{c,\Theta}$,
$w(S,P_{\theta(S)})=r(S)\geq c>\frac{3}{4}c+\frac{1}{12}$, which implies
$H(S,\theta(S))<1$ by Proposition 6.2(ii). Since $\overline{{\tenmsb
W}}_{c,\Theta}$ is compact, there exists a positive constant
$\varepsilon_{1}$, such that
(6.38) $H(S,\theta(S))\leq 1-\varepsilon_{1}\qquad\text{for all
}S\in\overline{{\tenmsb W}}_{c,\Theta}.$
By Lemma 5.3 (curvature estimates), if we denote
(6.39)
$\delta_{1}:=\frac{1}{4}\exp(-C_{6}C_{7}^{2}C_{9}^{2}\varepsilon_{1}^{-2})$
then there exists $R\in[4\delta_{1}R_{0},R_{0}]$, such that
(6.40) $\frac{R^{2}}{V(x_{0},R)}\int_{B_{R}(x_{0})}|B|^{2}*1\leq
C_{7}^{-2}C_{9}^{-2}\varepsilon_{1}^{2}.$
Letting $t=\theta(\bar{\gamma}_{R})$ and substituting (6.38) and (6.40) into
(6.36) gives
(6.41) $H\big{(}\gamma(x),\theta(\bar{\gamma}_{R})\big{)}\leq
H(\bar{\gamma}_{R},\theta(\bar{\gamma}_{R}))+C_{7}C_{9}(C_{7}^{-2}C_{9}^{-2}\varepsilon_{1}^{2})^{\frac{1}{2}}\leq
1-\varepsilon_{1}+\varepsilon_{1}=1$
for all $x\in B_{\frac{R}{4}}(x_{0})$. Hence by Proposition 6.2(ii),
$\gamma\big{(}B_{\delta_{1}R_{0}}(x_{0})\big{)}\subset\gamma\big{(}B_{\frac{R}{4}}(x_{0})\big{)}\subset\big{\\{}S\in\mathbb{G}_{n,m}:w(S,P_{\theta(\bar{\gamma}_{R})})\geq\frac{1}{12}+\frac{3}{4}c\big{\\}}.$
∎
Given
$\gamma\big{(}B_{\delta_{1}R_{0}}(x_{0})\big{)}\subset\big{\\{}S\in\mathbb{G}_{n,m}:w(S,P)\geq\frac{1}{12}+\frac{3}{4}c>\frac{1}{3}\big{\\}}$,
one can start an iteration as in [17] to get a-priori estimates for the Gauss
image:
###### Lemma 6.4.
Let $M^{n}$ be a submanifold in ${\tenmsb R}^{n+m}$ ($n\geq 3$) with parallel
mean curvature. Assume there is a distance function $d$ on $M$, the metric
ball $B_{4R_{0}}(x_{0})$ defined by $d$ satisfies the DVP-condition for some
$x_{0}\in M$ and $R_{0}\in(0,+\infty]$, and the Gauss image of
$B_{4R_{0}}(x_{0})$ is contained in a compact set $K\subset{\tenmsb W}_{1/3}$.
Then for arbitrary $\varepsilon>0$, there exists a constant
$\delta_{2}\in(0,1)$, depending only on $K,K_{1},K_{2},\varepsilon$, not
depending on $x_{0}$ and $R_{0}$, such that
(6.42) $w(\gamma(x),\gamma(x_{0}))\geq 1-\varepsilon\qquad\text{on
}B_{\delta_{2}R_{0}}(x_{0}).$
In particular, if $M$ is a minimal submanifold with $rank(\gamma)\leq 2$, then
the condition on the Gauss image can be relaxed to
$\gamma(B_{4R_{0}}(x_{0}))\subset K\subset{\tenmsb W}_{0}$.
Letting $R_{0}\rightarrow+\infty$ we arrive at a Bernstein type theorem.
###### Theorem 6.1.
Let $M^{n}$ be a submanifold in ${\tenmsb R}^{n+m}$ ($n\geq 3$) with parallel
mean curvature. Assume there is a distance function $d$ on $M$, such that $M$
satisfies the DVP-condition and the diameter of $M$ with respect to $d$ is
infinite; and there exist $P,Q\in\mathbb{G}_{n,m}$ that are S-orthogonal to
each other, such that $\big{(}w(\gamma(x),P),w(\gamma(x),Q)\big{)}$ is
contained in a compact subset $K$ of $\overline{{\tenmsb
D}}\backslash\big{(}\overline{{\tenmsb D}}_{1/3}\cup\\{(a,0):a\leq
0\\}\big{)}$ for all $x\in M$. Then $M$ has to be an affine linear subspace.
In particular, if $M$ is a minimal submanifold with $rank(\gamma)\leq 2$, then
the assumptions on the Gauss image can be replaced by
$\big{(}w(\gamma(x),P),w(\gamma(x),Q)\big{)}\in K\subset\overline{{\tenmsb
D}}\backslash\\{(a,0):a\leq 0\\}$.
The above general Theorem can be applied to graphic submanifolds as follows.
###### Theorem 6.2.
Let $z^{\alpha}=f^{\alpha}(x^{1},\cdots,x^{n}),\ \alpha=1,\cdots,m$, be smooth
functions defined everywhere in ${\tenmsb R}^{n}$ ($n\geq 3,m\geq 2$), such
that their graph $M=(x,f(x))$ is a submanifold with parallel mean curvature in
${\tenmsb R}^{n+m}$. Suppose that there exist $\beta_{0}<+\infty$ and
$\beta_{1}<3$, such that
(6.43) $\Delta_{f}:=\Big{[}\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\Big{]}^{\frac{1}{2}}\leq\beta_{0}.$
and
(6.44) $\Delta_{f}\leq\beta_{1}\Big{(}1+\big{(}\frac{\partial f^{1}}{\partial
x^{1}}\big{)}^{2}\Big{)}^{\frac{1}{2}}.$
Then $f^{1},\cdots,f^{m}$ have to be affine linear (representing an affine
$n$-plane).
###### Proof.
$F:{\tenmsb R}^{n}\rightarrow M$ defined by
$x\mapsto(x,f(x))$
is obviously a diffeomorphism. Thus $M$ can be viewed as an $n$-dimensional
Euclidean space equipped with metric $g=g_{ij}dx^{i}dx^{j}$. Here
$g_{ij}=\Big{\langle}F_{*}\frac{\partial missing}{\partial
x^{i}},F_{*}\frac{\partial missing}{\partial
x^{j}}\Big{\rangle}=\Big{\langle}\varepsilon_{i}+\frac{\partial
f^{\alpha}}{\partial
x^{i}}\varepsilon_{n+\alpha},\varepsilon_{j}+\frac{\partial
f^{\alpha}}{\partial
x^{j}}\varepsilon_{n+\alpha}\Big{\rangle}=\delta_{ij}+\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial x^{j}}$
with $\\{\varepsilon_{i},\varepsilon_{n+\alpha}\\}$ being the canonical
orthonormal basis of ${\tenmsb R}^{n+m}$. Let $Df:=(\frac{\partial
f^{\alpha}}{\partial x^{i}})$ be an $(n\times m)$-matrix valued function on
${\tenmsb R}^{n}$, then
$(g_{ij})=I_{n}+Df(Df)^{T}.$
Hence $(g_{ij})\geq I_{n}$. In conjunction with
$\det(g_{ij})=\Delta_{f}^{2}\leq\beta_{0}^{2}$, we can deduce that all the
eigenvalues of $(g_{ij})$ take values between $1$ and $\beta_{0}^{2}$. Denote
$d:M\times M\rightarrow{\tenmsb R}$
$d\big{(}F(x),F(y)\big{)}=|x-y|,$
then the diameter of $M$ is $+\infty$ and it has been shown in Remark 5.1 that
$M$ satisfies a DVP-condition.
Let $P,Q\in\mathbb{G}_{n,m}$ whose Plücker coordinates are
$\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$ and
$\varepsilon_{n+1}\wedge\varepsilon_{2}\wedge\cdots\wedge\varepsilon_{n}$,
respectively. Obviously, $P,Q$ are S-orthogonal to each other. Denote by
$\psi$ the Plücker embedding, then
$\psi\circ\gamma=\big{(}\varepsilon_{1}+\frac{\partial f^{\alpha}}{\partial
x^{1}}\varepsilon_{n+\alpha}\big{)}\wedge\cdots\wedge\big{(}\varepsilon_{n}+\frac{\partial
f^{\alpha}}{\partial x^{n}}\varepsilon_{n+\alpha}\big{)}$
and by a direct computation,
$\displaystyle w(\gamma,P)$ $\displaystyle=$
$\displaystyle\frac{\langle\psi\circ\gamma,P\rangle}{\langle\psi\circ\gamma,\psi\circ\gamma\rangle^{\frac{1}{2}}\langle
P,P\rangle^{\frac{1}{2}}}=\Delta_{f}^{-1},$ $\displaystyle w(\gamma,Q)$
$\displaystyle=$
$\displaystyle\frac{\langle\psi\circ\gamma,Q\rangle}{\langle\psi\circ\gamma,\psi\circ\gamma\rangle^{\frac{1}{2}}\langle
Q,Q\rangle^{\frac{1}{2}}}=\frac{\partial f^{1}}{\partial
x^{1}}\Delta_{f}^{-1}.$
Hence $\big{(}w(\gamma,P),w(\gamma,Q)\big{)}\in K\subset\overline{{\tenmsb
D}}\backslash\big{(}\overline{{\tenmsb D}}_{1/3}\cup\\{(a,0):a\leq
0\\}\big{)}$ if and only if
$\beta_{1}^{-2}\leq
w(\gamma,P)^{2}+w(\gamma,Q)^{2}=\Big{(}1+\big{(}\frac{\partial f^{1}}{\partial
x^{1}}\big{)}^{2}\Big{)}\Delta_{f}^{-2}$
for a constant $\beta_{1}<3$. This is equivalent to (6.44). So the Bernstein
type result follows from Theorem 6.1.
∎
###### Remark 6.2.
Obviously $\beta_{1}\Big{(}1+\big{(}\frac{\partial f^{1}}{\partial
x^{1}}\big{)}^{2}\Big{)}^{\frac{1}{2}}\geq\beta_{1}$ on the right hand side of
(6.44). Hence Corollary 6.2 improves Theorem 6.2 in [17]. We note that
$\frac{\partial f^{1}}{\partial x^{1}}$ in (6.44) can be replaced by
$\frac{\partial f^{\alpha}}{\partial x^{i}}$ for arbitrary $1\leq\alpha\leq m$
and $1\leq i\leq n$.
## 7\. Appendix
Let ${\tenmsb H}$ denote the quaternions with the standard basis $1,i,j,k$ and
${\tenmsb C}=\mathbb{R}+\mathbb{R}i,{\tenmsb H}={\tenmsb C}+{\tenmsb C}j.$ Let
$q=z_{1}-\bar{z}_{2}j\in{\tenmsb H}$ with $z_{1},z_{2}\in{\tenmsb C}$. Then
$\bar{q}=\bar{z}_{1}+\bar{z}_{2}j$ and
$\displaystyle qi\bar{q}$
$\displaystyle=(z_{1}-\bar{z}_{2}j)i(\bar{z}_{1}+\bar{z}_{2}j)$
$\displaystyle=(\bar{z}_{2}k+z_{1}i)(\bar{z}_{1}+\bar{z}_{2}j)$
$\displaystyle=\bar{z}_{2}k\bar{z}_{1}+z_{1}i\bar{z}_{1}+\bar{z}_{2}k\bar{z}_{2}j+z_{1}i\bar{z}_{2}j$
$\displaystyle=z_{1}\bar{z}_{2}k+|z_{1}|^{2}i-|z_{2}|^{2}i+z_{1}\bar{z}_{2}k$
$\displaystyle=(|z_{1}|^{2}-|z_{2}|^{2})i+2z_{1}\bar{z}_{2}k$
which coincides with the usual Hopf map
$\displaystyle\eta$ $\displaystyle:$ $\displaystyle{\tenmsb
H}\to\text{Im}\,{\tenmsb H}$ $\displaystyle\eta$ $\displaystyle=$
$\displaystyle((|z_{1}|^{2}-|z_{2}|^{2},\;2z_{1}\bar{z}_{2}):\mathbb{R}^{4}\to\mathbb{R}^{3}$
with $\eta(S^{3})\in S^{2}$. Let
$\zeta(x)=s(r)\eta\left(\frac{x}{r}\right)=s(r)r^{-2}\eta(x):=\tilde{s}(r)\eta(x)$
with $r=|x|$. It was shown in ([13] Theorem 3.2, p. 135) that
$x\to(x,\zeta(x))$
define a coassociative $4-$ submanifold in $\mathbb{R}^{7}$ invariant under
$S^{3}$, provided
$s(4s^{2}-5r^{2})^{2}=C,\qquad C\in\mathbb{R}.$
Those are area-minimazing smooth minimal submanifolds except in the case
$C=0.$ When $C=0$ then $s(r)=\frac{\sqrt{5}}{2}r$ and then
$\tilde{s}(r)=\frac{\sqrt{5}}{2r}$, the function
$\zeta(x):\mathbb{R}^{4}\to\mathbb{R}^{3}$ given by
$\zeta=\frac{\sqrt{5}}{2r}\eta(x)$
defines a cone over the entire $\mathbb{R}^{4}$. This was discovered by Lawson
and Osserman [18]. This LO-cone shows that Moser’s theorem that entire minimal
graphs of bounded slope are affine linear cannot be extended to the case of
dimension $4$ and codimension $3$. In this appendix, we compute some important
geometric quantities of this remarkable example.
Put
$x=(z_{1},z_{2})\in{\tenmsb C}^{2}=\mathbb{R}^{4},\quad
z_{1}=r_{1}e^{i\theta_{1}},\quad
z_{2}=r_{2}e^{i\theta_{2}},\quad|x|^{2}=r^{2}=r_{1}^{2}+r_{2}^{2}.$
The flat metric on $\mathbb{R}^{4}$ reads
$ds^{2}=dr_{1}^{2}+r_{1}^{2}d\theta_{1}^{2}+dr_{2}^{2}+r_{2}^{2}d\theta_{2}^{2}.$
The orthonormal basis on $T_{x}\mathbb{R}^{4}$ is given by
$\\{e_{0},e_{1},e_{2},e_{3}\\},$ where
$e_{0}=\frac{\partial}{\partial r}{}=\frac{r_{1}}{r}\frac{\partial}{\partial
r_{1}}+\frac{r_{2}}{r}\frac{\partial}{\partial r_{2}},\quad
e_{1}=\frac{r_{2}}{r}\frac{\partial}{\partial
r_{1}}-\frac{r_{1}}{r}\frac{\partial}{\partial r_{2}}$
$e_{2}=\frac{r_{2}}{r_{1}r}\frac{\partial}{\partial\theta_{1}}-\frac{r_{1}}{r_{2}r}\frac{\partial}{\partial\theta_{2}},\quad
e_{3}=\frac{1}{r}\left(\frac{\partial}{\partial\theta_{1}}+\frac{\partial}{\partial\theta_{2}}\right).$
Now, we have
$\eta(x)=\left(r_{1}^{2}-r_{2}^{2},\;2r_{1}r_{2}e^{i(\theta_{1}-\theta_{2})}\right):\mathbb{R}^{4}\to\mathbb{R}^{3}.$
Since
$\frac{\partial}{\partial
r_{1}}\tilde{s}(r)=\tilde{s}^{\prime}\frac{r_{1}}{r},\quad\frac{\partial}{\partial
r_{2}}\tilde{s}(r)=\tilde{s}^{\prime}\frac{r_{2}}{r},$
then
$\displaystyle\zeta_{*}e_{0}$
$\displaystyle=\left(\frac{r_{1}}{r}\frac{\partial}{\partial
r_{1}}+\frac{r_{2}}{r}\frac{\partial}{\partial
r_{2}}\right)(\tilde{s}\eta)=\left(\tilde{s}^{\prime}+\frac{2\tilde{s}}{r}\right)\eta,$
$\displaystyle\zeta_{*}e_{1}$
$\displaystyle=\frac{r_{2}}{r}\frac{\partial}{\partial
r_{1}}(\tilde{s}\eta)-\frac{r_{1}}{r}\frac{\partial}{\partial
r_{2}}(\tilde{s}\eta)=\frac{\tilde{s}}{r}\left(4r_{1}r_{2},\;2(r_{2}^{2}-r_{1}^{2})e^{i(\theta_{1}-\theta_{2})}\right).$
Since
$\displaystyle\zeta_{*}\frac{\partial}{\partial\theta_{1}}$
$\displaystyle=\frac{\partial}{\partial\theta_{1}}(\tilde{s}\eta)=\tilde{s}\left(0,\;2ir_{1}r_{2}e^{i(\theta_{1}-\theta_{2})}\right),$
$\displaystyle\zeta_{*}\frac{\partial}{\partial\theta_{2}}$
$\displaystyle=\frac{\partial}{\partial\theta_{2}}(\tilde{s}\eta)=\tilde{s}\left(0,\;-2ir_{1}r_{2}e^{i(\theta_{1}-\theta_{2})}\right),$
then
$\displaystyle\zeta_{*}e_{2}$
$\displaystyle=\frac{r_{2}}{r_{1}r}\zeta_{*}\frac{\partial}{\partial\theta_{1}}-\frac{r_{1}}{r_{2}r}\zeta_{*}\frac{\partial}{\partial\theta_{2}}=\tilde{s}r\left(0,\;2ie^{i(\theta_{1}-\theta_{2})}\right)$
$\displaystyle\zeta_{*}e_{3}$ $\displaystyle=0.$
Put
$\displaystyle\rho_{0}^{2}$
$\displaystyle=\frac{1}{1+|\zeta_{*}e_{0}|^{2}}=\frac{1}{1+(r^{2}\tilde{s}^{\prime}+2r\tilde{s})^{2}},\quad\rho_{1}^{2}=\frac{1}{1+|\zeta_{*}e_{1}|^{2}}=\frac{1}{1+4r^{2}\tilde{s}^{2}}$
$\displaystyle\rho_{2}^{2}$
$\displaystyle=\frac{1}{1+|\zeta_{*}e_{2}|^{2}}=\frac{1}{1+4r^{2}\tilde{s}^{2}},\quad\rho_{3}^{2}=\frac{1}{1+|\zeta_{*}e_{3}|^{2})}=1$
Then the Gauss map $\gamma$ for the coassociate $4-$submanifold is expressed
by
$(e_{0},\zeta_{*}e_{0})\wedge(e_{1},\zeta_{*}e_{1})\wedge(e_{2},\zeta_{*}e_{2})\wedge(e_{3},\zeta_{*}e_{3})$
and the correspoinding $W-$matrix relative to $e_{0}\wedge e_{1}\wedge
e_{2}\wedge e_{3}$ is
$W=\left(\begin{array}[]{cccc}\rho_{0}&0&0&0\\\ 0&\rho_{1}&0&0\\\
0&0&\rho_{2}&0\\\ 0&0&0&1\end{array}\right).$
The nonzero Jordan angles are
$\theta_{0}=\arccos\rho_{0},\;\theta_{1}=\arccos\rho_{1},\;\theta_{2}=\arccos\rho_{2}.$
In particular, the Jordan angles of the image under the Gauss map for the LO-
cone can be obtained by substituting $\tilde{s}=\frac{\sqrt{5}}{2}r^{-1}$ in
the above expressions. Those are the following constants
$\theta_{0}=\arccos\frac{2}{3},\quad\theta_{1}=\theta_{2}=\arccos\frac{\sqrt{6}}{6}.$
The $w-$function is identically $\frac{1}{9}$ and hence $v-$function, denoted
also by $\Delta_{f}$ for the graphic case, equals $9$. This fact was
originally verified by a computer program by Lawson-Osserman in [18]. The LO-
cone defined by $(f^{1},f^{2},f^{3})$ on $\mathbb{R}^{4}$, where
$\displaystyle f^{1}$
$\displaystyle=\frac{\sqrt{5}}{2}\frac{(x^{1})^{2}+(x^{2})^{2}-(x^{3})^{2}-(x^{4})^{2}}{\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2}}},$
$\displaystyle f^{2}$
$\displaystyle=\frac{\sqrt{5}}{2}\frac{2(x^{1}x^{3}+x^{2}x^{4})}{\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2}}},$
$\displaystyle f^{3}$
$\displaystyle=\frac{\sqrt{5}}{2}\frac{2(x^{1}x^{3}-x^{2}x^{4})}{\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2}}}.$
At $x^{3}\neq 0$ and $x^{1}=x^{2}=x^{3}=0,$ we have $\frac{\partial
f^{2}}{\partial x^{1}}=\sqrt{5}$ and $(1+(\frac{\partial f^{2}}{\partial
x^{1}})^{2})^{\frac{1}{2}}=\sqrt{6}.$
## References
* [1] W.Allard: On the first variation of a varifold. Ann.Math. 95 (1972), 417-491
* [2] J.L.M.Barbosa: An extrinsic rigidity theorem for minimal immersion from $S^{2}$ into $S^{n}$. J. Diff. Geom. 14(3) (1980), 355-368.
* [3] E. Bombieri, E. Giusti: Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math. 15(1972), 24-46.
* [4] P. Buser: A note on the isoperimetric constant. Ann. Scient. Ec. Norm. Sup. 15(1982), 213-230.
* [5] S. S. Chern and R. Osserman: Complete minimal surfaces in Euclidean $n-$space. J. d’Anal. Math. 19(1967), 15-34.
* [6] D. Fischer-Colbrie: Some rigidity theorems for minimal submanifolds of the sphere, Acta math. 145(1980), 29-46.
* [7] M. Giaquinta, E. Giusti: On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46.
* [8] M. Giaquinta, S. Hildebrandt: A priori estimates for harmonic mappings, J. reine angew. Math. 336 (1982), 124-164.
* [9] M. Grüter, K. Widman: The Green function for uniformly elliptic equations, Manuscripta Math. 37(1982), 303-342.
* [10] R. Gulliver, J. Jost: Harmonic maps which solve a free-boundary problem, J. reine angew. Math. 381 (1987), 61-89.
* [11] S. Hildebrandt, H. Kaul, K.O. Widman: An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), 1-16.
* [12] S. Hildebrandt, J. Jost, K. Widman: Harmonic mappings and minimal submanifolds, Invent. math. 62(1980), 269-298.
* [13] R. H. Harvey and B. Lawson: Calibrated geometry, Acta Math. 148 (1982), 47-157.
* [14] J. Jost: Generalized Dirichlet forms and harmonic maps, Calc. Var. PDE 5(1997), 1-19.
* [15] J. Jost, Y. L. Xin: Bernstein type theorems for higher codimension, Calc.Var. and Partial Differential Equations 9 (1999), 277-296.
* [16] J. Jost, Y. L. Xin and Ling Yang: The regularity of harmonic maps into spheres and applications to Bernstein problems. arXiv:0912.0447v1. to appear in JDG.
* [17] J. Jost, Y. L. Xin and Ling Yang: The Gauss image of entire graphs of higher codimension and Bernstein type theorems. arXiv: 1009.3901v1. to appear in Calc. Var. and and Partial Differential Equations.
* [18] H. B. Lawson and R. Osserman: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta math. 139(1977), 1-17.
* [19] J. Moser: On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14(1961), 577-591.
* [20] E. A. Ruh and J. Vilms: The tension field of Gauss maps, Trans. AMS. 149(1970), 569-573.
* [21] J. Simons: Minimal varieties in Riemannian manifolds. Ann. Math. 88 (1968), 62-105.
* [22] Yung-Chow Wong: Differential geometry of Grassmann manifolds, Proc. N.A.S. 57 (1967), 589-594.
* [23] Yuanlong Xin: Minimal submanifolds and related topics, World Scientic Publ. 2003.
* [24] Y. L. Xin: Bernstein type theorems without graphic condition, Asian J. Math. 9(1), (2005), 31-44.
* [25] Y. L. Xin: Curvature estimates for submanifolds with prescribed Gauss image and mean curvature, Calc. Var. and Partial Differential Equations 37(3-4) (2010), 385-405.
* [26] Y. L. Xin and Ling Yang: Convex functions on Grassmannian manifolds and Lawson-Osserman problem, Adv. Math. 219(4), (2008), 1298-1326.
|
arxiv-papers
| 2011-09-29T04:25:06 |
2024-09-04T02:49:22.577972
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Jost, Y. L. Xin and Ling Yang",
"submitter": "Ling Yang",
"url": "https://arxiv.org/abs/1109.6394"
}
|
1109.6525
|
# Symmetry breaking and manipulation of nonlinear optical modes in an
asymmetric double-channel waveguide
Rujiang Li1, Fei Lv1, Lu Li1 llz@sxu.edu.cn Zhiyong Xu2 1Institute of
Theoretical Physics, Shanxi University, Taiyuan 030006, China 2Nonlinear
Physics Centre, Research School of Physics and Engineering, The Australian
National University, Canberra ACT, 0200, Australia
###### Abstract
We study light beam propagation in a nonlinear coupler with an asymmetric
double-channel waveguide, and derive various analytical forms of optical
modes. The results show that the symmetry-preserving modes in a symmetric
double-channel waveguide is deformed due to the asymmetry of the two-channel
waveguide, and such a coupler supports yet the symmetry-breaking modes. The
dispersion relations reveal that the system with self-focusing nonlinear
response supports the degenerate modes, while for self-defocusing medium the
degenerate modes do not exist. Furthermore, nonlinear manipulation is
investigated by launching optical modes supported in double-channel waveguide
into a nonlinear uniform medium.
###### pacs:
42.65.Tg, 42.65.Jx, 42.65.Wi
## I introduction
Propagation of optical waves in waveguide arrays has become an important and
effective means to investigate various optical phenomena which have analogues
in many fields of physics Longhi . Special attention has been paid to
nonlinear surface waves and nonlinear guided waves in planar layered
structures M1 ; M2 ; M3 ; M4 ; M5 ; M6 ; M7 ; M8 ; M9 ; M10 ; M11 ; M12 and
nonlinear couplers Marcuse ; Yangcc ; Yangcc1 ; Chen ; Yasumoto , generation
and properties of solitons in nonlinear waveguide arrays Nat ; Lederer ; PO ;
YS ; Ablowitz ; YS4 and other nonlinear periodic systems, such as optically
induced lattices MS ; Neshev ; Yaroslov ; Zhangyp . The investigation of light
beam propagation in waveguide arrays attracts increasing attention due to
their potential applications in all-optical signal processing in fiber optic
networks and devices, the passive mode-locking by using waveguide array R2 ,
and the beam steering PRE53 ; ZhangOC ; Xu .
The behavior of light beam propagation in a coupler composed by two-channel
nonlinear waveguide gained particular attention because it can exhibit some of
universal properties in nonlinear periodic systems and nonlinear waveguide
arrays Yaroslov1 . It is shown that the coupler can support the symmetry-
preserving solutions which have the linear counterparts and the symmetry-
breaking solutions without any linear counterparts Kevrekidis ; Birnbaum ; Jia
, in which the spontaneous symmetry-breaking has been experimentally
demonstrated in optically induced lattices with a local double-well potential
Kevrekidis .
In this paper, we consider light beam propagation in an asymmetric double-
channel waveguide with Kerr-type nonlinear response, and derive various
analytical stationary solutions in detail. It is found that the asymmetric
double-channel waveguide can break the symmetric form of the symmetry-
preserving modes otherwise in the symmetric double-channel waveguide, and such
a coupler supports the symmetry-breaking modes. We also investigate how the
type of nonlinear response affects the existence and properties of nonlinear
optical modes in the asymmetric double-channel waveguide. The dispersion
relation shows that the degenerate modes exist in the system with self-
focusing nonlinear response, while for the coupler with self-defocusing
response the degenerate modes do not exist. In addition, based on these
optical modes supported in asymmetric double-channel waveguide we demonstrate
the control and manipulation of optical modes in different nonlinear media by
tuning the waveguide parameters.
The paper is organized as follows. In Section II, the model equation
describing beam propagation in a double-channel waveguide is derived. In
Section III, various analytical forms of optical modes are presented both in
self-focusing and self-defocusing media. Meanwhile, the dispersion relations
between the total energy and the propagation constant are discussed. In
Section IV, we study the nonlinear manipulation of optical modes in double-
channel waveguide. The conclusion is summarized in Section V.
## II Model equation and reductions
We consider a planar graded-index waveguide with refractive index
$n(z,x)=F(x)+n_{2}I(z,x).$ (1)
Here the first term on the right hand side presents a two-channel waveguide
with the different refractive index, namely, $F(x)=$ $n_{11}$ as
$-L_{0}/2-D_{0}<x<-L_{0}/2$, and $F(x)=n_{12}$ as $L_{0}/2<x<L_{0}/2+D_{0}$,
otherwise, $F(x)=$ $n_{0}$ ($<n_{11},n_{12}$), where $D_{0}$ and $L_{0}$
represent the width of waveguide and the separation between waveguides,
respectively; while $n_{0}$, $n_{11}$ and $n_{12}$ denote the refractive index
of cladding and waveguide, respectively. The second term denotes Kerr-type
nonlinearity, $I(z,x)$ is the optical intensity, and positive (negative) value
of nonlinear coefficient $n_{2}$ indicates self-focusing (self-defocusing)
medium. Under slowly varying envelope approximation, the nonlinear wave
equation governing beam propagation in such a waveguide with the refractive
index given by Eq. (1) can be written as
$i\frac{\partial\psi}{\partial
z}+\frac{1}{2k_{0}}\frac{\partial^{2}\psi}{\partial
x^{2}}+\frac{k_{0}\left[F(x)-n_{0}\right]}{n_{0}}\psi+\frac{k_{0}n_{2}}{n_{0}}\left|\psi\right|^{2}\psi=0,$
(2)
where $\psi(z,x)$ is the envelope function, and $k_{0}=2\pi n_{0}/\lambda$ is
wave number with $\lambda$ being wavelength of the optical source generating
the beam. By introducing the normalized transformations
$\psi(z,x)=(k_{0}\left|n_{2}\right|L_{D}/n_{0})^{-1/2}\varphi(\zeta,\xi)$,
$\xi=x/w_{0}$ and $\zeta=z/L_{D}$ with $L_{D}=2k_{0}w_{0}^{2}$, which
represents the diffraction length, we get the dimensionless form of Eq. (2) as
follows
$i\frac{\partial\varphi}{\partial\zeta}+\frac{\partial^{2}\varphi}{\partial\xi^{2}}+V(\xi)\varphi+\eta\left|\varphi\right|^{2}\varphi=0,$
(3)
where $\eta=n_{2}/\left|n_{2}\right|=\pm 1$ corresponds to self-focusing ($+$)
and self-defocusing ($-$) nonlinearity of the waveguides, respectively, and
$V(\xi)=2k_{0}^{2}w_{0}^{2}\left[F(w_{0}\xi)-n_{0}\right]/n_{0}$ is of the
form
$V(\xi)=\left\\{\begin{array}[c]{cc}V_{1},&-L/2-D<\xi<-L/2,\\\
V_{2},&L/2<\xi<L/2+D,\\\ 0,&\text{otherwise,}\end{array}\right.$ (4)
which describes the dimensionless two-channel waveguide structure with
different refractive index, where
$V_{1}=2k_{0}^{2}w_{0}^{2}(n_{11}-n_{0})/n_{0}$ and $V_{2}=$
$2k_{0}^{2}w_{0}^{2}(n_{12}-n_{0})/n_{0}$ being the modulation depth of the
refractive index of the left and right waveguide, and $L=L_{0}/w_{0}$ and
$D=D_{0}/w_{0}$ corresponding to scaled separation and width of waveguide,
respectively. Here, we use the typical waveguide parameters $D=3.5$, $L=5$,
$V_{2}=2.525$, and vary $V_{1}$. Figure 1 shows the profile of the two-channel
waveguide structure given by Eq. (4). It should be pointed out that such
structure can be realized experimentally FL . It is shown that the Eq. (3)
conserves the total energy flow
$P(\zeta)=\int_{-\infty}^{+\infty}\left|\varphi(\zeta,\xi)\right|^{2}d\xi=P_{0}$,
where $P_{0}$ is the dimensionless initial total energy.
Figure 1: The profile of a two-channel waveguide with different refractive
index.
Assuming that the stationary solution of Eq. (3) is of the form
$\varphi(\zeta,\xi)=u(\xi)\exp(i\beta\zeta)$, where $u(\xi)$ is a real
function, and $\beta$ is the propagation constant, and substituting it into
Eq. (3), we find that the function $u(\xi)$ obeys the following nonlinear
equation
$\frac{d^{2}u}{d\xi^{2}}+\left[V(\xi)-\beta\right]u+\eta u^{3}=0,$ (5)
where $\eta=\pm 1$ corresponds to self-focusing ($+$) and self-defocusing
($-$) nonlinearity of the waveguides, respectively. It should be pointed out
that Eq. (5) not only can describe the optical modes in the double-channel
waveguide structure with the different refractive index, but also can describe
one-dimensional Bose-Einstein condensate trapped in a finite asymmetry double
square well potential $-V(\xi)$. In particular, when $V_{1}=V_{2}$, the
corresponding optical modes in the symmetric double-channel waveguide
structure have been studied, and the results have shown that the coupler not
only supports symmetry-preserving modes but also symmetry-breaking modes Jia .
## III Optical modes
In this section, we will present the analytical solutions of Eq. (5) with the
potential (4) for $\eta=\pm 1$. Generally, the solutions of Eq. (5) can be
constructed in terms of the Jacobi elliptic functions depending on the values
of the variable $\xi$, and have the same propagation constants in different
regions. Within the double-channel waveguides of $-L/2-D<\xi<-L/2$ and
$L/2<\xi<L/2+D$, the solution of Eq. (5) is the oscillatory function, so it
can be selected in the form WDLi
$u_{1}(\xi;A,K,\delta)=A\operatorname*{sn}\left(K\xi+\delta,-\frac{\eta
A^{2}}{2K^{2}}\right),$ (6)
with $\beta=V_{1}-K^{2}+\eta A^{2}/2$ in the region of $-L/2-D<\xi<-L/2$ and
$\beta=V_{2}-K^{2}+\eta A^{2}/2$ in the region of $L/2<\xi<L/2+D$. In the
region of $\left|\xi\right|<L/2$, the solution of Eq. (5) has two different
Jacobi elliptic functions for the symmetric and the antisymmetric case,
respectively. For the symmetric case, the solution is WDLi
$u_{2}(\xi;B,Q,\sigma)=B\operatorname*{nc}\left(Q\xi+\sigma,1+\frac{\eta
B^{2}}{2Q^{2}}\right),$ (7)
with $\beta=Q^{2}+\eta B^{2}$; and for the antisymmetric case, the solution is
WDLi
$u_{2}(\xi;B,Q,\sigma)=B\operatorname*{sc}\left(Q\xi+\sigma,1+\frac{\eta
B^{2}}{2Q^{2}}\right),$ (8)
with $\beta=Q^{2}-\eta B^{2}/2$. It should be noted that those two solutions
are precise solutions of Eq. (5) for one node and no node within the region of
$\left|\xi\right|<L/2$. In other regions, the solution of Eq. (5) is required
to tend to zero as $\xi\rightarrow\pm\infty$, so it is taken as Jia
$u_{3}(\xi;b)=\frac{1}{be^{-\sqrt{\beta}\xi}+ce^{\sqrt{\beta}\xi}},$ (9)
with $\beta>0$. Substituting (9) into Eq. (5), one obtains $c=\eta/(8\beta
b)$.
Note here that although the modulus in the usual Jacobi elliptic function is
restricted from 0 to 1, this problem can be solved by the modular
transformation such that the modulus in the Jacobi elliptic functions given by
Eqs. (6-8) can take any positive or negative values in our investigations, as
shown in Refs. Byrd ; WDLi , so those solutions do not depend on nonlinearity
sign.
In the following, we show the analytical global solutions of Eq. (5). With the
help of Eqs. (6), (7) [or (8)], and (9), the solutions of Eq. (5) can be
written as
$u(\xi)=\left\\{\begin{array}[c]{ll}u_{3}(\xi;b_{1}),&\text{ \ }\xi<-L/2-D,\\\
u_{1}(\xi;A_{1},K_{1},\delta_{1}),&\text{ \ }-L/2-D<\xi<-L/2,\\\
u_{2}(\xi;B,Q,\sigma),&\text{ \ }\left|\xi\right|<L/2,\\\
u_{1}(\xi;A_{2},K_{2},\delta_{2}),&\text{ \ }L/2<\xi<L/2+D,\\\
u_{3}(\xi;b_{2}),&\text{ \ }\xi>L/2+D.\end{array}\right.$ (10)
The continuity conditions for $u$ and $\partial u/\partial\xi$ at the
boundaries of $\xi=\pm L/2$ and $\xi=\pm(L/2+D)$ require
$\displaystyle u_{3}\left(-L/2-D;b_{1}\right)$
$\displaystyle=u_{1}\left(-L/2-D;A_{1},K_{1},\delta_{1}\right),$
$\displaystyle\frac{du_{3}}{d\xi}\left(-L/2-D;b_{1}\right)$
$\displaystyle=\frac{du_{1}}{d\xi}\left(-L/2-D;A_{1},K_{1},\delta_{1}\right),$
$\displaystyle u_{1}\left(-L/2;A_{1},K_{1},\delta_{1}\right)$
$\displaystyle=u_{2}(-L/2;B,Q,\sigma),$
$\displaystyle\frac{du_{1}}{d\xi}\left(-L/2;A_{1},K_{1},\delta_{1}\right)$
$\displaystyle=\frac{du_{2}}{d\xi}(-L/2;B,Q,\sigma),$ $\displaystyle
u_{2}(L/2;B,Q,\sigma)$ $\displaystyle=u_{1}(L/2;A_{2},K_{2},\delta_{2}),$
$\displaystyle\frac{du_{2}}{d\xi}(L/2;B,Q,\sigma)$
$\displaystyle=\frac{du_{1}}{d\xi}(L/2;A_{2},K_{2},\delta_{2}),$
$\displaystyle u_{1}(L/2+D;A_{2},K_{2},\delta_{2})$
$\displaystyle=u_{3}(L/2+D;b_{2}),$
$\displaystyle\frac{du_{1}}{d\xi}(L/2+D;A_{2},K_{2},\delta_{2})$
$\displaystyle=\frac{du_{3}}{d\xi}(L/2+D;b_{2}),$ (11)
with $\beta=V_{1}-K_{1}^{2}+\eta A_{1}^{2}/2=V_{2}-K_{2}^{2}+\eta
A_{2}^{2}/2$, and $\beta=Q^{2}+\eta B^{2}$ for the symmetric case given by Eq.
(7) or $\beta=Q^{2}-\eta B^{2}/2$ for the antisymmetric case given by Eq. (8).
In Eq. (10), there are eleven parameters $A_{1}$, $K_{1}$, $\delta_{1}$,
$A_{2}$, $K_{2}$, $\delta_{2}$, $B$, $Q$, $\sigma$, $b_{1}$, and $b_{2}$,
which can be calculated by solving numerically Eqs. (11) with the conditions
that the propagation constants in different regions should be same. Once those
parameters are determined numerically, we can obtain the exact optical modes
for asymmetric double-channel nonlinear waveguides.
Figure 2: (Color Online) Various different optical modes existing in self-
defocusing medium ($\eta=-1$), where the dash-dotted red line is the optical
mode for $V_{1}=2.500$, the solid green line is the optical mode for
$V_{1}=2.525$, and the dashed blue line is the optical mode for $V_{1}=2.550$.
Here $\beta=2.00$ in (a) and (b), $\beta=0.85$ in (c) and (d). Figure 3:
(Color Online) Various different optical modes existing in self-focusing
medium ($\eta=1$), where the dash-dotted red line is the optical mode for
$V_{1}=2.500$, the solid green line is the optical mode for $V_{1}=2.525$, and
the dashed blue line is the optical mode for $V_{1}=2.550$. Here $\beta=2.30$
in (a) and (b), $\beta=1.10$ in (c) and (d).
Fig. 2 and Fig. 3 show several different optical modes in a nonlinear
asymmetric double-channel waveguide in the self-defocusing medium and the
self-focusing medium, respectively. These optical modes are induced from the
symmetry-preserving optical modes in the symmetric double-channel waveguide,
where for comparison, we also plotted the corresponding symmetry-preserving
optical modes in the symmetric double-channel waveguides in the same figure.
From Figs. 2 and 3, we found that the symmetry of the modes is broken due to
asymmetry of the two-channel waveguide, and the amplitude of the modes in the
lower refractive index waveguide is smaller than that in the higher refractive
index waveguide for the self-defocusing medium, while for the self-focusing
medium, the amplitude of the modes in the lower refractive index waveguide is
larger than that in the higher refractive index waveguide.
Figure 4: (Color Online) Several optical modes with different $\beta$ in a
nonlinear asymmetrical double-channel waveguide for the self-defocusing medium
($\eta=-1$). Here the parameters are $V_{1}=2.500$ and $V_{2}=2.525$. Figure
5: (Color Online) Several optical modes with different $\beta$ in a nonlinear
asymmetrical double-channel waveguide for the self-focusing medium ($\eta=1$).
Here the parameters are the same as in Fig. 4.
We also demonstrate the profiles of optical modes with dependence on the
propagation constant $\beta$. Fig. 4 and Fig. 5 present several corresponding
modes shown in Figs. 2 and 3 for different propagation constant $\beta$. From
them, one can see that, for self-defocusing media the profile of nonlinear
modes is shrunk, and the corresponding amplitude becomes smaller with an
increase of the propagation constant $\beta$, while for self-focusing case it
is opposite, namely, the profile of nonlinear modes becomes more prominent and
the corresponding amplitude becomes larger. This feature can be depicted by
the dispersion relations between the total energy $P_{0}$ and the propagation
constant $\beta$. As shown in Fig. 8 and Fig. 9, one can see that the total
energy decreases with the increase of the propagation constant for self-
defocusing media (see Fig. 8), whereas it is an increasing function of the
propagation constant for self-focusing case (see Fig. 9).
Figure 6: (Color Online) Several symmetry-breaking optical modes, where the
dash-dotted red lines are optical modes for $V_{1}=2.500$, the solid green
lines are optical modes for $V_{1}=2.525$, and the dashed blue lines are
optical modes for $V_{1}=2.550$. Here $\eta=-1$ in (a) and (c), $\eta=1$ in
(b) and (d) and $\beta=1.78$ in (a), $\beta=2.39$ in (b), $\beta=0.66$ in (c)
and $\beta=1.15$ in (d). Figure 7: (Color Online) Several symmetry-breaking
optical modes with different $\beta$ in a nonlinear asymmetrical double-
channel waveguide. Here the parameters are the same as in Fig. 6.
As discussed in Ref. Jia , besides the symmetry-preserving optical modes, a
double-channel waveguide also supports the symmetry-breaking optical modes,
and the corresponding optical modes in a nonlinear asymmetric double-channel
waveguide are presented in Fig. 6, in which we also plotted the corresponding
symmetry-breaking optical modes in a symmetric double-channel waveguide in the
same figure for comparison. One can find that the optical modes in a nonlinear
asymmetric double-channel waveguide are almost the same as the modes in a
symmetric one for the given $\beta$.
Similarly, the corresponding modes shown in Fig. 6 for different propagation
constant $\beta$ are presented in Fig. 7. It is shown that the profile of
nonlinear modes becomes shrunk with the increase of the propagation constant
$\beta$ for the self-defocusing case, while for the self-focusing case the
profile of nonlinear modes becomes more pronounced. This feature is depicted
by the dispersion relations between the total energy $P_{0}$ and the
propagation constant $\beta$. It should be pointed out that the modes shown in
Fig. 6 only exist in a small region, as shown in Fig. 8 and Fig. 9.
Figure 8: (Color Online) The dependence of the total energy $P_{0}$ on the
propagation constant $\beta$ for the modes existing in self-defocusing medium
($\eta=-1$). Here the parameters are $V_{1}=2.500$ and $V_{2}=2.525$. The
labeled shadow areas are enlarged in corresponding (a)-(b). And the labels 2a,
2b, $\cdots$, mean the corresponding modes shown in Fig. 2a, 2b, $\cdots$,
respectively. Figure 9: (Color Online) The dependence of the total energy
$P_{0}$ on the propagation constant $\beta$ for the modes existing in self-
focusing medium ($\eta=1$). Here the parameters are $V_{1}=2.500$ and
$V_{2}=2.525$. The labeled shadow areas are enlarged in corresponding (a)-(d).
And the labels 3a, 3b, $\cdots$, mean the corresponding modes shown in Fig.
3a, 3b, $\cdots$, respectively.
From the dispersion relations shown in Figs. 8 and 9, one can see that for the
self-defocusing medium there is no intersection between dispersion curves (see
Fig. 8 and the enlarged Figs. 8a and 8b), which indicates that there is no
degenerate modes existing, and the total energy of the mode shown in Fig. 2a
is the highest for a given propagation constant $\beta$. While for the self-
focusing medium the dispersion curves can intersect (see the enlarged Figs. 9c
and 9d), which implies that there exists two different modes with the same
total energy at the intersection point, namely, the degeneracy occurs at the
intersection point. Note that for our choices of the parameters, the
intersection points of the dispersion curves for the modes shown in Figs. 3a
and 3b, and Figs. 3c and 3d are about $2.5485$ and $1.3595$, respectively, and
the corresponding degenerate modes are shown by the green solid curves in Fig.
5\. Here, to distinguish the case of the intersection, we rotate the
dispersion curve for the modes shown in Fig. 3a (Fig. 3c) for an angle of
$\pi/6$ anticlockwise as the center of intersection point and the same angle
for dispersion curve of the mode shown in Fig. 3b (Fig. 3d) but rotate
clockwise, as shown in Figs. 9c and 9d.
Figure 10: (Color Online) The evolution of optical modes shown in Fig. 2 into
the self-focusing Kerr medium without channels, where $\eta^{\prime}=10$ in
(a) and (b), and $\eta^{\prime}=20$ in (c) and (d). Here the parameters are
$V_{1}=V_{2}=2.525$ and $\zeta_{0}=10$. The labels (a), (b), (c), and (d) mean
the corresponding modes shown in Fig. 2a, 2b, 2c, and 2d, respectively. Figure
11: (Color Online) The evolution of optical modes shown in Fig. 3 into the
self-focusing Kerr medium without channels, where $\eta^{\prime}=5$ in (a) and
(b), and $\eta^{\prime}=10$ in (c) and (d). Here the parameters are
$V_{1}=V_{2}=2.525$ and $\zeta_{0}=10$. The labels (a), (b), (c), and (d) mean
the corresponding modes shown in Fig. 3a, 3b, 3c, and 3d, respectively.
## IV The nonlinear manipulation of optical modes in double-channel waveguide
In this section, we will demonstrate the control and manipulation of optical
modes in reconfigurable nonlinear media. Thus our interest is to investigate
the evolution dynamics of optical beams existing in double-channel waveguide
propagating into a uniform nonlinear medium. In this case, the governing
equation can be generally written as
$i\frac{\partial\psi}{\partial
z}+\frac{1}{2k_{0}}\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{k_{0}\Delta
n(z,x)}{n_{0}}\psi=0,$ (12)
where the refractive index change $\Delta n(z,x)$ is a function of $z$ and
$x$, and $\Delta n(z,x)=n(z,x)-n_{0}$. Here we assume that when $0\leq z\leq
Z_{0}$, $n(z,x)$ is in the form of Eq. (1), $\Delta
n(z,x)=F(x)+n_{2}I(z,x)-n_{0}$; while for $z>Z_{0}$, $\Delta
n(z,x)=n_{2}^{\prime}I(z,x)-n_{0}$. Here, $n_{2}$ and $n_{2}^{\prime}$ are the
Kerr nonlinear coefficients of different media in the region of $0\leq z\leq
Z_{0}$ and $z>Z_{0}$, respectively. Thus, when $0\leq z\leq Z_{0}$, namely in
the region of $0\leq\zeta\leq\zeta_{0}$, Eq. (12) can be normalized to Eq.
(3), where $\zeta=z/L_{D}$ and $\zeta_{0}=Z_{0}/L_{D}$ with
$L_{D}=2k_{0}w_{0}^{2}$. At the same time, in the region of $z>Z_{0}$, namely
$\zeta>\zeta_{0}$, Eq. (12) is reduced to the dimensionless form as follows
$i\frac{\partial\varphi}{\partial\zeta}+\frac{\partial^{2}\varphi}{\partial\xi^{2}}+\eta^{\prime}\left|\varphi\right|^{2}\varphi=0,$
(13)
where $\eta^{\prime}=n_{2}^{\prime}/\left|n_{2}\right|$. Note that Eq. (13) is
different from Eq. (3), in which Eq. (3) includes a potential function
$V(\xi)$ given by Eq. (4), while Eq. (13) does not include and can describe
the dynamics of beams in Kerr media without any refractive index modulation.
In the following analysis, optical beams of different modes existing in
double-channel waveguide are injected into the uniform nonlinear medium after
propagating diffraction length of $\zeta_{0}$ in double-channel waveguide.
First we consider the situation that optical beams are injected from
symmetrical double-channel waveguide. For the optical modes shown in Fig. 2,
which exist in self-defocusing medium ($\eta=-1$), the numerical simulations
show that when $\eta^{\prime}<0$, these optical modes are diffracted quickly
after entering into a uniform Kerr medium. However, when $\eta^{\prime}>0$ and
is large enough, the evolution of optical beams exhibits different scenarios
in the Kerr medium without any channels, as shown in Fig. 10. Similarly, for
the optical modes shown in Fig. 3, which exist in self-focusing media
($\eta=1$) with double-channel waveguide, as shown in Fig. 11, our numerical
simulations show that the evolution of optical modes exhibit almost similar
properties with that in Fig. 10.
From Figs. 10 and 11, one can see that when the optical modes existing both in
self-defocusing and self-focusing media with double-channel waveguide are
injected into the self-defocusing medium without any channels, the beams
should be diffracted quickly. However, when the optical modes are injected
into the self-focusing medium without any channels and the corresponding
nonlinear coefficient $\eta^{\prime}$ is enough larger, the beams could be
manipulated effectively. In this situation, when the optical modes shown in
Fig. 2a and Fig. 3a are injected into self-focusing medium without any
channels the beams appear to attract and repel each other periodically, as
shown in Figs. 10a and 11a. While when the modes shown in Figs. 2b-2d and
Figs. 3b-3d are injected into self-focusing medium without any channels, the
beams repel each other, as shown in Figs. 10b-10d and Figs. 11b-11d. Note that
the escape angle are the same for the beams in Fig. 2b (Fig. 3b) due to the
symmetry of the double-channel waveguide.
Figure 12: (Color Online) The evolution of optical modes shown in Fig. 6 into
the self-focusing Kerr medium without channels, where $\eta^{\prime}=10$. Here
the parameters are $V_{1}=V_{2}=2.525$ and $\zeta_{0}=10$. The labels (a),
(b), (c), and (d) mean the corresponding modes shown in Fig. 6a, 6b, 6c, and
6d, respectively. Figure 13: (Color Online) Energy sharing (row 1) and escape
angles (row 2) of optical beams as a function of parameter $V_{1}$. In all
cases, the solid-blue and the dashed-red curves correspond to the left and
right beams, respectively. Here the parameter are $V_{2}=2.525$,
$\eta^{\prime}=10$ in (a) and (c) corresponding to the modes shown in Fig. 2b
(namely Fig. 11b) and $\eta^{\prime}=5$ in (b) and (d) corresponding to the
modes shown in Fig. 3b (namely Fig. 12b).
The evolution of optical modes shown in Fig. 6 is presented in Fig. 12, in
which Figs. 12a and 12c (Figs. 12b and 12d) demonstrate the evolution of
optical modes in the self-focusing medium without any channels with initial
input beams injected from self-defocusing (self-focusing) medium with double-
channel waveguide. As shown in Fig. 12a and 12b, one can see that optical
beams with a single hump can be compressed effectively. and as shown in Figs.
12c and 12d the optical modes with two peaks are separated during the
evolution due to the repulsive interaction force resulted from the phase
difference between the two peaks.
It should be pointed out that these results only take into account the optical
modes existing in the symmetrical double-channel waveguide. Then, one will
naturally ask what is the influence of the asymmetrical double-channel on the
evolution of optical modes. In order to understand this question, we launch
optical beams from an asymmetrical double-channel waveguide into the self-
focusing medium to observe their evolution by tuning the depth of left channel
of the waveguide for a fixed depth of the right channel, namely, the value of
$V_{1}$ varies from $2.500$ to $2.550$ for $V_{2}=2.525$. As an example, we
demonstrated the evolution dynamics for the modes shown in Fig. 2b and Fig.
3b. In Fig. 13, we present the dependence of the energy sharing, the ratio of
the energy carried by each component in the mode over the total energy, and
the escape angle, the angle of the each peak in the mode and the propagation
direction $\zeta$, on the value of $V_{1}$. As shown in Fig. 13, one can see
that the energy carried by each beam is different due to the asymmetry of the
double-channel waveguide [shown in Figs. 13a and 13b]. Also, one can see
clearly that the escape angles of the two beams take different values with the
change of the value $V_{1}$, which means that the beams can be controlled by
tuning the depth of the left channel of the waveguide.
## V Conclusion
We have studied light beam propagation in an asymmetric double-channel
waveguide in the form of a nonlinear coupler. A family of analytical solutions
with symmetric and antisymmetric forms has been obtained for both self-
focusing and self-defocusing nonlinear media, and the dispersion relations
between the total energy and the propagation constant has been discussed in
detail. Our results reveal that the system with self-focusing nonlinear
response supports the degenerate modes, while for self-defocusing medium the
degenerate modes do not exist. In addition, we explored new ways to steer
optical modes propagating from double-channel waveguide into a uniform self-
focusing medium. The compression of beam with single hump and split of beams
with two humps have been demonstrated by tuning the depth of the channel of
the waveguide. These properties may be applied in practical optical devices,
and be useful for optical processing, optical switching or optical routing.
## VI ACKNOWLEDGEMENT
The authors acknowledge fruitful discussions with Professor Yuri Kivshar. This
research is supported by the National Natural Science Foundation of China
grant 61078079, the Shanxi Scholarship Council of China grant 2011-010, and
the Australian Research Council.
## References
* (1) S. Longhi, Laser Photon. Rev. 3, 243 (2009).
* (2) W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
* (3) V. K. Fedyanin and D. Mihalache, Z. Phys. B47, 167 (1982); D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, Physica Scripta 29, 269 (1984); D. Mihalache, D. Mazilu, and H. Totia, Physica Scripta 30, 335 (1984).
* (4) N. N. Akhmediev, Zh. Eksp. Teor. Fiz. 83, 545 (1982) [Sov. Phys. JETP 56, 299 (1982)].
* (5) G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
* (6) N. N. Akhmediev, V. I. Korneyev, and Y. V. Kuzmenko, Zh. Eksp. Teor. Fiz. 88, 107 (1985).
* (7) F. Lederer and D. Mihalache, Solid State Commun. 59, 151 (1986); D. Mihalache, D. Mazilu, and F. Lederer, Opt. Commun. 59, 391 (1986).
* (8) D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, Opt. Lett. 12, 187 (1987).
* (9) D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 229 (1989).
* (10) U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Reports 194, 325 (1990).
* (11) A. D. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, in “Nonlinear surface electromagnetic phenomena”, Edited by H. E. Ponath and G. I. Stegeman, North-Holland, Amsterdam, 1991, pp. 73-287.
* (12) F. Lederer, L. Leine, R. Muschall, T. Peschel, C. Schmidt-Hattenberger, T. Trutschel, A. D. Boardman, C. Wachter, Opt. Commun. 99, 95 (1993).
* (13) A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 16, 21209 (2008).
* (14) D. Marcuse, Journal of Lightwave Technology 5, 113 (1987).
* (15) C. C. Yang, Opt. Lett. 16, 1641 (1991).
* (16) C. C. Yang and A. J. S. Wang, IEEE Journal of Quantum Electronics 28, 479 (1992).
* (17) Y. Chen, Optical and Quantum Electronics 24, 539 (1992).
* (18) K. Yasumoto, H. Maeda, and N. Maekawa, Journal of Lightwave Technology 14, 628 (1996).
* (19) D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature (London)424, 817 (2003).
* (20) F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Phys. Rep. 463, 1 (2008).
* (21) Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
* (22) H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998); D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
* (23) M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
* (24) Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, Phys. Rev. Lett. 98, 023901 (2007).
* (25) J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London)422, 147 (2003); J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
* (26) D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
* (27) Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 153903 (2004).
* (28) Y. P. Zhang and B. Wu, Phys. Rev. Lett. 102, 093905 (2009).
* (29) J. Proctor and J. Kutz, Opt. Lett. 30, 2013 (2005).
* (30) A. B. Aceves, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E53, 1172 (1996).
* (31) H. F. Zhang, J. Jia, S. T. Jia, and L. Li, Opt. Commun. 281, 4130 (2008).
* (32) Z. Xu, Mario I. Molina, and Yuri S. Kivshar, Phys. Rev. A80, 013817 (2009).
* (33) Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
* (34) P. G. Kevrekidis, Zhigang Chen, B. A. Malomed, D. J. Frantzeskakis, and M. I. Weinstein, Phys. Lett. A 340, 275 (2005).
* (35) Ze’ev Birnbaum and B. A. Malomed, Physica D 237, 3252 (2008).
* (36) J. F. Jia, Y. P. Zhang, W. D. Li, and L. Li, Opt. Commun. 283, 132 (2010).
* (37) W. D. Li, Phys. Rev. A74, 063612 (2006).
* (38) T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
* (39) P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. Springer-Verlag, New York, 1971.
|
arxiv-papers
| 2011-09-29T13:46:18 |
2024-09-04T02:49:22.594556
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rujiang Li, Fei Lv, Lu Li, and Zhiyong Xu",
"submitter": "Rujiang Li",
"url": "https://arxiv.org/abs/1109.6525"
}
|
1109.6582
|
# Electroweak penguin decays at LHCb
T. Blake on behalf of the LHCb collaboration CERN, Switzerland
###### Abstract
Promising ways to search for New Physics effects in radiative penguin decays
are in the angular analysis of $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$, in the
measurement of direct CP violation in $B_{d}\rightarrow K^{*0}\gamma$ and a
time dependent analysis of $B_{s}\rightarrow\phi\gamma$. All of these studies
are being pursued at LHCb. First results will be shown from the 2010 and early
2011 data, with particular emphasis on $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$.
## I Introduction
The decay processes $b\rightarrow s\ell^{+}\ell^{-}$ and $b\rightarrow
s\gamma$ are flavour changing neutral currents that are forbidden at tree
level in the Standard Model (SM). These processes can proceed via higher order
electroweak $Z^{0}$/$\gamma$ penguin or box diagrams. In extensions to the SM,
new virtual particles can enter in competing (loop order) diagrams, leading,
in the absence of a dominant SM tree process, to comparably large deviations
from SM predictions. These deviations may be: in enhancements (or supprssion)
of branching fractions; in angular distributions (e.g. $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$); CP or Isospin aymmetries. Rare decay processes provide
a complementary approach to direct searches at the general purpose detectors
and can provide sensitivity to new particles with masses of up-to
$\mathcal{O}(10-100\mathrm{\,Te\kern-1.00006ptV})$.
The angular analysis of $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$, based on
309$\mbox{\,pb}^{-1}$ of integrated luminosity collected by the LHCb
experiment in 2011, is described in Sec. II. More details on the angular
analysis of $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ with this data set can be
found in Ref. bib:kstmm . Initial studies of $B_{d}\rightarrow K^{*0}\gamma$
and $B_{s}\rightarrow\phi\gamma$ based on 88$\mbox{\,pb}^{-1}$ of integrated
collected in 2010 and 2011 are described in Sec. III. The search for the rare
decay $B_{s}\rightarrow\mu^{+}\mu^{-}$ is described in detail elsewhere in
these proceedings bib:bstomumu .
### I.1 The LHCb detector
The LHCb detector is a single-arm spectrometer designed to study $b$-hadron
decays with an acceptance for charged tracks with pseudorapidity, $\eta$, of
$2<\eta<5$. Primary proton-proton vertices (PVs), and secondary $B$-vertices
are identified in a silicon strip vertex detector (the VELO) that approaches
within 8$\rm\,mm$ of the LHC beam. Tracks from charged particles are
reconstructed in the vertex detector and a set of tracking stations and their
curvature in the dipole magnet allows momenta to be determined with a
precision of $\delta p/p=0.35$–$0.5\%$. Two Ring Imaging CHerenkov (RICH)
detectors allow for charged hadrons to be separated over a momentum range
$2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}<p<100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Muons wth momentum above 3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are
identified on the basis of the number of hits left in detectors interleaved
with an iron muon filter. Further details on the LHCb detector can be found in
Ref. DetectorPaper . Key for the rare electroweak penguin processes are the
excellent momentum and mass resolution provided by the long lever arm for
tracking, the primary-secondary vertex separation provided by the VELO and the
ability to reject a range of exclusive backgrounds provided by LHCb’s RICH
detectors.
### I.2 LHCb data taking performance in 2011
In the first three months of data taking in 2011, LHCb accumulated $\sim
300\mbox{\,pb}^{-1}$ of integrated luminosity at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. This luminosity was delivered by
the LHC at instantaneous luminosities of $3\times 10^{32}{\rm\,cm}^{-2}{\rm
s}^{-1}$, 50% above the original design luminosity of LHCb. At this
instantaneous luminosity, it is expected that LHCb will collect
$\mathcal{O}(1\mbox{\,fb}^{-1})$ of integrated luminosity in 2011.
## II The decay $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$
The decay $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ is an example of a
$b\rightarrow s\ell^{+}\ell^{-}$ process that can be a highly sensitive probe
of new right-handed currents and large contributions from new scalar or
pseudo-scalar couplings. In many models these new virtual particles give rise
to modifications in the distribution of the daughters of the $B_{d}$ that can
be probed through an angular analysis. To achieve ultimate sensitivity, the
long term goal of LHCb is to perform a full angular analysis of the decay
bib:kstarmumu:full . However, with more modest data sets, the focus is instead
on obseravables that are theoretically clean and can be extracted from simple
counting experiments or from simple fits to angular distributions. One such
observable, that is widely discussed in the literature, is the forward-
backward asymmetry of the muon system ($A_{FB}$). $A_{FB}$ varies with the
invariant mass-squared of the dimuon pair ($q^{2}$) and in the SM changes sign
at a well defined point, where the leading hadronic uncertainties cancel. In
many NP models the shape of $A_{FB}$ as a function of $q^{2}$ can be
dramatically altered. The variation of $A_{FB}(q^{2})$ in the SM and several
NP models in the low-$q^{2}$ region, from Ref. bib:altmannshofer , are shown
in Fig. 1
Recent measurements from BABAR bib:kstarmumu:babar , Belle bib:kstarmumu:belle
and CDF bib:kstarmumu:cdf have generated excitement as they appear to favour
a forward-backward asymmetry with the opposite sign to the SM prediction at
low-$q^{2}$ and no zero-crossing point.
Figure 1: The forward backward asymmetry of muons in $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ as a function of $q^{2}=m_{\mu^{+}\mu^{-}}^{2}$
predicted in the SM (solid-black line) and a variety of NP models. The SM
prediction and the NP models are described in detail in Ref. bib:altmannshofer
.
In 309$\mbox{\,pb}^{-1}$ LHCb observes $302\pm 20$ candidates in a $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ $K^{\pm}\pi^{\mp}\mu^{+}\mu^{-}$
mass window. Candidates were triggered by: a single high-$p_{T}$ muon in
LHCb’s Level 0 hardware trigger bib:level0 ; a single high impact parameter
and high-$p_{T}$ daughter bib:track in the first stage of a two stage
software trigger and ‘topologically’ by partially reconstructing the $B_{d}$
decay in the second stage of the software trigger bib:topo .
Offline, candidates are first selected by applying a loose pre-selection based
on the $B_{d}$ lifetime, daughter impact parameters and a requirement that the
$B_{d}$ points back to one of the primary vertices in the event. A
multivariate selected based on a Boosted Decision Tree (BDT) was then used to
further reduce combinatorial background. The BDT combined information on the
$B_{d}$ kinematics, $B_{d}$ vertex quality, and the kaon, pion and muon impact
parameter and particle identification. The BDT allowed a signal-to-background
ratio of three-to-one to be achieved across the $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ signal mass window. The
multivariate selection was trained using $B_{d}\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and background
from the upper mass sideband of a 36$\mbox{\,pb}^{-1}$ data set collected in
2010. These events are not used in the subsequent analysis. Care has also been
taken to avoid further biasing the angular distribution in the offline
selection.
Specific exclusive backgrounds from, e.g,
$B_{d}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-}\\{\pi^{-}\\})K^{*0}(\rightarrow
K^{+}\pi^{-}\\{\mu^{-}\\})$ where the pion from the $K^{*0}$ is misidentified
as a muon and a muon from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
as the pion, are removed by exchanging the mass assignments of the daughters
and cutting on the the resulting $B_{d}$ mass in combination with cuts on the
daughter particle identification. These selection criteria are almost 100%
efficient on genuine $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates and
reduce this type of background to a negligible level. In particular the
exclusive background from $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ where the
$K^{*0}$ is misidentified as a $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is reduced to 1% of the level of
the signal, diluting the measured $A_{FB}$ by 2%. This dilution is accounted
for in the results presented below. The regions around the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ are also removed
as they are dominated by different physics from $c\overline{c}$ loops (see
Fig. 2).
The forward backward-asymmetry ($A_{FB}$) of the muon system is extracted from
the angular distribution of the $\mu^{+}$ ($\mu^{-}$) helicity angle w.r.t.
the dimuon flight direction in the rest-frame of the $B_{d}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{d}$). The probability density
distribution for the cosine of this angle, $\cos\theta_{\ell}$ is given by:
$\frac{1}{\Gamma}\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}q^{2}}=\frac{3}{4}{F_{L}}(1-\cos^{2}\theta_{\ell})+\frac{3}{8}(1-{F_{L}})(1+\cos^{2}\theta_{\ell})+{A_{FB}}\cos\theta_{\ell}$
(1)
Figure 2: Scatter plot highlighting the correlation between
$m_{\mu^{+}\mu^{-}}$ and $m_{K\pi\,\mu\rm m}$ for candidates used in the
analysis. The vertical band illustrates the $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ signal mass window used in the
analysis. The horizontal bands illustrate the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi(2S)$ regions taht are
treated separately in the analysis. A clear $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal is visible across $m_{\mu^{+}\mu^{-}}$, centered
on the $B_{d}$ mass. Figure 3: The ${K^{\pm}\pi^{\mp}\mu\mu}$ mass
distribution of $B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates in six
$q^{2}$ bins. The solid line shows a fit to this distribution with a double-
Gaussian signal component (thin-green line) and Exponential background
component (dashed-red line).
which also depends on $F_{L}$, the fraction of longitudinal polarisation of
the $K^{*0}$. $F_{L}$ is constrained by simultaneously fitting the helicity
angle of the kaon ($\theta_{K}$). The probability density function for
$\cos\theta_{K}$ only depends on $F_{L}$ and is given by:
$\frac{1}{\Gamma}\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}\cos\theta_{K}\,\mathrm{d}q^{2}}=\frac{3}{2}{F_{L}}\cos^{2}\theta_{K}+\frac{3}{4}(1-{F_{L}})(1-\cos^{2}\theta_{K})$
(2)
The analysis is performed in six-$q^{2}$ bins, chosen for consistency to be
the same binning scheme used by previous experiments. A significant
$B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ signal is visible in each of the six
bins with an excellent signal-to-background ratio even in the lowest $q^{2}$
bin (see Fig. 3).
Unfortunately the extraction of $A_{FB}$ is complicated by a correlation
between $A_{FB}$ and $F_{L}$ that prevents the $A_{FB}$ from being large if
$F_{L}$ is also large. The allowed region of phase space, where Eq. 1 remains
positive and well defined, corresponds to
$\left|A_{FB}\right|\leq\frac{3}{4}(1-F_{L}$). To account for the ‘physical’
region a profile likelihood scan is made over the plane, with a flat prior
that the maximum likelihood must lie in the physical region. The statistical
uncertainty on the central value of $A_{FB}$ and $F_{L}$ is calculated by
integrating the likelihood to yield an (asymmetric) 68% confidence limt on
$A_{FB}$ and $F_{L}$.
The fit results for $A_{FB}$ and $F_{L}$ in six $q^{2}$ bins are shown in Fig.
4 along with the differential branching fraction of $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ as a function of $q^{2}$. The differential branching
fraction is extracted from fits to the $K^{\pm}\pi^{\mp}\mu^{+}\mu^{-}$ mass
distribution in the $q^{2}$ bins and normalised with respect to the branching
fraction of $B_{d}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$.
The systematic uncertainties on $A_{FB}$, $F_{L}$ and the differential
branching fraction are typically 30% of the statistical uncertainty. Across
$q^{2}$ a dominant contribution to the systematic uncertainty comes from the
understanding of data-derived corrections to the detector performance, which
is expected to improve with increased statistics. The choice of the background
angular and mass model in the fit for $A_{FB}$ and $F_{L}$ lead to a
systematic uncertainty at the level of 10-20% of the statistical uncertainty.
This model dependence will also improve with an enlarged data set.
The observed forward-backward asymmetry, longitudinal $K^{*0}$ polarisation
and differential branching fraction are consistent with a SM interpretation.
In particular the observed forward-backward asymmetry shows evidence for a
zero-crossing point which could be measured with a larger data set.
Figure 4: $A_{FB}$, $F_{L}$ and the differential branching fraction of
$B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates as a function of $q^{2}$ in
the six $q^{2}$ bins. The theory predictions are described in Ref.
Bobeth:2011gi
## III Radiative decays in LHCb
The long term goal of the LHCb radiative decay program is a measurement of the
right-handed component of the photon polarization, probed using a time
dependent analysis of $B_{s}\rightarrow\phi\gamma$ decays bib:zwicky . In the
shorter term, a large $B_{d}\rightarrow K^{*0}\gamma$ yield will enable LHCb
to make a world leading measurement of the $\mathcal{A}_{CP}(B_{d}\rightarrow
K^{*0}\gamma)$. The most precise single measurement of
$\mathcal{A}_{CP}(B_{d}\rightarrow K^{*0}\gamma)$ comes from the BABAR
experiment, based on 2400 $B_{d}$ candidates bib:babaracp . A large
$B_{s}\rightarrow\phi\gamma$ could allow an improved measurement of the
$B_{s}\rightarrow\phi\gamma$ branching fraction.
In 88$\mbox{\,pb}^{-1}$ collected by LHCb in 2010 and 2011, LHCb observes
$485\pm 43$ $B_{d}\rightarrow K^{*0}\gamma$ candidates and $60\pm 12$
$B_{s}\rightarrow\phi\gamma$ candidates. The $B_{s}\rightarrow\phi\gamma$
yield is the largest yield of $B_{s}$ radiative decays at a single experiment.
The ${K^{\pm}K^{\mp}\gamma}$ and ${K^{\pm}\pi^{\mp}\gamma}$ mass distributions
of these candidates is shown in Fig. 5. The reconstructed
${K^{\pm}K^{\mp}\gamma}$ and ${K^{\pm}\pi^{\mp}\gamma}$ signal mass resolution
is $\sim 125{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This is larger than the
Monte Carlo prediction of 100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, but
will improve with ongoing (time-dependent) Calorimeter calibration.
$B_{d}\rightarrow K^{*0}\gamma$ and $B_{s}\rightarrow\phi\gamma$ candidates
are triggered in LHCb’s Level-0 hardware trigger on the high-$E_{T}$ photon or
by the kaon or pion from the $\phi$ or $K^{*0}$ bib:level0 . In the first
stage of the software trigger the events are triggered by a high-$p_{T}$ and
high-IP track (from the $\phi$ or $K^{*0}$) or by a softer track and a
high-$E_{T}$ photon. In the second stage of the software trigger a full
exclusive reconstruction of the $B$ is performed which mirrors the offline
selection.
The offline selection is similar to the pre-selection of $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ described above in that the tracks from the $K^{*0}$ and
$\phi$ are required to have significant impact parameter and the $B$ is
required to point back to the PV. Additional requirements are then made of the
$\gamma$ $p_{T}$ ($\mbox{$p_{T}$}>2.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)
and on the $B$ $p_{T}$
($\mbox{$p_{T}$}>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and isolation to
improve the purity of the signal.
Figure 5: Reconstructed ${K^{\pm}\pi^{\mp}\gamma}$ (top) and
${K^{\pm}K^{\mp}\gamma}$ mass distributions in 88$\mbox{\,pb}^{-1}$ of
integrated luminosity. The solid line is a fit to the data with a Gaussian
signal distribution (green-dotted line) and exponential background
distribution (red-thin line)
Extrapolating to a 1$\mbox{\,fb}^{-1}$ data set, LHCb expects to collect a
data sample of $\mathcal{O}(5000)$ $B_{d}\rightarrow K^{*0}\gamma$ candidates
and $\mathcal{O}(700)$ $B_{s}\rightarrow\phi\gamma$ candidates, which would
also represent the largest sample of radiative $B_{d}$ decays at a single
experiment.
## IV Summary
In 2011 the performance of the LHC and LHCb has been excellent delivering
300$\mbox{\,pb}^{-1}$ of integrated luminosity in the first three months of
data taking. This data sample has enabled LHCb to make the world’s most
precise measurement of the $A_{FB}(q^{2})$ in $B_{d}\rightarrow
K^{*0}\mu^{+}\mu^{-}$, which shows a striking agreement with the SM
prediction. By the end of 2011, it is expected that LHCb will have recorded
1$\mbox{\,fb}^{-1}$ of integrated luminosity providing the worlds largest
samples of reconstructed $B_{d}\rightarrow K^{*0}\gamma$ and
$B_{s}\rightarrow\phi\gamma$ decays. This expanded data set will also enable
LHCb to make a first measurement of the forward-backward asymmetry zero-
crossing point and transverse asymmetries bib:kstarmumu:full in
$B_{d}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and to pursue a wide range of other
rare decay measurements.
###### Acknowledgements.
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Région Auvergne.
## References
* (1) The LHCb collaboration, CERN-LHCb-CONF-2011-038 (2011).
* (2) M-O. Bettler, these proceedings.
* (3) A. A. Alves Jr. et al., JINST 3 S08005 (2008).
* (4) U. Egede _et al._ , JHEP 1010:056 (2010).
* (5) W. Altmannshofer _et al._ , JHEP 0901:019 (2009).
* (6) B. Aubert _et al._ , Phys. Rev. D. 79 (2009).
* (7) J.-T. Wei _et al._ , Phys. Rev. Lett. 103 (2009).
* (8) T. Aaltonen _et al._ , Phys. Rev. Lett. 106 (2011).
* (9) The LHCb collaboration, CERN-LHCC-2003-031(2003).
* (10) V. V. Gligorov _et. al._ , CERN-LHCb-PUB-2011-003 (2011).
* (11) M. Williams _et. al._ , CERN-LHCb-PUB-2011-002 (2011).
* (12) C. Bobeth _et al._ , arXiv:1105.0376 (2011). U. Egede _et al._ , JHEP 0811:032 (2008).
* (13) R. Zwicky _et al._ , Phys. Lett. B. 664 (2008).
* (14) B. Aubert _et al._ , Phys. Rev. D. 103 (2009).
|
arxiv-papers
| 2011-09-29T16:33:37 |
2024-09-04T02:49:22.605773
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Blake",
"submitter": "Thomas Blake",
"url": "https://arxiv.org/abs/1109.6582"
}
|
1109.6583
|
# Approximate cloaking for the Helmholtz equation via transformation optics
and consequences for perfect cloaking
Hoai-Minh Nguyen 111Courant Institute, NYU, 251 Mercer Street, New York, NY
10012, USA, hoaiminh@cims.nyu.edu
(April 07, 2011)
###### Abstract
In this paper, we study approximate cloaking of active devices for the
Helmholtz equation in the whole space of dimension $2$ or $3$ using the scheme
introduced by Kohn, Shen, Vogelius, and Weinstein in [12]. More precisely, we
assess the degree of invisibility, determine the limit of the field inside the
cloaked and cloaking regions, and show that the scheme is unstable with
respect to the material parameters. As a consequence, we obtain some feasible
properties of “perfect” cloaking.
## 1 Introduction
Cloaking via change of variables was introduced by Greenleaf, Lassas, Uhlmann
[10] for electrical impedance tomography, Pendry, Schurig, and Smith [21] for
the Maxwell system, and Leonhardt [14] in the geometric optics setting. They
used a singular change of variables which blows up a point into a cloaked
region. This singular structure implies not only difficulties in practice, but
also in analysis. Some approaches [6], [25], [26] are proposed to tackle this
problem mathematically based on the notion of “weak” solution. To avoid using
the singular structure, regularized schemes have been proposed in [23], [2],
[28], [22], [12], and [7]. The reader can find more information and references
related to cloaking in the works mentioned above or in the reviews [9] and
[27].
In this paper, we study approximate cloaking of active devices for the
Helmholtz equation in the whole space of dimension $2$ or $3$ for the scheme
introduced by Kohn, Shen, Vogelius, and Weinstein in [12], where they used a
transformation which blows up a small region instead of a point into the
cloaked region. We assess the degree of invisibility, determine the limit of
the field inside the cloaked and cloaking regions, and show that the scheme is
unstable with respect to the material parameters. As a consequence, we obtain
some feasible properties of “perfect” cloaking. More precisely, let
$\varepsilon$ denote the parameter of regularization i.e. the ratio between
the diameter of the region blown up to the cloaked region $D$ and the diameter
of the cloaked region:
$a)$ In the $3d$ non-resonant case i.e. when the frequency is not an
eigenvalue of the Neumann problem in $D$, we show that the difference between
the field and the “push-forward” of the solution of the Helmholtz equation in
free space is of order $\varepsilon$ in any bounded region away from $D$
(Theorem 1). Concerning the limiting behavior of the approximate field in $D$,
we prove that it converges and the limit is the solution of the corresponding
Neumann problem in $D$ (Theorem 1 and Definition 1).
$b)$ In the $3d$ resonant case i.e. when the frequency is an eigenvalue of the
Neumann problem in $D$, under the assumption that the source inside the
cloaked region is orthogonal, with respect to the $L^{2}$-scalar product in
$D$, to all Neumann eigenfunctions 222This condition is necessary and
sufficient to have a solution of the Neumann problem. This is a consequence of
the Fredholm alternative theory. (this holds if the cloaked region is
passive), we also prove that the difference between the field and the “push-
forward” of the solution of the Helmholtz equation in free space is of order
$\varepsilon$ in any bounded region away from $D$, the approximate field
converges in $D$ and the limit is uniquely determined. However, this limit is
not only a solution of the Neumann problem in $D$, but also depends on the
value of the solution of the Helmholtz equation in free space at the point
where the maps are blown up in a quite involved manner (Theorem 1 and
Definition 1).
$c)$ In the $3d$ resonant case, without the assumption on the orthogonality of
the source mentioned in statement $b)$, the energy of the field inside $D$
explodes as $\varepsilon\to 0$; moreover, cloaking can be not achieved
(Proposition 2).
$d)$ In the $2d$-non resonant case (see the definition in Definition 2), we
show that the field converges to the “push-forward” of the solution of the
Helmholtz equation in free space in any bounded set away from $D$ with a rate
$1/|\ln\varepsilon|$ and the limiting behavior of the field in $D$ exhibits a
non-local structure. Therefore, the limit is generally not the solution of the
Neumann problem in $D$ (Theorem 2 and Definition 2).
$e)$ In the $2d$-resonant case (see the definition in Definition 2), we prove
that the energy inside the cloaked region can go to infinity and cloaking can
be not achieved (Proposition 3).
$f)$ Concerning the stability with respect to the material parameters inside
the cloaked region without a source, we show that cloaking is unstable with
respect to these parameters. Roughly speaking, there exist some fixed
parameters such that for each $\varepsilon$, one can perturb these parameters
by an amount of order $\varepsilon$ in $3d$ and $1/|\ln\varepsilon|$ in $2d$
so that the degree of visibility is of order 1 (Proposition 4).
###### Remark 1.
Property $f)$ does not contradict Properties $a)$ and $d)$ since Property $f)$
is only stated under a condition on the amount of the perturbation which does
not hold when the material parameters are fixed.
###### Remark 2.
A similar conclusion as in statement $f)$ for $2d$ bounded domains is
previously observed by Kohn et al. [11].
Our results in the $3d$ non-resonant case are compatible with what has been
mentioned in the literature: cloaking is achieved, the energy of the field is
finite in the cloaked region, and the limit of the field inside the cloaked
region is the solution of the Neumann problem see e.g. [6], [26]. However, our
results in the $3d$ resonant case are quite different. In the setting in [6],
the limit field could be any Neumann eigenfunction in the cloaked region for a
passive device. In [26], the author asserted that the field inside and outside
the cloaked region are completely decoupled from each other. It is discussed
in the literature that: cloaking for the Helmholtz equation is achieved and
the energy of the field inside the cloaked region is finite (see e.g. [6],
[4], [25], [26]). The $2d$ perfect cloaking has not been rigorously studied as
extensively as in the $3d$ one and it is often argued that the field in the
cloaked region is a solution of the Neumann problem.
We recall that the weak solution considered in [6] is only discussed in $3d$.
The degree of invisibility in the approximate cloaking problem is more widely
understood when one uses an appropriate lossy-layer. In this case, the same
estimates as above hold independent of the material parameters in $D$ (see
[11], [17], [19]) and there are explicit frequency dependent estimates which
are valid for all frequency (see [19]). The zero frequency case is less
complicated. This is studied in [12] (see also [18]) where no lossy-layer is
used and better estimates are obtained. Without a lossy-layer, the degree of
invisibility in $2d$ is discussed in [22] when the material parameters inside
the cloaked region are isotropic and homogeneous, and the approximate cloaking
is confirmed for the $3d$ non-resonant case in [8] (without an estimate of the
degree of invisibility). Recently, Greenleaf, Kurulev, Lassas, and Uhlmann [5]
observed that cloaking without shielding is possible (compare with our results
in the $3d$ resonant case).
Let us describe the problem more precisely. To illustrate the idea, let us
suppose that the cloaking region is the annular $\\{1<|x|<2\\}$ and the
cloaked region is the unit ball $B$ centered at the origin of $\mathbb{R}^{d}$
($d=2,3$). Using the scheme in [12], the parameters in the cloaking region are
given by
$a_{c},\sigma_{c}={F_{\varepsilon}}_{*}I,{F_{\varepsilon}}_{*}1,$
where $F_{\varepsilon}$ is the maps which blows up the ball $B_{\varepsilon}$
into $B_{1}$ given by
$F_{\varepsilon}=\left\\{\begin{array}[]{cl}x&\mbox{ if
}x\in\mathbb{R}^{d}\setminus B_{2},\\\\[12.0pt]
\displaystyle\Big{(}\frac{2-2\varepsilon}{2-\varepsilon}+\frac{|x|}{2-\varepsilon}\Big{)}\frac{x}{|x|}&\mbox{
if }x\in B_{2}\setminus B_{\varepsilon},\\\\[12.0pt]
\displaystyle\frac{x}{\varepsilon}&\mbox{ if }x\in
B_{\varepsilon}.\end{array}\right.$ (1.1)
Hereafter we use the standard notation
$F_{*}A(y)=\frac{DF(x)A(x)DF^{T}(x)}{\det DF(x)}\quad\mbox{and}\quad
F_{*}\Sigma(y)=\frac{\Sigma(x)}{\det DF(x)},\quad\mbox{ where }x=F^{-1}(y),$
(1.2)
for any matrix-valued function $A$ and any complex function $\Sigma$ and we
denote $D_{r}$ the set $\\{rx;x\in D\\}$ for any open bounded set $D$ of
$\mathbb{R}^{d}$ and for any $r>0$.
Let $a$ be a uniformly elliptic matrix-valued function defined in $B_{1}$,
$\sigma$ be a complex function defined in $B_{1}$, $f\in
L^{2}(\mathbb{R}^{d})$ such that
$0\leq\mathrm{ess}\,\inf\Im(\sigma)\leq\mathrm{ess}\,\sup\Im(\sigma)<+\infty$,
$0<\mathrm{ess}\,\inf\Re(\sigma)<\mathrm{ess}\,\sup\Re(\sigma)<+\infty$,
$\operatorname{supp}f\subset(B_{4}\setminus B_{3})\cup\overline{B_{1}}$. For
$k>0$, let $u,u_{c}\in H^{1}_{{}_{loc}}(\mathbb{R}^{d})$ be the unique
outgoing solutions of the equations
$\Delta u+k^{2}u=f\chi_{\\{|x|\geq 2\\}}\quad\mbox{ in }\mathbb{R}^{d},$ (1.3)
and
$\operatorname{div}(A_{c}\nabla u_{c})+k^{2}\Sigma_{c}u_{c}=f\quad\mbox{ in
}\mathbb{R}^{d},$ (1.4)
where $\chi_{\Omega}$ denotes the characteristic function of the set $\Omega$
for any $\Omega\subset\mathbb{R}^{d}$, and
$A_{c},\Sigma_{c}=\left\\{\begin{array}[]{cl}I,1&\mbox{ in
}\mathbb{R}^{d}\setminus B_{2},\\\\[12.0pt]
a_{c}={F_{\varepsilon}}_{*}I,\sigma_{c}={F_{\varepsilon}}_{*}1&\mbox{ in
}B_{2}\setminus B_{1},\\\\[12.0pt] a,\sigma&\mbox{ in
}B_{1}.\end{array}\right.$ (1.5)
In the following, whenever we talk about outgoing solutions to a Helmholtz
problem at frequency $k$, we mean solutions that satisfy
$\displaystyle\frac{\partial v}{\partial
r}-ikv=o(r^{-\frac{d-1}{2}})\quad\mbox{ as }r=|x|\rightarrow\infty,$
where $d=2$ or 3 is the dimension of space.
###### Remark 3.
Physically $a$, $\sigma$ are the material parameters in the cloaked region
$B_{1}$; $f$ is a given source outside the cloaking region, $u$ describes the
field corresponding to the source existing outside the cloaked and cloaking
regions in free space, and $u_{c}$ describes the field due to the source $f$
in the presence of the cloaking device and the cloaked object.
The effectiveness of the approximate cloak and the limit of $u_{c}$ as
$\varepsilon\to 0$ are described in Theorems 1 and 2 below. The proofs will be
presented in Sections 2 and 3.
###### Theorem 1.
Let $d=3$, $k>0$, and $0<\varepsilon<1$. Define 333In the following, $\eta$
denotes the unit normal vector on $\partial D$ directed to the complement of
$D$ for any smooth bounded open subset $D$ of $\mathbb{R}^{d}$.
$M:=\Big{\\{}\psi\in
H^{1}(B_{1});\;\operatorname{div}(a\nabla\psi)+k^{2}\sigma\psi=0\mbox{ in
}B_{1}\mbox{ and }a\nabla\psi\cdot\eta=0\mbox{ on }\partial B_{1}\Big{\\}}.$
(1.6)
In the resonant case i.e. when $M\not=\\{0\\}$, assume in addition that
$\displaystyle\int_{B_{1}}f\bar{\bf e}=0$ for any ${\bf e}\in M$ 444This
condition is necessary and sufficient to have a solution $v\in H^{1}(B_{1})$
of the system $\operatorname{div}(a\nabla v)+k^{2}\sigma v=f$ in $B_{1}$ and
$a\nabla v\cdot\eta=0$ on $\partial B_{1}$ by the Fredholm alternative theory
see e.g. [1] or [13].. Then for all
$K\subset\subset\mathbb{R}^{3}\setminus\overline{B}_{1}$,
$\|u_{c}-u\circ F_{0}^{-1}\|_{H^{1}(K)}\leq C\varepsilon\|f\|_{L^{2}},$
for some positive constant $C$ depending on $k$, $K$, $a$, and $\sigma$, but
independent of $\varepsilon$, $f$. We also have that $u_{c}\rightharpoonup
Cl_{3}(f)$ weakly in $H^{1}(B_{1})$. As a consequence
$\lim_{\varepsilon\to 0}\|u_{c}-Cl_{3}(f)\|_{L^{2}(B_{1})}=0.$
Here and in the following, $F_{0}(x):=\lim_{\varepsilon\to}F_{\varepsilon}(x)$
for $x\in\mathbb{R}^{d}\setminus\\{0\\}$ and for $d=2,3$.
The operator $Cl_{3}$ is defined in the following
###### Definition 1.
Let $d=3$. We define $Cl_{3}$ as follows:
1. i)
The non-resonant case: $M=\\{0\\}$. Define $Cl_{3}(f)=v$ where $v\in
H^{1}(B_{1})$ is the unique solution of the system
$\left\\{\begin{array}[]{ll}\operatorname{div}(a\nabla v)+k^{2}\sigma
v=f&\mbox{ in }B_{1},\\\\[12.0pt] \displaystyle a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial
B_{1}.\end{array}\right.$
2. ii)
The resonant case: $M\not=\\{0\\}$. Assume in addition that
$\displaystyle\int_{B_{1}}f\bar{\bf e}=0$ for any ${\bf e}\in M$. Consider the
triple of functions $(v_{\mathrm{ext}},v_{\mathrm{int}},w)\in
W^{1}(\mathbb{R}^{3}\setminus\overline{B}_{1})\times H^{1}(B_{1})\times
M^{\perp}$ which is the unique solution of the systems
$\left\\{\begin{array}[]{ll}\Delta v_{\mathrm{ext}}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla v_{\mathrm{int}})+k^{2}\sigma
v_{\mathrm{int}}=f&\mbox{ in }B_{1},\\\\[12.0pt] \operatorname{div}(a\nabla
w)+k^{2}\sigma w=0&\mbox{ in }B_{1},\end{array}\right.\mbox{ and
}\left\\{\begin{array}[]{ll}\displaystyle
v_{\mathrm{ext}}-v_{\mathrm{int}}=-u(0)&\mbox{ on }\partial B_{1},\\\\[12.0pt]
\displaystyle a\nabla v_{\mathrm{int}}\cdot\eta=0&\mbox{ on }\partial
B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
v_{\mathrm{ext}}}{\partial\eta}=a\nabla w\cdot\eta&\mbox{ on }\partial
B_{1}.\end{array}\right.$
Define
$Cl_{3}(f)=v_{\mathrm{int}}.$
Hereafter for a connected smooth open region $U$ of $\mathbb{R}^{3}$ with a
bounded complement (this includes $U=\mathbb{R}^{3}$), $W^{1}(U)$ is defined
as follows 555The space $W^{1}$, which is defined here in $3d$ and later in
$2d$ (see Definition 2), has been used in the study of the Laplace equation in
an unbounded domain e.g. [16].
$W^{1}(U)=\Big{\\{}\psi\in L^{1}_{loc}(U)\leavevmode\nobreak\
;\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{\psi(x)}{\sqrt{1+|x|^{2}}}\in
L^{2}(U)\mbox{ and }\nabla\psi\in L^{2}(U)\Big{\\}}.$ (1.7)
On the boundary $\partial D$ of any bounded open set $D\subset\mathbb{R}^{3}$,
$\phi\Big{|}_{\mathrm{ext}}$ and $\phi\Big{|}_{\mathrm{int}}$ denote the trace
of $\phi$ from the outside and the inside. For any closed subspace $M$ of
$H^{1}(D)$, we also denote the space $M^{\perp}$ by
$M^{\perp}=\Big{\\{}\psi\in
H^{1}(D);\;\int_{D}\Big{(}\nabla\psi\nabla\bar{\phi}+\psi\bar{\phi}\Big{)}\,dx=0\quad\forall\,\phi\in
M\Big{\\}}.$ (1.8)
###### Remark 4.
The uniqueness of the triple $(v_{\mathrm{ext}},v_{\mathrm{int}},w)$ is
established in Lemma 2 (Section 2). The existence of
$(v_{\mathrm{ext}},v_{\mathrm{int}},w)$ will follow from the proof of Theorem
1.
The following definition will be used in Theorem 2 concerning the $2d$
setting.
###### Definition 2.
In the $2d$ setting, the system is non-resonant if the following problem
$\left\\{\begin{array}[]{ll}\Delta w=0&\mbox{in }\mathbb{R}^{2}\setminus
B_{1},\\\\[12.0pt] \operatorname{div}(a\nabla w)+k^{2}\sigma w=0&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
w}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
w\cdot\eta\Big{|}_{\mathrm{int}}&\mbox{on }\partial B_{1}.\end{array}\right.$
(1.9)
only has the zero solution in $W^{1}(\mathbb{R}^{2})$. Otherwise, the system
is resonant. In the non-resonant case, we define $CL_{2}(f)=v$ the unique
solution in $W^{1}(\mathbb{R}^{2})$ of the system 666The existence of $v$ is a
consequence of the Fredholm alternative theory (see also part $ii)$ of Lemma
5)
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{in }\mathbb{R}^{2}\setminus
B_{1},\\\\[12.0pt] \operatorname{div}(a\nabla v)+k^{2}\sigma v=f&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
v}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}&\mbox{on }\partial B_{1}.\end{array}\right.$
Hereafter for a connected smooth open region $U$ of $\mathbb{R}^{2}$ with a
bounded complement (this include $U=\mathbb{R}^{2}$), $W^{1}(U)$ is defined as
follows
$W^{1}(U)=\Big{\\{}\psi\in
L^{1}_{loc}(U):\frac{\psi(x)}{\ln(2+|x|)\sqrt{1+|x|^{2}}}\in L^{2}(U)\mbox{
and }\nabla\psi\in L^{2}(U)\Big{\\}}.$ (1.10)
###### Theorem 2.
Let $d=2$, $k>0$, and $0<\varepsilon<1$. Assume that the system is non-
resonant. Then for all
$K\subset\subset\mathbb{R}^{2}\setminus\overline{B}_{1}$,
$\|u_{c}-u\circ
F_{0}^{-1}\|_{H^{1}(K)}\leq\frac{C}{|\ln\varepsilon|}\|f\|_{L^{2}},$
for some positive constant $C$ depending on $K$, $k$, $a$, and $\sigma$, but
independent of $\varepsilon$ and $f$. We also have that $u_{c}\rightharpoonup
Cl_{2}(f)$ weakly in $H^{1}(B_{1})$. As a consequence,
$\lim_{\varepsilon\to 0}\|u_{c}-Cl_{2}(f)\|_{L^{2}(B_{1})}=0.$
###### Remark 5.
From Theorem 2, the limit of the field in $D$ is $0$ when the cloaked region
is passive in the $2d$ non-resonant case.
We also show in the following proposition that if $k$ is small enough (the
smallness condition depends only on the bounds of $a$ and $\sigma$) then the
system is non-resonant. Proposition 1 will be proved in Section 3.
###### Proposition 1.
Let $0<c_{1}<c_{2}<0$. Assume that $c_{1}|\xi|^{2}\leq\langle
a\xi,\xi\rangle\leq c_{2}|\xi|^{2}$, $0\leq\Im(\sigma)\leq c_{2}$,
$c_{1}<\Re(\sigma)<c_{2}$ in $B_{1}$. There exists $k_{0}>0$, depending only
on $c_{1}$ and $c_{2}$, such that if $k<k_{0}$, then the system is non-
resonant.
The following proposition, which will be proved in Section 4, establishes the
results mentioned in statement $c)$.
###### Proposition 2.
Let $d=3$ and $k>0$. Assume that $M\not=0$ and fix an element ${\bf e}\in M$
such that $\|{\bf e}\|_{L^{2}(B_{1})}=1$. Let $u_{c}$ be the solution of (1.4)
with $f=0$ in $\mathbb{R}^{3}\setminus B_{1}$ and $f={\bf e}$ in $B_{1}$.
1. i)
We have
$\liminf_{\varepsilon\to 0}\varepsilon\|u_{c}\|_{H^{1}(B_{1})}>0.$
2. ii)
Assume in addition that ${\bf e}$ is radial, $a$ and $\sigma$ are isotropic
and homogeneous in $B_{1}$ i.e. $a=\lambda_{1}I$ and $\sigma=\lambda_{2}$ for
some positive constants $\lambda_{1}$ and $\lambda_{2}$. Then
$\liminf_{\varepsilon\to 0}\|u_{c}\|_{L^{2}(B_{4}\setminus B_{2})}>0.$
Concerning the $2d$-resonant case, we have the following proposition which
establishes the results in statement e) and is proved in Section 5.
###### Proposition 3.
Let $d=2$ and $k>0$. Assume that the system is resonant. Define
$N=\Big{\\{}\psi\in W^{1}(\mathbb{R}^{2});\,\psi\mbox{ satisfies
}\eqref{uniqueness2d}\Big{\\}}.$
Fix an element ${\bf e}\in N$ such that $\|{\bf e}\|_{L^{2}(B_{1})}=1$. Let
$u_{c}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ be the unique outgoing solution of
(1.4) with $f=0$ in $\mathbb{R}^{2}\setminus B_{1}$ and $f={\bf e}$ in
$B_{1}$.
1. i)
We have
$\liminf_{\varepsilon\to 0}\|u_{c}\|_{H^{1}(B_{1})}=+\infty.$
2. ii)
Assume in addition that ${\bf e}$ is radial, $a$ and $\sigma$ are isotropic
and homogeneous in $B_{1}$ i.e. $a=\lambda_{1}I$ and $\sigma=\lambda_{2}$ for
some positive constants $\lambda_{1}$ and $\lambda_{2}$. Then
$\liminf_{\varepsilon\to 0}\|u_{c}\|_{L^{2}(B_{4}\setminus B_{2})}>0.$
Concerning the instability of the approximate cloaking with respect to the
material parameters inside the cloaked region, we establish the following
result which is proved in Section 6:
###### Proposition 4.
Let $d=2,3$, $k>0$, and $\eta\in\mathbb{R}^{d}$ with $|\eta|=1$. There exist a
positive number $\sigma_{0}>0$ and a family of positive numbers
$(\sigma_{\varepsilon})$ such that
$\left\\{\begin{array}[]{cl}0<\liminf_{\varepsilon\to
0}\varepsilon^{-1}|\sigma_{\varepsilon}-\sigma_{0}|\leq\limsup_{\varepsilon\to
0}\varepsilon^{-1}|\sigma_{\varepsilon}-\sigma_{0}|<+\infty&\mbox{ if
}d=3,\\\\[12.0pt] 0<\liminf_{\varepsilon\to
0}|\ln\varepsilon||\sigma_{\varepsilon}-\sigma_{0}|\leq\limsup_{\varepsilon\to
0}|\ln\varepsilon||\sigma_{\varepsilon}-\sigma_{0}|<+\infty&\mbox{ if
}d=2,\end{array}\right.$ (1.11)
and
$\liminf_{\varepsilon\to 0}\|u_{c,s}\|_{L^{2}(B_{4}\setminus B_{2})}>0.$
Here $u_{c,s}\in H^{1}_{{}_{loc}}(\mathbb{R}^{d})$ is such that $u_{c,s}$
satisfies the outgoing condition and $u_{c}(x):=u_{c,s}(x)+e^{ik\eta\cdot x}$
is a solution of the equation
$\operatorname{div}(A_{c}\nabla u_{c})+k^{2}\Sigma_{c}u_{c}=0.$
Here $(A_{c},\Sigma_{c})$ is defined in (1.5) with $a=I$ and
$\sigma=\sigma_{\varepsilon}$.
###### Remark 6.
We recall that when the parameters $a$ and $\sigma$ are fixed and the cloaked
region is passive, cloaking is achieved in the $3d$ and $2d$ non-resonant
cases (see Theorems 1 and 2). Nevertheless Proposition 4 does not contradict
this fact since it holds under condition (1.11) which is invalid for fixed
$a_{\varepsilon}$ and $\sigma_{\varepsilon}$.
Our approach to Theorems 1 and 2 is based on the study of the effect of a
small inclusion. The study of approximate cloaking based on the effect of a
small inclusion were discussed in [12], [15], [18], [11], [17], [19]. It is
well-known that when material parameters inside a small inclusion are, roughly
speaking, bounded from below and above by positive constants, the effect of
the small inclusion is small (see e.g. [24]). Without this assumption, the
effect of the inclusion is not small (see e.g. [11], [17]) unless there is an
appropriate lossy-layer (see [11], [17], [19]). In our setting, the
boundedness assumption is violated and no lossy-layer is used. Nevertheless,
the effect of the small inclusion is still small (in the non-resonant case)
due to the special structure induced from (1.2). The starting point of our
approach relies on the following well-known fact:
###### Proposition 5.
Let $d\geq 2$, $k>0$, $A$ be a bounded matrix-valued function and $\Sigma$ be
a bounded complex function defined on $\mathbb{R}^{d}$, $h\in
L^{2}(\mathbb{R}^{d})$, and $F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be a
Lipschitz, surjective, and invertible with $F(x)=x$ on
$\mathbb{R}^{d}\setminus B_{2}$, and $\det DF>c$ a.e. $x\in\mathbb{R}^{d}$,
for some positive constant $c$. Then $u\in H^{1}_{{}_{loc}}(\mathbb{R}^{d})$
is a solution of
$\operatorname{div}(A\nabla u)+k^{2}\Sigma u=h\quad\mbox{ in }\mathbb{R}^{d}$
if and only if $v:=u\circ F^{-1}\in H^{1}_{{}_{loc}}(\mathbb{R}^{d})$ is a
solution of
$\operatorname{div}(F_{*}A\,\nabla v)+k^{2}F_{*}\Sigma\,v=F_{*}h\quad\mbox{ in
}\mathbb{R}^{d},$
where $F_{*}A$ and $F_{*}\Sigma$ are defined in (1.5), and $F_{*}h$ is
similarly defined as $F_{*}\Sigma$. Moreover $u=v$ outside $B_{2}$.
Finally we want to mention that the approximate cloaking for the wave equation
has been recently studied in [20] where an appropriate lossy-layer is used.
The paper is organized as follows. Section 2 is devoted to the proof of
Theorem 1. In Section 3 we prove Theorem 2 and Proposition 1. The proofs of
Propositions 2, 3, and 4 are presented in Sections 4, 5, and 6.
## 2 Proof of Theorem 1
It is clear from the definition of $(A_{c},\Sigma_{c})$ in (1.5) that
$A_{c}={F_{\varepsilon}}_{*}A_{\varepsilon}$ and
$\Sigma_{c}={F_{\varepsilon}}_{*}\Sigma_{\varepsilon}$ where
$A_{\varepsilon},\Sigma_{\varepsilon}=\left\\{\begin{array}[]{cl}I,1&\mbox{if
}x\in\mathbb{R}^{3}\setminus B_{\varepsilon},\\\\[12.0pt]
\displaystyle\frac{1}{\varepsilon}a(x/\varepsilon),\frac{1}{\varepsilon^{3}}\sigma(x/\varepsilon)&\mbox{if
}x\in B_{\varepsilon}.\end{array}\right.$
Applying Proposition 5, Theorem 1 is a consequence of the following
###### Theorem 3.
Let $k>0$ and $0<\varepsilon<1$. Let $u_{\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ be the unique solution of
$\left\\{\begin{array}[]{ll}\operatorname{div}(A_{\varepsilon}\nabla
u_{\varepsilon})+k^{2}\Sigma_{\varepsilon}u_{\varepsilon}={F_{\varepsilon}^{-1}}_{*}f&\mbox{
in }\mathbb{R}^{3},\\\\[12.0pt] u_{\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
In the resonant case, assume in addition that
$\displaystyle\int_{B_{1}}f\bar{\bf e}=0$ for all ${\bf e}\in M$. Then for all
$r>0$, there exists a constant $C=C(r,k,a,\sigma)$ which is independent of
$\varepsilon$ and $f$ such that
$\|u_{\varepsilon}-u\|_{H^{1}(B_{3r}\setminus B_{2r})}\leq
C\varepsilon\|f\|_{L^{2}}.$ (2.1)
Moreover, if $U_{\varepsilon}(x)=u_{\varepsilon}(\varepsilon x)$, then
$U_{\varepsilon}\rightharpoonup Cl_{3}(f)$ weakly in $H^{1}(B_{1})$.
Consequently,
$\lim_{\varepsilon\to 0}\|U_{\varepsilon}-Cl_{3}(f)\|_{L^{2}(B_{1})}=0.$
Here the operator $Cl_{3}$ is given in Definition 1.
The rest of this section will be devoted to the proof of Theorem 3.
### 2.1 Preliminaries
In this section we present some lemmas which will be used in the proof of
Theorem 3. Instead of dealing with the unit ball $B_{1}$, we will present
results for a smooth open subset $D$ of $\mathbb{R}^{3}$. We will also suppose
that $D\subset B_{1}$ and $\mathbb{R}^{3}\setminus D$ is connected. We first
“recall” the following result [17, Lemma 2.2] which will be useful in our
analysis.
###### Lemma 1.
Let $0<\varepsilon<1$, and $g_{\varepsilon}\in H^{\frac{1}{2}}(\partial D)$.
Assume that $v_{\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{3}\setminus\bar{D})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
v_{\varepsilon}+\varepsilon^{2}v_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{3}\setminus\bar{D},\\\\[12.0pt]
v_{\varepsilon}=g_{\varepsilon}&\mbox{on }\partial D,\\\\[12.0pt]
v_{\varepsilon}\mbox{ satisfies the outgoing condition}.\end{array}\right.$
1. $i)$
We have
$\|v_{\varepsilon}\|_{H^{1}(B_{r}\setminus\overline{D})}\leq
C_{r}\|g_{\varepsilon}\|_{H^{\frac{1}{2}}(\partial D)}\quad\forall\,r>5,$
(2.2)
and
$\varepsilon^{1/2}\|v_{\varepsilon}\|_{L^{2}(B_{4r/\varepsilon}\setminus
B_{r/\varepsilon})}\leq C_{r}\|g_{\varepsilon}\|_{H^{\frac{1}{2}}(\partial
D)},$ (2.3)
for some positive constants $C_{r}=C(r,D)$.
2. $ii)$
Assume in addition that $g_{\varepsilon}\rightharpoonup g$ weakly in
$H^{\frac{1}{2}}(\partial D)$ as $\varepsilon\to 0$. Then
$v_{\varepsilon}\rightharpoonup v$ weakly in
$H^{1}_{{}_{loc}}(\mathbb{R}^{3}\setminus\overline{D})$ where $v\in
W^{1}(\mathbb{R}^{3}\setminus\overline{D})$ 777The space $W^{1}$ is defined in
(1.7). is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] v=g&\mbox{ on }\partial
D.\end{array}\right.$ (2.4)
Proof. Inequalities (2.2), and (2.3) with $r=1$ are in [17, Lemma 2.2]
888There is a typo in [17, (2.4)] for $d=3$ where the term $\varepsilon^{1/2}$
is missing . They are consequences of the fact that the fundamental solution
of the Helmholtz equation converges to the fundamental solution of the Laplace
equation as the frequency goes to 0 in $3d$. The proof of (2.3) in the general
case follows in the same manner. Part $ii)$ follows from part $i)$ as follows.
From part $i)$, one may assume that $v_{\varepsilon}\rightharpoonup v$ weakly
in $H^{1}_{{}_{loc}}(\mathbb{R}^{3}\setminus\overline{D})$ (up to a
subsequence) and $v$ satisfies (2.4). Using the representation formula and the
the fact that the fundamental solution of the Helmholtz equation converges to
the fundamental solution of the Laplace equation as the frequency goes to 0 in
$3d$, one can prove that $v\in W^{1}(\mathbb{R}^{3}\setminus\overline{D})$.
Since (2.4) has a unique solution $v\in
W^{1}(\mathbb{R}^{3}\setminus\overline{D})$ (see e.g. [16, Theorem 2.5.14 on
page 64] ), the result holds for the whole family $(v_{\varepsilon})$. The
details of the proof are left to the reader. $\Box$
In what follows, $a$ denotes a real symmetric matrix-valued function and
$\sigma$ denotes a complex function defined on $D$. We also assume that $a$ is
uniformly elliptic and $\sigma$ satisfies
$0<\mathrm{ess}\,\inf\Re\sigma\leq\mathrm{ess}\,\sup\Re\sigma<+\infty$ and
$0\leq\mathrm{ess}\,\inf\Im\sigma\leq\mathrm{ess}\,\sup\Im\sigma<+\infty$. We
define
$M:=\Big{\\{}\psi\in
H^{1}(D);\;\operatorname{div}(a\nabla\psi)+k^{2}\sigma\psi=0\mbox{ in }D\mbox{
and }a\nabla\psi\cdot\eta=0\mbox{ on }\partial D\Big{\\}}.$ (2.5)
The following lemma establishes the uniqueness of $(v,w)$ in Definition 1.
This lemma is also used in the proof of Lemma 3.
###### Lemma 2.
Assume that the system is resonant i.e. $M\neq\\{0\\}$. Then there exists no
nonzero solution $(v,w)$ in $W^{1}(\mathbb{R}^{3})\times M^{\perp}$ of the
systems
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=0&\mbox{ in }D,\\\\[12.0pt] \operatorname{div}(a\nabla
w)+k^{2}\sigma w=0&\mbox{ in }D,\end{array}\right.\mbox{ and
}\left\\{\begin{array}[]{ll}\displaystyle a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial D,\\\\[12.0pt]
\displaystyle\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
w\cdot\eta=0&\mbox{ on }\partial D.\end{array}\right.$ (2.6)
Here $M$ and $M^{\perp}$ are respectively defined in (2.5) and (1.8).
Proof. Since $v\in W^{1}(\mathbb{R}^{3}\setminus\overline{D})$, it follows
from (2.6) that
$\int_{\mathbb{R}^{3}\setminus D}|\nabla v|^{2}=-\int_{\partial
D}\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}\bar{v}=-\int_{\partial
D}(a\nabla w\cdot\eta)\bar{v}.$ (2.7)
On the other hand, from (2.6), we have
$\int_{D}\langle a\nabla v,\nabla v\rangle-\int_{D}k^{2}\sigma|v|^{2}=0.$
This implies that $v=0$ on the set $\\{\Im\sigma>0\\}$. Thus we deduce from
(2.6) that
$\int_{\partial D}(a\nabla w\cdot\eta)\bar{v}=\int_{D}a\nabla
w\nabla\bar{v}-\int_{D}k^{2}\sigma w\bar{v}\\\ =\overline{\int_{D}a\nabla
v\nabla\bar{w}-\int_{D}k^{2}\sigma v\bar{w}}$ (2.8)
and
$\int_{D}a\nabla v\nabla\bar{w}-\int_{D}k^{2}\sigma v\bar{w}=\int_{\partial
D}(a\nabla v\cdot\eta\Big{|}_{\mathrm{int}})\bar{w}=0.$ (2.9)
A combination of (2.7), (2.8), and (2.9) yields $v=0$ in
$\mathbb{R}^{3}\setminus D$. It follows that, by (2.6) and the unique
continuation principle,
$v=0\quad\mbox{ in }\mathbb{R}^{3}$
From (2.6), $a\nabla w\cdot\eta=0$. Hence $w=0$ since $w\in M^{\perp}$ and
$\operatorname{div}(a\nabla w)+k^{2}\sigma w=0$. $\Box$
We now establish the crucial ingredient in the proof of Theorem 3.
###### Lemma 3.
Let $0<\varepsilon<1$ and $k>0$. Let $\theta_{\varepsilon}\in L^{2}(D)$,
$g_{\varepsilon}\in H^{-\frac{1}{2}}(\partial D)$, and $v_{\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ be the unique solution of the system
$\left\\{\begin{array}[]{ll}\Delta
v_{\varepsilon}+\varepsilon^{2}k^{2}v_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{3}\setminus\bar{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v_{\varepsilon})+k^{2}\sigma v_{\varepsilon}=\theta_{\varepsilon}&\mbox{in
}D,\\\\[12.0pt] \displaystyle\frac{\partial
v_{\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon}a\nabla
v_{\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=g_{\varepsilon}&\mbox{on
}\partial D\\\\[12.0pt] v_{\varepsilon}\mbox{ satisfies the outgoing
condition}.\end{array}\right.$
In the case $M\neq\\{0\\}$ 999$M$ is defined in (2.5)., assume in addition
that $\displaystyle\int_{B_{1}}\theta_{\varepsilon}\bar{\bf e}=0$ for all
${\bf e}\in M$. We have
$\|v_{\varepsilon}\|_{H^{1}(B_{5})}\leq
C\big{(}\|\theta_{\varepsilon}\|_{L^{2}(D)}+\|g_{\varepsilon}\|_{H^{-\frac{1}{2}}(\partial
D)}\big{)},$ (2.10)
for some positive constant $C$ depending only on $k$, $a$, $\sigma$, and $D$
but independent of $\varepsilon$, $\theta_{\varepsilon}$ and
$g_{\varepsilon}$. Assume that $\theta_{\varepsilon}\rightharpoonup\theta$
weakly in $L^{2}(D)$ and $g_{\varepsilon}\rightharpoonup g$ weakly in
$H^{-\frac{1}{2}}(\partial D)$ as $\varepsilon\to 0$. Then
$v_{\varepsilon}\rightharpoonup v$ weakly in $H^{1}(D)$. Consequently,
$\lim_{\varepsilon\to 0}\|v_{\varepsilon}-v\|_{L^{2}(D)}=0.$
Here
1. $i)$
in the non-resonant case: $M=\\{0\\}$, $v\in H^{1}(D)$ is the unique solution
of the system
$\left\\{\begin{array}[]{ll}\operatorname{div}(a\nabla v)+k^{2}\sigma
v=\theta&\mbox{ in }D,\\\\[12.0pt] \displaystyle a\nabla v\cdot\eta=0&\mbox{
on }\partial D.\end{array}\right.$ (2.11)
2. $ii)$
in the resonant case: $M\neq\\{0\\}$, $v$ is the first component of the pair
$(v,w)\in W^{1}(\mathbb{R}^{3})\times M^{\perp}$ which is the unique solution
of the systems
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=\theta&\mbox{ in }D,\\\\[12.0pt] \operatorname{div}(a\nabla
w)+k^{2}\sigma w=0&\mbox{ in
}D,\end{array}\right.\quad\mbox{and}\quad\left\\{\begin{array}[]{ll}\displaystyle
a\nabla v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial D,\\\\[12.0pt]
\displaystyle\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
w\cdot\eta=g&\mbox{ on }\partial D.\end{array}\right.$ (2.12)
Here $M^{\perp}$ is defined by (1.8).
###### Remark 7.
The uniqueness of $(v,w)$ follows from Lemma 2. The existence of $(v,w)$ will
be proved in the proof of Lemma 3.
Proof. We first prove that
$\|v_{\varepsilon}\|_{L^{2}(B_{5})}\leq
C\big{(}\|\theta_{\varepsilon}\|_{L^{2}(D)}+\|g_{\varepsilon}\|_{H^{-\frac{1}{2}}(\partial
D)}\big{)},$ (2.13)
by contradiction. Suppose that there exist $(\theta_{n})\subset L^{2}(D)$ (we
also assume that $\displaystyle\int_{D}\theta_{n}\bar{\bf e}=0$ for any ${\bf
e}\in M$ in the resonant case), $(g_{n})\subset H^{-\frac{1}{2}}(\partial D)$
and $\varepsilon_{n}\to 0$ such that
$\|v_{n}\|_{L^{2}(B_{5})}=1\quad\mbox{and}\quad\lim_{n\rightarrow\infty}\big{(}\|\theta_{n}\|_{L^{2}(D)}+\|g_{n}\|_{H^{-\frac{1}{2}}(\partial
D)}\big{)}=0.$ (2.14)
Here $v_{n}\in H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
v_{n}+\varepsilon_{n}^{2}k^{2}v_{n}=0&\mbox{in
}\mathbb{R}^{3}\setminus\bar{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v_{n})+k^{2}\sigma v_{n}=\theta_{n}&\mbox{in }D,\\\\[12.0pt]
\displaystyle\frac{\partial
v_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon_{n}}a\nabla
v_{n}\cdot\eta\Big{|}_{\mathrm{int}}=g_{n}&\mbox{on }\partial D\\\\[12.0pt]
v_{n}\mbox{ satisfies the outgoing condition}.\end{array}\right.$ (2.15)
Multiplying system (2.15) by $\bar{v}_{n}$ (the conjugate of $v_{n}$) and
integrating the expression obtained over $B_{4}$, we have
$\int_{B_{4}\setminus D}|\nabla
v_{n}|^{2}-\varepsilon_{n}^{2}k^{2}\int_{B_{4}\setminus
D}|v_{n}|^{2}+\frac{1}{\varepsilon_{n}}\int_{D}\langle a\nabla v_{n},\nabla
v_{n}\rangle-\frac{1}{\varepsilon_{n}}\int_{D}k^{2}\sigma|v_{n}|^{2}\\\
=\int_{\partial B_{4}}\frac{\partial v_{n}}{\partial
r}\bar{v}_{n}-\int_{\partial
D}g_{n}\bar{v}_{n}-\frac{1}{\varepsilon_{n}}\int_{D}\theta_{n}\bar{v}_{n}.$
(2.16)
Applying Lemma 1, we deduce from (2.14) and (2.16) that
$\int_{D}|\nabla v_{n}|^{2}\leq C,$
which implies, since $\|v_{n}\|_{L^{2}(D)}\leq 1$,
$\|v_{n}\|_{H^{\frac{1}{2}}(\partial D)}\leq C.$ (2.17)
Using Lemma 1, from (2.14), (2.15), and (2.17), we have
$\|v_{n}\|_{H^{1}(B_{r})}\leq C_{r}\quad\forall\,r>0.$ (2.18)
Case 1: The non-resonant case: $M=\\{0\\}$.
From (2.18) and part $ii)$ of Lemma 1, one may assume that
$v_{n}\rightharpoonup v$ weakly in $H^{1}_{{}_{loc}}(\mathbb{R}^{3})$,
$v_{n}\to v$ in $L^{2}(B_{5})$ for some $v\in W^{1}(\mathbb{R}^{3})$ such that
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=0&\mbox{ in }D,\\\\[12.0pt] a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial D.\end{array}\right.$
Since $M=\\{0\\}$, it follows that
$v=0\quad\mbox{ in }D.$ (2.19)
Hence
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] v=0&\mbox{ on }\partial
D.\end{array}\right.$
Since $v\in W^{1}(\mathbb{R}^{3}\setminus\overline{D})$ (by Lemma 1), it
follows that (see e.g. [16, Theorem 2.5.14 on page 64])
$v=0\quad\mbox{ in }\mathbb{R}^{3}\setminus D.$ (2.20)
Combining (2.19) and (2.20) yields $v=0$. We have a contradiction since
$\|v\|_{L^{2}(B_{5})}=\lim_{n\to\infty}\|v_{n}\|_{L^{2}(B_{5})}=1$.
Case 2: The resonant case: $M\not=\\{0\\}$.
Since $\displaystyle\int_{D}\theta_{n}\bar{\bf e}=0$ for any ${\bf e}\in M$,
it follows from the Fredholm alternative theory (see e.g. [1] or [13]) that
there exists a unique solution $v_{1,n}\in M^{\perp}$ of the system
$\left\\{\begin{array}[]{ll}\operatorname{div}(a\nabla v_{1,n})+k^{2}\sigma
v_{1,n}=\theta_{n}&\mbox{ in }D,\\\\[12.0pt] a\nabla v_{1,n}\cdot\eta=0&\mbox{
on }\partial D.\end{array}\right.$ (2.21)
Moreover, we have
$\|v_{1,n}\|_{H^{1}(D)}\leq C\|\theta_{n}\|_{L^{2}(D)},$ (2.22)
for some positive constant $C$ independent of $\theta_{n}$. Let $v_{2,n}$ be
the projection of $v_{n}-v_{1,n}$ into $M$ i.e. $v_{2,n}\in M$ and
$v_{n}-v_{1,n}-v_{2,n}\in M^{\perp}$. Set
$w_{n}=\frac{1}{\varepsilon_{n}}(v_{n}-v_{1,n}-v_{2,n})\;(\in M^{\perp}).$
(2.23)
Then
$a\nabla v_{n}\cdot\eta\Big{|}_{\mathrm{int}}=\varepsilon_{n}a\nabla
w_{n}\cdot\eta\quad\mbox{ on }\partial D.$ (2.24)
We deduce from (2.15) and the definition of $M$ that
$\left\\{\begin{array}[]{cl}\operatorname{div}(a\nabla w_{n})+k^{2}\sigma
w_{n}=0&\mbox{ in }D,\\\\[12.0pt] \displaystyle a\nabla
w_{n}\cdot\eta=\frac{\partial
v_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-g_{n}&\mbox{ on }\partial
D.\end{array}\right.$ (2.25)
Combining (2.14), (2.18), and (2.25) yields
$\|a\nabla w_{n}\cdot\eta\|_{H^{-\frac{1}{2}}(\partial
D)}\leq\Big{\|}\frac{\partial
v_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}\Big{\|}_{H^{-\frac{1}{2}}(\partial
D)}+\|g_{n}\|_{H^{-\frac{1}{2}}(\partial D)}\leq C.$
Since $\operatorname{div}(a\nabla w_{n})+k^{2}\sigma w_{n}=0$ in $D$ and
$w_{n}\in M^{\perp}$, it follows that
$\|w_{n}\|_{H^{1}(D)}\leq C.$ (2.26)
From (2.18), (2.26), and the fact that $w_{n}\in M^{\perp}$, one may assume
that
$\left\\{\begin{array}[]{l}v_{n}\rightharpoonup v\mbox{ weakly in
}H^{1}_{{}_{loc}}(\mathbb{R}^{3}),\\\\[12.0pt] w_{n}\rightharpoonup w\in
M^{\perp}\mbox{ weakly in }H^{1}(D).\end{array}\right.$
By Lemma 1, it follows from (2.14), (2.15), (2.24), and (2.25) that $(v,w)\in
W^{1}(\mathbb{R}^{3})\times M^{\perp}$ is a solution of the system
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=0&\mbox{ in }D,\\\\[12.0pt] \operatorname{div}(a\nabla
w)+k^{2}\sigma w=0&\mbox{ in
}D,\end{array}\right.\quad\mbox{and}\quad\left\\{\begin{array}[]{ll}\displaystyle
a\nabla v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial D,\\\\[12.0pt]
\displaystyle\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
w\cdot\eta&\mbox{ on }\partial D.\end{array}\right.$
We deduced from Lemma 2 that $v=0$. We have a contradiction since
$\|v\|_{L^{2}(B_{5})}=1$.
From Cases 1 and 2, (2.13) is proved. Hence we obtain (2.10) by using (2.13)
and applying the same arguments used to get (2.18). The conclusion of second
part is a consequence of (2.10) and can be processed as follows. From (2.10),
one may assume that $v_{\varepsilon}\rightharpoonup v$ weakly in $H^{1}(D)$
(up to a subsequence). In the non-resonant case, $v\in H^{1}(D)$ is a solution
of the system (2.11). Since (2.11) has a unique solution $v\in H^{1}(D)$, the
conclusion in this case holds for the whole family. In the resonant case,
applying the same decomposition as in (2.23) and using the same facts as in
(2.15), (2.24), and (2.25), one may assume that (up to a subsequence)
$\left\\{\begin{array}[]{l}v_{\varepsilon}\rightharpoonup v\in
W^{1}(\mathbb{R}^{3})\mbox{ weakly in
}H^{1}_{{}_{loc}}(\mathbb{R}^{3}),\\\\[12.0pt] w_{\varepsilon}\rightharpoonup
w\in M^{\perp}\mbox{ weakly in }H^{1}(D),\end{array}\right.$
and $(v,w)\in W^{1}(\mathbb{R}^{3})\times M^{\perp}$ satisfies system (2.12).
Since system (2.12) has a unique solution in $(v,w)\in
W^{1}(\mathbb{R}^{3})\times M^{\perp}$ (by Lemma 2), the conclusion holds for
the whole family. $\Box$
### 2.2 Proof of Theorem 3.
We follow the method used in the proof of [17, Theorem 2.1]. Let
$u_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ be the unique solution
of
$\left\\{\begin{array}[]{ll}\Delta
u_{1,\varepsilon}+k^{2}u_{1,\varepsilon}=f\leavevmode\nobreak\
(={F_{\varepsilon}^{-1}}_{*}f)&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
u_{1,\varepsilon}=0&\mbox{ in }B_{\varepsilon}\\\\[12.0pt]
u_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
Define
$w_{1,\varepsilon}=u_{1,\varepsilon}-u,$
and
$w_{2,\varepsilon}=u_{\varepsilon}-u_{1,\varepsilon}.$
We claim that
$\|w_{1,\varepsilon}\|_{H^{1}(B_{4r}\setminus B_{r})}\leq
C_{r}\varepsilon\|f\|_{L^{2}}$ (2.27)
and
$\|w_{2,\varepsilon}\|_{H^{1}(B_{4r}\setminus B_{r})}\leq
C_{r}\varepsilon\|f\|_{L^{2}}.$ (2.28)
Proof of Claim (2.27). From the definition of $w_{1,\varepsilon}$, it follows
that $w_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ and
$w_{1,\varepsilon}$ satisfies
$\left\\{\begin{array}[]{ll}\Delta
w_{1,\varepsilon}+k^{2}w_{1,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
w_{1,\varepsilon}=-u&\mbox{ in }\partial B_{\varepsilon}\\\\[12.0pt]
w_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
(2.29)
Define $W_{1,\varepsilon}(x)=w_{1,\varepsilon}(\varepsilon x)$. Then
$W_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ and
$W_{1,\varepsilon}$ satisfies
$\left\\{\begin{array}[]{ll}\Delta
W_{1,\varepsilon}+\varepsilon^{2}k^{2}W_{1,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
W_{1,\varepsilon}=-u(\varepsilon\cdot)&\mbox{ in }\partial B_{1}\\\\[12.0pt]
W_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
(2.30)
Since $\|u(\varepsilon\cdot)\|_{H^{\frac{1}{2}}(\partial
B_{1})}\leq\|f\|_{L^{2}}$, by Lemma 1, we have
$\varepsilon\int_{B_{4r/\varepsilon}\setminus
B_{r/\varepsilon}}|W_{1,\varepsilon}|^{2}\leq C_{r}\|f\|_{L^{2}}^{2}.$
By a change of variables,
$\int_{B_{4r}\setminus B_{r}}|w_{1,\varepsilon}|^{2}\leq
C_{r}\varepsilon^{2}\|f\|_{L^{2}}^{2}.$ (2.31)
Since $\Delta w_{1,\varepsilon}+k^{2}w_{1,\varepsilon}=0$ in
$\mathbb{R}^{3}\setminus\bar{B}_{\varepsilon}$, by the regularity theory of
elliptic equations, Claim (2.27) follows from (2.31).
Proof of Claim (2.28). It is clear that $w_{2,\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
w_{2,\varepsilon}+k^{2}w_{2,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
\operatorname{div}(A_{\varepsilon}\nabla
w_{2,\varepsilon})+k^{2}\Sigma_{\varepsilon}w_{2,\varepsilon}={F_{\varepsilon}^{-1}}_{*}f&\mbox{
in }B_{\varepsilon},\\\\[12.0pt] \displaystyle\frac{\partial
w_{2,\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-A_{\varepsilon}\nabla
w_{2,\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=-\frac{\partial
u_{1,\varepsilon}}{\partial\eta}&\mbox{ on }\partial
B_{\varepsilon},\\\\[12.0pt] w_{2,\varepsilon}\mbox{ satisfies the outgoing
condition.}&\end{array}\right.$
Define $W_{2,\varepsilon}(x)=w_{2,\varepsilon}(\varepsilon x)$. Then
$W_{2,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ and
$W_{2,\varepsilon}$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
W_{2,\varepsilon}+\varepsilon^{2}k^{2}W_{2,\varepsilon}=0&\mbox{in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla W_{2,\varepsilon})+k^{2}\sigma
W_{2,\varepsilon}=f&\mbox{in }B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
W_{2,\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon}a\nabla
W_{2,\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=-\varepsilon\frac{\partial
u_{1,\varepsilon}}{\partial\eta}(\varepsilon x)&\mbox{on }\partial
B_{1},\\\\[12.0pt] W_{2,\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
Since $u_{1,\varepsilon}=w_{1,\varepsilon}+u$ and
$W_{1,\varepsilon}=w_{1,\varepsilon}(\varepsilon\cdot)$, it follows that
$\varepsilon\frac{\partial u_{1,\varepsilon}}{\partial\eta}(\varepsilon
x)=\frac{\partial
W_{1,\varepsilon}}{\partial\eta}(x)+\varepsilon\frac{\partial
u}{\partial\eta}(\varepsilon x)\quad\mbox{ on }\partial B_{1}.$ (2.32)
From (2.30) and Lemma 1, we obtain
$\Big{\|}\varepsilon\frac{\partial
u_{1,\varepsilon}}{\partial\eta}(\varepsilon
x)\Big{\|}_{H^{-\frac{1}{2}}(\partial B_{1})}\leq C\|f\|_{L^{2}}.$
We deduce from Lemma 3 that
$\|W_{2,\varepsilon}\|_{H^{1}(B_{5})}\leq C\|f\|_{L^{2}}.$
Applying Lemma 1, we have
$\varepsilon\int_{B_{4r/\varepsilon}\setminus
B_{r/\varepsilon}}|W_{2,\varepsilon}|^{2}\leq C_{r}\|f\|_{L^{2}}^{2}.$
By a change of variables,
$\int_{B_{4r}\setminus B_{r}}|w_{2,\varepsilon}|^{2}\leq
C_{r}\varepsilon^{2}\|f\|_{L^{2}}^{2}.$
Claim (2.28) now follows from the regularity theory of elliptic equations and
the fact that $\Delta w_{2,\varepsilon}+k^{2}w_{2,\varepsilon}=0$ in
$\mathbb{R}^{3}\setminus\bar{B}_{\varepsilon}$.
Thus Claims (2.27) and (2.28) are proved. Since
$u_{\varepsilon}-u=w_{1,\varepsilon}+w_{2,\varepsilon}$, (2.1) follows.
The rest of the proof goes as follows. Since $u(\varepsilon\cdot)\rightarrow
u(0)$, it follows from (2.30) and Lemma 1 that
$W_{1,\varepsilon}\rightharpoonup W_{1}$ weakly in
$H^{1}_{{}_{loc}}(\mathbb{R}^{3}\setminus\overline{B}_{1})$ where $W_{1}\in
W^{1}(\mathbb{R}^{3}\setminus\overline{B}_{1})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta W_{1}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt] W_{1}=-u(0)&\mbox{ in
}\partial B_{1}.\end{array}\right.$ (2.33)
Case 1: The non-resonant case: $M=\\{0\\}$. By Lemma 3,
$W_{2,\varepsilon}\rightharpoonup\hat{v}$ weakly in $H^{1}(B)$ where $\hat{v}$
is the unique solution of the system
$\left\\{\begin{array}[]{ll}\operatorname{div}(a\nabla\hat{v})+k^{2}\sigma\hat{v}=f&\mbox{
in }B_{1},\\\\[12.0pt] \displaystyle
a\nabla\hat{v}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial
B_{1}.\end{array}\right.$
Therefore, the conclusion follows in this case.
Case 2: The resonant case: $M\neq\\{0\\}$. By Lemma 3, it follows from (2.32)
that $W_{2,\varepsilon}\rightharpoonup\hat{v}$ weakly in
$H^{1}_{{}_{loc}}(\mathbb{R}^{3})$ where $\hat{v}$ is the first component of
the pair $(\hat{v},\hat{w})\in W^{1}(\mathbb{R}^{3})\times M^{\perp}$ which is
the unique solution of the systems
$\left\\{\begin{array}[]{ll}\Delta\hat{v}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla\hat{v})+k^{2}\sigma\hat{v}=f&\mbox{ in
}B_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla\hat{w})+k^{2}\sigma\hat{w}=0&\mbox{ in
}B_{1},\end{array}\right.\mbox{ and }\left\\{\begin{array}[]{ll}\displaystyle
a\nabla\hat{v}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{ on }\partial
B_{1},\\\\[12.0pt]
\displaystyle\frac{\partial\hat{v}}{\partial\eta}\Big{|}_{\mathrm{ext}}+\frac{\partial
W_{1}}{\partial\eta}=a\nabla\hat{w}\cdot\eta&\mbox{ on }\partial
B_{1}.\end{array}\right.$ (2.34)
Hence $\hat{v}$ depends on $u(0)$ through $W_{1}$ (see the transmission
condition in (2.34) and the system of $W_{1}$ in (2.33)). Define
$v_{\mathrm{ext}}=\hat{v}+W_{1}$ for
$x\in\mathbb{R}^{3}\setminus\overline{B}_{1}$, $v_{\mathrm{int}}=\hat{v}$ if
$x\in B_{1}$, and $w=\hat{w}$ if $x\in B_{1}$. It follows from (2.33) and
(2.34) that the triple $(v_{\mathrm{ext}},v_{\mathrm{int}},w)\in
W^{1}(\mathbb{R}^{3}\setminus\overline{B}_{1})\times H^{1}(B_{1})\times
M^{\perp}$ is the unique solution of the systems
$\left\\{\begin{array}[]{ll}\Delta v_{\mathrm{ext}}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla v_{\mathrm{int}})+k^{2}\sigma
v_{\mathrm{int}}=f&\mbox{ in }B_{1},\\\\[12.0pt] \operatorname{div}(a\nabla
w)+k^{2}\sigma w=0&\mbox{ in }B_{1},\end{array}\right.\mbox{ and
}\left\\{\begin{array}[]{ll}\displaystyle
v_{\mathrm{ext}}-v_{\mathrm{int}}=-u(0)&\mbox{ on }\partial B_{1},\\\\[12.0pt]
\displaystyle a\nabla v_{\mathrm{int}}\cdot\eta=0&\mbox{ on }\partial
B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
v_{\mathrm{ext}}}{\partial\eta}=a\nabla w\cdot\eta&\mbox{ on }\partial
B_{1}.\end{array}\right.$
Since $U_{\varepsilon}=W_{2,\varepsilon}$ in $B_{1}$ and
$W_{2,\varepsilon}\rightharpoonup\hat{v}=v_{\mathrm{int}}=Cl_{3}(f)$ in
$H^{1}(B_{1})$, we obtain the conclusion in this case. $\Box$
## 3 Proofs of Theorem 2 and Proposition 1
According to Proposition 5, Theorem 2 is a consequence of the following
###### Theorem 4.
Let $d=2$, $k>0$, and $0<\varepsilon<1$. Define
$A_{\varepsilon},\Sigma_{\varepsilon}=\left\\{\begin{array}[]{cl}I,1&\mbox{if
}x\in\mathbb{R}^{2}\setminus B_{\varepsilon},\\\\[12.0pt] \displaystyle
a(x/\varepsilon),\frac{1}{\varepsilon^{2}}\sigma(x/\varepsilon)&\mbox{if }x\in
B_{\varepsilon}.\end{array}\right.$
Let $u_{\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ be the unique
solution of
$\left\\{\begin{array}[]{ll}\operatorname{div}(A_{\varepsilon}\nabla
u_{\varepsilon})+k^{2}\Sigma_{\varepsilon}u_{\varepsilon}=F_{\varepsilon}^{-1}*f&\mbox{
in }\mathbb{R}^{2},\\\\[12.0pt] u_{\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
Assume the system is non-resonant 101010The non-resonant system is defined in
Definition 2. We have
$\|u_{\varepsilon}-u\|_{H^{1}(B_{4r}\setminus
B_{r})}\leq\frac{C}{|\ln\varepsilon|}\|f\|_{L^{2}},$ (3.1)
for some $C=C(r,k,a,\sigma)$. Moreover, if
$U_{\varepsilon}(x)=u_{\varepsilon}(\varepsilon x)$, then
$U_{\varepsilon}\rightharpoonup Cl_{2}(f)$ weakly in $H^{1}(B_{1})$.
Consequently,
$\lim_{\varepsilon\to 0}\|U_{\varepsilon}-Cl_{2}(f)\|_{L^{2}(B_{1})}=0.$
In the rest of this section, we present the proofs of Theorem 4 and
Proposition 1.
### 3.1 Preliminaries
In this section we present some lemmas which will be used in the proof of
Theorem 4. Instead of dealing with the unit ball $B_{1}$, we will present
results for a smooth open subset $D$ of $\mathbb{R}^{2}$. We will also assume
that $D\subset B_{1}$ and $\mathbb{R}^{2}\setminus D$ is connected. We first
“recall” the following result [17, Lemma 2.2] which will be useful in our
analysis.
###### Lemma 4.
Let $0<\varepsilon<1$, and $g_{\varepsilon}\in H^{\frac{1}{2}}(\partial D)$.
Assume that $\mathbb{R}^{2}\setminus D$ is connected and $v_{\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
v_{\varepsilon}+\varepsilon^{2}v_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{D},\\\\[12.0pt]
v_{\varepsilon}=g_{\varepsilon}&\mbox{on }\partial D\\\\[12.0pt]
v_{\varepsilon}\mbox{ satisfies the outgoing condition}.\end{array}\right.$
1. $i)$
We have
$\|v_{\varepsilon}\|_{H^{1}(B_{r}\setminus D)}\leq
C_{r}\|g_{\varepsilon}\|_{H^{\frac{1}{2}}(\partial D)}\quad\forall\,r>5,$
(3.2)
and
$\|v_{\varepsilon}\|_{L^{2}(B_{4r/\varepsilon}\setminus
B_{r/\varepsilon})}\leq\frac{C_{r}}{\varepsilon|\log\varepsilon|}\|g_{\varepsilon}\|_{H^{\frac{1}{2}}(\partial
D)},$ (3.3)
for some positive constants $C_{r}=C(r,k,D)$.
2. $ii)$
Assume that $g_{\varepsilon}\rightharpoonup g$ weakly in
$H^{\frac{1}{2}}(\partial D)$. Then $v_{\varepsilon}\rightharpoonup v$ weakly
in $H^{1}_{{}_{loc}}(\mathbb{R}^{2})$, where $v\in
W^{1}(\mathbb{R}^{2}\setminus D)$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{D},\\\\[12.0pt] v=g&\mbox{ on }\partial
D.\end{array}\right.$ (3.4)
Proof. Inequalities (3.2), and (3.3) with $r=1$ are in [17, Lemma 2.2]
111111There is a typo in [17, (2.4)] for $d=2$ where the term
$\varepsilon^{1/2}$ must be replaced by $\varepsilon$.. The proof of (3.3) in
the general case follows in the same manner. To prove part $ii)$ we process as
follows. Since $(v_{\varepsilon})$ is bounded in
$H^{1}_{{}_{loc}}(\mathbb{R}^{2}\setminus D)$, one may assume that
$v_{\varepsilon}\rightharpoonup v$ weakly in
$H^{1}_{{}_{loc}}(\mathbb{R}^{2}\setminus D)$ (up to a subsequence). Then
$v\in W^{1}(\mathbb{R}^{2}\setminus\bar{D})$. This fact is not stated in [17,
Lemma 2.2] however the proof is already there (see [17, (2.20) and (2.22)]).
It is clear that $v$ satisfies (3.4). Since (3.4) has a unique solution $v\in
W^{1}(\mathbb{R}^{2}\setminus\bar{D})$ (see e.g. [16, Theorem 2.5.14 and the
remarks on page 64]) the conclusion holds for the whole sequence. The details
of the proof are left to the reader. $\Box$
The following lemma plays an important role in the proof of Theorem 4.
###### Lemma 5.
Let $0<\varepsilon<1$, $k>0$, $\theta_{\varepsilon}\in L^{2}(D)$,
$g_{\varepsilon}\in H^{-\frac{1}{2}}(\partial D)$, $a$ be a real symmetric
matrix-valued function, and $\sigma$ be a complex function defined on $D$.
Assume that $a$ is uniformly elliptic,
$0<\mathrm{ess}\,\inf\Re\sigma\leq\mathrm{ess}\,\sup\Re\sigma<+\infty$ and
$0\leq\mathrm{ess}\,\inf\Im\sigma\leq\mathrm{ess}\,\sup\Im\sigma<+\infty$, and
the system is non-resonant 121212The definition of the non-resonance in this
case is the same as the case corresponding to the unit ball $B_{1}$ i.e. if
$w\in W^{1}(\mathbb{R}^{2})$ satisfies (1.9) where $B_{1}$ is replaced by $D$,
then $w=0$.. Let $v_{\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ be the
unique solution of
$\left\\{\begin{array}[]{ll}\Delta
v_{\varepsilon}+\varepsilon^{2}k^{2}v_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v_{\varepsilon})+k^{2}\sigma v_{\varepsilon}=\theta_{\varepsilon}&\mbox{in
}D,\\\\[12.0pt] \displaystyle\frac{\partial
v_{\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
v_{\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=g_{\varepsilon}&\mbox{on
}\partial D\\\\[12.0pt] v_{\varepsilon}\mbox{ satisfies the outgoing
condition}.\end{array}\right.$
Then
1. $i)$
We have
$\|v_{\varepsilon}\|_{H^{1}(B_{5}\setminus D)}\leq
C(\|\theta_{\varepsilon}\|_{L^{2}(D)}+\|g_{\varepsilon}\|_{H^{-\frac{1}{2}}(\partial
D)}),$
for some positive constant $C$ depending only on $k$, $a$, $\sigma$ and $D$
but independent of $\varepsilon$, $\theta_{\varepsilon}$, and
$g_{\varepsilon}$.
2. $ii)$
Assume in addition that $\theta_{\varepsilon}\rightharpoonup\theta$ weakly in
$L^{2}(D)$ and $g_{\varepsilon}\rightharpoonup g$ weakly in
$H^{-\frac{1}{2}}(\partial D)$ as $\varepsilon\to 0$. Then
$v_{\varepsilon}\rightharpoonup v$ weakly in $H^{1}(D)$. Consequently,
$\lim_{\varepsilon\to 0}\|v_{\varepsilon}-v\|_{L^{2}(D)}=0.$
Here $v\in W^{1}(\mathbb{R}^{2})$ is the unique solution of the system
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=\theta&\mbox{in }D,\\\\[12.0pt] \displaystyle\frac{\partial
v}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}=g&\mbox{on }\partial D.\end{array}\right.$
(3.5)
Proof. We first prove that
$\|v_{\varepsilon}\|_{L^{2}(B_{5})}\leq
C\big{(}\|\theta_{\varepsilon}\|_{L^{2}(D)}+\|g_{\varepsilon}\|_{H^{-\frac{1}{2}}(\partial
D)}\big{)},$ (3.6)
by contradiction. Suppose that there exist $(\theta_{n})\subset L^{2}(D)$,
$(g_{n})\subset H^{-\frac{1}{2}}(\partial D)$ and $(\varepsilon_{n})$ such
that $(\varepsilon_{n})$ converges to $0$,
$\|v_{n}\|_{L^{2}(B_{5})}=1\quad\mbox{and}\quad\lim_{n\rightarrow\infty}\big{(}\|\theta_{n}\|_{L^{2}(D)}+\|g_{n}\|_{H^{-\frac{1}{2}}(\partial
D)}\big{)}=0.$ (3.7)
Here $v_{n}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ is the unique solution of the
system
$\left\\{\begin{array}[]{ll}\Delta
v_{n}+\varepsilon_{n}^{2}k^{2}v_{n}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v_{n})+k^{2}\sigma v_{n}=\theta_{n}&\mbox{in }D,\\\\[12.0pt]
\displaystyle\frac{\partial v_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
v_{n}\cdot\eta\Big{|}_{\mathrm{int}}=g_{n}&\mbox{on }\partial D\\\\[12.0pt]
v_{n}\mbox{ satisfies the outgoing condition}.\end{array}\right.$ (3.8)
Applying Lemma 4 and the regularity theory of elliptic equations, we deduce
from (3.7) that
$\|v_{n}\|_{H^{1}(B_{r}\setminus B_{3})}\leq C_{r}\quad\forall\,r>5.$ (3.9)
Multiplying system (3.8) with $\bar{v}_{n}$ (the conjugate of $v_{n}$) and
integrating the expression obtained over $B_{4}$, we have
$\int_{B_{4}\setminus D}|\nabla
v_{n}|^{2}-\varepsilon_{n}^{2}k^{2}\int_{B_{4}\setminus
D}|v_{n}|^{2}+\int_{D}\langle a\nabla v_{n},\nabla
v_{n}\rangle-\int_{D}k^{2}\sigma|v_{n}|^{2}\\\\[12.0pt]
=-\int_{D}\theta_{n}\bar{v}_{n}+\int_{\partial B_{4}}\frac{\partial
v_{n}}{\partial r}\bar{v}_{n}-\int_{\partial D}g_{n}\bar{v}_{n}.$
From (3.7) and (3.9), it follows that
$\int_{B_{4}}|\nabla v_{n}|^{2}\leq C.$ (3.10)
Combining (3.7), (3.9), and (3.10) yields
$\|v_{n}\|_{H^{1}(B_{r})}\leq C_{r}\quad\forall\,r>0.$ (3.11)
Hence without loss of generality, one may assume that $v_{n}\rightharpoonup v$
weakly in $H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ and $v_{n}\to v$ in
$L^{2}_{{}_{loc}}(\mathbb{R}^{2})$, where $v\in W^{1}(\mathbb{R}^{2})$ (by
Lemma 4) is a solution of the system
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{in
}\mathbb{R}^{2}\setminus\bar{D},\\\\[12.0pt] \operatorname{div}(a\nabla
v)+k^{2}\sigma v=0&\mbox{in }D,\\\\[12.0pt] \displaystyle\frac{\partial
v}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial D.\end{array}\right.$
Since the system is non-resonant, $v=0$. We have a contradiction since
$\|v\|_{L^{2}(B_{5})}=\lim_{n\to\infty}\|v_{n}\|_{L^{2}(B_{5})}=1.$
Hence (3.6) is proved. Using the same argument to obtain (3.11), we deduce
part $i)$ of the conclusion from (3.6). Part $ii)$ of the conclusion follows
from part $i)$ and the uniqueness of system (3.5). $\Box$
### 3.2 Proof of Theorem 4.
The proof of Theorem 4 is quite similar to the one of Theorem 3. However
instead of using Lemmas 1, 2, and 3, we apply Lemmas 4 and 5. For the
convenience of the reader, we present the proof.
Let $u_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ be the unique
solution of
$\left\\{\begin{array}[]{ll}\Delta
u_{1,\varepsilon}+k^{2}u_{1,\varepsilon}=f\leavevmode\nobreak\
(={F_{\varepsilon}^{-1}}_{*}f)&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
u_{1,\varepsilon}=0&\mbox{ in }B_{\varepsilon}\\\\[12.0pt]
u_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
Define
$w_{1,\varepsilon}=u_{1,\varepsilon}-u,$
and
$w_{2,\varepsilon}=u_{\varepsilon}-u_{1,\varepsilon}.$
We claim that
$\|w_{1,\varepsilon}\|_{H^{1}(B_{4r}\setminus
B_{r})}\leq\frac{C_{r}}{|\ln\varepsilon|}\|f\|_{L^{2}}$ (3.12)
and
$\|w_{2,\varepsilon}\|_{H^{1}(B_{4r}\setminus
B_{r})}\leq\frac{C_{r}}{|\ln\varepsilon|}\|f\|_{L^{2}}.$ (3.13)
Step 1: Proof of Claim (3.12). From the definition of $w_{1,\varepsilon}$, it
follows that $w_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ and
$w_{1,\varepsilon}$ satisfies
$\left\\{\begin{array}[]{ll}\Delta
w_{1,\varepsilon}+k^{2}w_{1,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
w_{1,\varepsilon}=-u&\mbox{ in }\partial B_{\varepsilon}\\\\[12.0pt]
w_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
Define $W_{1,\varepsilon}(x)=w_{1,\varepsilon}(\varepsilon x)$. Then
$W_{1,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ and
$W_{1,\varepsilon}$ satisfies
$\left\\{\begin{array}[]{ll}\Delta
W_{1,\varepsilon}+\varepsilon^{2}k^{2}W_{1,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
W_{1,\varepsilon}=-u(\varepsilon\cdot)&\mbox{ in }\partial B_{1}\\\\[12.0pt]
W_{1,\varepsilon}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
(3.14)
Since $\|u(\varepsilon\cdot)\|_{H^{\frac{1}{2}}(\partial
B_{1})}\leq\|f\|_{L^{2}}$, by Lemma 4, we have
$\varepsilon^{2}\int_{B_{4r/\varepsilon}\setminus
B_{r/\varepsilon}}|W_{1,\varepsilon}|^{2}\leq\frac{C_{r}}{|\ln\varepsilon|^{2}}\|f\|_{L^{2}}^{2}.$
By a change of variables,
$\int_{B_{4r}\setminus
B_{r}}|w_{1,\varepsilon}|^{2}\leq\frac{C_{r}}{|\ln\varepsilon|^{2}}\|f\|_{L^{2}}^{2}.$
(3.15)
Since $\Delta w_{1,\varepsilon}+k^{2}w_{1,\varepsilon}=0$ in
$\mathbb{R}^{2}\setminus\bar{B}_{\varepsilon}$, by the regularity theory of
elliptic equations, Claim (3.12) follows from (3.15).
Step 2: Proof of Claim (3.13). It is clear that $w_{2,\varepsilon}\in
H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
w_{2,\varepsilon}+k^{2}w_{2,\varepsilon}=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{\varepsilon},\\\\[12.0pt]
\operatorname{div}(A_{\varepsilon}\nabla
w_{2,\varepsilon})+k^{2}\Sigma_{\varepsilon}w_{2,\varepsilon}={F_{\varepsilon}^{-1}}_{*}f&\mbox{
in }B_{\varepsilon},\\\\[12.0pt] \displaystyle\frac{\partial
w_{2,\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-A_{\varepsilon}\nabla
w_{2,\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=-\frac{\partial
u_{1,\varepsilon}}{\partial\eta}&\mbox{ on }\partial
B_{\varepsilon},\\\\[12.0pt] w_{2,\varepsilon}\mbox{ satisfies the outgoing
condition.}&\end{array}\right.$
Define $W_{2,\varepsilon}(x)=w_{2,\varepsilon}(\varepsilon x)$. Then
$W_{2,\varepsilon}\in H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ and
$W_{2,\varepsilon}$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta
W_{2,\varepsilon}+\varepsilon^{2}k^{2}W_{2,\varepsilon}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla W_{2,\varepsilon})+k^{2}\sigma
W_{2,\varepsilon}=f&\mbox{in }B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
W_{2,\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
W_{2,\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=-\varepsilon\frac{\partial
u_{1,\varepsilon}}{\partial\eta}(\varepsilon x)&\mbox{on }\partial
B_{1},\\\\[12.0pt] W_{2,\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
Since $u_{1,\varepsilon}=w_{1,\varepsilon}+u$ and
$W_{1,\varepsilon}=w_{1,\varepsilon}(\varepsilon\cdot)$, it follows that
$\varepsilon\frac{\partial u_{1,\varepsilon}}{\partial\eta}(\varepsilon
x)=\frac{\partial
W_{1,\varepsilon}}{\partial\eta}(x)+\varepsilon\frac{\partial
u}{\partial\eta}(\varepsilon x)\quad\mbox{ on }\partial B_{1}.$ (3.16)
From (3.14) and Lemma 4, we obtain
$\Big{\|}\varepsilon\frac{\partial
u_{1,\varepsilon}}{\partial\eta}(\varepsilon
x)\Big{\|}_{H^{-\frac{1}{2}}(\partial B_{1})}\leq C\|f\|_{L^{2}}.$
We deduce from Lemma 5 that
$\|W_{2,\varepsilon}\|_{H^{1}(B_{5})}\leq C\|f\|_{L^{2}}.$
Applying Lemma 4, we have
$\varepsilon^{2}\int_{B_{4r/\varepsilon}\setminus
B_{r/\varepsilon}}|W_{2,\varepsilon}|^{2}\leq\frac{C_{r}}{|\ln\varepsilon|^{2}}\|f\|_{L^{2}}^{2}.$
By a change of variables,
$\int_{B_{4r}\setminus
B_{r}}|w_{2,\varepsilon}|^{2}\leq\frac{C_{r}}{|\ln\varepsilon|^{2}}\|f\|_{L^{2}}^{2}.$
Claim (3.13) now follows from the regularity theory of elliptic equations and
the fact that $\Delta w_{2,\varepsilon}+k^{2}w_{2,\varepsilon}=0$ in
$\mathbb{R}^{2}\setminus\bar{B}_{\varepsilon}$.
Thus Claims (3.12) and (3.13) are proved. Since
$u_{\varepsilon}-u=w_{1,\varepsilon}+w_{2,\varepsilon}$, (3.1) follows.
The rest of the proof now follows from the following facts:
1. $a)$
$W_{1,\varepsilon}\rightharpoonup W_{1}=-u(0)$ weakly in
$H^{1}_{{}_{loc}}(\mathbb{R}^{2}\setminus\overline{B}_{1})$ since $W_{1}\in
W^{1}(\mathbb{R}^{2}\setminus\overline{B}_{1})$ is the unique solution of
$\left\\{\begin{array}[]{ll}\Delta W_{1}=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt] W_{1}=-u(0)&\mbox{ in
}\partial B_{1}.\end{array}\right.$ (3.17)
by Lemma 4.
2. $b)$
$W_{2,\varepsilon}\rightharpoonup v$ weakly in
$W^{1}_{{}_{loc}}(\mathbb{R}^{2})$ where $v$ is the unique solution of the
system
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{ in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla v)+k^{2}\sigma v=f&\mbox{ in }B_{1},\\\\[12.0pt]
\displaystyle\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
v\cdot\eta&\mbox{ on }\partial B_{1},\end{array}\right.$
since $W_{1}=-u(0)$ in $\mathbb{R}^{2}\setminus B_{1}$ by statement a), and
(3.16) holds.
3. $c)$
$U_{\varepsilon}=W_{2,\varepsilon}$ in $B_{1}$ and $v=Cl_{2}(f)$.
$\Box$
### 3.3 Proof of Proposition 1
Proposition 1 is a consequence of the following lemma.
###### Lemma 6.
Let $k>0$, $0<c_{1}<c_{2}<\infty$, $\theta\in L^{2}(B_{1})$ and $g\in
H^{\frac{1}{2}}(\partial B_{1})$. Assume that $c_{1}|\xi|^{2}\leq\langle
a\xi,\xi\rangle\leq c_{2}|\xi|^{2}$, $0\leq\Im(\sigma)\leq c_{2}$,
$c_{1}<\Re(\sigma)<c_{2}$ on $B_{1}$. Then there exists $k_{0}>0$ depending
only on $c_{1}$, and $c_{2}$ such that if $0<k<k_{0}$ then there exists a
unique solution $v\in W^{1}(\mathbb{R}^{2})$ of the system
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla v)+k^{2}\sigma v=\theta&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
v}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}+g&\mbox{on }\partial
B_{1}.\end{array}\right.$ (3.18)
Proof. We only prove the uniqueness of $v$. The existence of $v$ follows from
the uniqueness. For this end, it suffices to prove that $v=0$ if $v\in
W^{1}(\mathbb{R}^{2})$ is a solution of
$\left\\{\begin{array}[]{ll}\Delta v=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla v)+k^{2}\sigma v=0&\mbox{in }B_{1},\\\\[12.0pt]
\displaystyle\frac{\partial v}{\partial\eta}\Big{|}_{\mathrm{ext}}=a\nabla
v\cdot\eta\Big{|}_{\mathrm{int}}&\mbox{on }\partial B_{1}.\end{array}\right.$
(3.19)
Since $v\in W^{1}(\mathbb{R}^{2})$, it follows from (3.19) that (see e.g. [16]
or [18, Proposition 2])
$\int_{\partial B_{1}}\frac{\partial
v}{\partial\eta}\Big{|}_{\mathrm{ext}}=0\;$ (3.20)
and
$\int_{\mathbb{R}^{2}\setminus B_{1}}|\nabla v|^{2}+\int_{B_{1}}\langle
a\nabla v,\nabla v\rangle-\int_{B_{1}}k^{2}\sigma|v|^{2}=0.$ (3.21)
Integrating the equation $\operatorname{div}(a\nabla v)+k^{2}\sigma v=0$ on
$B_{1}$ and using (3.20), we have
$\int_{B_{1}}\sigma v=0,$ (3.22)
which implies, by a standard compactness argument,
$\|\nabla v\|_{L^{2}(B_{1})}\geq C\|v\|_{L^{2}(B_{1})},$ (3.23)
for some positive constant $C$ depending only on $c_{1}$ and $c_{2}$.
Combining (3.21) and (3.23) yields
$\int_{\mathbb{R}^{2}\setminus B_{1}}|\nabla v|^{2}+\int_{B_{1}}\langle
a\nabla v,\nabla v\rangle=0$
if $0<k<k_{0}$ for some $k_{0}>0$. This implies that $v$ is constant. Hence
$v=0$ by (3.22). $\Box$
## 4 Proof of Proposition 2
Define $U_{\varepsilon}=u_{\varepsilon}(\varepsilon x)$ where
$u_{\varepsilon}=u_{c}\circ F_{\varepsilon}$. Then $U_{\varepsilon}$ is the
solution of
$\left\\{\begin{array}[]{ll}\Delta
U_{\varepsilon}+\varepsilon^{2}k^{2}U_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla U_{\varepsilon})+k^{2}\sigma U_{\varepsilon}={\bf
e}&\mbox{in }B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon}a\nabla
U_{\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial
B_{1},\\\\[12.0pt] U_{\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
Step 1: Proof of part $i)$. It suffices to prove
$\liminf_{\varepsilon\to
0}\varepsilon\|U_{\varepsilon}\|_{H^{\frac{1}{2}}(\partial B_{1})}>0,$ (4.1)
since $u_{c}=U_{\varepsilon}$ in $B_{1}$. We will prove (4.1) by
contradiction. Suppose that there exists $\varepsilon_{n}\to 0$ such that
$\lim_{n\to\infty}\varepsilon_{n}\|U_{n}\|_{H^{\frac{1}{2}}(\partial
B_{1})}=0,$
where $U_{n}$ is the solution of
$\left\\{\begin{array}[]{ll}\Delta
U_{n}+\varepsilon_{n}^{2}k^{2}U_{n}=0&\mbox{in
}\mathbb{R}^{3}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla U_{n})+k^{2}\sigma U_{n}={\bf e}&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon_{n}}a\nabla
U_{n}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial B_{1},\\\\[12.0pt]
U_{n}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
By Lemma 1, we have
$\lim_{n\to\infty}\varepsilon_{n}\|U_{n}\|_{H^{1}(B_{r}\setminus\overline{B}_{1})}=0\quad\forall\,r>1,$
which implies, since $\Delta U_{n}+\varepsilon_{n}^{2}k^{2}U_{n}=0$ in
$\mathbb{R}^{3}\setminus\overline{B}_{1}$,
$\lim_{n\to\infty}\varepsilon_{n}\Big{\|}\frac{\partial
U_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}\Big{\|}_{H^{-\frac{1}{2}}(\partial
B_{1})}=0.$
Hence, since $\displaystyle\frac{\partial
U_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-\frac{1}{\varepsilon_{n}}a\nabla
U_{n}\cdot\eta\Big{|}_{\mathrm{int}}=0$ on $\partial B_{1}$, we have
$\lim_{n\to\infty}\Big{\|}a\nabla
U_{n}\cdot\eta\Big{|}_{\mathrm{int}}\Big{\|}_{H^{-\frac{1}{2}}(\partial
B_{1})}=0.$ (4.2)
Multiplying the equation $\operatorname{div}(a\nabla U_{n})+k^{2}\sigma
U_{n}={\bf e}$ by $\bar{\bf e}$, integrating the expression obtained on
$B_{1}$, and using the fact that ${\bf e}\in M$, we have
$\int_{\partial B_{1}}\big{(}a\nabla U_{n}\cdot\eta\big{)}\bar{\bf
e}=\int_{B_{1}}|{\bf e}|^{2}=1.$ (4.3)
Combining (4.2) and (4.3), we have a contradiction.
Step 2: Proof of part $ii)$. Since ${\bf e}$ is radial, $a=\lambda_{1}I$, and
$\sigma=\lambda_{2}$ for some positive constants $\lambda_{1}$ and
$\lambda_{2}$, $U_{\varepsilon}$ is radial. This implies $\displaystyle
U_{\varepsilon}(x)=c_{\varepsilon}\frac{e^{ik\varepsilon|x|}}{|x|}$ if $|x|>1$
for some positive constant $c_{\varepsilon}$. Hence the conclusion of part
$ii)$ follows from (4.1) and the fact that
$u_{c}(x)=U_{\varepsilon}(x/\varepsilon)$ for $x\in B_{4}\setminus B_{2}$.
$\Box$
## 5 Proof of Proposition 3
Define $U_{\varepsilon}=u_{\varepsilon}(\varepsilon x)$ where
$u_{\varepsilon}=u_{c}\circ F_{\varepsilon}$. Then $U_{\varepsilon}$ is the
solution of
$\left\\{\begin{array}[]{ll}\Delta
U_{\varepsilon}+\varepsilon^{2}k^{2}U_{\varepsilon}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla U_{\varepsilon})+k^{2}\sigma U_{\varepsilon}={\bf
e}&\mbox{in }B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{\varepsilon}}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
U_{\varepsilon}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial
B_{1},\\\\[12.0pt] U_{\varepsilon}\mbox{ satisfies the outgoing
condition}.&\end{array}\right.$
Step 1: Proof of part $i)$. It suffices to prove
$\liminf_{\varepsilon\to 0}\|U_{\varepsilon}\|_{H^{1}(B_{1})}=\infty,$ (5.1)
since $u_{c}=U_{\varepsilon}$ in $B_{1}$. We will prove (5.1) by
contradiction. Suppose that there exists $\varepsilon_{n}\to 0$ such that
$\sup_{n\in\mathbb{N}}\|U_{n}\|_{H^{1}(B_{1})}<+\infty,$ (5.2)
where $U_{n}$ is the solution of
$\left\\{\begin{array}[]{ll}\Delta
U_{n}+\varepsilon_{n}^{2}k^{2}U_{n}=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla U_{n})+k^{2}\sigma U_{n}={\bf e}&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{n}}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
U_{n}\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial B_{1},\\\\[12.0pt]
U_{n}\mbox{ satisfies the outgoing condition}.&\end{array}\right.$
By Lemma 4, we have
$\sup_{n\in\mathbb{N}}\|U_{n}\|_{H^{1}(B_{r}\setminus\overline{B}_{1})}<+\infty\quad\forall\,r>1.$
It follows from (5.2) that
$\sup_{n\in\mathbb{N}}\|U_{n}\|_{H^{1}(B_{r})}<+\infty\quad\forall\,r>1.$
Applying Lemma 4, we may assume that $U_{n}\rightharpoonup U$ in
$H^{1}_{{}_{loc}}(\mathbb{R}^{2})$ for some $U\in W^{1}(\mathbb{R}^{2})$ which
satisfies
$\left\\{\begin{array}[]{ll}\Delta U=0&\mbox{in
}\mathbb{R}^{2}\setminus\overline{B}_{1},\\\\[12.0pt]
\operatorname{div}(a\nabla U)+k^{2}\sigma U={\bf e}&\mbox{in
}B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U}{\partial\eta}\Big{|}_{\mathrm{ext}}-a\nabla
U\cdot\eta\Big{|}_{\mathrm{int}}=0&\mbox{on }\partial B_{1},\\\\[12.0pt]
\end{array}\right.$ (5.3)
However, system (5.3) has no solution in $W^{1}(\mathbb{R}^{2})$ since ${\bf
e}\in N$. We have a contradiction.
Step 2: Proof of part $ii)$. For notational ease, we will assume that
$\lambda_{1}=1$ and $\lambda_{2}=\lambda$. Since ${\bf e}$ is radial, $a=I$,
and $\sigma=\lambda$, $U_{\varepsilon}$ is radial. Thus $U_{\varepsilon}$ can
be written under the form
$U_{\varepsilon}=c_{\varepsilon}H^{(1)}_{0}(\varepsilon kr)\quad\mbox{ for
}r=|x|\geq 1.$
This implies
$\frac{\partial U_{\varepsilon}}{\partial\eta}=c_{\varepsilon}\varepsilon
k\partial_{r}H^{(1)}_{0}(\varepsilon k)\quad\mbox{ for }r=|x|=1.$ (5.4)
Since ${\bf e}$ is radial, ${\bf e}\in
W^{1}(\mathbb{R}^{2}\setminus\overline{B}_{1})$, and $\Delta{\bf e}=0$ in
$\mathbb{R}^{2}\setminus\overline{B}_{1}$, it follows that ${\bf e}$ is
constant in $\mathbb{R}^{2}\setminus B_{1}$. Hence
$\frac{\partial{\bf e}}{\partial\eta}=0\quad\mbox{ on }\partial B_{1}.$ (5.5)
It is clear that
$\Delta U_{\varepsilon}+k^{2}\lambda U_{\varepsilon}={\bf e}\mbox{ in
}B_{1}\quad\mbox{ and }\quad\Delta{\bf e}+k^{2}\lambda{\bf e}=0\mbox{ in
}B_{1}.$
Multiplying the first equation by $\bar{\bf e}$, the second equation by
$\bar{U}_{\varepsilon}$ and integrating the obtained expressions on $B_{1}$,
we have
$\int_{B_{1}}\nabla U_{\varepsilon}\nabla\bar{\bf e}-k^{2}\lambda
U_{\varepsilon}\bar{\bf e}=-\int_{B_{1}}|{\bf e}|^{2}+\int_{\partial
B_{1}}\frac{\partial U_{\varepsilon}}{\partial\eta}\bar{\bf e},$
and
$\int_{B_{1}}\nabla{\bf e}\nabla\bar{U}_{\varepsilon}-k^{2}\lambda{\bf
e}\bar{U}_{\varepsilon}=\int_{\partial B_{1}}\frac{\partial{\bf
e}}{\partial\eta}\bar{U}_{\varepsilon}=0.$
(in the last equality, we used (5.5)). This implies
$1=\int_{B_{1}}|{\bf e}|^{2}=\int_{\partial B_{1}}\frac{\partial
U_{\varepsilon}}{\partial\eta}\bar{\bf e}.$
It follows from (5.4) that
$\int_{\partial B_{1}}\bar{\bf e}c_{\varepsilon}\varepsilon
k\partial_{r}H^{(1)}_{0}(\varepsilon k)=1.$
We recall that, see e.g., [3, Chapter 3],
$\lim_{r\rightarrow 0}\frac{rdH_{0}^{(1)}(r)}{dr}=-\frac{2}{i\pi},$ (5.6)
which yields
$|c_{\varepsilon}|\geq c,$
for some positive constant $c$. Hence
$|U_{\varepsilon}(x/\varepsilon)|\geq c|H^{(1)}_{0}(k|x|)|\quad\mbox{ for
}2\leq|x|\leq 4.$
Since $u_{c}(x)=U_{\varepsilon}(x/\varepsilon)$ for $x\in B_{4}\setminus
B_{2}$, the conclusion of part $ii)$ follows. $\Box$
## 6 Proof of Proposition 4
This section is devoted to the proof of Proposition 4. It suffices to prove
the following proposition.
###### Proposition 6.
Let $d=2,3$, $k>0$, and $\sigma_{0}>0$ be such that
$j_{0}^{\prime}(k\sigma_{0})=0$ if $d=3$ and $J_{0}^{\prime}(k\sigma_{0})=0$
if $d=2$. Then there exists a family of positive numbers
$(\sigma_{\varepsilon})$ such that
$\left\\{\begin{array}[]{cl}0<\liminf_{\varepsilon\to
0}\varepsilon^{-1}|\sigma_{\varepsilon}-\sigma_{0}|\leq\limsup_{\varepsilon\to
0}\varepsilon^{-1}|\sigma_{\varepsilon}-\sigma_{0}|<+\infty&\mbox{ if
}d=3,\\\\[12.0pt] 0<\liminf_{\varepsilon\to
0}|\ln\varepsilon||\sigma_{\varepsilon}-\sigma_{0}|\leq\limsup_{\varepsilon\to
0}|\ln\varepsilon||\sigma_{\varepsilon}-\sigma_{0}|<+\infty&\mbox{ if
}d=2,\end{array}\right.$
and
$\|u_{c,s}\|_{L^{2}(B_{4}\setminus
B_{2})}\geq\left\\{\begin{array}[]{cl}\|h_{0}\|_{L^{2}(B_{4}\setminus
B_{2})}&\mbox{ if }d=3,\\\\[6.0pt] \|H_{0}\|_{L^{2}(B_{4}\setminus
B_{2})}&\mbox{ if }d=2.\end{array}\right.$
Here $u_{c,s}\in H^{1}_{{}_{loc}}(\mathbb{R}^{d})$ is such that $u_{c,s}$
satisfies the outgoing condition and if $u_{c}:=u_{c,s}+u_{c,i}$ with
$u_{c,i}=\left\\{\begin{array}[]{cl}j_{0}(k|x|)&\mbox{ if }d=3,\\\\[6.0pt]
J_{0}(k|x|)&\mbox{ if }d=2,\end{array}\right.$
then $u_{c}$ is a solution of the equation
$\operatorname{div}(A_{c}\nabla u_{c})+k^{2}\Sigma_{c}u_{c}=0,$
where $(A_{c},\Sigma_{c})$ is defined in (1.5) with $a=I$ and
$\sigma=\sigma_{\varepsilon}$.
Hereafter $h_{0}$ denotes the spherical Hankel function of the first kind of
order $0$, $j_{0}=\Re(h_{0})$, and $y_{0}=\Im(h_{0})$, and $H_{0}$ denotes the
Hankel function of the first kind of order $0$, $J_{0}=\Re(H_{0})$, and
$Y_{0}=\Im(H_{0})$.
Proof. Set $u_{\varepsilon}=u_{c}\circ F_{\varepsilon}$ and
$u_{\varepsilon,s}=u_{\varepsilon}-u_{c,i}$ where $F_{\varepsilon}$ is given
in (1.1). According to Proposition 5, $u_{\varepsilon}=u_{c}$ and
$u_{\varepsilon,s}=u_{c,s}$ in $\mathbb{R}^{d}\setminus B_{2}$,
$u_{\varepsilon,s}$ satisfies the outgoing condition, and $u_{\varepsilon}$ is
the solution of the equation
$\operatorname{div}(A_{\varepsilon}\nabla
u_{\varepsilon})+k^{2}\Sigma_{\varepsilon}u_{\varepsilon}=0,$
where
$A_{\varepsilon},\Sigma_{\varepsilon}=\left\\{\begin{array}[]{cl}I,1&\mbox{if
}x\in\mathbb{R}^{d}\setminus B_{\varepsilon},\\\\[12.0pt]
\displaystyle\frac{I}{\varepsilon^{d-2}},\frac{\sigma_{\varepsilon}}{\varepsilon^{d}}&\mbox{if
}x\in B_{\varepsilon}.\end{array}\right.$
Define $U_{\varepsilon}(x)=u_{\varepsilon}(\varepsilon x)$ and
$U_{\varepsilon,s}(x)=u_{\varepsilon,s}(\varepsilon
x)=U_{\varepsilon}(x)-u_{c,i}(\varepsilon x)$ for $x\in\mathbb{R}^{d}$. Then
$U_{\varepsilon,s}(x)=u_{c,s}(\varepsilon x)$ for $|x|>2/\varepsilon$,
$U_{\varepsilon}$ satisfies the equation
$\operatorname{div}(\tilde{A}_{\varepsilon}\nabla
U_{\varepsilon})+k^{2}\tilde{\Sigma}_{\varepsilon}U_{\varepsilon}=0,$
and $U_{\varepsilon,s}$ satisfies the outgoing condition. Here
$\tilde{A}_{\varepsilon},\tilde{\Sigma}_{\varepsilon}=\left\\{\begin{array}[]{cl}I,\varepsilon^{2}&\mbox{if
}x\in\mathbb{R}^{d}\setminus B_{1},\\\\[12.0pt]
\displaystyle\frac{I}{\varepsilon^{d-2}},\frac{\sigma_{\varepsilon}}{\varepsilon^{d-2}}&\mbox{if
}x\in B_{1}.\end{array}\right.$
Step 1: $d=3$. It is clear that
$U_{\varepsilon,i}=j_{0}(k\varepsilon|x|),\quad
U_{\varepsilon,s}=\alpha_{0}h_{0}(k\varepsilon|x|)\quad\mbox{ for }|x|>1,$
and
$U_{\varepsilon,t}=\beta_{0}j_{0}(k_{\varepsilon}|x|)\quad\mbox{ for }|x|<1,$
where $k_{\varepsilon}=k\sigma_{\varepsilon}$,
$U_{\varepsilon,t}:=U_{\varepsilon,s}+U_{\varepsilon,i}=U_{\varepsilon}$ in
$B_{1}$, for some $\alpha_{0},\beta_{0}$. Using the transmission conditions,
namely,
$\left\\{\begin{array}[]{ll}U_{\varepsilon,s}+U_{\varepsilon,i}=U_{\varepsilon,t}&\mbox{
on }\partial B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{\varepsilon,s}}{\partial r}+\frac{\partial U_{\varepsilon,i}}{\partial
r}=\frac{1}{\varepsilon}\frac{\partial U_{\varepsilon,t}}{\partial r}&\mbox{
on }\partial B_{1},\end{array}\right.$
we have
$\left\\{\begin{array}[]{ll}\alpha_{0}h_{0}(k\varepsilon)+j_{0}(k\varepsilon)=\beta_{0}j_{0}(k_{\varepsilon})&\mbox{
on }\partial B_{1},\\\\[12.0pt] \alpha_{0}k\varepsilon
h_{0}^{\prime}(\varepsilon)+k\varepsilon
j_{0}^{\prime}(\varepsilon)=\frac{1}{\varepsilon}k_{\varepsilon}\beta_{0}j_{0}^{\prime}(k_{\varepsilon})&\mbox{
on }\partial B_{1}.\end{array}\right.$
It follows that
$\alpha_{0}=-\frac{k\varepsilon
j_{0}^{\prime}(k\varepsilon)j_{0}(k_{\varepsilon})-\frac{1}{\varepsilon}k_{\varepsilon}j_{0}(k\varepsilon)j_{0}^{\prime}(k_{\varepsilon})}{k\varepsilon
h_{0}^{\prime}(k\varepsilon)j_{0}(k_{\varepsilon})-\frac{1}{\varepsilon}k_{\varepsilon}h_{0}(k\varepsilon)j_{0}^{\prime}(k_{\varepsilon})}.$
(6.1)
Since $y_{0}(t)=\cos t/t$, we have
$k\varepsilon^{2}\frac{y_{0}^{\prime}(k\varepsilon)}{y_{0}(k\varepsilon)}=k\varepsilon^{2}\Big{(}-\frac{1}{k\varepsilon}-\frac{\sin(k\varepsilon)}{\cos(k\varepsilon)}\Big{)}=-\varepsilon-k\varepsilon^{2}\sin(k\varepsilon)/\cos(k\varepsilon).$
Let $\sigma_{\varepsilon}$ be such that
$k_{\varepsilon}=k\sigma_{\varepsilon}$ converges to $k\sigma_{0}$ and
$\frac{j_{0}^{\prime}(k_{\varepsilon})}{j_{0}(k_{\varepsilon})}=-\varepsilon-k\varepsilon^{2}\sin(k\varepsilon)/\cos(k\varepsilon).$
Then the complex part in the denominator of the RHS of (6.1) equals 0. On the
other hand, the real part in the denominator of the RHS of (6.1) equals the
numerator of the RHS of (6.1). Hence it follows from (6.1) that
$\alpha_{0}=-1,$
which implies the conclusion in the $3d$ case since
$u_{c,s}(x)=U_{\varepsilon,s}(x/\varepsilon)$ for $|x|>2$.
Step 2: $d=2$. It is clear that
$U_{\varepsilon,i}=J_{0}(k\varepsilon|x|),\quad
U_{\varepsilon,s}=\alpha_{0}H_{0}(k\varepsilon|x|)\quad\mbox{ for }|x|>1,$
and
$U_{\varepsilon,t}=\beta_{0}J_{0}(k_{\varepsilon}|x|)\quad\mbox{ for }|x|<1,$
where $k_{\varepsilon}=k\sigma_{\varepsilon}$,
$U_{\varepsilon,t}:=U_{\varepsilon,s}+U_{\varepsilon,i}=U_{\varepsilon}$ in
$B_{1}$, for some $\alpha_{0},\beta_{0}$. Using the transmission conditions,
namely,
$\left\\{\begin{array}[]{ll}U_{\varepsilon,s}+U_{\varepsilon,i}=U_{\varepsilon,t}&\mbox{
on }\partial B_{1},\\\\[12.0pt] \displaystyle\frac{\partial
U_{\varepsilon,s}}{\partial r}+\frac{\partial U_{\varepsilon,i}}{\partial
r}=\frac{\partial U_{\varepsilon,t}}{\partial r}&\mbox{ on }\partial
B_{1},\end{array}\right.$
we have
$\left\\{\begin{array}[]{ll}\alpha_{0}H_{0}(k\varepsilon)+J_{0}(k\varepsilon)=\beta_{0}J_{0}(k_{\varepsilon})&\mbox{
on }\partial B_{1},\\\\[12.0pt] \alpha_{0}k\varepsilon
H_{0}^{\prime}(k\varepsilon)+k\varepsilon
J_{0}^{\prime}(k\varepsilon)=k_{\varepsilon}\beta_{0}J_{0}^{\prime}(k_{\varepsilon})&\mbox{
on }\partial B_{1},\end{array}\right.`$
Thus it follows that
$\alpha_{0}=-\frac{k\varepsilon
J_{0}^{\prime}(k\varepsilon)J_{0}(k_{\varepsilon})-k_{\varepsilon}J_{0}(k\varepsilon)J_{0}^{\prime}(k_{\varepsilon})}{k\varepsilon
H_{0}^{\prime}(k\varepsilon)J_{0}(k_{\varepsilon})-k_{\varepsilon}H_{0}(k\varepsilon)J_{0}^{\prime}(k_{\varepsilon})}.$
(6.2)
Since $Y_{0}(t)=\frac{2}{\pi}\ln(t/2)$, we have
$k\varepsilon\frac{Y_{0}^{\prime}(k\varepsilon)}{Y_{0}(k\varepsilon)}=\frac{1}{\ln(k\varepsilon/2)}.$
Let $\sigma_{\varepsilon}$ be such that
$k_{\varepsilon}=k\sigma_{\varepsilon}\to k\sigma_{0}$ and
$`\frac{k_{\varepsilon}J_{0}^{\prime}(k_{\varepsilon})}{J_{0}(k_{\varepsilon})}=\frac{1}{\ln(k\varepsilon/2)}.$
Then as in the $3d$ case, $\alpha_{0}=-1$, and the conclusion in the $2d$ case
follows from the fact that $u_{c,s}(x)=U_{\varepsilon,s}(x/\varepsilon)$ for
$|x|>2$. $\Box$
Acknowledgments. The author thanks Bob Kohn for stimulating discussions and
useful comments on the manuscript.
## References
* [1] H. Brezis, _Analyse Fonctionnelle. Théorie et applications_ , Mathématiques appliquées pour la maîtrise, Dunod, 2002.
* [2] W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, _Optical cloaking with metamaterials_ , Nature Photonics 1 (2007), 224–227.
* [3] D. Colton and R. Kress, _Inverse acoustic and electromagnetic scattering theory_ , second ed., Applied Mathematical Sciences, vol. 98, Springer-Verlag, Berlin, 1998.
* [4] S. A. Cummer, P. I. Popa, D. Schurig, D. R. Smith, J. B. Pendry, M. Rahm, and A. F. Starr, _Scattering theory derivation of $3d$ acoustic cloaking shell_, Phys. Rev. Lett. 100 (2008), 024301.
* [5] A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, _Cloaking vs. shielding in transformation optics_ , preprint.
* [6] , _Full-wave invisibility of active devices at all frequencies_ , Comm. Math. Phys. 275 (2007), 749–789.
* [7] , _Improvement of cylindrical cloaking with the SHS lining_ , Opt. Exp. 15 (2007), 12717.
* [8] , _Isotropic transformation optics: approximate acoustic and quantum cloaking_ , New Jour. of Physics 10 (2008), 115024–115051.
* [9] , _Invisibility and inverse problems_ , Bull. Amer. Math. Soc. 46 (2009), 55–97.
* [10] A. Greenleaf, M. Lassas, and G. Uhlmann, _On nonuniqueness for Calderon’s inverse problem_ , Math. Res. Lett. 10 (2003), 685–693.
* [11] R. V. Kohn, D. Onofrei, M. S. Vogelius, and M. I. Weinstein, _Cloaking via change of variables for the Helmholtz equation_ , Comm. Pure Appl. Math. 63 (2010), 973–1016.
* [12] R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, _Cloaking via change of variables in electric impedance tomography_ , Inverse Problems 24 (2008), 015016.
* [13] P. Lax, _Functional analysis_ , Pure and Applied Mathematics, Wiley-Interscience, 2002.
* [14] U. Leonhardt, _Optical conformal mapping_ , Science 312 (2006), 1777–1780.
* [15] H. Liu, _Virtual reshaping and invisibility in obstacle scattering_ , Inverse Problems 25 (2009), 045006.
* [16] J. C. Nedelec, _Acoustic and Electromagnetic Equations_ , Applied Mathematical Sciences, vol. 144, Springer-Verlag, Berlin, 2001.
* [17] H-M. Nguyen, _Cloaking for the Helmholtz equation in the whole spaces_ , Comm. Pure Appl. Math. (2010), to appear.
* [18] H-M. Nguyen and M. S. Vogelius, _A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009).
* [19] , _Full Range Scattering Estimates and their Application to Cloaking_ , (2010), submitted.
* [20] , _Approximate cloaking for the wave equation via change of variables_ , (2011), preprint.
* [21] J. B. Pendry, D. Schurig, and D. R. Smith, _Controlling electromagnetic fields_ , Science 312 (2006), 1780–1782.
* [22] Z. Ruan, M. Yan, C. M. Neff, and M. Qiu, _Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations_ , Phys. Rev. Lett. 99 (2007), 113903.
* [23] D. Schurig, J. J. Mock, J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, _Metamaterial electromagnetic cloak at microwave frequencies_ , Science 314 (2006), 1133628.
* [24] M. S. Vogelius and D. Volkov, _Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter_ , M2AN Math. Model. Numer. Anal. 34 (2000), 723–748.
* [25] R. Weder, _A rigorous analysis of high-order electromagnetic invisibility cloaks_ , J. Phys. A: Math. Theor. 41 (2008), 065207.
* [26] , _The boundary conditions for point transformed electromagnetic invisibility cloaks_ , J. Phys. A: Math. Theor. 41 (2008), 415401.
* [27] B. Wood, _Metamaterials and invisibility_ , C. R. Physique 10 (2009), 379–390.
* [28] M. Yan, Z. Ruan, and M. Qiu, _Cylindrical invisibility cloak with simplified material parameters is inherently visible_ , Phys. Rev. Lett. 99 (2007), 233901.
|
arxiv-papers
| 2011-09-29T16:35:08 |
2024-09-04T02:49:22.612963
|
{
"license": "Public Domain",
"authors": "Hoai-Minh Nguyen",
"submitter": "Hoai Minh Nguyen",
"url": "https://arxiv.org/abs/1109.6583"
}
|
1109.6805
|
# A Search for the Higgs Boson in the $H\rightarrow\gamma\gamma$ Channel with
CMS DPF-2011 Proceedings
Christopher Palmer Department of Physics, University of California, San
Diego, CA, USA
###### Abstract
We report on a search for SM Higgs Boson in the Higgs to two photons decay
channel conducted by the CMS experiment with the data accumulated during the
2010 and 2011 running of the LHC at sqrt(s) = 7 TeV.
## I Introduction - $H\rightarrow\gamma\gamma$ in a Nutshell
We are motivated to explore this signal because of its narrow resonance over a
smoothly falling background. Despite the enormous background and the somewhat
low number of expected events, this decay channel is extremely relevant to the
favored, low-mass Higgs search as long as the resolution of the peak can be
measured with reasonable resolution.
Our background can be categorized in two components: irreducible and
reducible. Irreducible background is more difficult to eliminate because these
are events which have two real photons in them. Reducible background events
are typically poorly reconstructions electrons or jets which are mistakenly
taken to be photons. In the final selection invariant mass plot ( Figure 1)
one can see the various components from each.
Figure 1: Final invariant mass selection with simulation.
## II Necessary CMS Design/ECAL
### II.1 Compact Muon Solenoid (CMS)
CMS is a general purpose detector. It is designed to detect and reconstruct
numerous physics objects (e.g. photons, electrons, muons, hadronic jets, etc).
From this general design we are compatible of seeking out new physics (e.g.
Higgs and SUSY). Below ( Figure 2) is a cross sectional schematic of the
detectored with the relevant components highlighted (i.e. the trackers and the
electromagnetic calorimeter (ECAL)).
Figure 2: Cross sectional schematic of CMS.
### II.2 CMS - Electromagnetic Calorimeter (ECAL)
The ECAL is made of $\sim$76K $PbWO_{4}$ crystals in the barrel
($|\eta|<1.48$) and the endcap ($1.48<|\eta|<3.$) of CMS. It was reconstructed
in order to determine the energy of photons and electrons to high precision.
The design resolution of unconverted photons in the barrel with energy greater
than 100 GeV was $\sim$0.5%
There are two main critical issues that impact the resolution that CMS is able
to determine. One is the calibration of the crystals. Two techniques, which
are exploited, are $\pi_{0}\rightarrow\gamma\gamma$ events for inter-crystal
calibration and $Z\rightarrow e^{+}e^{-}$ for global energy scale calibration.
The other critical issue is transparency corrections for radiation damage. An
integrated laser system measures the transparency of crystals and that
information is used to correct the transparency loss. Below is a schematic of
the ECAL ( Figure 4) and a photo of one of the crystals ( Figure 3).
Figure 3: A photograph of one of the crystals from the ECAL. Figure 4: A
schmatic of CMS’s ECAL.
## III Analysis Strategy
### III.1 DiPhoton Selection
Photon identification is important for removal of reducible background. The
primary tools used to distinguish between signal and background are isolation,
cluster shape and electron veto. The cuts set in these photon identification
variables are set separately for in categories defined by the photon being in
the barrel or one of the endcaps and by whether or not the photon converted to
an electron-positron before before reaching the ECAL. The efficiency versus
$\eta$ is plotted below for these four categories ( Figure 5). Also, note that
our selection contains minimum thresholds on the transverse momentum of the
two photons (40 and 30 GeV).
Figure 5: Efficiency of selection in categories as a function of $\eta$ with
simulated Higgs’ photons.
### III.2 Vertex Selection
In our data, numerous proton-proton interactions occur each time bunches of
protons pass through each other. If the wrong vertex is chosen to reconstruct
the photons then the energy of the photons and the mass of the di-photon will
be incorrect. Effectively smearing the mass of the Higgs signal.
We use the following quantities to increase our probability of determining the
correct vertex. For unconverted photons we use
$\displaystyle\sum\limits_{Tracks}P_{T}^{2}$, projection of tracks onto
$\gamma\gamma$ and the balance between $\gamma\gamma$ and vertex’s tracks. For
converted $\gamma$’s the conversion-tracks are used to point back to vertex.
The efficiency of this method is plotted below as a function of di-photon
transverse momentum ( Figure 6).
Figure 6: Efficiency of vertex selection within 1 cm as function of Higgs’
transverse momentum.
## IV Efficiencies from Data
Making the basic assumption that electrons and photons have similar shower
properties we use the tag and probe method with $Z\rightarrow e^{+}e^{-}$
electrons to determine the efficiency of our photon selection. First events
with robustly identified electrons are selected and we seek an additional
electron passing a minimal transverse energy requirement, the probe. The
photon reconstructed with the same energy deposit has the photon cuts applied
to it and efficiency of passing events to selected events is measured in each
category. Below is a table of these efficiencies. Please note the majority of
our photons are in the unconverted, barrel category, which is more than 90%
efficient.
To measure the efficiency of the electron veto cuts, photons from
$Z\rightarrow\mu^{+}\mu^{-}\gamma$ events have the selection applied.
Efficiencies are listed in the table below.
Table 1: Tag and Probe Efficiency Category | $\epsilon_{data}$ (%) | $\epsilon_{MC}$ (%) | $\epsilon_{data}$/$\epsilon_{MC}$
---|---|---|---
All cuts except electron rejection (from $Z\rightarrow e^{+}e^{-}$)
1 | 91.77$\pm$0.14 | 92.43$\pm$0.07 | 0.993$\pm$0.002
2 | 72.67$\pm$0.43 | 71.89$\pm$0.08 | 1.011$\pm$0.007
3 | 80.33$\pm$0.47 | 80.04$\pm$0.18 | 1.004$\pm$0.008
4 | 57.80$\pm$1.26 | 55.09$\pm$0.15 | 1.049$\pm$0.025
Electron rejection cut (from $Z\to\mu\mu\gamma$)
1 | 99.78${}^{+0.13}_{-0.16}$ | 99.59${}^{+0.13}_{-0.17}$ | 1.002${}^{+0.002}_{-0.002}$
2 | 98.77${}^{+0.59}_{-0.73}$ | 97.70${}^{+0.32}_{-0.37}$ | 1.011${}^{+0.007}_{-0.008}$
3 | 99.32${}^{+0.51}_{-1.02}$ | 99.29${}^{+0.30}_{-0.42}$ | 1.000${}^{+0.006}_{-0.011}$
4 | 93.0${}^{+2.1}_{-2.3}$ | 93.34${}^{+0.79}_{-0.86}$ | 0.996${}^{+0.024}_{-0.027}$
Table 2: Electron Veto Efficiency Both photons in barrel
---
2 Unconverted | 1,2 Converted
100.00${}^{+0.00}_{-0.01}$% | 99.53$\pm$0.04%
One or more in endcap
2 Unconverted | 1,2 Converted
100.00${}^{+0.00}_{-0.02}$% | 98.86$\pm$0.07%
## V Resolution Results
ECAL resolution measured from $Z\rightarrow e^{+}e^{-}$ ( Figure 7)is applied
to simulated Higgs’ $\gamma$’s ( Figure 8). The simulated Higgs’ $\gamma$’s
with data resolution are used in signal modeling for CL limits. Suboptimal
transparency loss corrections may be responsible for degraded resolution. An
example of the measured resolution and its application to the best Higgs’
photons category are below.
Figure 7: Resolution from $Z\rightarrow e^{+}e^{-}$ events in data. Figure 8:
Signal resolution in $H\rightarrow\gamma\gamma$ estimated from data.
## VI Cateogories and Limit Setting
### VI.1 Event Classes Used for CL Evaluation
There are eight event classes in which confidence levels are computed. There
are both photons in the barrel or either in an endcap times both converted or
either unconverted times high/low diphoton transverse momentum
classifications. Below is a table which contains the fraction of signal and
background in each event class.
| Both $\gamma$’s in barrel | One or more in endcap
---|---|---
| 2 Unconverted | 1,2 Converted | 2 Unconverted | 1,2 Converted
$\mathbf{P_{T}^{\gamma\gamma}<}$ 40 GeV/c | | | |
Signal | 0.209 | 0.271 | 0.094 | 0.116
Background | 0.167 | 0.263 | 0.129 | 0.203
Signal $\sigma_{effective}$ (GeV/$c^{2}$) | 1.58 | 2.33 | 3.14 | 3.60
$\mathbf{P_{T}^{\gamma\gamma}>}$ 40 GeV/c | | | |
Signal | 0.102 | 0.122 | 0.035 | 0.051
Background | 0.043 | 0.079 | 0.043 | 0.074
Signal $\sigma_{effective}$ (GeV/$c^{2}$) | 1.37 | 2.12 | 2.95 | 3.26
Second order polynomial fits are performed in each of the eight categories on
the data. The resulting fit is the background distribution used for toy
experiments. The two both barrel, both unconverted categories are below (
Figure 9 and Figure 10).
Figure 9: High $P_{T}^{\gamma\gamma}$, both high $R_{9}$, barrel-barrel .
Figure 10: Low $P_{T}^{\gamma\gamma}$, both high $R_{9}$, barrel-barrel .
### VI.2 Systematics
Below ( Figure VI.2) is a table that summarizes the systematics that are
applied to the signal models. Since the background model is from data, no
systematics are applied to it.
### VI.3 Evaluated Limits
Confidence Level (CL) limits are determined in two ways with extremely
consistent results. Our official limits are set with the modified frequentist
approach (CLs) using profile likelihood. As a cross check with all use the
Bayesian method with flat prior in cross section. In the following two plots
show SM exclusions between cross sections of between 0.06 and 0.26 pb ( Figure
VI.3) and between 1.9 and 7.0 $*\sigma_{SM}$ ( Figure VI.3). While the
following plot shows Fermiophobic Higgs cross section excluded between 0.04
and 0.18 pb ( Figure VI.3) and its mass constrained to be greater than 111
GeV.
## VII Conclusions/Outlook
Our analysis is defined by our use of photon selection in categories and our
use of various methods to select the best vertex possible. Data is used to
determine the resolution of our signal (via $Z\rightarrow e^{+}e^{-}$ events)
model and our background model is determined directly from the fit of the data
in event classes. The CL evaluation in event classes improves our sensitivity
to both the SM and Fermiophobic Higgs.
We are keen to improve this analysis by taking as much data as possible and by
improving the resolution of CMS’s ECAL. Improving resolution could be the
signal greatest improvement to the SM analysis and work is ongoing in this
field.
###### Acknowledgements.
Vivek Sharma and Christoph Paus have put forth great effort in coordinating
the Higgs effort within CMS over the past year. The Higgs result from CMS
would not have been achieved without them. Christopher Seez and Paolo
Meridiani have been directing the research specifically outlined in this
proceeding and should be commended for their efforts and the group’s
achievements. Finally, I would like to personally thank my advisor Jim Branson
and Marco Pieri for their direction of my research.
## References
* (1) CMS PAS HIG-11-010 ($H\rightarrow\gamma\gamma$ PAS) - dsweb.cern.ch/record/1369553/files/HIG-11-010-pas.pdf
* (2) CMS PAS HIG-11-011 (Higgs combination PAS - cdsweb.cern.ch/record/1370076/files/HIG-11-011-pas.pdf
* (3) Other public plots - https://twiki.cern.ch/twiki/bin/view/CMSPublic/Hig11010TWiki
|
arxiv-papers
| 2011-09-30T12:19:57 |
2024-09-04T02:49:22.628047
|
{
"license": "Public Domain",
"authors": "Christopher Palmer",
"submitter": "Christopher Palmer",
"url": "https://arxiv.org/abs/1109.6805"
}
|
1109.6820
|
# Properties of proper rational numbers
Konstantine Zelator
Mathematics, Statistics, and Computer Science
212 Ben Franklin Hall
Bloomsburg University of Pennsylvania
400 East 2nd Street
Bloomsburg, PA 17815
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
kzelator@.bloomu.edu
e-mails: konstantine zelator@yahoo.com
## 1 Introduction
The set of rational numbers can be thought of as the disjoint union of two of
its main subsets: the set of integers and the set of proper rationals.
Definition 1: A proper rational number is a rational number which is not an
integer.
The aim of this work is simple and direct. Namely, to explore some of the
basic or elementary properties of the proper rationals.
We will make use of the standard notation $(u,w)$ denoting the greatest common
divisor of two integers $u$ and $w$. Also, the notation $u|w$ to denote that
$u$ is a divisor of $w$.
###### Proposition 1.
Let $r$ be a proper rational number. Then $r$ can be written in the form,
$r=\frac{c}{b}$ where $c$ and $b$ are relatively prime integers; $(c,b)=1$,
and with $b\geq 2$.
###### Proof.
Since $r$ is a proper rational, it cannot, by definition, be zero. Hence
$r=\frac{A}{B}$, for some positive integers $A$ and $B$; if $r>0$. If, on the
other hand, $r<0$, then $r=-\frac{A}{B}$, $A,B$ being positive integers. Let
$d=(A,B)$, then $A=da$, $B=db$, for relative prime positive integers $a$ and
$b$. We have, $r=\frac{A}{B}=\frac{da}{db}=\frac{a}{b}$, for $r>0$. Clearly,
$b$ cannot equal 1, for then $r$ would equal $a$, an integer, contrary to the
fact that $r$ is a proper rational. Hence, $b\geq 2$.
If, on the other hand, $r<0$, $r=\frac{-A}{B}=\frac{-a}{b}=\frac{c}{b}$ with
$c=-a$, and $b\geq 2$. ∎
Definition 2: A proper rational number $r$ is said to be in standard form if
it is written as $r=\frac{c}{b}$, where $c$ and $b$ are relatively prime
integers and $b\geq 2$
## 2 The reciprocal of a proper rational
We state the following result without proof. We invite the interested reader
tofill in the details.
###### Theorem 1.
Let $r=\frac{c}{b}$ be a proper rational in standard form.
1. (i)
If $|c|=1$, the reciprocal $\frac{1}{r}=\frac{b}{c}$ is an integer.
2. (i)
If $c\geq 2$, the reciprocal $\frac{1}{r}$ is a proper positive rational.
3. (iii)
If $|c|\geq 2$ and $c<0$, the reciprocal $\frac{1}{r}$ is negative proper
rational with the standard form being $\frac{1}{r}=\frac{d}{|c|}$, where
$d=-b$.
## 3 An obvious property
Is the sum of a proper rational and an integer always a proper rational? The
answer is a rather obvious yes.
###### Theorem 2.
Suppose that $r=\frac{c}{b}$ is a proper rational in standard form; and $d$
and integer. Then the sume $r+d$ is a proper rational.
###### Proof.
If, to the contrary, $r+d=i$, for some $i\in{\mathbb{Z}}$, then $r=i-d$, an
integer contradicting the fact that $r$ is a proper rational. ∎
## 4 A lemma from number theory
We will make repeated use of the very well known, and important, lemma below.
For a proof of this lemma, see reference [1]. It can be found in just about
every elementary number theory book.
###### Lemma 1.
(Euclid’s lemma)
1. (i)
(Standard version) Let $m,n,k$ be positive integers such that $m$ is a divisor
of the product $n\cdot k$; and suppose that $(m,n)=1$. Then $m$ is a divisor
of $k$.
2. (ii)
(Extended version) Let $m,n,k$ be non-zero integers such that $m|nk$ and
$(m,n)=1$. Then $m|k$.
## 5 A slightly less obvious property
When is the product of a proper rational with an integer, an integer? A proper
rational?
###### Theorem 3.
Let $r=\frac{c}{b}$ be a proper rational in standard form and $i$ an integer.
1. (a)
The product $r\cdot i$ is an integer if, and only if, $b|i$.
2. (b)
The product $r\cdot i$ is a proper rational if, and only if, $b$ is not a
divisor of $i$.
###### Proof.
(b) This part is logically equivalent to part (a).
(a) If $b$ divides $i$, then $i=b\cdot q$, an integer.
so we have $r\cdot i=\frac{cbq}{b}=c\cdot q$, an integer. Now the converse.
Suppose that $r\cdot i$ is an integer $t$: $r\cdot i=t$, which yields,
$c\cdot i=b\cdot t$ (1)
Since $r$ is a proper rational, $(b,c)=1$ by defintion. Equation (1) shows
that $b|c\cdot i$; and since $(b,c)=1$. Lemma 1 implies that $b$ must divide
$i$. We are done. ∎
## 6 The sum of two proper rationals
An interesting equation arises. When is the sum of two proper rationals also a
proper rational? When is it an integer? There is no obvious answer here.
###### Theorem 4.
Let $r_{1}=\frac{c_{1}}{b_{1}}$ and $r_{2}=\frac{c_{2}}{b_{2}}$ be proper
rationals in standard form. Then,
1. (i)
The sum $r_{1}+r_{2}$ is an integer if, and only if, $b_{1}=b_{2}$ and $b_{1}$
is a divisor of the sum $c_{1}+c_{2}$.
2. (ii)
The sum $r_{1}+r_{2}$ is a proper rational if, and only if, either $b_{1}\neq
b_{2}$ or $b_{1}=b_{2}$ but with $b_{1}$ not being a divisor of $c_{1}+c_{2}$.
###### Proof.
(ii) This part is logically equivalent to part (i).
(i) If $b_{1}=b_{2}$ and $b_{1}|(c_{1}+c_{1})$, then
$r_{1}+r_{2}=\frac{c_{1}+c_{2}}{b_{1}}$, is obviously an integer. Next, let us
prove the converse statement.
Suppose that $r_{1}+r_{2}=i$, an integer. Some routine algebra produces
$c_{1}b_{2}+c_{2}b_{1}=ib_{1}b_{2}$ (2)
or equivalently
$c_{1}b_{2}=b_{1}(ib_{2}-c_{2}).$ (3)
According to (3), $b_{1}|c_{1}b_{2}$; and since $(b_{1},c_{1})=1$, Lemma 1
implies that $b_{1}|b_{2}$. A similar argument, using equation (2), once more
establishes that $b_{2}|b_{1}$. Clearly, since the two positive integers
$b_{1}$ and $b_{2}$ are divisors of each other, they must be equal;
$b_{1}=b_{2}$ (an easy exercise in elementary number theory). From
$b_{1}=b_{2}$ and (2), we obtain $c_{1}+c_{2}=i\cdot b_{i}$; and thus it is
clear that $b_{1}|(c_{1}+c_{2})$. ∎
## 7 The product of two proper rationals
###### Theorem 5.
Let $r_{1}=\frac{c_{1}}{b_{1}}$ and $r_{2}=\frac{c_{2}}{b_{2}}$ be proper
rationals in standard form.
1. (a)
The product $r_{1}r_{2}$ is an integer if, and only if, $b_{1}|c_{2}$ and
$b_{2}|c_{1}$.
2. (b)
The product $r_{1}r_{2}$ is a proper rational if, and only if, $b_{1}$ is not
a divisor of $c_{2}$; or $b_{2}$ is not a divisor of $c_{1}$.
###### Proof.
(b) This part is logically equivalent to part (a).
(a) Suppose that $b_{1}|c_{2}$ and $b_{2}|c_{1}$; then $c_{2}=b_{1}a$ and
$c_{1}=b_{2}d$ where $a$ and $d$ are (non-zero) integers.
We have
$r_{1}r_{2}=\frac{c_{1}c_{2}}{b_{1}b_{2}}=\frac{adb_{1}b_{2}}{b_{1}b_{2}}=ad$,
an integer.
Conversely, suppose that $r_{1}r_{2}=i$, an integer. Then
$c_{1}c_{2}=ib_{1}b_{2}$ (4)
Since $(b_{1},c_{1})=1=(c_{2},b_{2})$, (4), in conjunction with Lemma 1, imply
that $b_{1}|c_{2}$ and $b_{2}|c_{1}$. We are done. ∎
## 8 One more result and its corollary
In Theorem 4 part (i), gives us the precise conditions for the sum of two
proper rationals to be an integer. Likewise, Theorem 5 part (a) gives us the
exact conditions for the product of two proper rationals to be an integer.
Naturally, the following question arises. Can we find two proper rational
numbers whose sum is an integer; and also whose product is an integer? Theorem
7 provides an answer in the negative. Theorem 7 is a direct consequence of
Theorem 6 below.
###### Theorem 6.
If both the sum and the product of two rational numbers are integers, then so
are the two rationals, integers.
###### Proof.
Let $r_{1},r_{2}$ be the two rationals, and suppose that
$\left\\{\begin{array}[]{rcl}r_{1}+r_{2}&=&i_{1}\\\ r_{1}r_{2}&=&i_{2}\\\
i_{1},i_{2}\in{\mathbb{Z}}\end{array}\right\\}$ (5)
If either of $r_{1},r_{2}$ is an integer, then the first equation in (5)
implies that the other one is also an integer. So we are done in this case.
So, assume that neither of $r_{1},r_{2}$ is an integer; which means that they
are both proper rationals. Let then $r_{1}=\frac{c_{1}}{b_{1}},\
r_{2}=\frac{c_{2}}{b_{2}}$ be the standard forms of $r_{1}$ and $r_{2}$. That
is, $(c_{1},b_{1})=1=(c_{2},b_{2}),\ b_{1}\geq 2,\ b_{2}\geq 2$ and, of
course, $c_{1}c_{2}\neq 0$.
Combining this information with (5), we get
$\left\\{\begin{array}[]{rcl}c_{1}b_{2}+c_{2}b_{1}&=&i_{1}b_{1}b_{2}\\\
c_{1}c_{2}&=&i_{2}b_{1}b_{2}\end{array}\right\\}$ (6)
From the first equation in (6) we obtain
$c_{1}b_{2}=b_{1}(i_{1}b_{2}-c_{2}),$
which shows that $b_{1}|c_{1}b_{2}$.
This, combined with $(c_{1},b_{1})=1$ and Lemma 1 allow us to deduce that
$b_{1}|b_{2}$. Similarly, using the first equation in (6), we infer that
$b_{2}|b_{1}$ which implies $b_{1}=b_{2}$. Hence, the second equation of (6)
gives,
$c_{1}c_{2}=i_{2}b_{1}^{2}$ (7)
By virtue of $(b_{1},c_{1})=(b_{1},c_{2})=1$, equation (7) implies $b_{1}=1;\
b_{1}=b_{2}=1$. Therefore $r_{1}$ and $r_{2}$ are integers. ∎
We have the immediate corollary.
###### Theorem 7.
There exist no two proper rationals both of whose sum and product are
integers.
## 9 A closing remark
Theorem 7 can also be proved by using the well known Rational Root Theorem for
polynomials with integer coefficients. The Rational Root Theorem implies that
if a monic (i.e., leading coefficient is $1$) polynomial with integer
coefficients has a rational root that root must be an integer. Every rational
of such a monic polynomia must be an integer (equivalently, each of its real
roots, if any, must be either an irrational number or an integer). Thus, in
our case, the rational numbers $r_{1}$ and $r_{2}$ are the roots of the monic
trinomial, $t(x)=(x-r_{1})(x-r_{2})=x^{2}-i_{1}x+i_{2}$; a monic quadratic
polynomial with integer coefficients $-i_{1}$ and $i_{2}$. Hence, $r_{1}$ and
$r_{2}$ must be integers.
For more details, see reference [1].
## References
* [1] 1 Kenneth H. Rose, Elementary Numbers Theory and Its Applications, fifth edition, 2005, Pearson-Addison-Wesley. For Lemma 1 (Lemma 3.4 in the above book), see page 109 for the Rational Root Theorem, see page 115.
|
arxiv-papers
| 2011-09-29T18:19:29 |
2024-09-04T02:49:22.633217
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1109.6820"
}
|
1109.6831
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-016 CERN-PH-EP-2011-151
Measurements of the Branching fractions for $B_{(s)}\rightarrow
D_{(s)}\pi\pi\pi$ and $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi\pi\pi$
The LHCb Collaboration 111Authors are listed on the following pages.
Branching fractions of the decays $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ relative to $H_{b}\rightarrow H_{c}\pi^{-}$ are
presented, where $H_{b}$ ($H_{c}$) represents $\bar{B}^{0}$ ($D^{+}$), $B^{-}$
($D^{0}$), $\bar{B}_{s}^{0}$ ($D_{s}^{+}$) and $\Lambda_{b}^{0}$
($\Lambda_{c}^{+}$). The measurements are performed with the LHCb detector
using 35 $\rm pb^{-1}$ of data collected at $\sqrt{s}=7$ TeV. The ratios of
branching fractions are measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-})}=2.38\pm
0.11\pm 0.21$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-})}=1.27\pm 0.06\pm 0.11$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-})}=2.01\pm 0.37\pm 0.20$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-})}=1.43\pm
0.16\pm 0.13.$
We also report measurements of partial decay rates of these decays to excited
charm hadrons. These results are of comparable or higher precision than
existing measurements.
R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z.
Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P.
Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,
R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A.
Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11,
J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C.
Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A.
Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8,
G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M.
van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T.
Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V.
Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52,
T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J.
Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H.
Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J.
Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7, M. Calvi20,j,
M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho
Akiba23, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N.
Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M.
Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P.
Collins37, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42,
G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P.
David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C. Deplano15, O. Deschamps5,
F. Dettori15,d, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, S. Donleavy48,
A. Dosil Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R.
Dzhelyadin34, C. Eames49, S. Easo45, U. Egede49, V. Egorychev30, S.
Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L.
Eklund47, Ch. Elsasser39, D.G. d’Enterria35,o, D. Esperante Pereira36, L.
Estève43, A. Falabella16,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C.
Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S.
Filippov32, C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M.
Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J.
Garofoli52, J. Garra Tico43, L. Garrido35, C. Gaspar37, N. Gauvin38, M.
Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C.
Göbel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M.
Grabalosa Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34,
T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-
Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7,
V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks48, W. Hofmann10, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40, M.
Karacson37, T.M. Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38,
B. Khanji6, Y.M. Kim46, M. Knecht38, S. Koblitz37, P. Koppenburg23, A.
Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25,
R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J.
Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37,
J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N.
Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A.
Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A.
Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B.
Maynard52, A. Mazurov32,16,37, G. McGregor50, R. McNulty12, C. Mclean14, M.
Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A.
Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P. Morawski25, R.
Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26,
M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, J.
Nardulli45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-
Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N. Nikitin31, A. Oblakowska-Mucha26,
V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, B. Pal52, J. Palacios39, M.
Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C. Parkes47,37,
C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49,
G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36,
A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, B. Pie
Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R. Plackett47, S. Playfer46, M.
Plo Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V.
Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M.
Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S.
Roiser37, V. Romanovsky34, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, R. Santinelli37, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I.
Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O.
Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P.
Skottowe43, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, K. Sobczak5, F.J.P.
Soler47, A. Solomin42, F. Soomro49, B. Souza De Paula2, B. Spaan9, A.
Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S.
Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, N.
Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T.
Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51,45, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, A. Ukleja27, P. Urquijo52, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36,
S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42,
A. Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25,37, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51,
Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, L. Zhang52, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avan$c$cats (ICREA), Barcelona,
Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Over the last two decades, a wealth of information has been accumulated on the
decays of $b$-hadrons. Measurements of their decays have been used to test the
CKM mechanism [1] for describing weak decay phenomena in the Standard Model,
as well as provide measurements against which various theoretical approaches,
such as HQET [2] and the factorization hypothesis, can be compared. While many
decays have been measured, a large number remain either unobserved or poorly
measured, most notably in the decays of $B^{0}_{s}$ mesons and
$\Lambda_{b}^{0}$ baryons. Among the largest hadronic branching fractions are
the decays $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$, where $H_{b}$
($H_{c}$) represents $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
($D^{+}$), $B^{-}$ ($D^{0}$), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ ($D_{s}^{+}$) and
$\Lambda_{b}^{0}$ ($\Lambda_{c}^{+}$). The first three branching fractions
were determined with only 30-40% accuracy, and the
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ branching
fraction was unmeasured.
Beyond improving our overall understanding of hadronic $b$ decays, these
decays are of interest because of their potential use in CP violation studies.
It is well known that the Cabibbo-suppressed decays $B^{-}\rightarrow DK^{-}$
[3, 4, 5] and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}$ [6, 7] provide clean measurements of the weak phase
$\gamma$ through time-independent and time-dependent rate measurements,
respectively. Additional sensitivity can be obtained by using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ [8]
decays. As well as these modes, one can exploit higher multiplicity decays,
such as $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow DK^{*0}$,
$B^{-}\rightarrow DK^{-}\pi^{+}\pi^{-}$ [9] and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}\pi^{\pm}\pi^{\mp}$. Moreover, the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ has been used to measure $\Delta m_{s}$ [10],
and with a sufficiently large sample, provides a calibration for the flavor-
mistag rate for the time-dependent analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}\pi^{\pm}\pi^{\mp}$.
The first step towards exploiting these multi-body decays is to observe them
and quantify their branching fractions. The more interesting Cabibbo-
suppressed decays are $O(\lambda^{3})$ in the Wolfenstein parameterization
[11], and therefore require larger data samples. Here, we present measurements
of the Cabibbo-favored $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ decays.
The leading amplitudes contributing to these final states are shown in Fig. 1.
Additional contributions from annihilation and $W$-exchange diagrams are
suppressed and are not shown here. Note that for the $B^{-}$ and
$\Lambda_{b}^{0}$ decays, unlike the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, there is potential for
interference between diagrams with similar magnitudes. In Ref. [12], it is
argued that this interference can explain the larger rate for
$B^{-}\rightarrow D^{0}\pi^{-}$ compared to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$. Thus, it
is interesting to see whether this is also true when the final state contains
three pions.
Figure 1: Feynman diagrams for $H_{b}\rightarrow H_{c}\pi^{-}$ and
$H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ decays. Figs. (a) and (b) show
external tree diagrams, (c) and (d) show color-suppressed tree diagrams
($B^{-}$ and $\Lambda_{b}^{0}$ only), and (e) shows the Cabibbo-suppressed
external tree diagram, only accessible to $B^{0}$ meson.
In this paper, we report measurements of the $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ branching fractions, relative to $H_{b}\rightarrow
H_{c}\pi^{-}$. We also report on the partial branching fractions,
$H_{b}\rightarrow H_{c}^{*}\pi^{-},~{}H_{c}^{*}\rightarrow
H_{c}\pi^{+}\pi^{-}$, where $H_{b}$ is either $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, $B^{-}$, or $\Lambda_{b}^{0}$,
and $H_{c}^{*}$ refers to $D_{1}(2420)^{+,0}$, $D_{2}^{*}(2460)^{0}$,
$\Lambda_{c}(2595)^{+}$, or $\Lambda_{c}(2625)^{+}$. We also present results
on the partial rates for
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}(2544)^{0,++}\pi^{\pm}\pi^{-}$. Charge
conjugate final states are implied throughout.
## 2 Detector and Trigger
The data used for this analysis were collected by the LHCb experiment during
the 2010 data taking period and comprise about 35 $\rm pb^{-1}$ of integrated
luminosity. LHCb has excellent capabilities to trigger on and reconstruct
bottom and charm hadrons. The most important element of the detector for this
analysis is a charged particle tracking system that covers the forward angular
region from about $15-350$ mrad and $15-250$ mrad in the horizontal and
vertical directions, respectively. It includes a 21 station, one-meter long
array of silicon strip detectors (VELO) that come within 8 mm of the LHC
beams, a 4 Tm dipole magnetic field, followed by three multi-layer tracking
stations (T-stations) downstream of the dipole magnet. Each T-station is
composed of a four layer silicon strip detector (IT) in the high occupancy
region near the beam pipe, an eight layer straw tube drift chamber (OT)
composed of 5 mm diameter straws outside this high occupancy region. Just
upstream of the dipole magnet is a four-layer silicon strip detector (TT).
Overall, the tracking system provides an impact parameter (IP) resolution of
$\sim 16\mu$m + 30$\mu$m/$p_{T}$ (transverse momentum, $p_{T}$ in GeV/$c$),
and a momentum resolution that ranges from $\sigma_{p}/p\sim 0.4\%$ at 3
GeV/$c$ to $\sim 0.6\%$ at 100 GeV/$c$. Two Ring Imaging Cherenkov Counters
(RICH) provide a kaon identification efficiency of $\sim$95% for a pion fake
rate of a few percent, integrated over the momentum range from 3$-$100
GeV/$c$. Downstream of the second RICH is a Preshower/Scintillating Pad
Detector (PS/SPD), and electromagnetic (ECAL) and hadronic (HCAL)
calorimeters. Information from the ECAL/HCAL is used to form the hadronic
triggers. Finally, a muon system consisting of five stations is used for
triggering on and identifying muons.
To reduce the 40 MHz crossing rate to about 2 kHz for permanent storage, LHCb
uses a two-level trigger system. The first level of the trigger, L0, is
hardware based and searches for either a large transverse energy cluster
($E_{T}>3.6$ GeV) in the calorimeters, or a single high $p_{T}$ or di-muon
pair in the muon stations. Events passing L0 are read out and sent to a large
computing farm, where they are analyzed using a software-based trigger. The
first level of the software trigger, called HLT1, uses a simplified version of
the offline software to apply tighter selections on charged particles based on
their $p_{T}$ and minimal IP to any primary vertex (PV), defined as the
location of the reconstructed $pp$ collision(s). The HLT1 trigger relevant for
this analysis searches for a single track with IP larger than 125 $\mu$m,
$p_{T}>1.8$ GeV/$c$, $p>12.5$ GeV/$c$, along with other track quality
requirements. Events that pass HLT1 are analyzed by a second software level,
HLT2, where the event is searched for 2, 3, or 4-particle vertices that are
consistent with $b$-hadron decays. Tracks are required to have $p>5$ GeV/$c$,
$p_{T}>0.5$ GeV/$c$ and IP $\chi^{2}$ larger than 16 to any PV, where the
$\chi^{2}$ value is obtained assuming the IP is equal to zero. We also demand
that at least one track has $p_{T}>1.5$ GeV/$c$, a scalar $p_{T}$ sum of the
track in the vertex exceed 4 GeV/$c$, and that the corrected mass222The
corrected mass is defined as $M_{\rm cor}=\sqrt{M^{2}+p_{\rm trans}^{2}}$,
where $M$ is the invariant mass of the 2, 3 or 4-track candidate (assuming the
kaon mass for each particle), and $p_{\rm trans}$ is the momentum imbalance
transverse to the direction of flight, defined by the vector that joins the
primary and secondary vertices. is between 4 and 7 GeV/$c^{2}$. These HLT
trigger selections each have an efficiency in the range of 80$-$90% for events
that pass typical offline selections for a large range of $B$ decays. A more
detailed description of the LHCb detector can be found in Ref. [13].
Events with large occupancy are known to have intrinsically high backgrounds
and to be slow to reconstruct. Therefore such events were suppressed by
applying global event cuts (GECs) to hadronically triggered decays. These GECs
included a maximum of 3000 VELO clusters, 3000 IT hits, and 10,000 OT hits. In
addition, hadron triggers were required to have less than 900 or 450 hits in
the SPD, depending on the specific trigger setting.
## 3 Candidate Reconstruction and Selection
Charged particles likely to come from a $b$-hadron decay are first identified
by requiring that they have a minimum IP $\chi^{2}$ with respect to any PV of
more than 9. We also require a minimum transverse momentum, $p_{T}>300$
MeV/$c$, except for $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ decays,
where we allow (at most) one track to have $200<p_{T}<300$ MeV/$c$. Hadrons
are identified using RICH information by requiring the difference in log-
likelihoods ($\Delta LL$) of the different mass hypotheses to satisfy $\Delta
LL(K-\pi)>-5$, $\Delta LL(p-\pi)>-5$ and $\Delta LL(K-\pi)<12$, for kaons,
protons and pions, respectively. These particle hypotheses are not mutually
exclusive, however the same track cannot enter more than once in the same
decay chain.
Charm particle candidates are reconstructed in the decay modes
$D^{0}\rightarrow K^{-}\pi^{+}$, $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$,
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ and $\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+}$. The candidate is associated to one of the PVs in the event
based on the smallest IP $\chi^{2}$ between the charm particle’s reconstructed
trajectory and all PVs in the event. A number of selection criteria are
imposed to reduce backgrounds from both prompt charm with random tracks as
well as purely combinatorial background. To reduce the latter, we demand that
each candidate is well separated from the associated PV by requiring that its
flight distance (FD) projected onto the $z$-axis is larger than 2 mm, the FD
$\chi^{2}>49$333This is the $\chi^{2}$ with respect to the FD=0 hypothesis.,
and that the distance in the transverse direction ($\Delta R$) is larger than
100 $\mu$m. Background from random track combinations is also suppressed by
requiring the vertex fit $\chi^{2}$/ndf$<8$, and $p_{T}>1.25$ GeV/$c$ (1.5
GeV/$c$ for $D^{+}_{(s)}$ in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$.)
To reduce the contribution from prompt charm, we require that the charm
particle has a minimal IP larger than 80 $\mu$m and IP $\chi^{2}>12.25$ with
respect to its associated PV. For $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$, we
employ tighter particle identification requirements on the kaons, namely
$\Delta LL(K-\pi)>0$, if the $K^{+}K^{-}$ invariant mass is outside a window
of $\pm$20 MeV/$c^{2}$ of the $\phi$ mass [15]. Lastly, we require the
reconstructed charm particles masses to be within 25 MeV/$c^{2}$ of their
known values.
The bachelor pion for $H_{b}\rightarrow H_{c}\pi^{-}$ is required to have
$p_{T}>0.5$ GeV/$c$, $p>5.0$ GeV/$c$ and IP $\chi^{2}>16$. For the 3$\pi$
vertex associated with the $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$
decays, we apply a selection identical to that for the charm particle
candidates, except we only require the $p_{T}$ of the $3\pi$ system to be
larger than 1 GeV/$c$ and that the invariant mass is in the range from
$0.8~{}{\rm GeV}/c^{2}<M(\pi\pi\pi)<3.0$ GeV/$c^{2}$.
Beauty hadrons are formed by combining a charm particle with either a single
pion candidate (for $H_{b}\rightarrow H_{c}\pi^{-}$) or a $3\pi$ candidate
(for $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$.) The $b$-hadron is
required to have a transverse momentum of at least 1 GeV/$c$. As with the
charm hadron, we require it is well-separated from its associated PV, with FD
larger than 2 mm, FD $\chi^{2}>49$ and $\Delta R>100$ $\mu$m. We also make a
series of requirements that ensure that the $b$-hadron candidate is consistent
with a particle produced in a proton-proton interaction. We require the
candidate to have IP$<$90 $\mu$m, IP $\chi^{2}<16$, and that the angle
$\theta$ between the $b$-hadron momentum and the vector formed by joining the
associated PV and the decay vertex satisfies $\cos\theta>0.99996$. To ensure a
good quality vertex fit, we require a vertex fit $\chi^{2}/$ndf$<6$ (8 for
$H_{b}\rightarrow H_{c}\pi^{-}$.)
To limit the timing to process high occupancy events, we place requirements on
the number of tracks444Here, tracks refer to charged particles that have
segments in both the VELO and the T-stations. in an event. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$,
the maximum number of tracks is 180, and for
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ and $B^{-}\rightarrow
D^{0}\pi^{-}$ it is 120. These selections are 99% and 95% efficient,
respectively, after the GECs. The $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ selection requires fewer than 300 tracks, and thus
is essentially 100% efficient after the GECs.
Events are required to pass the triggers described above. This alone does not
imply that the signal $b$-hadron decay was directly responsible for the
trigger. We therefore also require that one or more of the signal $b$-hadron
daughters is responsible for triggering the event. We thus explicitly select
events that Triggered On the Signal decay (TOS) at L0, HLT1 and HLT2. For the
measurements of excited charm states, where our yields are statistically
limited, we also make use of L0-triggers that Triggered Independently of the
Signal decay (TIS). In this case, the L0 trigger is traced to one or more
particles other than those in the signal decay.
Lastly, we note that in $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$
candidate events, between 4% and 10% have multiple candidates (mostly two) in
the same event. In such cases we choose the candidate with the largest
transverse momentum. This criterion is estimated to be $(75\pm 20)$% efficient
for choosing the correct candidate. For $H_{b}\rightarrow H_{c}\pi^{-}$
multiple candidates occur in less than 1% of events, from which we again
choose the one with the largest $p_{T}$.
### 3.1 Selection Efficiencies
Selection and trigger efficiencies are estimated using Monte Carlo (MC)
simulations. The MC samples are generated with an average number of
interactions per crossing equal to 2.5, which is similar to the running
conditions for the majority of the 2010 data. The $b$-hadrons are produced
using pythia [16] and decayed using evtgen [17]. The $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ decays are produced using a cocktail for the
$\pi\pi\pi$ system that is $\sim$2/3
$a_{1}(1260)^{-}\rightarrow\rho^{0}\pi^{-}$ and about 1/3 non-resonant
$\rho^{0}\pi^{-}$. Smaller contributions from $D_{1}^{0}(2420)\pi$ and
$D_{2}^{*0}(2460)\pi$ are each included at the 5% level to $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ and 2% each for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$. For
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$, we include
contributions from $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$, which
contribute 9% and 7% to the MC sample. The detector is simulated with geant4
[18], and the event samples are subsequently analyzed in the same way as data.
We compute the total kinematic efficiency, $\epsilon_{\rm kin}$ from the MC
simulation as the fraction of all events that pass all reconstruction and
selection requirements. These selected events are then passed through a
software emulation of the L0 trigger, and the HLT software used to select the
data, from which we compute the trigger efficiency ($\epsilon_{\rm trig}$).
The efficiencies for the decay modes under study are shown in Table 1. Only
the relative efficiencies are used to obtain the results in this paper.
Table 1: Summary of efficiencies for decay channels under study. Here, $\epsilon_{\rm kin}$ is the total kinematic selection efficiency, $\epsilon_{\rm trig}$ is the trigger efficiency, and $\epsilon_{\rm tot}$ is their product. The uncertainties shown are statistical only. Decay | $\epsilon_{\rm kin}$ | $\epsilon_{\rm trig}$ | $\epsilon_{\rm tot}$
---|---|---|---
| (%) | (%) | (%)
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}\pi^{+}\pi^{-}$ | $0.153\pm 0.003$ | $22.6\pm 0.5$ | $0.0347\pm 0.0011$
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$ | $0.275\pm 0.007$ | $27.4\pm 0.6$ | $0.0753\pm 0.0019$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ | $0.137\pm 0.003$ | $24.9\pm 0.7$ | $0.0342\pm 0.0012$
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ | $0.110\pm 0.005$ | $24.0\pm 0.7$ | $0.0264\pm 0.0008$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ | $0.882\pm 0.014$ | $20.8\pm 0.3$ | $0.184\pm 0.004$
$B^{-}\rightarrow D^{0}\pi^{-}$ | $1.54\pm 0.02$ | $27.4\pm 0.3$ | $0.421\pm 0.007$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$ | $0.868\pm 0.010$ | $23.1\pm 0.2$ | $0.201\pm 0.003$
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ | $0.732\pm 0.015$ | $24.7\pm 0.4$ | $0.181\pm 0.004$
## 4 Reconstructed Signals in Data
The reconstructed invariant mass distributions are shown in Figs. 2 and 3 for
the signal and normalization modes, respectively. Unbinned likelihood fits are
performed to extract the signal yields, where the likelihood functions are
given by the sums of signal and several background components. The signal and
background components are shown in the figures. The signal contributions are
each described by the sum of two Gaussian shapes with equal means. The
relative width and fraction of the wider Gaussian shape with respect to the
narrower one are constrained to the values found from MC simulation based on
agreement with data in the large yield signal modes. This constraint is
included with a 10-12% uncertainty (mode-dependent), which is the level of
agreement found between data and MC simulation. The absolute width of the
narrower Gaussian is a free parameter in the fit, since the data shows a
slightly worse ($\sim$10%) resolution that MC simulation.
For $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ decays, there are peaking backgrounds from
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ just below the $B_{s}^{0}$ mass. We therefore fix
their core Gaussian widths as well, based on the resolutions found in data for
the kinematically similar $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ decays, scaled by 0.93, which is the ratio of
expected widths obtained from MC simulation.
Figure 2: Invariant mass distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ (top left), $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ (top right), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ (bottom left), and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ (bottom
right). Fits showing the signal and background components are indicated, and
are described in the text.
Figure 3: Invariant mass distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ (top
left), $B^{-}\rightarrow D^{0}\pi^{-}$ (top right), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
(bottom left), and $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ (bottom
right). Fits showing the signal and background components are indicated, and
are described in the text.
A number of backgrounds contribute to these decays. Below the $b$-hadron
masses there are generally peaking background structures due to partially
reconstructed $B$ decays. These decays include $B_{(s)}\rightarrow
D_{(s)}^{*}\pi(\pi\pi)$, with a missed photon, $\pi^{0}$ or $\pi^{+}$, as well
as $B_{(s)}\rightarrow D_{(s)}\rho^{-}$, where the $\pi^{0}$ is not included
in the decay hypothesis. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and
$B^{-}\rightarrow D^{0}\pi^{-}$ decays, the shapes of these backgrounds are
taken from dedicated signal MC samples. The double-peaked background shape
from partially-reconstructed $D^{*}\pi$ decays is obtained by fitting the
background MC sample to the sum of two Gaussian shapes with different means.
The difference in their means is then fixed, while their average is a free
parameter in subsequent fits to the data. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$,
the shape of the partially-reconstructed $D^{*}\pi\pi\pi$ background is not as
easily derived since the helicity amplitudes are not known. This low mass
background is also parametrized using a two-Gaussian model, but we let the
parameters float in the fit to the data. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$, we obtain the background shape from a large
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}X$
inclusive MC sample. Less is known about the $\Lambda_{b}^{0}$ hadronic decays
that would contribute background to the
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ invariant
mass spectra. For
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$, we see no
clear structure due to partially-reconstructed backgrounds. For
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$, there does appear to be
structure at about 5430 MeV/$c^{2}$, which may be due to
$\Lambda_{c}^{+}\rho^{-}$. The enhancement is described by a single Gaussian
above the combinatoric background, which, given the limited number of events,
provides a good description of this background.
There are also so-called reflection backgrounds, where fully reconstructed
signal decays from one $b$-hadron decay mode produce peaking structures in the
invariant mass spectra of other decay modes when one of the daughter particles
is misidentified. For $B\rightarrow D\pi^{-}(\pi^{+}\pi^{-})$, there are
reflections from $B\rightarrow DK^{-}(\pi^{+}\pi^{-})$ Cabibbo-suppressed
decays, where the kaon is misidentified as a pion. Due to the Cabibbo
suppression and the excellent RICH performance, their contributions are
limited to the 1% level. The shape of this misidentification background is
taken from MC simulation and is constrained to be $(1\pm 1)\%$ of the signal
yield.
For the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ decays, there are reflection backgrounds from
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ modes, when either of the $\pi^{+}$ from the
$D^{+}$ decay is misidentified as a $K^{+}$. This cross-feed background is
evaluated in two ways. First, we take our $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$) data, which have very loose particle
identification (PID) requirements on the pions, and apply the kaon PID
selection to them. If either of the two pions pass, and the recomputed
($KK\pi$) mass is within the $D_{s}^{+}$ mass window, the candidate is counted
as a reflection background. Using this technique, we find $(5.3\pm 0.4)\%$
[$(6.3\pm 0.6)\%$] of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ [$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$] signal decays reflect into the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
[$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$] signal region. In the second method, we apply
a $\pi$-faking-$K$ misidentification matrix (in bins of $p$ and $p_{T}$),
obtained from a $D^{*+}$ data calibration sample to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ (or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$) signal MC sample, followed by the $D_{s}^{+}$
mass window requirement (after replacing the pion mass with the kaon mass.)
The results of this second procedure are $(4.4\pm 0.3)\%$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ and
$(5.2\pm 0.4)\%$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$, both of which are consistent with the first
method. We therefore constrain the peaking background from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$) into $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$) to be $(4.0\pm 1.5)\%$ ($(5.0\pm 2.0)\%$),
where the Gaussian constraint is conservatively assigned a 40% relative
uncertainty. The shape of this peaking background is obtained from MC
simulation and is well-described by a single Gaussian of mean 5350 MeV/$c^{2}$
and width 30 MeV/$c^{2}$. This shape is in good agreement with what is
observed in data.
The second reflection background to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$) is
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$
($\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$), where the
proton from the $\Lambda_{c}$ decay is misidentified as a kaon. This is
similar to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ reflection,
except here the $\Lambda_{b}^{0}$ yield is significantly smaller, obviating
the need for making an explicit $\Delta LL(K-p)$ requirement to reject
protons. The $\Lambda_{b}^{0}$ reflection background is evaluated using the
first technique as described above leading to reflection rates of $(15\pm
3)\%$ for $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ into $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $(20\pm 4)\%$ for
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ into $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$. We conservatively assign a 20% uncertainty on
this rate based on the agreement between data and MC simulation. The
asymmetric shape of this background is described by the simulation, which is
consistent with the shape observed in data. The combinatorial background is
modeled with an exponential distribution. The fits are superimposed on the
data in Figs. 2 and 3, and the fitted yields are summarized in Table 2.
The ratios of branching ratios are given by:
$\displaystyle{{\cal{B}}(H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(H_{b}\rightarrow
H_{c}\pi^{-})}={Y^{\rm sig}/\epsilon_{\rm tot}^{\rm sig}\over Y^{\rm
norm}/\epsilon_{\rm tot}^{\rm norm}}$
where the $Y$ factors are the observed yields in the signal and normalization
modes, and $\epsilon_{\rm tot}$ are the total selection efficiencies.
Table 2: Summary of yields for the branching fraction computation. Uncertainties are statistical only. Decay | Yield | Decay | Yield
---|---|---|---
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}\pi^{+}\pi^{-}$ | $1150\pm 43$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ | $2745\pm 66$
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$ | $950\pm 41$ | $B^{-}\rightarrow D^{0}\pi^{-}$ | $4244\pm 90$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ | $138\pm 23$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$ | $434\pm 32$
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ | $174\pm 18$ | $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ | $853\pm 36$
## 5 Systematic Uncertainties
Several sources contribute uncertainty to the measured ratios of branching
fractions. Because we are measuring ratios of branching fractions, most, but
not all of the potential systematics cancel. Here, we discuss only the non-
cancelling uncertainties. With regard to the reconstruction of the
$H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ and $H_{b}\rightarrow
H_{c}\pi^{-}$ decays, the former has two additional pions which need to pass
our selections, and the $3\pi$ system needs to pass the various vertex-related
selection criteria. The track reconstruction efficiency and uncertainty are
evaluated by measuring the ratio of fully reconstructed $J/\psi$’s to all
$J/\psi$’s obtained from an inclusive single muon trigger, where only one of
the muons is required to be reconstructed. After reweighting the efficiencies
to match the kinematics of the signal tracks, the uncertainty is found to be
2% per track, which leads to a 4% uncertainty in the branching fraction
ratios. The IP resolution in data is about 20% worse than in the simulation,
leading to (i) a larger efficiency for tracks to pass the IP-related cuts (as
well as larger background), and (ii) a lower efficiency to pass the vertex
$\chi^{2}$ selections, for data relative to the value predicted by simulation.
The first of these is studied by reducing the IP $\chi^{2}$ requirement in
simulation by 20%, and the second by smearing the vertex $\chi^{2}$
distribution in simulation until it agrees with data. The combined correction
is found to be $1.02\pm 0.03$.
Another potential source of systematic uncertainty is related to the
production and decay model for producing the $H_{c}\pi\pi\pi$ final state. We
have considered that the $p_{T}$ spectrum of the pions in the 3$\pi$ system
may be different between simulation and data. To estimate the uncertainty, we
reweight the MC simulation to replicate the momentum spectrum of the lowest
momentum pion (amongst the pions in the $3\pi$ vertex.) We find that the total
efficiency using the reweighted spectra agrees with the unweighted spectra to
within 3%. We have also investigated the effect of differences in the $p_{T}$
spectra of the charm particle, and find at most a 1% difference. Our candidate
selection is limited to the mass region $M(\pi\pi\pi)<3$ GeV/$c^{2}$. Given
that the phase space population approaches zero as $M(\pi\pi\pi)\rightarrow
3.5$ GeV/$c^{2}$ (i.e., $M_{B}-M_{D}$) and that the simulation reasonably
reproduces the $\pi^{-}\pi^{+}\pi^{-}$ mass spectrum, we use the simulation to
assess the fraction of the $\pi\pi\pi$ mass spectrum beyond 3 GeV/$c^{2}$. We
find the fraction of events above 3 GeV/$c^{2}$ is $(3.5-4.5)$% for the decay
modes under study. We apply a correction of $1.04\pm 0.02$, where we have
assigned half the correction as an estimate of the uncertainty. In total, the
correction for production and decay models is $1.04\pm 0.04$.
As discussed in Sec. 3, we choose only one candidate per event. The efficiency
of this selection is estimated by comparing the signal yield in multiple-
candidate events before and after applying the best candidate selection. The
selection is estimated to be $(75\pm 20)\%$ efficient. In the
$H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ the multiple candidate rate
varies from 4% to 10%, so we have corrections that vary from 1.01 to 1.03. For
$H_{b}\rightarrow H_{c}\pi^{-}$, this effect is negligible. The corrections
for each mode are given in Table 3.
For the trigger efficiency, we rely on signal MC simulations to emulate the
online trigger. The stability of the relative trigger efficiency was checked
by reweighting the $b$-hadron $p_{T}$ spectra for both the signal and
normalization modes, and re-evaluating the trigger efficiency ratios. We find
maximum differences of 2% for L0, 1% for HLT1 and 1% for HLT2, (2.4% total)
which we assign as a systematic uncertainty.
Fitting systematics are evaluated by varying the background shapes and
assumptions about the signal parameterization for both the $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ and $H_{b}\rightarrow H_{c}\pi^{-}$ modes and re-
measuring the yield ratios. For the combinatorial background, using first and
second order polynomials leads to a 3% uncertainty on the relative yield.
Reflection background uncertainties are negligible, except for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$,
where we find deviations as large as 5% when varying the central value of the
constraints on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$
reflections by $\pm$1 standard deviation. We have checked our sensitivity to
the signal model by varying the constraints on the width ratio and core
Gaussian area fraction by one standard deviation (2%). We also include a
systematic uncertainty of 1% for neglecting the small radiative tail in the
fit, which is estimated by comparing the yields between our double Gaussian
signal model and the sum of a Gaussian and Crystal Ball [20] line shape. Taken
together, we assign a 4% uncertainty to the relative yields. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ branching fraction ratio, the
total fitting uncertainty is 6.4%.
Another difference between the $H_{b}\rightarrow H_{c}\pi^{-}$ and
$H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ selection is the upper limit on
the number of tracks. The efficiencies of the lower track multiplicity
requirements can be evaluated using the samples with higher track multiplicity
requirements. Using this technique, we find corrections of $0.95\pm 0.01$ for
the $B^{-}$ and $\Lambda_{b}^{0}$ branching fraction ratios, and $0.99\pm
0.01$ for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ branching fraction ratios.
We have also studied the PID efficiency uncertainty using a $D^{*+}$
calibration sample in data. Since the PID requirements are either common to
the signal and normalization modes, or in the case of the bachelor pion(s),
the selection is very loose, the uncertainty is small and we estimate a
correction of $1.01\pm 0.01$. We have also considered possible background from
$H_{b}\rightarrow H_{c}D_{s}^{-}$ which results in a correction of $0.99\pm
0.01$.
All of our MC samples have a comparable number of events, from which we incur
3-4% uncertainty in the efficiency ratio determinations. The full set of
systematic uncertainties and corrections are shown in Table 3. In total, the
systematic uncertainty is $\sim$9%, with correction factors that range from
1.01 to 1.07.
Table 3: Summary of corrections and systematic uncertainties to the ratio of branching fractions ${\cal{B}}(H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-})/{\cal{B}}(H_{b}\rightarrow H_{c}\pi^{-})$. Source | central value $\pm$ syst. error
---|---
| $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ | $B^{-}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ | $\Lambda_{b}$
Track Reconstruction | $1.00\pm 0.04$
IP/Vertex Resolution | $1.02\pm 0.03$
Production/Decay Model | $1.04\pm 0.04$
Best Cand. Selection | $1.02\pm 0.02$ | $1.01\pm 0.01$ | $1.02\pm 0.02$ | $1.03\pm 0.02$
Trigger Efficiency | $1.00\pm 0.02$
Fitting | $1.00\pm 0.04$ | $1.00\pm 0.04$ | $1.00\pm 0.06$ | $1.00\pm 0.04$
Cut on #Tracks | $0.99\pm 0.01$ | $0.95\pm 0.01$ | $0.99\pm 0.01$ | $0.95\pm 0.01$
PID | $1.01\pm 0.01$
$H_{c}D_{s}^{+}$ background | $0.99\pm 0.01$
MC Statistics | $1.00\pm 0.04$ | $1.00\pm 0.03$ | $1.00\pm 0.04$ | $1.00\pm 0.04$
Total Correction | 1.07 | 1.01 | 1.07 | 1.03
Total Systematic (%) | 8.8 | 8.4 | 10.1 | 9.2
## 6 Results for $\boldsymbol{H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}}$
The results for the ratios of branching ratios are
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-})}=2.38\pm
0.11\pm 0.21$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-})}=1.27\pm 0.06\pm 0.11$ (1) $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-})}=2.01\pm 0.37\pm 0.20$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-})}=1.43\pm
0.16\pm 0.13,$
where the first uncertainty is statistical and the second is systematic. These
measurements are all substantially more precise than the current world average
values. Naively, one might have expected the four branching fraction ratios to
be nearly equal. The observed differences may be explained in terms of the
contributing Feynman diagrams. From Fig. 1, we see that the primary
contribution to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}(\pi^{+}\pi^{-})$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}(\pi^{+}\pi^{-})$ is from a single decay diagram, an external
tree diagram. On the other hand the $B^{-}\rightarrow
D^{0}\pi^{-}(\pi^{+}\pi^{-})$ and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}(\pi^{+}\pi^{-})$ amplitudes
receive contributions from both external and color-suppressed tree diagrams.
This would suggest that the interference tends to be more constructive in
$B^{-}\rightarrow D^{0}\pi^{-}$ and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ than in $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ respectively.
The role of the various contributing topological amplitudes and the strong
phases in $B\rightarrow D\pi$ is discussed in the literature [12]. In general
we see the branching fractions for the $H_{c}\pi\pi\pi$ final states are at
least as large or even twice as large as the single-$\pi$ bachelor states.
## 7 Kinematic Distributions and Mass Spectra in the
$\boldsymbol{\pi^{-}\pi^{+}\pi^{-}}$ System
Since we rely on MC simulation to estimate signal efficiencies, we now compare
a few distributions between signal MC simulation and data. The higher signal
yield $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ decay modes are used, and for each we perform a
sideband subtraction, where the signal region includes candidates within 50
MeV/$c^{2}$ of the $B^{0}$ mass, ($m_{B^{0}}$) [15], and the sidebands
$60<|M-m_{B^{0}}|<110$ MeV/$c^{2}$. For both data and simulation, we require
events to pass any L0 trigger, and signal candidates must satisfy the HLT1 and
HLT2 triggers described in Sec. 2. Clearly, two of the most important
quantities used in our candidate selection are the $p_{T}$ and IP of the
daughters from the $D^{+}$ and the recoiling pion(s). Figure 4 compares the
$p_{T}$ and IP distributions of the $D^{+}$ daughters in data to those from
signal MC simulation. Figure 5 shows the corresponding comparisons for the
recoiling pion(s) in the respective $B$ decay. Overall, the agreement between
data and MC simulation is very good.
Figure 4: Comparisons of the $p_{T}$ and IP spectra for the daughters from the
$D^{+}$ in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}$ [(a) and (b)], and from the $D^{+}$ in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ [(c) and (d)]; Points with error bars are data and
the solid lines are simulation.
Figure 5: Comparisons of the $p_{T}$ and IP spectra for the bachelor pion in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$
[(a) and (b)], and for the 3 pions in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ [(c) and (d)]. Points with error bars are data and
the solid lines are simulation.
It is also interesting to examine the $\pi^{-}\pi^{+}\pi^{-}$ invariant mass
spectra for the four signal decay modes. Here, we use the sPlot method [19] to
obtain the underlying signal spectra, based on the event-by-event $b$-hadron
mass signal and background probabilities. The $\pi^{-}\pi^{+}\pi^{-}$ mass
spectra are shown in Fig. 6, along with signal MC shapes that are normalized
to the same yield as data. We also show several resonant contributions:
$D_{1}(2420)^{+}$ (2%), $D_{1}(2420)^{0}$ and $D_{2}^{*}(2460)^{0}$ (14% in
total), $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ (9% total), and
$\Sigma_{c}^{0}$ and $\Sigma_{c}^{++}$ (12% total), where the quantities in
parentheses are the normalizations relative to the total (see Sec. 8.) A
prominent structure at low mass, consistent with the $a_{1}(1260)^{-}$ is
evident for all decay modes, along with a long tail extending to 3
GeV/$c^{2}$. In all cases, the 3$\pi$ mass spectrum appears shifted toward
lower mass as compared to the MC simulation. The simulated value for the
$a_{1}(1260)^{-}$ mass is 1230 MeV/$c^{2}$, which is equal to the central
value given in the PDG [15] of $(1230\pm 40)$ MeV/$c^{2}$. Besides having a
large uncertainty, the mass as obtained by experiment may be process-
dependent, so it is difficult to draw any definitive conclusion from this
shift. Since both the reconstruction and trigger efficiency are flat through
this mass region, this small shift in mass does not introduce any significant
systematic uncertainty in the branching fraction measurement.
Figure 6: Invariant mass of the $3\pi$ system in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ (top left), $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ (top right), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ (bottom left) and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ (bottom
right) decays. The data are the points with error bars and the simulations are
the solid lines and shaded regions.
We have also looked at the di-pion invariant masses within the 3$\pi$ system,
shown for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$(a,b) and $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$(c,d) in Fig. 7. Contributions from the narrow
excited charm states, which are discussed in Sec. 8, are excluded. In all
cases, in the low $M(\pi^{-}\pi^{+}\pi^{-})$ mass region, we see a dominant
$\rho^{0}\pi^{-}$ contribution, consistent with the $a_{1}(1260)^{-}$
resonance. In the higher $M(\pi\pi\pi)$ regions there appears to be an
additional resonant structure, consistent with the $f_{2}(1270)$ state, in
addition to the $\rho^{0}$ contribution. Similar spectra are found for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-}$ and
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ (not shown.)
The $f_{2}(1270)$ has been previously seen in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{*+}\pi^{-}\pi^{+}\pi^{-}$ [21]. The like-sign di-pion invariant mass
spectra do not show any resonant features.
Figure 7: $\pi^{+}\pi^{-}$ invariant mass (2 combinations per $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate) in the $3\pi$ system
for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ when (a) $M(\pi^{-}\pi^{+}\pi^{-})<1.5$
GeV/$c^{2}$ and (b) $M(\pi^{-}\pi^{+}\pi^{-})\geq 1.5$ GeV/$c^{2}$. The
corresponding plots for $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$ are
shown in (c) and (d).
## 8 Contributions from Excited Charm Hadrons
Within the $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ final state, we
search for $D_{1}(2420)$, $D_{2}^{*}(2460)$, $\Lambda_{c}(2595)^{+}$,
$\Lambda_{c}(2625)^{+}$ and $\Sigma_{c}^{0,++}$, which may decay to $D$ or
$\Lambda_{c}^{+}$ with an accompanying $\pi^{\pm}$ or $\pi\pi$ pair. To search
for $H_{c}^{*}\rightarrow H_{c}\pi^{+}\pi^{-}$ intermediate states, we select
events in the $b$-hadron signal region ($\pm 60$ MeV/$c^{2}$ around the
nominal mass) and compute the invariant mass difference $\Delta
M_{\pi\pi}\equiv M(H_{c}\pi^{+}\pi^{-})-M(H_{c})$ (two combinations per
$b$-hadron candidate.) For the
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi^{\pm}\pi^{-},~{}\Sigma_{c}^{0,++}\rightarrow\Lambda_{c}^{+}\pi^{\pm}$,
we use $\Delta M_{\pi}\equiv M(H_{c}\pi^{\pm})-M(H_{c})$ in a similar way (one
(two) $\Sigma_{c}^{++}$ ($\Sigma_{c}^{0}$) candidates per $\Lambda_{b}^{0}$
decay.) We also have looked in the upper mass sidebands, and the $\Delta
M_{\pi\pi}$ and $\Delta M_{\pi}$ distributions are consistent with a smooth
background shape with no signal component. We look at all data, irrespective
of trigger, to establish signal significances, but for the branching fraction
measurement, we use the same trigger requirements described in Sec. 7. We
choose only one candidate per event using the same criteria as discussed
previously. We normalize the rates to the respective inclusive
$H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$ decay, using the same trigger
selection as above. We show only the $\Delta M_{\pi\pi}$ and $\Delta M_{\pi}$
distributions after the specified trigger, since the distributions before the
trigger are quite similar, except they typically have 25$-$30% larger yields
than the ones shown.
The $\Delta M_{\pi\pi}$ distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{+}$ are shown in Fig. 8
and the $\Delta M_{\pi}$ for $\Lambda_{b}^{0}$ are shown in Fig. 9. For
$B_{s}^{0}$, the size of the data sample is insufficient to observe the
excited $D_{s}$ states in these hadronic decays.
Signal yields are determined using unbinned extended maximum likelihood fits.
Starting with $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ (Fig. 8(a)),
we see an excess at $\Delta M_{\pi\pi}\sim 560$ MeV/$c^{2}$, consistent with
the $D_{1}(2420)^{+}$. We fit the distribution to the sum of a signal Breit-
Wigner shape convoluted with a Gaussian resolution, and an exponential
background shape. The full width is fixed to 25 MeV/$c^{2}$ [15] and the mass
resolution is set to 7.5 MeV/$c^{2}$ based on simulation. The fitted yield is
$33\pm 8$ events and the fitted mean is $(562\pm 4)$ MeV/$c^{2}$, consistent
with the expected value. If the width is allowed to float, we find $(22.7\pm
8.0({\rm stat}))$ MeV/$c^{2}$, also in agreement with the world average. Prior
to applying the specific trigger selection, we find $40\pm 9$ signal events,
corresponding to a statistical significance of 6.8 standard deviations (for
one degree of freedom) as determined from the difference in log-likelihoods,
$\sqrt{-2\Delta LL}$, where the difference is taken between the signal yield
taken as a free parameter and fixed to zero.
Figure 8: Invariant mass difference $M(D\pi^{-}\pi^{+})-M(D)$, for (a) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ signal candidates, (b) $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ signal candidates, (c) $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ through a $D^{*+}$ intermediate state, and (d)
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$ not through a $D^{*+}$
intermediate state. The signal components are the white region (and lightly
shaded regions for $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$), and the
background component is the darker shaded region.
The $\Delta M_{\pi\pi}$ distributions for $B^{-}$ displayed in Fig. 8(b) show
not only the $D_{1}(2420)^{0}$, but also a shoulder at $\sim$600 MeV/$c^{2}$,
consistent with the $D_{2}^{*}(2460)^{0}$. Hence, we allow for both
$D_{1}(2420)^{0}$ and $D_{2}^{*}(2460)^{0}$ signal components, and fix their
full widths to the PDG values [15] of 20.4 MeV/$c^{2}$ and 42.9 MeV/$c^{2}$,
respectively. The means and yields are left as free parameters in the fit. The
fitted $D_{1}(2420)^{0}$ and $D_{2}^{*}(2460)^{0}$ yields are $124\pm 14$ and
$49\pm 12$, with masses that are consistent with the expected values. The
respective signal yields before the trigger requirement are $165\pm 17$ and
$63\pm 15$ events, with corresponding statistical significances of 10.5 and
5.5 standard deviations for the $D_{1}(2420)^{0}$ and $D_{2}^{*}(2460)^{0}$,
respectively. These $B^{0}$ and $B^{-}$ decays have also been observed by
Belle [22].
We have also measured the relative fractions of $D_{1}(2420)^{0}$ and
$D_{2}^{*}(2460)^{0}$ that do or do not decay through $D^{*+}$ by taking the
subset of candidates with $M(D^{0}\pi^{+})-M(D^{0})\leq 150$ MeV/$c^{2}$ or
$M(D^{0}\pi^{+})-M(D^{0})>150$ MeV/$c^{2}$, respectively. The corresponding
$\Delta M_{\pi\pi}$ distributions are shown in Fig. 8(c) and Fig. 8(d). A fit
is made to the data as discussed previously, and the yields are summarized in
Table 4.
For $\Lambda_{b}^{0}$ (see Fig. 9(a)), we find two well-separated peaks in the
$\Delta M_{\pi\pi}$ distribution, one at $\sim$307 MeV/$c^{2}$, and a second
at $\sim$340 MeV/$c^{2}$, consistent with the expected values for the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$, respectively. The full
width of the $\Lambda_{c}(2595)^{+}$ is fixed to the PDG value of 3.6
MeV/$c^{2}$, and the mass resolution for each peak is fixed to 2.0
MeV/$c^{2}$, as determined from simulation. The fitted signal yields are
$9.7\pm 3.5$ and $9.3\pm 3.2$ for the $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$, respectively. Before the trigger, we find signal
yields of $10.6\pm 3.8$ for $\Lambda_{c}(2595)^{+}$ and $15.7\pm 4.1$ for
$\Lambda_{c}(2625)^{+}$, corresponding to statistical significances of 4.3 and
6.6 standard deviations. Thus we have evidence for
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2595)^{+}\pi^{-}$ and observation of
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2625)^{+}\pi^{-}$. The systematic
uncertainties do not change this conclusion. These decays have also been
reported by CDF [23], but are not yet published. The fitted $\Delta
M_{\pi\pi}$ values of $(306.7\pm 1.1)$ MeV/$c^{2}$ and $(341.7\pm 0.6)$
MeV/$c^{2}$, for the $\Lambda_{c}(2625)^{+}$ and $\Lambda_{c}(2625)^{+}$,
respectively, are consistent with the known mass differences [15] for these
excited states.
Figure 9: Intermediate resonances contributing to the
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ decay. Shown
are distributions for (a)
$M(\Lambda_{c}^{+}\pi^{-}\pi^{+})-M(\Lambda_{c}^{+})$, with
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ contributions, (b)
$M(\Lambda_{c}^{+}\pi^{\pm})-M(\Lambda_{c}^{+})$ (3 combinations per
$\Lambda_{b}^{0}$ candidate) (c)
$M(\Lambda_{c}^{+}\pi^{-})-M(\Lambda_{c}^{+})$ (2 combinations per
$\Lambda_{b}^{0}$ candidate), and (d)
$M(\Lambda_{c}^{+}\pi^{+})-M(\Lambda_{c}^{+})$ (1 combination per
$\Lambda_{b}^{0}$ candidate), showing the intermediate $\Sigma_{c}$ states.
The line is a fit as described in the text, and the shaded region is the
fitted background.
We also observe the decays
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi^{\mp}\pi^{-}$, with
$\Sigma_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ or
$\Sigma_{c}^{++}\rightarrow\Lambda_{c}^{+}\pi^{+}$. The $\Delta M_{\pi}$
distributions are shown in Fig. 9(b-d) for both $\Sigma_{c}^{0}$ and
$\Sigma_{c}^{++}$ candidates, $\Sigma_{c}^{0}$ candidates only (c), and (d)
$\Sigma_{c}^{++}$ candidates only. The data are fit to the sum of a Breit-
Wigner shape convolved with a Gaussian resolution function and a smooth
threshold function. The full width is fixed to 2.2 MeV/$c^{2}$ [15] in all
cases, and the $\Delta M_{\pi}$ resolution is fixed to 1 MeV/$c^{2}$ based on
simulation. The combined $\Sigma_{c}^{0}$ and $\Sigma_{c}^{++}$ signal has a
statistical significance of 6.0 standard deviations. The $\Sigma_{c}^{0}$ and
$\Sigma_{c}^{++}$ signals have statistical significances of 4.9 and 3.5,
respectively. These decays have also been seen by CDF [23].
Table 4 summarizes the yields for the various excited charm states for both
the full data sample and after the trigger selection as well as the yields in
the normalizing modes (after trigger selection.)
Table 4: Summary of yields for the signal and normalization modes. Below $D_{1}$ and $D_{2}^{*}$ refer to the $D_{1}(2420)$ and $D_{2}^{*}(2460)$ mesons, respectively. Decay | $H_{c}^{*}\pi(\pi)$ Signal Yields | $H_{c}\pi^{-}\pi^{+}\pi^{-}$
---|---|---
| All | Trig. Sel | Trig. Sel
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{1}^{+}\pi^{-},~{}D_{1}^{+}\rightarrow D^{+}\pi^{-}\pi^{+}$ | $41\pm 8$ | $33\pm 7$ | $1741\pm 55$
$B^{-}\rightarrow D_{1}^{0}\pi^{-},~{}D_{1}^{0}\rightarrow D^{0}\pi^{-}\pi^{+}$ | $165\pm 17$ | $126\pm 14$ | $1386\pm 51$
$B^{-}\rightarrow D_{1}^{0}\pi^{-},~{}D_{1}^{0}\rightarrow D^{*+}\pi^{-}$ | $111\pm 14$ | $75\pm 12$ | $1386\pm 51$
$B^{-}\rightarrow D_{1}^{0}\pi^{-},~{}D_{1}^{0}\rightarrow D^{0}\pi^{-}\pi^{+},~{}{\rm non-}D^{*}$ | $57\pm 10$ | $52\pm 9$ | $1386\pm 51$
$B^{-}\rightarrow D_{2}^{*0}\pi^{-},~{}D_{2}^{*0}\rightarrow D^{0}\pi^{-}\pi^{+}$ | $66\pm 15$ | $49\pm 12$ | $1386\pm 51$
$B^{-}\rightarrow D_{2}^{*0}\pi^{-},~{}D_{2}^{*0}\rightarrow D^{*+}\pi^{-}$ | $46\pm 12$ | $34\pm 10$ | $1386\pm 51$
$B^{-}\rightarrow D_{2}^{*0}\pi^{-},~{}D_{2}^{*0}\rightarrow D^{0}\pi^{-}\pi^{+},~{}{\rm non-}D^{*}$ | $23\pm 9$ | $18\pm 8$ | $1386\pm 51$
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2595)^{+}\pi^{-}$ | $10.6\pm 3.8$ | $9.7\pm 3.5$ | $312\pm 23$
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2625)^{+}\pi^{-}$ | $15.7\pm 4.1$ | $9.3\pm 3.2$ | $312\pm 23$
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi^{\mp}\pi^{-}$ | $29.3\pm 7.0$ | $24.9\pm 6.2$ | $312\pm 23$
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0}\pi^{-}\pi^{+}$ | $19.6\pm 5.7$ | $16.2\pm 5.0$ | $312\pm 23$
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{++}\pi^{-}\pi^{-}$ | $10.1\pm 4.0$ | $9.3\pm 3.7$ | $312\pm 23$
The branching ratios for these modes are computed using:
${{\cal{B}}(H_{b}\rightarrow
H_{c}^{*}\pi(\pi))\times{\cal{B}}(H_{c}^{*}\rightarrow
H_{c}\pi(\pi))\over{\cal{B}}(H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-})}={N_{\rm signal}\over N_{\rm
norm}}\left(\epsilon_{\rm sel}^{\rm rel}\times\epsilon_{\rm trig|sel}^{\rm
rel}\right)^{-1}$ (2)
where $H_{c}^{*}$ refers to one of the observed excited charm states, $N_{\rm
signal}$ and $N_{\rm norm}$ are the number of reconstructed decays in the
signal and normalization modes after the trigger requirement, $\epsilon_{\rm
sel}^{\rm rel}$ is the reconstruction and selection efficiency relative to the
normalization mode, and $\epsilon_{\rm trig|sel}^{\rm rel}$ is the relative
trigger efficiency. All efficiencies are given for the mass region $0.8~{}{\rm
GeV}/c^{2}<M(\pi^{-}\pi^{+}\pi^{-})<3$ GeV/$c^{2}$.
The relative reconstruction, selection and trigger efficiencies, shown in
Table 5, are evaluated using MC simulations. The $D_{1}(2420)^{0}$ and
$D_{2}^{*}(2460)^{0}$ are each assumed to decay 70% through
$D^{*+}\pi^{-}\rightarrow D^{0}\pi^{+}\pi^{-}$ and 30% non-resonant
$D^{0}\pi^{+}\pi^{-}$. The $D_{1}(2420)^{+}$ is taken to be 100% non-resonant
$D^{+}\pi^{-}\pi^{+}$. The $\Lambda_{c}(2595)^{+}$ decay is simulated as 36%
$\Sigma_{c}^{0}\pi^{+}$, 36% $\Sigma_{c}^{++}\pi^{-}$ and 28% non-resonant
$\Lambda_{c}^{+}\pi^{-}\pi^{+}$. The $\Lambda_{c}(2625)^{+}$ decay is assumed
to be 100% non-resonant $\Lambda_{c}^{+}\pi^{-}\pi^{+}$. The
$\Sigma_{c}(2544)$ baryons are simulated non-resonant in phase space.
The relative efficiencies agree qualitatively with our expectations based on
the kinematics and proximity to threshold for these excited charm states. The
differences in the relative efficiency between the pairs of excited charm
states for a given $b$-hadron species are negligible compared to the
uncertainty from our limited MC event sample, and we use the average relative
efficiency for each pair of decays.
Table 5: Summary of the relative reconstruction and selection efficiencies ($\epsilon_{\rm sel}^{\rm rel}$) and trigger efficiencies ($\epsilon_{\rm trig|sel}^{\rm rel}$) for the excited charm hadron intermediate states with respect to the inclusive $H_{c}\pi^{-}\pi^{+}\pi^{-}$ final states. Below $D_{1}$ and $D_{2}^{*}$ refer to $D_{1}(2420)$ and $D_{2}^{*}(2460)$, respectively. The uncertainties shown are statistical only. Decay | $\epsilon_{\rm sel}^{\rm rel}$ | $\epsilon_{\rm trig|sel}^{\rm rel}$ | $\epsilon_{\rm total}^{\rm rel}$
---|---|---|---
| (%) | (%) | (%)
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{1}^{+}\pi^{-}$ | $0.83\pm 0.06$ | $1.05\pm 0.09$ | $0.87\pm 0.10$
$B^{-}\rightarrow(D_{1}^{0},~{}D_{2}^{*0})\pi^{-}$ | $0.70\pm 0.04$ | $1.24\pm 0.07$ | $0.86\pm 0.07$
$B^{-}\rightarrow(D_{1}^{0},~{}D_{2}^{*0})\pi^{-}({\rm via}~{}D^{*})$ | $0.66\pm 0.05$ | $1.29\pm 0.08$ | $0.84\pm 0.08$
$B^{-}\rightarrow(D_{1}^{0},~{}D_{2}^{*0})\pi^{-}({\rm non-}D^{*})$ | $0.78\pm 0.06$ | $1.15\pm 0.10$ | $0.91\pm 0.11$
$\Lambda_{b}^{0}\rightarrow(\Lambda_{c}(2595),~{}\Lambda_{c}(2625)^{+})\pi^{-}$ | $0.52\pm 0.03$ | $1.30\pm 0.07$ | $0.67\pm 0.06$
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi\pi,~{}\Sigma_{c}^{0,++}\rightarrow\Lambda_{c}^{+}\pi^{\mp}$ | $0.67\pm 0.05$ | $1.10\pm 0.13$ | $0.75\pm 0.10$
The dominant sources of systematic uncertainty are the limited MC sample sizes
and the fit model. Starting with the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, the uncertainty due to limited MC
statistics is 11%. For the fit model, the largest source of uncertainty is
from a possible $D_{2}^{*}(2460)^{+}\pi^{-},~{}D_{2}^{*}(2460)^{+}\rightarrow
D^{+}\pi^{-}\pi^{+}$ contribution. If this contribution is included in the fit
using a Breit-Wigner shape with mean and width taken from the PDG [15], the
returned signal yield is $0^{+7}_{-0}$. If we assume isospin symmetry, and
constrain this fraction (relative to $D_{1}(2420)$) to be $(40\pm 11)\%$, the
ratio found for the $B^{-}$ decay, the fitted $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D_{1}(2420)^{+}\pi^{-},~{}D_{1}(2420)^{+}\rightarrow D^{+}\pi^{-}\pi^{+}$
signal yield is $26\pm 6$ events. We take this as a one-sided uncertainty of
${}^{+0\%}_{-21\%}$. Sensitivity to the background shape is estimated by using
a second order polynomial for the background (3%). The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass sidebands, which have a
$D_{1}(2420)^{+}$ fitted yield of $2^{+3}_{-2}$ events from which we
conservatively assign as a one-sided systematic uncertainty of
${}^{+0\%}_{-6\%}$. For the signal decays, 4% of events have
$M(\pi^{-}\pi^{+}\pi^{-})>3$ GeV/$c^{2}$, whereas for the $D_{1}(2420)^{+}$,
we find a negligible fraction fail this requirement. We therefore apply a
correction of $0.96\pm 0.02$, where we have taken 50% uncertainty on the
correction as the systematic error. The systematic uncertainty on the yield in
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ normalizing mode is 3%. We thus arrive at a total
systematic error on the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
branching fraction ratio of ${}^{+12}_{-25}\%$.
For the $B^{-}$, we have a similar set of uncertainties. They are as follows:
MC sample size (8%), background model (1%, 2%), $D_{1}(2420)^{0}$ width (2%,
4%), $D_{2}^{*}(2460)^{0}$ width (1%, 3%), where the two uncertainties are for
the ($D_{1}(2420)^{0}$, $D_{2}^{*}(2460)^{0}$) intermediate states. We have
not accounted for interference, and have assumed it is negligible compared to
other uncertainties. A factor of $0.98\pm 0.01$ is applied to correct for the
fraction of events with $M(\pi^{-}\pi^{+}\pi^{-})>3$ GeV/$c^{2}$. Including a
3% uncertainty on the $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$ yield, we
find total systematic errors of 9% and 10% for the $D_{1}(2420)^{0}$ and
$D_{2}^{*}(2460)^{0}$ intermediate states, respectively. For the $D^{*}$ sub-
decays, the total systematic uncertainties are 10% and 11% for
$B^{-}\rightarrow D_{1}(2420)^{0}\pi^{-},~{}D_{1}(2420)^{0}\rightarrow
D^{*+}\pi^{-}$ and $B^{-}\rightarrow
D_{2}^{*}(2460)^{0}\pi^{-},~{}D_{2}^{*}(2460)^{0}\rightarrow D^{*+}\pi^{-}$,
respectively. For final states not through $D^{*}$, we find a total systematic
uncertainty of 13% for both intermediate states. In all cases, the dominant
systematic uncertainty is the limited number of MC events.
For the $\Lambda_{b}^{0}$ branching fraction ratios, we attribute uncertainty
to limited MC sample sizes (8%), the $\Lambda_{c}^{+}(2595)$ width
(${}^{+9\%}_{-5\%}$),
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}$ signal yield
(3%), and apply a correction of $0.96\pm 0.02$ for the ratio of yields with
$M(\pi^{-}\pi^{+}\pi^{-})>3$ GeV/$c^{2}$. In total, the systematic
uncertainties on the $\Lambda_{c}^{+}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$
partial branching fractions are ${}_{-10\%}^{+13\%}$ and $\pm 10\%$,
respectively.
For the $\Sigma_{c}^{0,++}$ intermediate states, the systematic uncertainties
include 14% from finite MC statistics, and 4% from the $\Sigma_{c}^{0,++}$
width. For the $\Sigma_{c}^{0,++}$ simulation, 10% of decays have
$M(\pi^{-}\pi^{+}\pi^{-})>3$ GeV/$c^{2}$, compared to 4% for the normalizing
mode. We therefore apply a correction of $1.06\pm 0.03$ to the ratio of
branching fractions. All other uncertainties are negligible in comparison. We
thus arrive at a total systematic uncertainty of 16%.
The final partial branching fractions are
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D_{1}^{-}\pi^{+},~{}D_{1}^{-}\rightarrow D^{+}\pi^{-}\pi^{+})\over\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}}$ $\displaystyle=(2.1\pm 0.5_{-0.5}^{+0.3})\%$
$\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{1}^{0}\pi^{+},~{}D_{1}^{0}\rightarrow D^{0}\pi^{-}\pi^{+})\over
B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}}$ $\displaystyle=(10.3\pm 1.5\pm
0.9)\%$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{1}^{0}\pi^{+},~{}D_{1}^{0}\rightarrow D^{*+}\pi^{-})\over B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}}$ $\displaystyle=(9.3\pm 1.6\pm 0.9)\%$
$\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{1}^{0}\pi^{+},~{}D_{1}^{0}\rightarrow D^{0}\pi^{-}\pi^{+})_{{\rm
non-}D^{*}}\over B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(4.0\pm 0.7\pm 0.5)\%$
$\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{2}^{*0}\pi^{+},~{}D_{2}^{*0}\rightarrow D^{0}\pi^{-}\pi^{+})\over
B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}}$ $\displaystyle=(4.0\pm 1.0\pm
0.4)\%$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{2}^{*0}\pi^{+},~{}D_{2}^{*0}\rightarrow D^{*+}\pi^{-})\over
B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}}$ $\displaystyle=(3.9\pm 1.2\pm
0.4)\%$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D_{2}^{*0}\pi^{+},~{}D_{2}^{*0}\rightarrow D^{0}\pi^{-}\pi^{+})_{{\rm
non-}D^{*}}\over B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(1.4\pm 0.6\pm 0.2)\%$
$\displaystyle<3.0\%{\rm~{}at~{}90\%~{}C.L.}$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2595)^{+}\pi^{+},~{}\Lambda_{c}(2595)^{+}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+})\over\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(4.4\pm 1.7_{-0.4}^{+0.6})\%$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}(2625)^{+}\pi^{+},~{}\Lambda_{c}(2625)^{+}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+})\over\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(4.3\pm 1.5\pm 0.4)\%$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi^{\mp}\pi^{-},~{}\Sigma_{c}^{0,++}\rightarrow\Lambda_{c}^{+}\pi^{\mp})\over\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(11.4\pm 3.1\pm 1.8)\%$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0}\pi^{+}\pi^{-},~{}\Sigma_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-})\over\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(7.4\pm 2.4\pm 1.2)\%$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{++}\pi^{-}\pi^{-},~{}\Sigma_{c}^{++}\rightarrow\Lambda_{c}^{+}\pi^{+})\over\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=(4.2\pm 1.8\pm 0.7)\%,$
where the first uncertainties are statistical and the second are systematic.
For the modes with $D^{*+}$, we include a factor ${\cal{B}}(D^{*+}\rightarrow
D^{0}\pi^{+})=(0.677\pm 0.005)$ [15] to account for unobserved $D^{*+}$
decays. The first four and the sixth of these decays have been previously
measured by Belle [22] with comparable precision. To compare our results to
those absolute branching fractions, we multiply them by the relative $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ [$B^{-}$] branching fractions in
Eq. 1, and then in turn by ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-})=(2.68\pm
0.13)\times 10^{-3}$ [${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{-})=(4.84\pm
0.15)\times 10^{-3}$.] The resulting absolute branching fractions are
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D_{1}(2420)^{-}\pi^{+},~{}D_{1}(2420)^{-}\rightarrow
D^{+}\pi^{-}\pi^{+})=(1.3\pm 0.3_{-0.3}^{+0.2})\times 10^{-4}$
$\displaystyle{\cal{B}}(B^{-}\rightarrow
D_{1}(2420)^{0}\pi^{+},~{}D_{1}(2420)^{0}\rightarrow
D^{0}\pi^{-}\pi^{+})=(6.3\pm 0.9\pm 0.9)\times 10^{-4}$
$\displaystyle{\cal{B}}(B^{-}\rightarrow
D_{1}(2420)^{0}\pi^{+},~{}D_{1}(2420)^{0}\rightarrow D^{*+}\pi^{-})=(5.8\pm
1.0\pm 0.9)\times 10^{-4}$ $\displaystyle{\cal{B}}(B^{-}\rightarrow
D_{1}(2420)^{0}\pi^{+},~{}D_{1}(2420)^{0}\rightarrow
D^{0}\pi^{+}\pi^{-})_{{\rm non-}D^{*}}=(2.5\pm 0.4\pm 0.4)\times 10^{-4}$
$\displaystyle{\cal{B}}(B^{-}\rightarrow
D_{2}^{*}(2460)^{0}\pi^{+},~{}D_{2}^{*}(2460)^{0}\rightarrow
D^{*+}\pi^{-})=(2.5\pm 0.7\pm 0.4)\times 10^{-4}$
where the uncertainties are statistical and total systematic, respectively.
The corresponding values obtained by Belle are: $(0.89^{+0.23}_{-0.35})\times
10^{-4}$, $(6.5^{+1.1}_{-1.2})\times 10^{-4}$, $(6.8\pm 1.5)\times 10^{-4}$,
$(1.9^{+0.5}_{-0.6})\times 10^{-4}$, and $(1.8\pm 0.5)\times 10^{-4}$ [15,
22]. Our results are consistent with, and of comparable precision to, those
measurements.
Preliminary results on the
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}(2595)^{+}\pi^{-}$,
$\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}(2625)^{+}\pi^{-}$ and
$\Lambda_{b}^{0}\rightarrow\Sigma_{c}^{0,++}\pi^{\mp}\pi^{-}$ decays have been
reported by CDF [23]. Our values are consistent with these (unpublished)
results.
## 9 Summary
In summary, we have measured the branching fractions for $H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-}$ decays relative to $H_{b}\rightarrow
H_{c}\pi^{-}$. The ratio of branching fractions are measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-})}=2.38\pm
0.11\pm 0.21$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-})}=1.27\pm 0.06\pm 0.11$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-})}=2.01\pm 0.37\pm 0.20$
$\displaystyle{{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-})}=1.43\pm
0.16\pm 0.13.$
At low 3$\pi$ mass, these decays appear to be dominated by the $a_{1}(1260)$
resonance. We have also measured several partial decay rates through excited
charm states. The yields of $H_{b}\rightarrow H_{c}\pi^{-}\pi^{+}\pi^{-}$
relative to $H_{b}\rightarrow H_{c}\pi^{-}$ are in the range of 20$-$40%. If
the relative rates in the Cabibbo-suppressed decays, such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}\pi^{\pm}\pi^{\mp}$ and $B^{-}\rightarrow
DK^{-}\pi^{+}\pi^{-}$ relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp}$
and $B^{-}\rightarrow DK^{-}$, respectively, are comparable, they could be
useful for measuring the weak phase $\gamma$.
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
* [2] E. Eichten and B. R. Hill, Phys. Lett. B234, 511 (1990); N. Isgur and M. B. Wise, Phys.Lett. B232, 113 (1989); H. Georgi, Phys. Lett. B240, 447 (1990); E. Eichten and B. R. Hill, Phys. Lett. B243, 427 (1990); B. Grinstein, Nucl. Phys. B339, 253 (1990).
* [3] I. Dunietz, Phys. Lett. B270, 75 (1991); I. Dunietz, Z. Phys. C56, 129 (1992); D. Atwood, G. Eilam, M. Gronau, and A. Soni, Phys. Lett. B341, 372 (1995); D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78, 3257 (1997).
* [4] M. Gronau and D. London, Phys. Lett. B253, 483 (1991); M. Gronau and D. Tyler, Phys. Lett. B265, 172 (1991).
* [5] A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D68 054018 (2003).
* [6] R. Aleksan, I. Dunietz, and B. Kayser, Z. Phys. C54, 653 (1992).
* [7] I. Dunietz, Phys. Rev. D52, 3048 (1995).
* [8] C. S. Kim and S. Oh, Eur. Phys. J. C21, 495 (2001).
* [9] M. Gronau, Phys. Lett. B557, 198-206 (2003).
* [10] Measurement of $\Delta m_{s}$ in the Decay $B_{s}^{0}\rightarrow D_{s}^{-}((K^{-}K^{+}\pi^{-})(3)\pi$, [LHCb Collaboration], LHCb-CONF-2011-005 (2011).
* [11] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).
* [12] C.-W. Chiang and J. Rosner, Phys. Rev. F67, 074013 (2003); C. S. Kim et al , Phys. Lett B621, 259-268 (2004).
* [13] A. A. Alves Jr. et al. [LHCb Collaboration], JINST 3, S08005 (2008).
* [14] M. Williams et. al., LHCb Public Document, LHCb-PUB-2011-002.
* [15] K. Nakamura et al., J. Phys. G37, 075021 (2010).
* [16] T. Sj$\ddot{\rm o}$strand, S. Mrenna and P. Skands, JHEP 0605, 026 (2006).
* [17] D. J. Lange, Nucl. Instrum. Meth. A462, 152 (2001).
* [18] S. Agostinelli et al. [GEANT4 Collaboration], Nucl. Instrum. Meth. A506, 250 (2003).
* [19] M. Pivk and F. Le Diberder, Nucl Instrum. Meth A555, 356 (2005).
* [20] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances. PhD thesis, Institute of Nuclear Physics, Krakow, 1986. DESY-F31-86-02.
* [21] D. Monorchio, Study of the properties of the $a_{1}$ meson produced in the $B\rightarrow D^{*-}a_{1}^{+}$ at the BABAR Experiment, PhD thesis, Universita Degli Studi Di Napoli, 2005.
* [22] K. Abe et al. [Belle Collaboration] Phys. Rev. Lett. 94, 221805 (2005); Phys. Rev. D69, 112002 (2004).
* [23] P. Azzurri et al. [CDF Collaboration], in Proceedings of Lepton Photon 2009 Conference, Hamburg, Germany, 17-22 Aug. 2009, p 434, edited by T. Behnke and J. Mnich [arXiv:0912.4380].
|
arxiv-papers
| 2011-09-30T14:22:03 |
2024-09-04T02:49:22.639657
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Anderson, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A. Artamonov, M.\n Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad,\n T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, S. Brisbane, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea,\n A. Bursche, J. Buytaert, S. Cadeddu, J.M. Caicedo Carvajal, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo,\n M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, P. Collins, F. Constantin, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, S. Donleavy, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L. Est\\'eve, A.\n Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, N. Gauvin, M. Gersabeck,\n T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, W. Hofmann, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse,\n R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei,\n M. Karacson, T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, Y.M. Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, S. Kukulak, R. Kumar, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, J. Luisier, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde,\n R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A.\n Milanes, M.-N. Minard, S. Monteil, D. Moran, P. Morawski, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli, I. Nasteva, M.\n Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, V. Niess, N.\n Nikitin, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B. Pal,\n J. Palacios, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes,\n C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N.\n Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A.\n Petrella, A. Petrolini, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, J. Prisciandaro,\n V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, K. Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues,\n P. Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, B.\n Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, H.P.\n Skottowe, T. Skwarnicki, A.C. Smith, N.A. Smith, K. Sobczak, F.J.P. Soler, A.\n Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, N. Styles, V.K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, M.T. Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja, P.\n Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, K. Vervink, B. Viaud, I.\n Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Voong, A.\n Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D.R. Ward, A.D. Webber,\n D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P.\n Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A.\n Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zavertyaev, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev,\n A. Zvyagin",
"submitter": "Steven R. Blusk",
"url": "https://arxiv.org/abs/1109.6831"
}
|
1109.6915
|
arxiv-papers
| 2011-09-30T17:55:24 |
2024-09-04T02:49:22.653205
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhiwei Yang",
"submitter": "Zhiwei Yang",
"url": "https://arxiv.org/abs/1109.6915"
}
|
|
1110.0049
|
# Quasi-compactons in inverted nonlinear photonic crystals
Yongyao Li1,2 yongyaoli@gmail.com Boris A. Malomed3,4 Jianxiong Wu1 Wei Pang5
Sicong Wang2 Jianying Zhou2 stszjy@mail.sysu.edu.cn 1Department of Applied
Physics, South China Agricultural University, Guangzhou 510642, China
2State Key Laboratory of Optoelectronic Materials and Technologies,
Sun Yat-sen University, Guangzhou 510275, China
3 Department of Physical Electronics, School of Electrical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
4 ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park,08860
Castelldefels (Barcelona), Spain
5 Department of Experiment Teaching, Guangdong University of Technology,
Guangzhou 510006, China.
###### Abstract
We study large-amplitude one-dimensional solitary waves in photonic crystals
featuring competition between linear and nonlinear lattices, with minima of
the linear potential coinciding with maxima of the nonlinear pseudopotential,
and vice versa (inverted nonlinear photonic crystals, INPhCs), in the case of
the saturable self-focusing nonlinearity. Such crystals were recently
fabricated using a mixture of SU-8 and Rhodamine-B optical materials. By means
of numerical methods and analytical approximations, we find that large-
amplitude solitons are broad sharply localized stable pulses (quasi-
compactons, QCs). With the increase of the total power, $P$, the QC’s centroid
performs multiple switchings between minima and maxima of the linear
potential. Unlike cubic INPhCs, the large-amplitude solitons are mobile in the
medium with the saturable nonlinearity. The threshold value of the kick
necessary to set the soliton in motion is found as a function of $P$.
Collisions between moving QCs are considered too
###### pacs:
42.65.Tg; 42.70.Qs;05.45.Yv
## I Introduction and the model
Photonic and matter waves propagating under the combined action of linear and
nonlinear lattice (LL and NL) potentials exhibit a plenty of dynamics non-
soliton . In particular, solitons in such media is a topic of considerable
current interest, see recent review RMP_Kar and original works 1D ; Thaw
devoted to the studies of one-dimensional (1D) solitons. Other works were
dealing with solitons 2D ; Abd and localized vortices 2Dvort in two-
dimensional (2D) versions of such systems, which represent, in particular,
photonic crystal fibers PCF and 2D photonic crystals PC made of nonlinear
materials. The evolution of solitons in these systems obeys the nonlinear
Schrödinger equation (NLSE), in which the LL and NL are represented,
respectively, by a usual periodic potential and by a periodic
_pseudopotential_ LCQian ; RMP_Kar , which is induced by a spatially periodic
modulation of the local nonlinearity coefficient. In optics, the NL represents
the mismatch between the nonlinearity of the host material and the stuff
filling voids of the photonic-crystal-fibers structure, which may be air,
another solid material all-solid , or a liquid crystal LC . The same model, in
the form of the Gross-Pitaevskii equation, applies to matter waves in a Bose-
Einstein condensate (BEC) which is trapped in a combination of a linear
periodic potential, created by an optical OL or magnetic magnetic lattice,
and a pseudopotential lattice, that may be induced by a periodic modulation of
the local nonlinearity provided by a properly patterned RMP_Kar external
magnetic FR-magn or optical fields FR-magn .
Recently, a combination of _competing_ $\pi$-out-of-phase-juxtaposed LL and
NL, with maxima of the refractive index coinciding with minima of the local
strength of the self-focusing nonlinearity and vice versa, was considered in
several works Kar12 ; LYY1 ; LYY2 . This medium may be naturally called an
_inverted nonlinear photonic crystal_ (INPhC). It was reported that INPhCs
could be fabricated by means of the technique based on direct laser writing in
silica Blomer . It was also predicted that a similar setting can be created in
a virtual form, using the electromagnetically induced transparency acting on
dopant atoms periodically distributed in a passive matrix LYY2 . Due to the
competition between the LL and NL, solitons in INPhCs may feature specific
power-dependent properties, such as double symmetry breaking LYY2 . While
previous studies of INPhCs were dealing with the __ Kerr (cubic) nonlinearity,
in this work we consider solitary waves in the system with saturable
nonlinearity, which occurs in various optical media. The extension of the
analysis for this nonlinearity is natural, whilst solitons in INPhCs feature
the strong sensitivity to the power.
First, we propose an experiment setup to realize an INPhC with a saturable
nonlinearity. Recent works reported the creation of resonantly absorbing
waveguide arrays and imaginary-part photonic crystals, that feature a
spatially periodic modulation of the absorption coefficient, built on the
basis of the SU-8 polymer (a commonly used transparent negative photoresist)
doped with Rhodamine B (RhB, a dye featuring saturable absorption) MNF ; LJT ,
see Fig. 1 for a schematic setup.
Figure 1: (Color online) (a)The one-dimensional nonlinear photonic crystal,
with the blue and gray areas depicting the nonlinear and linear stripes,
respectively. (b) The wavelength dependence of the absorption coefficient and
refractive-index variation, $\alpha$ and $\Delta n$ (solid and dashed lines,
respectively) in the SU-8 - RhB mixture.
A similar pattern can be used for our purposes. In the paraxial approximation,
the light propagation in the array obeys the spatial-domain NLSE for the local
amplitude of the electromagnetic field, $u(z,x)$, where $z$ and $x$ are the
propagation distance and transverse coordinate:
$iu_{z}=-{\frac{1}{2k}}u_{xx}-{\frac{(k_{0}\delta
n+i{\alpha/2})R(x)}{1+|u|^{2}}}u.$ (1)
Here, $R(x)$ is the array’s structure function which is to be taken as per
Fig. 1, $k=k_{0}n$, where $n=1.62$ is the refractive index of pure SU-8,
$k_{0}=2\pi/\lambda$, and $\lambda$ is the wavelength of light in vacuum.
Further, $\alpha$ and $\delta n$ are the absorption coefficient and
_refractive index difference_ between the SU-8 - RhB mixture and pure SU-8,
respectively, which are coupled by the Kramers-Kronig relation, and are
plotted versus $\lambda$ in Fig. 1 LJT . According to experimental data LJT ,
in the relevant range of far blue shift from the absorption peak
($\lambda\approx 460$ nm), $\alpha/2\simeq(1/20)k_{0}\left|\delta n\right|$,
which allows one to neglect the dissipative term in Eq. (1), simplifying it to
$iu_{z}=-{\frac{1}{2}}u_{xx}+{\frac{V(x)}{1+|u|^{2}}}u,$ (2)
where $k$ was removed by rescaling of $x$, and $V(x)\equiv-\left(k_{0}\delta
n\right)R(x).$
In the system under consideration, light experiences strong saturable self-
focusing induced by RhB in the mixed material, while the nonlinearity of pure
SU-8 is negligible. On the other hand, the refractive index of the mixture is
smaller than in pure SU-8, i.e., $\delta n<0$ at $\lambda=460$ nm, hence
function $V(x)$ in Eq. (2) takes positive values, and the waveguide array
meets the condition of the $\pi$ phase shift between the spatial modulations
on the linear and nonlinear local characteristics of the medium, thus
realizing the INPhC with the saturable nonlinearity.
The objective of this work is to study the existence, stability, mobility, and
interactions of solitons in the INPhC model based on Eq. (2). In Sec. II, we
study properties of the solitons by means of numerical simulations and
analytical approximations. It will be demonstrated that they feature sharp
localization, i.e., a quasi-compacton (QC) shape. Following the increase of
the total power, $P$, the QC switches its position between maxima and minima
of the linear refractive index. In Sec. III, we study mobility of the QCs,
imposing the phase tilt onto them, i.e., suddenly multiplying the wave form by
$\exp(i\eta x)$. A critical tilt (alias kick), $\eta_{c}(P)$, beyond which the
compacton starts to move, is found. The dependence $\eta_{c}(P)$ features
variations with the same period in $P$ as the above-mentioned switching of the
quiescent solitons. Collisions between moving compactons are also studied in
Sec. III by means of direct simulations. The paper is concluded by Sec. IV.
## II Quasi-compactons: numerical and analytical results
### II.1 Numerical simulations
Modulation function $V(x)$ in Eq. (2) corresponding to Fig. 1 is a piecewise-
constant one, of the Kronig-Penny type. In this paper, we approximate $V(x)$
by the first term of its harmonic decomposition, assuming that contributions
of higher harmonics are negligible for sufficiently broad solitons:
$V(x)=\left(V_{0}/2\right)\left[1-\cos\left(2\pi x{/d}\right)\right],$ (3)
where $d$ and $V_{0}>0$ are the modulation period and depth. The scaling is
set by fixing $d\equiv 20$, which leaves $V_{0}$ as a free parameter. In this
section, we present numerical results for $V_{0}=0.02,~{}0.03$ and $0.04$,
which adequately illustrate the generic situation. Below, we focus on solitons
with a sufficiently large amplitude, as for small amplitudes the truncated
expansion of the saturable nonlinearity amounts to the previously studied Kerr
model Kar12 ; LYY1 ; LYY2 .
Typical numerical results for solitons solution at different values of the
total power, $P=\int_{-\infty}^{+\infty}|u(x)|^{2}dx$, are displayed in Fig.
2. The stationary profiles, presented in Figs. 2-2, were generated by dint of
the the imaginary-time-propagation method Chiofalo . It is concluded that the
profiles are strongly localized, making the solitons QCs (quasi-compactons) [a
solution fully localized within a finite (_compact_) interval of $y$ is
usually called a compacton Rosenau ; JHH ]. On the other hand, the soliton
broadens with the increase of $P$. At relatively small values of the total
power, e.g., $P=400$, the center of the soliton is located at a minimum of
$V(x)$ [Fig. 2]; then, at $P=800$ [Fig. 2], the position of the soliton is
switched to a maximum of $V(x)$, and at $P=2000$ it switches back to the
minimum [Fig. 2].
Figure 2: (Color online) Numerically found soliton profiles (blue solid lines)
for the modulation function (shown by the black solid line) with depth
$V_{0}=0.02$ (recall the modulation period is scaled to be $d\equiv 20$). (a)
At total power $P=400$, the center of the soliton is located at the minimum of
$V(x)$. (b) At $P=800$, the soliton shifts to the maximum of $V(x).$ (c) At
$P=2000$, the solitons shifts back to the minimum of $V(x)$. Panels (d), (e)
and (f) illustrate the stability of the solitons via direct simulations of
their perturbed evolution.
To quantify power-dependent properties of the solitons, we define the center-
of-mass coordinate and average width of the soliton as follows:
$\displaystyle
X_{\mathrm{mc}}(P)={P}^{-1}\int_{-\infty}^{+\infty}x|U(x,P)|^{2}dx$
$\displaystyle
W_{\mathrm{a}}(P)={P}^{-1}\int_{-\infty}^{+\infty}(x-x_{mc})^{2}|U(x,P)|^{2}dx,$
(4)
where $U(x,P)$ is the soliton solution with power $P$. Using the integration
in imaginary time, the data were collected for the solitons from $P=100$ to
$10000$, with modulation depths $V_{0}=0.02,~{}0.03$ and $0.04$. Real-time
simulations of the evolution of these solutions in the framework of Eq. (2)
demonstrate that they all are stable against perturbations.
Numerically found dependences (4) are displayed in Fig. 3. The upper panels,
Fig. 3(a-c), show that the soliton’s center of mass switches (as said above)
between the minimum ($x=0$) and maximum ($x=-10$) of the modulation function
$V(x)$. The period of the switching, $\Delta P$, increases with the increase
of modulation depth $V_{0}$. Further, Figs. 3-2 show that the average width of
the soliton grows with the increase of $P$ and decrease of $V_{0}$.
Figure 3: (Color online) The plots in panels (a1)-(c1) display the coordinate
of the soliton’s center of mass, $X_{\mathrm{mc}}$, as a function of $P$, for
$V_{0}=0.02,~{}0.03$ and $0.04$, respectively. (a2)-(c2). The critical kick
which initiates the motion of the soliton, $\eta_{\mathrm{c}}$, as a function
of $P$ for $V_{0}=0.02,~{}0.03$ and $0.04$. (d)-(f) The average width of the
soliton as a function of $P$ for $V_{0}=0.02,0.03$ and $0.04$.
### II.2 Analytical approximations
In saturable nonlinear media, large-amplitude solitons are always broad.
Accordingly, to develop an analytical approximation, we assume a soliton with
the width much larger than $d$. Then, in the zero-order approximation, Eq. (2)
is replaced by the equation with the averaged potential, $V(x)\rightarrow
V_{0}/2$,
$iu_{z}=-\frac{1}{2}u_{xx}+\frac{V_{0}}{2\left(1+|u|^{2}\right)}u.$ (5)
Equation (5) gives rise to soliton solutions, $u(z,x)=e^{ikz}U(x-\xi)$, with
the coordinate of the soliton’s center $\xi$, and real function $U(y)$ obeying
the following equation:
$\frac{d^{2}U}{dy^{2}}=2kU+\frac{V_{0}U}{1+U^{2}}\equiv-\frac{dW}{dU},$ (6)
where $y\equiv x-\xi$. The effective potential in Eq. (6),
$W(U)=\allowbreak
kU^{2}+\frac{V_{0}}{2}\ln\left(\frac{1+U_{0}^{2}}{1+U^{2}}\right),$ (7)
formally corresponds to the Newton’s equation of motion for a particle with
coordinate $U(y)$ ($y$ plays the role of time) and unitary mass. Then, the
shape of the soliton is determined by the respective energy equation,
$\left(\frac{dU}{dy}\right)^{2}+2k\left(U_{0}^{2}-U^{2}\right)+V_{0}\ln\left(\frac{1+U_{0}^{2}}{1+U^{2}}\right)=0,$
(8)
where $U_{0}$ is the amplitude of the soliton (the largest value of $U$). As
usual, the soliton trajectory corresponds to the solution of Eq. (8) starting
from $U=0$ at $y=-\infty$, bouncing back from the potential well at $U=U_{0}$,
$y=0$, and returning to $U=0$ at $y\rightarrow+\infty$. Setting in Eq. (8)
$U=dU/dy=0$, one can find a relation between $k$ and $U_{0}$. Being interested
in solutions with large amplitudes, we assume here $U_{0}^{2}\gg 1$, which
yields
$k\approx-\left(V_{0}/2\right)U_{0}^{-2}\ln\left(U_{0}^{2}\right),$ (9)
$U_{\min}^{2}\approx U_{0}^{2}/\ln\left(U_{0}^{2}\right),$ (10)
$U_{\min}$ being the coordinate of the minimum of potential (7). Equation (10)
implies $U_{\min}^{2}\ll U_{0}^{2}$, i.e., the minimum of the potential is
located much closer to $U=0$ than the largest value $U_{0}$. This fact suggest
a possibility to use the following approximation for solving Eq. (8): at the
first stage of the approximation, we drop the logarithmic term in Eq. (8)
altogether, i.e., we replace the equation by its “primitive version”,
$\left(\frac{dU}{dy}\right)^{2}+2k\left(U_{0}^{2}-U^{2}\right)=0,$ (11)
whose obvious solution is
$U(y)=U_{0}\cos\left(\sqrt{-2k}y\right)\approx
U_{0}\cos\left(\frac{\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}}{U_{0}}y\right),$
(12)
where Eq. (9) was used to replace $k$ by the $U_{0}$.
At the second stage of the approximation, we recall that the particle does not
perform periodic oscillations, as formally follows from the expression (12),
but it starts the motion from $U=0$ at $y=-\infty$, and returns to $U=0$ at
$y\rightarrow+\infty$, as the soliton solution must do, see above. This means
that approximation (12) is usable in interval
$|y|~{}<\frac{L_{\mathrm{tot}}}{2}\equiv\frac{\pi}{2\sqrt{-2k}}=\frac{\pi
U_{0}}{2\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}},$ (13)
When the moving particle approaches edges of interval (13), the logarithmic
term in Eq. (8), which was neglected in Eq. (11), “suddenly” becomes
important, leading to the stoppage of the particle. Thus, the full
approximation in the large-amplitude limit, $U_{0}\gg 1$, amounts to
representing the soliton as the QC (quasi-compacton):
$U(y)\approx\left\\{\begin{array}[]{c}U_{0}\cos\left(\frac{\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}}{U_{0}}y\right),~{}\mathrm{at}~{}~{}|y|~{}<L_{\mathrm{tot}}/2,\\\
0,~{}\mathrm{at}~{}~{}|y|~{}>L_{\mathrm{tot}}/2.\end{array}\right.$ (14)
where $L_{\mathrm{tot}}$, given by Eq. (13), plays the role of the total width
of the QC. Note that the large amplitude $U_{0}$ of the soliton implies that
width (13) is also large, i.e., the QC is a sharply localized but wide
localized pattern. In the present approximation, the total power of the
soliton can be calculated as follows:
$P\equiv\int_{-\infty}^{+\infty}\left|U(y)\right|^{2}dy\approx\frac{1}{2}L_{\mathrm{tot}}U_{0}^{2}=\frac{\pi
U_{0}^{3}}{2\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}}.$ (15)
Further, using the fact that $P$ and $U_{0}^{2}$ are large, Eq. (15) can be
approximately inverted, to give the peak power (squared amplitude) of the QC
as a function of its total power:
$U_{0}^{2}\approx\left(\frac{2P}{\pi}\sqrt{\frac{2V_{0}}{3}\left(\ln
P\right)}\right)^{2/3}.$ (16)
Next,using Eq. (13), it is also possible to approximately express the full
width of the compacton in terms of the total power:
$L_{\mathrm{tot}}\approx\left(\frac{3\pi^{2}}{V_{0}}\frac{P}{\ln
P}\right)^{1/3}.$ (17)
Figure 4: (Color online) (a) The comparison of the numerical results for the
peak power of the quasi-compactons with the analytical prediction given by Eq.
(16). (b) The full width of the quasi-compacton, as predicted by Eq. (17). In
this figure, $V_{0}=0.02$.
Relations (16) and (17) are plotted in Fig. 4 for different values of $P$. In
particular, Fig. 4(a) demonstrates that the analytical prediction for the peak
power is consistent with the numerical results. On the other hand, the values
of the width given by Eq. (17) [see Fig. 4(b)] are larger than their
numerically found counterparts presented in Fig. 3(d). However, one should
take into regard that width (17) is “all-inclusive” (total), while the width
shown in 3(b) is the average one (definitely far smaller than the total
width). Note also the similarity in the $P$-dependence of the width in Figs.
3(d) and 4(b).
To explain the position switchings of the QC, one may use the effective energy
(potential) of the interaction of the QC with the spatial modulation,
corresponding to the term $\sim\cos\left(2\pi x/d\right)$ in expression (3).
This energy can be defined as follows, with regard to the definition of
$y\equiv x-\xi$ in Eq. (6):
$\displaystyle
E_{\mathrm{int}}(\xi)=-V_{0}\int_{-\infty}^{+\infty}dx\cos\left(\frac{2\pi
x}{d}\right)\ln\left[1+U^{2}(x)\right]$ (18) $\displaystyle\approx$
$\displaystyle-V_{0}\int_{\xi-
L_{\mathrm{tot}}/2}^{\xi+L_{\mathrm{tot}}/2}dx\cos\left(\frac{2\pi
x}{d}\right)\ln\left[1+U^{2}(x)\right].$
In this expression, it is taken into account that QC (14) occupies,
approximately, a finite region of $x$, $\xi-
L_{\mathrm{tot}}/2<x<\xi+L_{\mathrm{tot}}/2$. After three integrations by
parts, expression (18) can be approximately calculated as follows,
$E_{\mathrm{int}}(\xi)\approx
K_{d}\ln(U_{0}^{2})\sin\left(\frac{\pi^{2}U_{0}}{d\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}}\right)\cos\left(\frac{2\pi}{d}\xi\right),$
(19)
where $K_{d}\equiv V_{0}^{2}d^{3}/\pi^{3}$. Expression (19) predicts that the
minimum of the interaction energy is located at points $\xi=nd$ with integer
$n$ (at minima of the modulation function) for values of $U_{0}$ such that
$\sin\left(\pi^{2}U_{0}/\left[d\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}\right]\right)<0$,
and at points $\xi=\left(n+1/2\right)d$ (at maxima of the modulation) in the
opposite case,
$\sin\left(\pi^{2}U_{0}/\left[d\sqrt{V_{0}\ln\left(U_{0}^{2}\right)}\right]\right)>0$.
Because $U_{0}$ grows as the function of $P$, as per Eq. (16), this means
that, with the increase of $P$, the position of the energy minimum must indeed
switch between adjacent extrema of the modulation function. The fact that
$V_{0}$ appears in the denominator of the argument of the sinusoidal factor
explains why the period of the switching increases with $V_{0}$.
## III Mobility of the quasi-compactons
### III.1 Driving the soliton by the phase tilt
Mobility of solitons under the combined action of LL and NL is a problem which
was considered in a number of different settings Sakaguchi ; Zhou ; Kar3 ;
Oster ; RMP_Kar . To set the QC in motion, we follow the standard approach,
suddenly kicking a quiescent one, i.e., multiplying the respective solution,
$U(x)$, by the phase-tilt factor: $U(x,\eta)\rightarrow\exp(i\eta x)U(x)$,
where $\eta$ is the strength of kick.
The simulated propagation of the QC, induced by kicking the standing one,
$U(x,P=2000)$, is displayed in Fig. 5, for two different values of $\eta$. It
is observed that $\eta=0.020(\pi/d)$ causes only oscillations of the soliton,
without any progressive motion. However, a slightly higher kick,
$\eta=0.025(\pi/d)$, is sufficient to set the same soliton in persistent
motion. These results imply that there must be a threshold (critical) value of
the kick, $\eta_{\mathrm{c}}$, such that the kick with
$\eta>\eta_{\mathrm{c}}$ makes the given QC a traveling soliton. For $P=2000$,
it is $\eta_{\mathrm{c}}=0.021\left(\pi/d\right)$.
Figure 5: (Color online) Direct simulations of the compacton with $P=2000$,
initiated by the application of the phase tilt with $\eta=0.020(\pi/d)$ (a)
and $\eta=0.025(\pi/d)$ (b). The lattice depth is $V_{0}=0.02$.
The critical kick for the QCs is shown as a function of the total power in
Figs. 3(a2)-3(c2), for $V_{0}=0.02,0.03$, and $0.04$. The general trend to the
decrease of $\eta_{\mathrm{c}}$ with the increase of $P$ is explained by the
broadening of the soliton with the increase of the total power in the case of
the saturable nonlinearity, and ensuing enhancement of the solitons’ mobility
Oster . A specific feature of the INPhC is that oscillations of
$\eta_{\mathrm{c}}$ are superimposed on top of the gradual decay of the
critical kick, the period of the oscillations coinciding with that of the
switching of the center-of-mass position. Figures 3-3 also demonstrate that
$\eta_{\mathrm{c}}$ decreases with the decreases of modulation depth $V_{0}$.
The latter trend is correlated with the fact that, as seen in Figs. 3-3, the
width of the solitons increases, making them more mobile, with the decrease of
$V_{0}$ (as well as with the increase of $P$).
Further, Fig. 6 shows that the momentum of the moving soliton, which is
defined as
$\displaystyle\langle p(z)\rangle=-i\int dxU(x,z,\eta)^{\ast}U_{x}(x,z,\eta),$
(20)
is, naturally, a linear function of the initial kick, $\eta$. Further
simulations (not shown here) demonstrate that the QC are _not_ destroyed even
by a very strong kick, far exceeding $\eta_{\mathrm{c}}$.
Figure 6: The momentum of the soliton with $P=2000$, measured at $z=1000$, as
a function of initial kick $\eta$.
It is relevant to compare the mobility of large-amplitude solitons in the
INPhCs models with the saturable and cubic self-focusing. In the latter case,
Eq. (2) is replaced by Thaw
$iu_{z}=-{\frac{1}{2}}u_{xx}+V(x)(1-|u|^{2})u,$ (21)
where we chose the modulation function, $V(x)$, in the same form as in Eq.
(3), with $V_{0}=0.02$ and $d=20$. A typical example of simulations of the
mobility of solitons in this model is displayed in Figs. 7(a)-(c), for three
kicks, $\eta=0.15{\pi/d},0.25{\pi/d}$, and $0.35{\pi/d}$. In particular, Fig.
7 shows that $\eta=0.15{\pi/d}$ give rise to oscillations of the soliton,
without depinning. Further, it is observed in Fig. 7 that the stronger kick,
$\eta=0.25{\pi/d}$, transforms the soliton into an oscillatory beam with a
fuzzy shape, but, still, it was not set in motion. Finally, Fig. 7
demonstrates that the strongest kick, $\eta=0.35{\pi/d}$, completely destroys
the soliton (recall the destruction never happens in the model with the
saturable nonlinearity). Therefore, there must exist critical values of the
kick separating the stable solitons, fuzzy beams, and complete destruction.
These borders are plotted in the top panel of Fig. 7, in the range of powers
$50<P<100$ (the average width of the static soliton in the same region is
plotted, versus $P$, in the bottom panel). Thus, the large-amplitude solitons
in the INPhC model with the cubic self-focusing are not mobile at all. On the
other hand, for dynamics of small-amplitude solitons in the models with the
saturable and cubic nonlinearities are similar, due to the obvious expansion,
$\left(1+|u|^{2}\right)^{-1}\approx 1-|u|^{2}$ (results for this case are not
shown here in detail, as they are less interesting).
Figure 7: (Color online) (a) Direct simulations of the evolution of the
soliton with total power $P=80$ in the model with the cubic nonlinearity [Eq.
(21)], initiated by the kick with strengths $\eta=0.15(\pi/d)$ (a),
$0.25(\pi/d)$ (b), and $\eta=0.35(\pi/d)$ (c). (d) The top plot:
$\eta_{1\mathrm{c}}$ is the border between stable solitons and fuzzy beams;
$\eta_{2\mathrm{c}}$ is the border between the fuzzy beams and completely
destroyed ones. The bottom plot shows the average width of the quiescent
soliton versus its total power $P$.
### III.2 Collisions between moving quasi-compactons
Collisions is a natural way to test interactions between solitons. Here we
present some numerical results for collisions between QCs in the INPhC with
the saturable nonlinearity. The initial state at $z=0$ is taken as a set of
two far separated kicked solitons:
$u(x)=U(x+x_{0},P)e^{i\eta_{1}(x+x_{0})}+U(x-x_{0},P)e^{-i\eta_{2}(x-x_{0})},$
(22)
where $U(x\pm x_{0},P)$ are the QC pulses with power $P$, which are centered
at $x=\mp x_{0}$, and $\eta_{1,2}$ are the kicks applied to them. Simulations
of Eq. (2) with initial conditions (22) are displayed in Fig. 8.
Figure 8: (Color online) Direct simulation of the collisions initiated by
kicks $\eta_{1}=0$, $\eta_{2}=0.1(\pi/d)$ (a); $\eta_{1}=0$,
$\eta_{2}=0.2(\pi/d)$ (b); $\eta_{1}=0.1(\pi/d)$, $\eta_{2}=0.1(\pi/d)$ (c);
$\eta_{1}=0.2(\pi/d)$, $\eta_{2}=0.2(\pi/d)$ (d). Other parameters are
$P=400$, $x_{0}=120$, and $V_{0}=0.02$. As in the rest of the paper, the
scaling is fixed by $d\equiv\allowbreak 20$.
The simulations demonstrate the repulsive interaction between the colliding
QC. In Figs. 8 and 8, they bounce back without overlapping, because the
collision velocity is small (nevertheless, one of the solitons suffers a
conspicuous perturbation in Fig. 8). In Figs. 8 and 8, the larger velocities
give rise to the overlap between the colliding QCs, which results in the
strong inelasticity of the collision in the case displayed in Fig. 8.
## IV Conclusions
The objective of this work was to consider 1D solitons in the INPhC (inverted
nonlinear photonic crystal) with competing LL and NL (linear and nonlinear
lattices). Unlike recently studied models with the cubic nonlinearity, we here
consider the saturable self-focusing. The 1D crystal of this type was recently
fabricated, using the SU-8 polymer material periodically doped with Rhodamine
B, which lends the medium the saturable nonlinearity. Combining numerical
methods and analytical approximations, we have demonstrated that broad
solitons are sharply localized in this setting, thus taking the shape of the
QCs (quasi-compactons). With the increase of the total power, $P$, the QC
remains a stable object, which switches its position between the linear and
nonlinear layers, which form the INPhC. The width of the soliton increases
with $P$, which is a manifestation of the saturable nonlinearity. The large
width of the QC makes it a mobile object, unlike solitons in the INPhC with
the cubic nonlinearity. The threshold value of the transverse kick (phase
tilt), which is necessary to set the QC in motion, was found as a function of
$P$. The threshold value gradually decays with the increase of $P$. Collisions
between two moving QCs were are also studied, by means of direct simulations.
The results reported in this work may be applied to the design of all-optical
data-processing schemes. In particular, the power-controlled switch of the
spatial soliton between adjacent layers, as well as the high mobility of these
solitons, may be quite relevant properties, in this context.
This work may be extended in other directions. It particular, 2D waveguide
arrays, based on the RhB-SU-8 mixtures, can be fabricated MNF ; LJT , which
suggests to consider spatial solitons in two-dimensional INPhC with the
saturable nonlinearity.
Y.L. appreciates useful discussions with Dr. M. Feng. This work was supported
by the Chinese agencies NKBRSF (grant No. G2010CB923204), NSFC(grant No.
11104083) and CNNSF(grant No. 10934011).
## References
* (1) Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996); E. Lidorikis, Q. Li, and C. M. Soukoulis, ibid. 54, 10249 (1996); D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999); B. Maes, P. Bienstman, and R. Baets, J. Opt. Soc. Am. B 22, 613 (2005).
* (2) Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
* (3) A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001); Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006); Y. Kominis, Phys. Rev. E. 73, 06619 (2006); Y. Kominis and K. Hizanidis, Opt. Exp. 16, 12124 (2008); Z. Rapti, P. G. Kevrekidis, V. V. Konotop, and C. K. R. T. Jones, J. Phys. A., 40, 14151 (2007).
* (4) T. Mayteevarunyoo and B. A. Malomed, J. Opt. Soc. Am. B 25, 1854 (2008).
* (5) P. Xie, Z.-Q. Zhang, and X. Zhang, Phys. Rev. E 67, 026607 (2003); A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, Opt. Exp. 11, 452 (2003).
* (6) H. L. F. da Luz, F. Kh. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, Phys. Rev. A 82, 043618 (2010).
* (7) A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, Opt. Exp. 12, 817 (2004); Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, Phys. Rev. Lett. 95, 123902 (2005).
* (8) P. St. J. Russell, J. Lightwave Tech. 24, 4729 (2006); S. Arismar Cerqueira, Jr., Rep. Prog. Phys. 73, 024401 (2010).
* (9) J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press: Princeton, 2008); M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystals Guiding (Cambridge University Press: Cambridge, 2009).
* (10) L. C. Qian, M. L. Wall, S. Zhang, Z. Zhou, and H. Pu, Phys. Rev. A 77, 013611 (2008); T. Mayteevarunyoo, B. A. Malomed, and G. Dong, ibid. 78, 053601 (2008).
* (11) F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, Opt. Lett. 29, 2369 (2004); G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, Opt. Exp. 13, 8452 (2005); A. Fuerbach, P. Steinvurzel, J. A. Bolger, A. Nulsen, and B. J. Eggleton, Opt. Lett. 30, 830 (2005).
* (12) T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, Opt. Exp. 11, 2589 (2003); F. Du , Y. Q. Lu, and S. T. Wu, Appl. Phys. Lett. 85, 2181 (2004); M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, IEEE Phot. Tech. Lett. 17, 819 (2005); C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Królikowski, A. Bjarklev, and Y. S. Kivshar, Opt. Exp. 15, 12145 (2007).
* (13) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998); M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Nature 415, 39 (2002).
* (14) S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, J. Phys. B 39, 847 (2006).
* (15) S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998); S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 85, 1795 (2000); E. Timmermans, P. Tommasini, M. S. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999).
* (16) P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996); M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, Phys. Rev. Lett. 93, 123001 (2004).
* (17) Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett., 33, 1747 (2008); ibid. 33, 2173 (2008).
* (18) Y. Li, B. A. Malomed, M. Feng, and J. Zhou, Phys. Rev. A, 82, 063813 (2010); C. Hang and V. V. Konotop, _ibid._ 81, 053849 (2010). J. Wu, M. Feng, W. Pang, S. Fu and Y. Li, J. Nonlinear Opt. Phys. (JNOP), 22, 193(2011).
* (19) Y. Li, B. A. Malomed, M. Feng, and J. Zhou, Phys. Rev. A, 83, 053823 (2011). J. Wu, M. Feng, W. Pang, S. Fu, and Y. Li, J. Nonlinear Opt. Phys. (JNOP), 20, 193 (2011).
* (20) A. Szameit,J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Exp., 14, 6055 (2006). D. Blömer, A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Exp., 14, 2151 (2006).
* (21) M. Feng, Y. Liu, Y. Li, X. Xie, and J. Zhou, Opt. Exp. 19, 7222 (2011).
* (22) J. Li, B. Liang, Y. Liu, P. Zhang, J. Zhou, S. O. Klimonsky, A. S. Slesarev, Y. D. Tretyakov, L. O’Faolain, and T. F. Krauss, Adv. Mater. 22, 1 (2010).
* (23) M. L. Chiofalo, S. Succi, and M. P. Tosi, Phys. Rev. E, 62, 7438 (2000). J. Yang, and T. I. Lakoba, Stud. Appl. Math., 120, 265 (2008). J. Yang, and T. I. Lakoba, Stud. Appl. Math., 118, 153 (2007). Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, Phys. Rev. A., 80, 043824 (2009).
* (24) P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
* (25) J. He and X. Wu, Chaos, Sol. Fractals 29, 108 (2006).
* (26) H. Sakaguchi, and B. A. Malomed, Phys. Rev. E., 70, 066613 (2005).
* (27) J. Zhou, C. Xue, Y. Qi, and S. Lou, Phys. Lett. A. 372, 4395 (2008).
* (28) Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt., 52, 63 (2009).
* (29) M. Öster, M. Johansson and A. Eriksson, Phys. Rev. E. 67, 056606 (2003); R. Vicencio and M. Johansson, ibid. 73, 046602 (2006); U. Naether, R. A. Vicencio, and M. Johansson, ibid. 83, 036602 (2011).
|
arxiv-papers
| 2011-09-30T23:09:19 |
2024-09-04T02:49:22.663435
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongyao Li, Boris A. Malomed, Jianxiong Wu, Wei Pang, Sicong Wang and\n Jianying Zhou",
"submitter": "Yongyao Li",
"url": "https://arxiv.org/abs/1110.0049"
}
|
1110.0177
|
11institutetext: Department of Computer Science, University of Auckland,
Private Bag 92019, Auckland, New Zealand 22institutetext: Department of
Chemistry, University of York, YO10 5DD, York, UK
# A Nuclear Magnetic Resonance Implementation of a Classical Deutsch-Jozsa
Algorithm
Alastair A. Abbott 11 aabb009@aucklanduni.ac.nz Matthias Bechmann 22
matthias.bechmann@york.ac.uk Cristian S. Calude 11 cristian@cs.auckland.ac.nz
Angelika Sebald 22 angelika.sebald@york.ac.uk
###### Abstract
Nuclear magnetic resonance (NMR) has been widely used as a demonstrative
medium for showcasing the ability for quantum computations to outperform
classical ones. A large number of such experiments performed have been
implementations of the Deutsch-Jozsa algorithm. It is known, however, that in
some cases the Deutsch-Jozsa problem can be solved classically using as many
queries to the black-box as in the quantum solution. In this paper we describe
experiments in which we take the contrasting approach of using NMR as a
classical computing medium, treating the nuclear spin vectors classically and
utilising an alternative embedding of bits into the physical medium. This
allows us to determine the actual Boolean function computed by the black-box
for the $n=1,2$ cases, as opposed to only the nature (balanced or constant) as
conventional quantum algorithms do. Discussion of these experiments leads to
some clarification of the complications surrounding the comparison of
different quantum algorithms, particularly black-box type algorithms.
###### keywords:
quantum computing, NMR, Deutsch-Jozsa, de-quantisation
## 1 Introduction
Nuclear magnetic resonance (NMR) experiments are generally conducted using
bulk samples and hence the manipulating radiofrequency (rf) pulses and the
detection signal have to be regarded in the context of an ensemble average of
the underlying nuclear magnetic spin dynamics. Theoretically, this situation
is successfully dealt with by a density matrix approach. Since, however, the
idea of quantum computation is based on the concept of being able to
manipulate the spin dynamics on the basis of pure quantum spin states, there
have been various attempts at implementing quantum computation algorithms
using the experimental conditions and restrictions of NMR by adopting pseudo-
pure spin state based approaches.
The most commonly implemented quantum algorithm, both in NMR and in general,
is the one due to Deutsch and Josza [8, 23, 9, 10, 14, 20, 26, 3, 24, 19, 21,
11, 29, 25, 16, 15, 18]. The various NMR implementations differ by: the
underlying spin quantum numbers $S$ ($S=1/2$ or $S>1/2$); the initial spin
states (thermal equilibrium state or pseudo pure state); the algorithmic
implementation of the problem (Collins [8] or Cleve [7] defining the number of
qubits necessary to operate a given DJ problem size).
In the context of computation, NMR has in the past been exclusively used for
implementing quantum computations. However, it also has potential as a
classical computing medium, where the rich state space can be fully utilised
to perform classical operations [28]. The Deutsch-Jozsa problem, long touted
as a simple and key example of the ability of quantum computing to outperform
classical computing, has more recently been shown to be _de-quantisable_ in
some cases—i.e., efficient classical solutions can be formulated [1, 6].
In this paper, we describe the implementation of the $n=1$ and $n=2$ de-
quantised solutions for the DJ problem in a classical NMR computation. The
process of implementing this solution highlights key aspects of quantum
algorithms and computation, and we discuss these in detail. In particular, we
emphasise the separation between three nested ‘layers’ of any quantum
algorithm: the problem formulation, the algorithm formulation, and the
physical implementation. In general these levels are independent, but certain
conditions on the relationship between levels must be satisfied. Specifically,
a particular algorithm applies only to a specific problem formulation, and for
each algorithm a choice of embedding into the physical medium must be made in
order to implement it. Further, for ‘oracle’ or ‘black-box’ problems such as
the Deutsch-Jozsa problem the comparison of different formulations of the
problem requires discussion of the ability to embed the black-box from one
formulation into the other. All these issues are subtle and require further
discussion.
## 2 Problem Formulation
The standard formulation of the Deutsch-Jozsa problem [13] is as follows. Let
$f:\\{0,1\\}^{n}\to\\{0,1\\}$, and suppose we are given a black-box computing
$f$ with the guarantee that $f$ is either constant (i.e. for all
$x_{1},x_{2}\in\\{0,1\\}^{n}$ we have $f(x_{1})=f(x_{2})$) or balanced (i.e.
$f(x)=0$ for exactly half of all possible inputs $x\in\\{0,1\\}^{n}$). The
problem is to determine, in as few black-box calls as possible, whether $f$ is
constant or balanced. The obvious classical algorithm must examine one more
than half the input bit-strings and thus requires $2^{n-1}+1$ black-box calls,
while the quantum solution requires only one.
There is, however, an important subtle issue: the classical and quantum
problems are slightly different. In one case we are given a _classical_ black-
box $C_{f}$ computing $f$, and in the other we are given a _unitary_ black-box
$U_{f}$, operating in a $2^{n+1}$ dimensional Hilbert space
$\mathcal{H}_{2^{n+1}}$, computing $U_{f}\ket{x}\ket{y}=\ket{x}\ket{y\oplus
f(x)}$ where ‘$\oplus$’ denotes addition modulo 2. As such, it is bending the
truth a little to say that ‘ _the_ problem’ can be solved more efficiently
quantum mechanically than classically.
The possibility in the $n=1,2$ cases to de-quantise the quantum solution to
give an equally good classical algorithm [1, 6] results from working with a
formulation of the problem in which we are given a higher dimensional
classical black-box. In this case, complex numbers are used as ‘complex
bits’—a classical analogue of a qubit—and the black-box $C_{f}$ operates (in
the $n=2$ case) as follows:
$C_{f}\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}=C_{f}\begin{pmatrix}a_{1}+b_{1}i\\\
a_{2}+b_{2}i\end{pmatrix}=\begin{pmatrix}(-1)^{f(00)}\left[a_{1}+(-1)^{f(00)\oplus
f(10)}b_{1}i\right]\\\ a_{2}+(-1)^{f(10)\oplus f(11)}b_{2}i\end{pmatrix}.$ (1)
In general, a particular algorithm, be it classical or quantum, solves a
particular formulation of the problem; i.e., it determines if a black-box of a
_specific type_ which _computes in some reasonable form_ $f$ is balanced or
constant. Hence, it seems at least some of the apparent difference in powers
of the classical and quantum solutions comes from the slightly different
formulation of the problems, i.e., the different ‘powers’ of the black-boxes.
A comment should be made about what it means to compute $f$ in some reasonable
form. Since quantum computing requires unitarity, the simplest and perhaps
most natural way to compute $f$ is with an $f$-controlled-NOT gate; indeed
this was the original method proposed by Deutsch [12]. One thus needs to be
careful of claims that the separate output qubit for $U_{f}$ is not needed [8]
and that the alternative quantum black-box
$U_{f}^{\prime}\ket{x}=(-1)^{f(x)}\ket{x}$ can equally well be used. Rather,
it seems that $U_{f}^{\prime}$ does not reasonably compute $f$ as absolute
phase factors have no physical significance and it is hence impossible to
characterise which boolean function $f$ the black-box ‘computes’ by trying
various inputs. The same issue is not present, however, in the classical de-
quantised solution because phase factors are measurable in this case, and thus
$C_{f}$ can be seen to compute $f$, albeit in a slightly non-standard way.
### 2.1 Black-box embeddings
The quantum black-box (represented by $U_{f}$) is often considered an
embedding of the classical black-box computing $f$ [30]; if this were true it
would be more reasonable to view the quantum solution as solving the original
problem. This, however, is a misunderstanding which helps hide the differences
between the classical and quantum formulations of the problem. For this to be
an embedding we would require a map
$e:\\{0,1\\}^{n}\hookrightarrow\mathcal{H}_{g(2^{n+1})}$ (where
‘$\hookrightarrow$’ denotes an embedding) which preserves the structure of the
computed function. That is, for $x\in\\{0,1\\}^{n}$, $e(f(x))=U_{f}(e(x))$. In
other words, the computational states we assign to represent the binary bits 0
and 1 must behave as expected under $U_{f}$ given that $U_{f}$ should compute
the function $f$.
However, the requirement of the unitarity of $U_{f}$ makes such an embedding
impossible. This can be verified by considering any constant boolean $f$: such
a function is not bijective, so no bijective $U_{f}$ preserving the required
structure can exist. Because no unitary embedding is possible, it seems more
suitable to consider the quantum solution as a method to solve an analogue of
the classical problem, rather than a more efficient solution to the classical
problem. One is forced to conclude that the typical claims comparing the
quantum and classical solution are, in fact, not valid. Any comparison of the
problems should take into account the differences in complexity of the black-
boxes [2].
Interestingly, there is an embedding between the quantum black-box $U_{f}$ and
the de-quantised black-box $C_{f}$, so it is not as unreasonable to compare
the solutions using these black-boxes as it is to compare the quantum and one-
dimensional classical solutions. By realising that the original classical
solution and the de-quantised classical solution are not solving the same
problem, we see there is no explicit contradiction with claims that
$2^{n-1}+1$ black-box calls is the best that can be done in the original
classical problem [27].
## 3 Algorithm Formulation
Given a particular formulation of the problem, the algorithm formulation
involves determining the input for the black-box, and what operations are
required to determine the nature of $f$ from the output of the black-box.
In the standard quantum solution [7, 13], we initially prepare our system in
the state $\ket{00}\ket{1}$, and then operate on it with a three-qubit
Hadamard gate, $H^{\otimes 3}$, to get:
$H^{\otimes
3}\ket{00}\ket{1}=\frac{1}{2}\sum_{x\in\\{0,1\\}^{2}}\ket{x}\ket{-}=\ket{++}\ket{-}.$
(2)
After applying the $f$-controlled-NOT gate $U_{f}$ we have
$\begin{split}U_{f}\frac{1}{2}\sum_{x\in\\{0,1\\}^{2}}\ket{x}\ket{-}&=\sum_{x\in\\{0,1\\}^{2}}\frac{(-1)^{f(x)}}{2}\ket{x}\ket{-}.\end{split}$
(3)
By applying a final 3-qubit Hadamard gate to project this state onto the
computational basis we obtain the state
$(-1)^{f(00)}\ket{f(00)\oplus f(10)}\otimes\ket{f(10)\oplus f(11)}\ket{1}.$
(4)
If we measure both the first and second qubits we can determine the nature of
$f$: if both qubits are measured as $0$, then $f$ is constant, otherwise $f$
is balanced. This result is correct with probability one.
The de-quantised solution works in a similar way, but uses complex numbers as
two-dimensional complex bits. Using the black-box $C_{f}$ defined previously,
the algorithm proceeds as follows. We set $z=z_{1}=z_{2}=1+i$, apply $C_{f}$
and multiply by $z$ to project onto the computational basis to obtain the
result:
$\frac{z}{2}\times C_{f}\begin{pmatrix}z\\\
z\end{pmatrix}=\frac{1}{2}\times\begin{cases}\begin{pmatrix}(-1)^{f(00)}z^{2}\\\
z^{2}\end{pmatrix}=\begin{pmatrix}(-1)^{f(00)}i\\\ i\end{pmatrix}&\text{if $f$
is constant,}\\\ &\\\ \begin{pmatrix}(-1)^{f(00)}z\bar{z}\\\
z^{2}\end{pmatrix}=\begin{pmatrix}(-1)^{f(00)}\\\ i\end{pmatrix}&\\\
\begin{pmatrix}(-1)^{f(00)}z\bar{z}\\\
z\bar{z}\end{pmatrix}=\begin{pmatrix}(-1)^{f(00)}\\\ 1\end{pmatrix}&\text{if
$f$ is balanced.}\\\ \begin{pmatrix}(-1)^{f(00)}z^{2}\\\
z\bar{z}\end{pmatrix}=\begin{pmatrix}(-1)^{f(00)}i\\\ 1\end{pmatrix}&\\\
\end{cases}$ (5)
By checking both of the resulting complex numbers, we can determine whether
$f$ is balanced or constant with certainty. If both complex numbers are
imaginary then $f$ is constant, otherwise it is balanced. In fact, the ability
to determine if the output numbers are negative or positive allows us to
determine the value of $f(00)$ and thus which Boolean function $f$ is; the
quantum algorithm is incapable of doing this [27].
The ability to de-quantise the $n=1$ and $n=2$ solutions suggests that, at
least in these cases, the power of the quantum algorithm comes from exploiting
the two-dimensionality of the black-box, rather than from quantum mechanical
effects [1].
## 4 Implementation of classical NMR computing
Figure 1: Creating and observing a NMR signal. After reaching thermal
equilibrium magnetisation in the external magnetic field, the magnetisation
vector $S$ is flipped to the $xy$-plane of the laboratory frame of reference
by applying a radio-frequency (rf) pulse of suitable duration. After the rf
pulse, the NMR signal is detected in the $xy$-plane in the form of a time-
domain signal, the free induction decay (FID) which is recorded by a receiver
coil, here assumed to be placed in the $x$-direction. The FID is converted
into the frequency-domain spectrum by a Fourier transform.
For the implementation of the de-quantised algorithm we use real, two-
dimensional vectors to represent our complex bits (only the direction of the
vector is of particular importance); this is equivalent to using complex
numbers, but more convenient for use in this implementation. In this
representation our basis bits corresponding to the classical 0 and 1 become
$(1,0)$ and $(0,1)$ respectively. For the implementation we must embed our
bits into the physical medium; to do so, we use the nuclear magnetisation spin
vectors for this embedding. Specifically, we embed the complex bits into the
$xy$-plane of the rotating frame of reference in a NMR experiment as is
illustrated in Fig. 1. We only consider uncoupled spin species with spin
quantum number $S=1/2$, and we have the advantage that every direction of the
magnetisation vector in the $xy$-plane is distinguishable. Working solely with
uncoupled spin species, their dynamics are fully described by a classical
model of magnetisation vectors subjected to a range of differnt rotations
(pulses) [22, 4]. Hence, at no point do we make explicit use of the quantum-
mechanical properties of nuclear spin systems. In particular, we take (row-
vectors represent complex bits, column vectors are the nuclear spin vectors;
we omit normalisation factors for convenience):
$(1,0)\rightarrow I_{-45}=\begin{pmatrix}1\\\
\frac{1}{\sqrt{2}}(1-i)\end{pmatrix},\qquad(0,1)\rightarrow
I_{+45}=\begin{pmatrix}1\\\ \frac{1}{\sqrt{2}}(1+i)\end{pmatrix}.$
Combinations of the complex bits are naturally taken by the vector addition
(i.e. the embedding is linear) of the corresponding spin vectors (e.g.
$(1,1)\rightarrow I_{x}=\tbinom{1}{1}$). This mapping is shown in Fig. 2. We
emphasise that while the same nuclear spin state vectors are used as in
conventional quantum computing experiments, we choose a different embedding of
our algorithm into this spin-state vector space which is more flexible for use
with complex bits.
Figure 2: The embedding from classical two-dimensional bits onto the nuclear
magnetisation vectors in the $xy$-plane. Assuming signal detection along the
$x$-direction, the phases of the corresponding NMR spectra are shown, together
with the basis bits $(0,1)$ and $(1,0)$ (see text).
Our sample consists of 99.8% deuterated chloroform with a small amount of H2O
added. The 1H spins in the residual CHCl3 and H2O in this mixture are used for
the implementations. The top row in Fig. 3 depicts a conventional 1H NMR
spectrum of the sample. The implementation for $n=1$ only requires one spin
species to be present. This is most easily achieved by using selective
excitation pulses, centred at the resonance frequency of the CHCl3 1H NMR
resonance. The 1H NMR spectrum obtained by selective excitation, together with
the corresponding excitation profile of the selective pulses used are shown in
the middle and bottom traces of Fig. 3.
Figure 3: 1H NMR spectrum ($\omega_{0}/2\pi=-600.13$ MHz) of 99.8% deuterated
chloroform with a small amount of H2O added (top trace). The middle trace
shows the 1H NMR spectrum after application of a selective 90° pulse, centered
around the resonance frequency of the CHCl3 1H NMR signal. The bottom trace
displays the excitation profile of the selective pulses used.
### 4.1 The $n=1$ implementation
The $n=1$ implementation relies on the form of the black-box $C_{f}$ operating
as
$C_{f}((a_{1},b_{1}))=((-1)^{f(0)}a_{1},(-1)^{f(1)}b_{2}).$
The NMR pulse sequence for implementing this for our sample is shown in Fig.
4. The sequence starts with a (selective) $\pi/2$ pulse, flipping the (CHCl3)
1H magnetisation vector from the initial equilibrium $z$-direction into the
$xy$-plane, followed by a sequence of two $\pi$ pulses applied to the
$xy$-magnetisation vector as required to implement the black-box before the
resulting signal is detected in the form of a FID.
Figure 4: The NMR pulse sequence for implementation of the $n=1$ case; the
flip angles effected by a rf pulse are given in fractions of $\pi$, the
symbols $A$, $B$ refer to the control parameters of the black-box, curved
shapes of rf pulses indicate selective pulses [17].
The four possible boolean functions for $n=1$ may be written $f_{AB}$ where
$A=f(0)$ and $B=f(1)$. The four corresponding black-boxes can be uniformly
implemented as the following set of $\pi$ pulses, applied to our basis bits
(see Fig. 2):
$(\pi)_{45}^{A}(\pi)_{-45}^{B}.$
We see that the physical embedding of the black-box fulfils the requirement
that the function $f$ is reasonably computed due to the ability to distinguish
all directions of magnetisation vectors in the $xy$-plane, and thus all valid
complex bits. In particular, applying the black-boxes for inputs $(1,0)$ or
$(0,1)$ yields $((-1)^{f(0)},0)$ and $(0,(-1)^{f(1)})$ respectively as
desired. The effect of the four black-boxes on the ‘basis bits’ is shown in
Table 1, both in terms of complex bits and spin vectors.
$A=f(0)$ | $B=f(1)$ | $C_{f}((1,0))$ | $C_{f}((0,1))$
---|---|---|---
0 | 0 | $(\phantom{-}1,0)$, $I_{-45}$ | $(0,\phantom{-}1)$, $I_{45}$
0 | 1 | $(\phantom{-}1,0)$, $I_{-45}$ | $(0,-1)$, $I_{-135}$
1 | 0 | $(-1,0)$, $I_{135}$ | $(0,\phantom{-}1)$, $I_{45}$
1 | 1 | $(-1,0)$, $I_{135}$ | $(0,-1)$, $I_{-135}$
Table 1: The effect of the four black-boxes on the basis-bit inputs for $n=1$.
For the actual algorithm, the de-quantised solution can be simplified by
dropping the use of any equivalent of the final Hadamard operation; there is
no need to project the result onto the basis states since non-basis states are
equivalently detectable. In fact, the choice of $I_{\pm 45}$ states as our
‘basis bits’ (rather than $I_{x}$ and $I_{y}$) is convenient because this
leaves the system in $I_{x}$ or $I_{y}$ states after the black-box, and these
states are particularly easy to distinguish (see Fig. 2).
The effect of the algorithm with each black-box is shown in Table 2. The
corresponding spectra relating to the computation are shown in Fig. 5, along
with the action of the black-box on the basis-state inputs. In particular for
a constant $f$ we have $I_{\pm x}$ spectra, and for balanced we have $I_{\pm
y}$ spectra. Further, because we can distinguish $I_{+x}$ spectra from
$I_{-x}$ spectra (and $I_{+y}$ spectra from $I_{-y}$ spectra) we can easily
determine which function $f$ the black-box represents.
$A=f(0)$ | $B=f(1)$ | Initial State | $(\pi/2)_{y}$ | $C_{f}=(\pi)_{45}^{A}(\pi)_{-45}^{B}$
---|---|---|---|---
0 | 0 | $I_{z}$ | $(1,1)$, $I_{x}$ | $(\phantom{-}1,\phantom{-}1)$, $I_{x}$
0 | 1 | $I_{z}$ | $(1,1)$, $I_{x}$ | $(\phantom{-}1,-1)$, $I_{-y}$
1 | 0 | $I_{z}$ | $(1,1)$, $I_{x}$ | $(-1,\phantom{-}1)$, $I_{y}$
1 | 1 | $I_{z}$ | $(1,1)$, $I_{x}$ | $(-1,-1)$, $I_{-x}$
Table 2: The algorithm run with each of the four black-boxes for $n=1$. Figure
5: 1H NMR spectra, CHCl3 resonance, implementing the $n=1$ case. The top two
rows show the action of the black-box on the basis-state input. The bottom row
shows the computation itself, i.e. with (1,1) input.
### 4.2 The $n=2$ implementation
The idea is that the $n=2$ case should be implemented as a natural extension
of the $n=1$ case by expanding it to include two different spin species. The
function to be implemented is:
$C_{f}((a_{1},b_{1})(a_{2},b_{2}))=\left(a_{1}(-1)^{f(00)},b_{1}(-1)^{f(10)}\right)\left(a_{2},b_{2}(-1)^{f(10)\oplus
f(11)}\right).$
With the same mapping of bits as for the $n=1$ case, the natural extension of
the embedding is to have the black-box, now defined by the three parameters
$A=f(00)$, $B=f(10)$ and $C=f(10)\oplus f(11)$, implemented as:
$(\pi)_{1,45}^{A}(\pi)_{1,-45}^{B}(\pi)_{2,-45}^{C}.$
With this we can see that the first spin is treated as for the $n=1$ case, and
the second spin requires only a single pulse (or none, depending on $f$). This
form of the black-box, however, is only valid if one assumes ideal pulses. In
reality, we now deal with two different spin species with different resonance
frequencies, hence the receiver can only be on resonance with one species at a
time. In addition, all rf pulses have non-vanishing durations (in particular
the rather ‘soft’ selective pulses with durations of the order of ms) during
which evolution of magnetisation occurs. Hence, the pulse sequence of the
physical implementation as shown in Fig. 6 has to account for non-ideal
behaviour in order to produce an equivalent result to the one obtained from
the idealised mathematical description.
Figure 6: The NMR pulse sequence for implementing the $n=2$ case with two
different spin species, 1 and 2; $\tau^{(C)}$ denotes a pre-acquisition delay
of suitable duration.
The NMR experiment starts with selective pulses centred on spin species 1
(exactly as for the $n=1$ case before), followed by a selective $\pi/2$ pulse
centred on spin species 2, and a delay of duration $\tau^{(C)}$. If the
receiver is kept on resonance with species 1 throughout, it is off resonance
for species 2. Accordingly, after the selective $\left(\pi/2\right)_{2}$ pulse
the magnetisation of this species will precess in the $xy$-plane, and by this
acquire a phase difference relative to the on-resonance signal of species 1\.
This precession frequency depends on the difference in resonance frequency
between the two spin species and hence one can easily calculate the correct
duration of the delay $\tau^{(C)}$ that corresponds to a dephasing by $3\pi/2$
from the starting $I_{x,2}$ condition to $I_{-y,2}$. At this point in time,
the situation is exactly equivalent to the ideal-pulse scenario and the data
acquisition is started.
The pulse sequence depicted in Fig. 6 is the simplest form in which the
required idealised black-box can be carried out in a real NMR experiment. For
example, one might have started the pulse sequence with a non-selective
(‘hard’) $\pi/2$ pulse covering both spin species simultaneously. That would
be a common preparation step for a spin system, creating a common initial
state from which to work. However, this would only complicate matters as now
one would have to take into account dephasing of species 2 magnetisation
during all selective pulses applied to species 1, and one would have to apply
the corresponding phase corrections to any pulses applied to species 2 later
in the sequence. We do not have to start from one common initial state;
instead our physical implementation avoids all unnecessary such corrections
and calculations by simply using selective pulses only and preparing species 2
‘just in time’ such that only the duration of the pre-acquisition delay needs
to be calculated.
As for the $n=1$ case, it is easily verified that once again the requirements
for the black-box to reasonably compute $f$ are satisfied: on a given ‘basis’
input an even number of sign changes indicates a function value of $0$ on this
input, and odd number a value of $1$. The results of the black-box on basis
inputs is shown in Table 3, and the result of the algorithm is shown in Table
4. The corresponding input/output spectra are shown, as for the $n=1$ case, in
Fig. 7.
A B C | $f$ | $C_{f}((1,0)(1,0))$ | $C_{f}((1,0)(0,1))$ | $C_{f}((0,1)(1,0))$ | $C_{f}((0,1)(0,1))$
---|---|---|---|---|---
0 0 0 | $f_{0000}$ | $(\phantom{-}1,0)(1,0)$ | $(\phantom{-}1,0)(0,\phantom{-}1)$ | $(0,\phantom{-}1)(1,0)$ | $(0,\phantom{-}1)(0,\phantom{-}1)$
0 0 1 | $f_{0101}$ | $(\phantom{-}1,0)(1,0)$ | $(\phantom{-}1,0)(0,-1)$ | $(0,\phantom{-}1)(1,0)$ | $(0,\phantom{-}1)(0,-1)$
0 1 0 | $f_{0011}$ | $(\phantom{-}1,0)(1,0)$ | $(\phantom{-}1,0)(0,\phantom{-}1)$ | $(0,-1)(1,0)$ | $(0,-1)(0,\phantom{-}1)$
0 1 1 | $f_{0110}$ | $(\phantom{-}1,0)(1,0)$ | $(\phantom{-}1,0)(0,-1)$ | $(0,-1)(1,0)$ | $(0,-1)(0,-1)$
1 0 0 | $f_{1100}$ | $(-1,0)(1,0)$ | $(-1,0)(0,\phantom{-}1)$ | $(0,\phantom{-}1)(1,0)$ | $(0,\phantom{-}1)(0,\phantom{-}1)$
1 0 1 | $f_{1001}$ | $(-1,0)(1,0)$ | $(-1,0)(0,-1)$ | $(0,\phantom{-}1)(1,0)$ | $(0,\phantom{-}1)(0,-1)$
1 1 0 | $f_{1111}$ | $(-1,0)(1,0)$ | $(-1,0)(0,\phantom{-}1)$ | $(0,-1)(1,0)$ | $(0,-1)(0,\phantom{-}1)$
1 1 1 | $f_{1010}$ | $(-1,0)(1,0)$ | $(-1,0)(0,-1)$ | $(0,-1)(1,0)$ | $(0,-1)(0,-1)$
Table 3: The effect of the eight black-boxes on the basis-bit inputs for $n=2$. A | B | C | $f$ | $(\pi/2)_{y}$ | $C_{f}=(\pi)^{A}_{45,1}(\pi)^{B}_{-45,1}(\pi)^{C}_{-45,2}$
---|---|---|---|---|---
0 | 0 | 0 | $f_{0000}$ | $(1,1)(1,1)$ | $(\phantom{-}1,\phantom{-}1)(1,\phantom{-}1)$
0 | 0 | 1 | $f_{0101}$ | $(1,1)(1,1)$ | $(\phantom{-}1,\phantom{-}1)(1,-1)$
0 | 1 | 0 | $f_{0011}$ | $(1,1)(1,1)$ | $(\phantom{-}1,-1)(1,\phantom{-}1)$
0 | 1 | 1 | $f_{0110}$ | $(1,1)(1,1)$ | $(\phantom{-}1,-1)(1,-1)$
1 | 0 | 0 | $f_{1100}$ | $(1,1)(1,1)$ | $(-1,\phantom{-}1)(1,\phantom{-}1)$
1 | 0 | 1 | $f_{1001}$ | $(1,1)(1,1)$ | $(-1,\phantom{-}1)(1,-1)$
1 | 1 | 0 | $f_{1111}$ | $(1,1)(1,1)$ | $(-1,-1)(1,\phantom{-}1)$
1 | 1 | 1 | $f_{1010}$ | $(1,1)(1,1)$ | $(-1,-1)(1,-1)$
Table 4: The algorithm run with each of the eight black-boxes for $n=2$. While
the idealised version of the black-box pulses for species 2 is quoted here,
the physically implemented pulse sequence gives identical final results.
Figure 7: 1H NMR spectra, CHCl3 and H2O resonances, implementing the $n=2$
case. The top four rows show the action of the black-box on the basis-state
input. The bottom row shows the computation itself, i.e. with (1,1) (1,1)
input.
## 5 Summary
The experiments described in this paper not only show the successful
implementation of the de-quantised Deutsch-Jozsa algorithm for $n=1,2$, but
also bring attention to some important points regarding quantum computations.
The experiments confirm that the Deutsch-Jozsa problem can indeed be solved
classically in some cases using as many black-box calls as the standard
quantum algorithm. During the process of developing our implementation some
issues about the comparisons of algorithms for different formulations of the
problem were raised and discussed in detail. We note that, while we explicitly
demonstrate the ability to solve the $n=1,2$ cases for what is essentially an
alternative (but equivalent to the quantum) problem formulation, the ability
to do so for higher values of $n$ reduces to the ability to de-quantise the
quantum solution for these cases, which is believed not to be possible [1].
These experiments reiterate the utility of NMR as a classical computing
substrate [28]. Specifically, the use of an alternative embedding from the
problem space onto the uncoupled nuclear spin vectors allowed us to perform
the classical algorithm and even determine the specific function $f$ computed.
This is a more general result regarding the ensemble computational nature of
NMR computing, which allows any spin-direction in the $xy$-plane to be (in
theory) resolvable. It is further possible that this alternative embedding
could be utilised for quantum computations involving only real-valued
coefficients. Since such computations are universal [5], it is plausible that
this could be used to perform ensemble quantum computations in which the state
amplitudes themselves can be measured directly.
## 6 Experimental
99.8% deuterated $\mathrm{CHCl}_{3}$ was obtained from Aldrich Chemicals.
${}^{1}\mathrm{H}$ NMR spectra of a $\mathrm{CHCl}_{3}$ /
$\mathrm{H}_{2}\mathrm{O}$ mixture were recorded on Bruker Avance II 600 NMR
spectrometer (see Fig. 3), corresponding to a ${}^{1}\mathrm{H}$ Larmor
frequency of $-600.13$ MHz. On-resonant 90° pulse durations were $2.5$ ms and
Gaussian selective pulses at a resolution of 1000 points utilised cut-offs of
1%. All spectra were recorded in single scans, allowing recycle delays of at
least 11 s between experiments.
## 7 Acknowledgments
We gratefully acknowledge the support of this work by the Deutsche
Forschungsgemeinschaft (MB and AS) and EPSRC (AA, MB and AS). We thank Susan
Stepney for comments which helped improve the paper.
## References
* [1] A. A. Abbott. (2011). The Deutsch-Jozsa problem: De-quantisation and entanglement. Natural Computing, to appear.
* [2] A. A. Abbott and C. S. Calude. (2010). Understanding the quantum computational speed-up via de-quantisation. EPTCS, 26:1–12.
* [3] Arvind. (2001). Quantum entanglement and quantum computational algorithms. Pramana—Journal of Physics, 56:357–365.
* [4] M. Bechmann, A. Sebald, and S. Stepney. Boolean logic gate design principles in unconventional computers: an NMR case study. accepted: Int. J. Unconventional Computing.
* [5] E. Bernstein and U. Vazirani. (1997). Quantum complexity theory. SIAM Journal on Computing, 26(5):1411–1473.
* [6] C. S. Calude. (Jun 2007). De-quantizing the solution of Deutsch’s problem. International Journal of Quantum Information, 5(3):409–415.
* [7] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. (Jan 1997). Quantum algorithms revisited. Proceedings of the Royal Society of London Series A, 1998(454):339–354.
* [8] D. Collins, K. W. Kim, and W. C. Holton. (Sep 1998). Deutsch-Jozsa algorithm as a test of quantum computation. Physical Review A, 58(3):R1633–R1636.
* [9] D. Collins, K. W. Kim, W. C. Holton, H. Sierzputowska-Gracz, and E. O. Stejskal. (Jul 2000). NMR quantum computation with indirectly coupled gates. Physical Review A, 62(2):022304.
* [10] D. G. Cory, R. Laflamme, E. Knill, L. Viola, T. F. Havel, N. Boulant, G. Boutis, E. Fortunato, S. Lloyd, R. Martinez, C. Negrevergne, M. Pravia, Y. Sharf, G. Teklemariam, Y. S. Weinstein, and W. H. Zurek. (2000). NMR based quantum information processing: Achievements and prospects. Fortschritte der Physik, 48(9-11):875–907.
* [11] R. Das and A. Kumar. (SEP 2003). Use of quadrupolar nuclei for quantum-information processing by nuclear magnetic resonance: Implementation of a quantum algorithm. Physical Review A, 68(3):032304.
* [12] D. Deutsch. (Jan 1985). Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London Series A, 400:97–117.
* [13] D. Deutsch and R. Jozsa. (Jan 1992). Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London Series A, 439:553–558.
* [14] K. Dorai, Arvind, and A. Kumar. (Mar 2000). Implementing quantum-logic operations, pseudopure states, and the Deutsch-Jozsa algorithm using noncommuting selective pulses in NMR. Physical Review A, 61(4):042306.
* [15] A. F. Fahmy, R. Marx, W. Bermel, and S. J. Glaser. (AUG 2008). Thermal equilibrium as an initial state for quantum computation by NMR. Physical Review A, 78(2, Part A):022317.
* [16] J. Fitzsimons, L. Xiao, S. C. Benjamin, and J. A. Jones. (JUL 20 2007). Quantum information processing with delocalized qubits under global control. Physical Review Letters, 99(3):030501.
* [17] R. Freeman. (1998). Shaped radiofrequency pulses in high resolution NMR. Progress in Nuclear Magnetic Resonance Spectroscopy, 32:59–106.
* [18] T. Gopinath and A. Kumar. (2008). Implementation of controlled phase shift gates and Collins version of Deutsch-Jozsa algorithm on a quadrupolar spin-7/2 nucleus using non-adiabatic geometric phases. Journal of Magnetic Resonance, 193(2):168–176.
* [19] A. R. Kessel and N. M. Yakovleva. (DEC 2002). Implementation schemes in NMR of quantum processors and the Deutsch-Jozsa algorithm by using virtual spin representation. Physical Review A, 66(6):062322.
* [20] J. Kim, J.-S. Lee, S. Lee, and C. Cheong. (Jul 2000). Implementation of the refined Deutsch-Jozsa algorithm on a three-bit NMR quantum computer. Physical Review A, 62(2):022312.
* [21] A. Kumar, K. V. Ramanathan, T. S. Mahesh, N. Sinha, and K. V. R. Murali. (AUG 2002). Developments in quantum information processing by nuclear magnetic resonance: Use of quadrupolar and dipolar couplings. Pramana—Journal of Physics, 59(2, Sp. Iss. SI):243–254. 2nd Winter Institute on Foundations of Quantum Theory and Quantum Optics, Kolkata, India, JAN 02-11, 2002.
* [22] M. H. Levitt. (2008). Spin Dynamics: Basics of Nuclear Magnetic Resonance. John Wiley & Sons, 2nd edition.
* [23] N. Linden, H. Barjat, and R. Freeman. (1998). An implementation of the Deutsch-Jozsa algorithm on a three-qubit NMR quantum computer. Chemical Physics Letters, 296(1-2):61–67.
* [24] T. S. Mahesh, K. Dorai, Arvind, and A. Kumar. (2001). Implementing logic gates and the Deutsch-Jozsa quantum algorithm by two-dimensional NMR using spin- and transition-selective pulses. Journal of Magnetic Resonance, 148(1):95–103.
* [25] O. Mangold, A. Heidebrecht, and M. Mehring. (OCT 2004). NMR tomography of the three-qubit Deutsch-Jozsa algorithm. Physical Review A, 70(4):042307.
* [26] R. Marx, A. F. Fahmy, J. M. Myers, W. Bermel, and S. J. Glaser. (Jun 2000). Approaching five-bit NMR quantum computing. Physical Review A, 62(1):012310.
* [27] N. D. Mermin. (2007). Quantum Computer Science. Cambridge University Press.
* [28] M. Roselló-Merino, M. Bechmann, A. Sebald, and S. Stepney. (2010). Classical computing in nuclear magnetic resonance. International Journal of Unconventional Computing, 6(3–4):163–195.
* [29] D. X. Wei, J. Luo, X. P. Sun, X. Z. Zeng, X. D. Yang, M. L. Liu, and S. W. Ding. (FEB 2003). NMR experimental realization of seven-qubit D-J algorithm and controlled phase-shift gates with improved precision. Chinese Science Bulletin, 48(3):239–243.
* [30] C. P. Williams. (2011). Explorations in Quantum Computing. Springer-Verlag, New York, Inc., second edition.
|
arxiv-papers
| 2011-10-02T12:19:32 |
2024-09-04T02:49:22.674682
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alastair A. Abbott, Matthias Bechmann, Cristian S. Calude, Angelika\n Sebald",
"submitter": "Matthias Bechmann",
"url": "https://arxiv.org/abs/1110.0177"
}
|
1110.0180
|
# An efficient algorithm to find a set of nearest elements in a mesh
Gleb Novitchkov
email: gleb.novitchkov@gmail.com
###### Contents
1. 1 Introduction
2. 2 Definition
1. 2.1 Mesh and the representation of the elements
3. 3 The algorithm
4. 4 Parallelization
5. 5 Conclusion
## 1 Introduction
Here we present an algorithm that find a list of elements neighboring some
given element in a linear time. More precisely, if there are $N_{\rm elem}$
elements in the mesh, the runtime of the algorithm is $\mathcal{O}(N_{\rm
elem})$.
## 2 Definition
###### Definition 2.1.
By _element_ we mean a 3-simplex $\Delta^{3}$ imbedded in $\mathbb{R}^{3}$.
Essentially an element is a tetrahedron in $\mathbb{R}^{3}$.
Goal: given an element of the mesh, we want to find a set of elements that are
near this element.
###### Definition 2.2.
Given an element $E$, we call an element $F$ a vertex-near element of $E$ if
$E$ and $F$ share common vertex; we call an element $F$ a edge-near element of
$E$ if $E$ and $F$ share common edge; we call an element $F$ a face-near
element of $E$ if $E$ and $F$ share common face. An element $F$ is near
element of $E$ if $F$ is either face-near, edge-near, of vertex-near element
of $E$.
### 2.1 Mesh and the representation of the elements
A mesh is specified by the cloud of $n$ points, or _nodes_ ,
$\\{p_{0},p_{1},\dots,p_{n-1}\\}$, where $p_{i}=(x_{i},y_{i},z_{i})$. An
element $E_{i}$ is specified by the four nodes,
$E_{i}=\\{p_{i_{0}},p_{i_{1}},p_{i_{2}},p_{i_{3}}\\}$.
## 3 The algorithm
The idea of the algorithm is to do the histogramming of the nodes.
Step 1.
(initialization) For $N_{\rm node}$ nodes allocate array $L$ of lists, the
length of the array is $N_{\rm node}$.
Step 2.
(histogramming) For each element $i_{\rm elem}$, $0\leq i_{\rm elem}\leq
N_{\rm elem}-1$, do:
Step 2.1
For each node $p_{j_{i}}$, $0\leq i\leq 3$ of element $i_{\rm elem}$ add index
$i_{\rm elem}$ to the lists $L[{j_{0}}]$, $L[{j_{1}}]$, $L[{j_{2}}]$,
$L[{j_{3}}]$. In C++ notation, it should be: L[$j_{0}$].push_back($i_{\rm
elem}$), L[$j_{1}$].push_back($i_{\rm elem}$), L[$j_{2}$].push_back($i_{\rm
elem}$), L[$j_{3}$].push_back($i_{\rm elem}$).
Step 3.
(Finding neighboring elements) Given element
$E_{i}=(j_{0},j_{1},j_{2},j_{3})$, list of elements neighboring $E_{i}$ is
given by the union of the lists $L[{j_{0}}]$, $L[{j_{1}}]$, $L[{j_{2}}]$, and
$L[{j_{3}}]$.
Step 4.
Deallocate array $L$ of lists.
## 4 Parallelization
Algorithm admits easy parallelization by OpenMP, one only has to take care to
use critical section for Step 2.1: adding element index to the lists should be
done inside the critical section.
## 5 Conclusion
Missing details will be provided later.
|
arxiv-papers
| 2011-10-02T12:32:32 |
2024-09-04T02:49:22.682795
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gleb Novichkov",
"submitter": "Gleb Novichkov",
"url": "https://arxiv.org/abs/1110.0180"
}
|
1110.0217
|
# Reciprocal Properties of Pythagorean triangles
Konstantine Zelator
Mathematics, Statistics, and Computer Science
212 Ben Franklin Hall
Bloomsburg University of Pennsylvania
400 East 2nd Street
Bloomsburg, PA 17815
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
kzelator@.bloomu.edu
e-mails: konstantine zelator@yahoo.com
## 1 Introduction
Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal
published by the Canadian Mathematical Society for undergraduate university
students and teachers, as well as high school teachers and students. It
publishes material ranging from K-10 to K-14 grade levels. The problems are
divided into four categories: skoliad problems, mayhem problems, mathematical
olympiad problems, and general problems. In the April, 2009 issue of the
journal (Vol. 35, No; see reference [1]), the following problem was published
(Mayhem problem M390), quote: “A Pythagorean triangle is a right-handed angled
triangle with all three sides of integer length. Let $a$ and $b$ be the legs
of a Pythagoream triangle and $h$ be altitude to the hypotenuse. Determine all
triangles for which,
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{h}=1{\rm{}^{\prime\prime}},\ {\rm end\
of\ quote}.$
A solution was published in the February 2010 issue of the journal (see [2]).
It was shown that the only Pythagorean triangle with the above property is the
triangle with leg lengths $3$ and $4$; and hypotenuse length $5$. Our solution
to the same problem appears in the proof of part (iii) of Theorem 2, found in
Section 5 of this article.
The above problem provides the motivation behind this work. Throughout, we
will use the standard notation $(u,w)$ to denote the greatest common divisor
of two integers $u$ and $v$.
The following definition creates conceptual basis for this work.
Definition 1: Let $(a,b,c)$ be a Pythagorean triple with $a$ and $b$ being the
leg lengths; $c$ the hypotenuse length. Also let $h$ be the altitude to the
hypotenuse. Let $v,k,l$ be positive integers with $k$ and $l$ being relatively
prime; $(k,l)=1$. Then, the Pythagorean triple is to have the reciprocal
property $R(v,k,l)$ if the positive integers $a,b,h,v,k,l$ satisfy the
condition or equation,
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{v}{h}=\dfrac{k}{l}$
According to Definition 1 and the solution to the above problem, the
Pythagorean triple that has the reciprocal property $(R(1,1,1)$, is the triple
$(3,4,5)$.
Let us outline the organization of this article.
In Section 2, we state the well known parametric formulas that generate the
entire set of Pythagorean triples. We provide two references, [3] and [4].
Section 3 contains three results from number theory. The first one, Lemma 1,
is very well known and it is generally known as Euclid’s Lemma. We use Lemma 1
to prove Lemma 2 which is in turn used in the proof for Theorem 4 (Section 5).
The third result in Section 3 is Theorem 1 which is very well known. It gives
the general parametric solution to a linear diophantine equation in two
variables; which, in turn, is used in establishing Theorem 5. Section 4
contains the key equation of this article. The results of this work are
expressed in Theorems 2, 3, 4, and 5, all found in Section 5.
## 2 Pythagorean triples
Definition 2 A triple $(a,b,c)$ of three positive integers $a,b,c$ is said to
be Pythagorean with hypotenuse length $c$ if $a^{2}+b^{2}=c^{2}$.
A positive integer triple $(a,b,c)$ is Pythagorean if, and only if,
$a=d(m^{2}-n^{2}),\ b=d(2mn),\ c=d(m^{2}+n^{2})$; ($a$ and $b$ may be
switched). where $m,n,d$ are positive integers such that $m>n$ (and so $m\geq
2$) $(m,n)=1$ (i.e., $m$ and $n$ are relative prime), and $m+n\equiv 1({\rm
mod}\,2)$ (i.e., one of $m,n$ is even; the other odd) If $d=1$, then triple
$(a,b,c)$ is said to be primitive.
(1)
## 3 Three results from number theory
For a proof of Lemma 1, see reference [3].
###### Lemma 1.
(Euclid’s lemma) Suppose that $\alpha,\beta,\gamma$ are positive integers such
that $\alpha$ divides the product $\beta\gamma$; with $\alpha$ and $\beta$
being relatively prime. Then $\alpha$ is a divisor of $\gamma$.
###### Lemma 2.
Let $\dot{q}$ and $r>q$ be two relatively prime positive integers with
different parities; one of them being even, the other odd. Also, assume that
$q$ is not divisible by $3$. That is, $q\not\equiv 0({\rm mod}\,3)$ and that
$r\not\equiv q({\rm mod}\,3)$. Then,
$\left(3r^{2}+q^{2}+2rq,\ \ 2rq(r^{2}-q^{2})\right)=1$
###### Proof.
To establish the result, we will show that no prime divisor of the product
$2rq(r-q)(r+q)$ divides the integer $3r^{2}+q^{2}+2rq$. First observe that by
virtue of the fact that one of $r,q$ is odd, while the other is even. The
integer $3r^{2}+q^{2}+2rq$ is clearly odd.
Now, let $p$ be an odd prime divisor of the product $2rq(r-q)(r+q)$. The prime
$p$ must divide one of the four factors $r,q,\ r-q,$ or $r+q$. If $p$ divides
$r$ and $p$ also divides $3r^{2}+q^{2}+2rq$, then it follows that $p$ must
divide $q^{2}$; and since $p$ is a prime, this implies that $p$ is a divisor
of $q$. So $p$ divides both $r$ and $q$ violating the condition $(r,q)=1$,
i.e., the hypothesis that $r$ and $q$ are relatively prime. Now, if $p$
divides $q$ and $p$ divides $3r^{2}+q^{2}+2rq$, a straight forward calculation
implies that $p$ divides $3r^{2}$. So, $p$ is a common divisor of both $q$ and
$3r^{2}$. But $q$ is not divisible by $3$ so $p\neq 3.$ Since $p$ divides
$3r^{2}$ and $(p,3)=1$, Lemma 1 implies that $p$ must divide $r^{2}$, and
hence, $p$ divides $r$. Once again, this is contrary to the hypothesis
$(r,q)=1$. Next, suppose that $p$ divides $r-q$ or that $p$ divides $r+q$. In
other words, $r\equiv\pm q({\rm mod}\,p)$. Consequently,
$\begin{array}[]{rcl}3r^{2}+q^{2}+2rq&\equiv&4q^{2}\pm 2q^{2}({\rm
mod}\,p);\\\ \\\ 3r^{2}+q^{2}+2rq&\equiv&6q^{2}\ {\rm or}\ 2q^{2}({\rm
mod}\,p)\end{array}$
So, if $p$ also divides $3r^{2}+q^{2}+2rq$, then, if the minus sign holds in
the above congruences, $p$ must also divide $2q^{2}$. But $p$ is odd, so this
implies that $p$ divides $q^{2}$, and so $p$ divides $q$. But then
$r\equiv-q({\rm mod}\,p)$ implies that $p$ divides $r$ as well; contrary to
the condition $(r,q)=1$.
Finally, if the plus sign holds in the above congruences, it follows that $p$
must divide $6q^{2}$. If $p$ divides $q^{2}$, then we obtain the same
contradiction as before (in the minus sign case) which means (the only
possibility left) that $p$ must divide $6$; and thus (since $p$ is odd),
$p=3$. But, then since the plus sign holds, we get $r\equiv q({\rm mod}\,3)$
contrary to the hypothesis that $r\not\equiv q({\rm mod}\,3)$. The proof is
complete. ∎
The theorem below is very well known. For a reference, see [3].
###### Theorem 1.
Let $A$ and $B$ be integers, not both zero, and $D$ their greatest common
divisor, $D=(A,B)$, and $C$ an integer. Consider the two variable linear
diophantine equation $Ax+By=C$.
1. (i)
If $D$ is not a divisor of $C$, then the above equation has not integer
solution.
2. (ii)
If $D$ is a divisor of $C$, then the entire integer solution set can be
described by the parametric formulas, $x=x_{0}+\frac{B}{D}t,\
y=y_{0}-\frac{A}{D}t$ where $t$ can be any integer and $\\{x_{0},y_{0}\\}$ is
a particular integer solution.
## 4 The key equation
Let $(a,b,c)$ be a Pythagorean triple and $v,k,l$ given positive integers with
$(k,l)=1$. If the triple $(a,b,c)$ has the reciprocal property $R(v,k,l)$, we
then have
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{v}{h}=\dfrac{k}{l}$ (2)
where $h$ is the altitude to the hypotenuse. We have Area
$A=\frac{1}{2}ab=\frac{1}{2}hc$, which gives
$h=\dfrac{ab}{c}$ (3)
From (2) and (3) we obtain
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{vc}{ab}=\dfrac{k}{l}$
which further gives
$l(b+a+vc)=k\,ab.$ (4)
Combining (4) with the parametrical formulas in (1) yields
$l\left[(v+1)m^{2}+(v-1)n^{2}+2mn\right]=d\cdot k\cdot(2mn)(m^{2}-n^{2})$ (5)
which is the key equation.
## 5 Theorems 2, 3, 4, 5 and their proofs
###### Theorem 2.
1. (i)
Let $(a,b,c)$ be a Pythagorean triple described by the parametric formulas in
(1). If $(a,b,c)$ has the reciprocal property $R(v,k,1)$, then it is necessary
that $m$ is a divisor of $v-1$ and $n$ a divisor of $v+1$.
2. (ii)
There exists no Pythagorean triple that has the reciprocal property
$R(2,k,1)$.
3. (iii)
(Our solution to problem M390) The only Pythagorean triple which has
reciprocal property $R(1,1,1)$ is the triple $(3,4,5)$.
4. (iv)
If $k\geq 2$, there exists no Pythagorean triple which has the reciprocal
property $R(1,k,l)$.
5. (v)
Let $v,k,l$ be positive integers with $(k.l)=1$, $l$ odd and $v$ even. Then,
there exists no Pythagorean triple which has the reciprocal property
$R(v,k,l)$.
###### Proof.
1. (i)
Since $l=1$, the key equation (5) implies
$(v+1)m^{2}+(v-1)n^{2}+2mn=d\cdot k\cdot(2mn)(m^{2}-n^{2})$
which gives
$(v-1)n^{2}=m\left[2dkn(m^{2}-n^{2})-2n-m(v+1)\right].$
Since $(m,n)=1=(m,n^{2})$, the last equation and Lemma 1 imply that $m$ must
be a divisor of $v-1$. Likewise, from the above equation we get
$(v+1)m^{2}=n\left[2dkm(m^{2}-n^{2})-2m-n(v-1)\right].$
A similar argumentm as in the previous case, establishes that $n$ must be a
divisor of $v+1$.
2. (ii)
If $v=2$ and $l=1$, then it follows from part (i) that $m$ must be a divisor
of $v-1=2-1=1$ which is impossible since $m\geq 2$ (see (1)).
3. (iii)
For $v=k=l=1$, equation (5) takes the form
$\begin{array}[]{rcl}2m^{2}+2mn&=&d(2mn)(m^{2}-n^{2});\\\ \\\
2m(m+n)&=&d(2mn)(m-n)(m+n);\\\ \\\ 1&=&dn(m-n).\end{array}$ (6)
Since $m,n,$ and $m-n$ are positive integers (in view of $m>n\geq 1$). The
last equation (6) implies $d=n=m-n=1$ which gives $d=1,\ n=1,\ m=2$. So, by
(1) we obtain $(a,b,c)=(3,4,5)$
4. (iv)
For $v=1$, equation (5) takes the form
$\begin{array}[]{rcl}l(2m^{2}+2mn)&=&d\cdot k(2mn)(m^{2}-n^{2});\\\ \\\
2ml(m+n)&=&d\cdot k(2mn)(m-n)(m+n);\\\ \\\ l&=&dk(m-n)\end{array}$ (7)
Since $k$ is relatively prime to $l$ and $k\geq 2$, clearly, equation is
impossible or contradictory.
5. (v)
Since $v$ is even, both integers $v-1$ and $v+1$ are odd. Since $m$ and $n$
have different parities (i.e., one is odd and the other even), it follows that
$(v+1)m^{2}+(v-1)n^{2}+2mn$ is an odd integer; and since $l$ is odd. The left-
hand side of equation (5) is then odd, while the right-hand side is even,
creating a contradiction. So, no Pythagorean triple can have a reciprocal
property $R(v,k,l)$ with $v$ being even and $l$ being odd.
∎
###### Theorem 3.
Let $k$ be a positive integer and $v$ a positive integer such that both $v-1$
and $v+1$ are primes (i.e., twin primes). Then, there exists no Pythagorean
triangle or triple that has the reciprocal property $R(v,k,1)$.
###### Proof.
If, to the contrary such a triangle or triple exists, then equation (5) with
$l=1$ gives
$(v+1)m^{2}+(v-1)n^{2}+2mn=d\cdot k\cdot(2mn)(m^{2}-n^{2})$ (8)
We put $v-1=p$, a prime; and so $v+1=p+2$, also a prime. By part (i), we know
that the positive integer $m$ must be a divisor of $v-1=p$; and since $p$ is a
prime and $m\geq 2$, it follows that $m=p$. Likewise, from part (i), we know
that $n$ must divide $v+1=p+2$. But, $p+2$ is a prime. So, either $n=1$ or
$n=p+2$ which cannot be the case since $m>n$, and so $p>n$. Hence, the only
remaining possbility is $n=1$.
We have $m=p,\ n=1,\ v-1=p$, and $v+1=p+2$. Accordingly, by equation (8) we
get
$\begin{array}[]{rcl}(p+2)p^{2}+p+2p&=&d\cdot 2\cdot p(p-1)(p+1)k;\\\ \\\
(p+2)p+3&=&2dk(p^{2}-1).\end{array}$
After some algebra
$(2dk-1)p^{2}-2p-(3+2dk)=0$ (9)
Equation (9) demonstrates that the prime $p$ is one of the two roots or zeros
of the quadratic trinomial $t(x)=(2dk-1)x^{2}-2x-(3+2dk)$.
Since this trinomial has integer coefficients and an integer root, the other
root must be a rational number (since the sum and the product of the roots are
both rational). This, in turn, necessitates that the discriminant must be a
perfect or integer square:
$4+4(2dk-1)(3+2dk)=T^{2},$
for some non-negative integer $T$. Obviously, $T$ must be even, $T=2t$ for
some $t\in{\mathbb{Z}},\ t\geq 0$.
We obtain
$\begin{array}[]{rcl}1+(2dk-1)(3+2dk)&=&t^{2};\\\ \\\
4dk(dk+1)-2&=&t^{2}.\end{array}$ (10)
However, $t^{2}\equiv 1$ or $0({\rm mod}\,4)$, depending on whether $t$ is odd
or even. Hence, the left-hand side of (10) is congruent to $2$ modulo $4$.
While the right-hand side is congruent to $1$ or $0$ modulo $4$, a
contradiction. The proof is complete. ∎
###### Theorem 4.
Let $(a,b,c)$ be a Pythagorean triangle or triple with the two (out of three)
generator positive integers $m$ and $n$ in (1), satisfying the additional
conditions $n\not\equiv 0({\rm mod}\,3)$ and $m\not\equiv n({\rm mod}\,3)$.
Under this additional hypothesis the following holds true. The triple
$(a,b,c)$ has the reciprocal property $R(2,1,l)$ if, and only if, the positive
integers $d$ (third generator in the formulas in (1)) and $l$ are of the form,
$\begin{array}[]{rcll}d&=&t\cdot(2mn)(m^{2}-n^{2})(3m^{2}+n^{2}+2mn)&{\rm
and}\\\ \\\ l&=&t(2mn)(m^{2}-n^{2})\end{array}$
where $t$ is a positive integer.
###### Proof.
For $v=2$ and $k=1$, the key equation (5) takes the form
$l(3m^{2}+n^{2}+2mn)=d(2mn)(m^{2}-n^{2}).$ (11)
A straight forward calculation shows that if $l$ and $d$ have the required
form, then equation (11) is satisfied. Now, the converse. Assume that (11) is
satisfied. Since $n\not\equiv 0({\rm mod}\,3),\ m\not\equiv n({\rm mod}\,3)$,
and also by (1), $m>n$ and one of $m,n$ is even, the other odd. We infer from
Lemma 2 that the positive integer $2mn(m^{2}-n^{2})$ is relatively prime to
the positive integer $3m^{2}+n^{2}+2mn$. Since it divides the product on the
left-hand side of (11), we deduce that $2mn(m^{2}-n^{2})$ must be a divisor of
$l$.
Hence, $l=t(2mn)(m^{2}-n^{2})$, for some positive integer $t$. Substituting
for $l=t(2mn)(m^{2}-n^{2})$, in (11), produces
$d=t(2mn)(m^{2}-n^{2})(3m^{2}+n^{2}+2mn)$. ∎
###### Theorem 5.
The Pythagorean triple $(3,4,5)$ has the reciprocal property $R(v,k.l)$
precisely when the three positive integers $v,k,$ and $l$ belong to one of six
groups.
$\begin{array}[]{llll}{\rm Group\ 1:}&v=1+12t,&k=1+5t,&l=1\\\ {\rm Group\
2:}&v=1+6t,&k=2+5t,&l=2\\\ {\rm Group\ 3:}&v=1+4t,&k=3+5t,&l=3\\\ {\rm Group\
4:}&v=1+3t,&k=4+5t,&l=4\\\ {\rm Group\ 5:}&v=1+2t,&k=6+5t,&l=6\\\ {\rm Group\
6:}&v=1+t,&k=12+5t,&l=12\end{array}$
where $t$ can be any non-negative integer in all six groups.
###### Proof.
Going back to (1), we see that for the Pythagorean triple $(3,4,5)$ we have
$d=1,\ m=2$, and $n=1$. Therefore, the key equation (5) yields,
$l(7+5v)=12k.$ (12)
By virtue of $(l,k)=1$, equation (12) and Lemma 1 imply that $l$ must be a
divisor of $12$. Thus,
$l=1,2,3,4,6,\ {\rm or}\ 12.$
For each of these six values of $l$, a linear diophantine equation the the
variables $v$ and $k$ is obtained.
We tabulate these below by including in each case an easily obtained (by
inspection) particular solution$\\{v_{0},k_{0}\\}$
$\begin{array}[]{|l|c|l|}\hline\cr&{\rm Linear\ diophantine\ equation}&{\rm
Particular\ solution}\\\ \hline\cr l=1&-5v+12k=7&v_{0}=1,\ k_{0}=1\\\
\hline\cr l=2&-5v+6k=7&v_{0}=1,\ k_{0}=2\\\ \hline\cr l=3&-5v+4k=7&v_{0}=1,\
k_{0}=3\\\ \hline\cr l=4&-5v+3k=7&v_{0}=1,\ k_{0}=4\\\ \hline\cr
l=5&-5v+2k=7&v_{0}=1,\ k_{0}=6\\\ \hline\cr l=6&-5v+k=7&v_{1}=1,\ k_{0}=12\\\
\hline\cr\end{array}$
Note that in each in the above table, the greatest common divisor of $D$ of
the coefficients is always $1$. Applying Theorem 1 we obtain the following
parametric solutions.
$\begin{array}[]{ll}{\rm For}\ l=1,&v=1+12t,\ k=1+5t\\\ {\rm For}\
l=2,&v=1+6t,\ k=2+5t\\\ {\rm For}\ l=3,&v=1+4t,\ k=3+5t\\\ {\rm For}\
l=4,&v=1+3t,\ k=4+5t\\\ {\rm For}\ l=6,&v=1+2t,\ k=6+5t\\\ {\rm For}\
l=12,&v=1+t,\ k=12+5t\end{array}$
According to Theorem 1, $t$ can be any integer in the general solution.
However, in our case $v$ and $k$ are positive integers. A cursory examination
shows that if $t$ is a negative integer in each of the above six cases, at
least one of $v$ and $k$ will be zero or negative. Thus, the parameter $t$
cannot take any negative values. On the other hand, in all six groupings, for
each non-negative value of $t$, the resulting values of $v$ and $k$ are both
positive. ∎
Remark: If one cannot find, by inspection, a particular solution
$\\{x_{0},y_{0}\\}$ to a linear diophantine equation (Theorem 1). There is the
Euclidean algorithm that guarantees that such a solution can be found.
## References
* [1] 1 Crux Mathematicorum with Mathematical Mayhem, 35, No. 3, April, 2009, Mayhem problem M390, page 134. ISSN 1706-8142 (Print)
ISSN 1496-4309 (Online)
* [2] 2 Crux Mathematicorum with Mathematical Mayhem, 36, No. 1, February, 2010. Solution to Mayhem problem M390, page 11.
* [3] 3 Rosen, Kenneth H., Elementary Numbers Theory and its Applications, fifth edition, Pearson, Addison-Wesley, 2005. ISBN 0-321-23707-2 For Lemma 1(Lemma 3.4 in the above book), see page 109 for Pythagorean Triples, see pages 510-514.
* [4] 4 Sierpinski, W., Elementary Theory of Numbers, original edition, Warsaw, 1964, 488 pp (no ISBN in the original edition). More recent edition, Elsevier Publishing (1988). After that, UMI Books on Demand, ProQuest Company. ISBN 0-598-52758-3 For a description and derivation of Pythagorean triples, see pages 38-42.
|
arxiv-papers
| 2011-10-02T19:16:21 |
2024-09-04T02:49:22.690028
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1110.0217"
}
|
1110.0270
|
Add-ons for Lattice Boltzmann Methods: Regularization, Filtering and Limiters
R.A. Brownlee, J. Levesley, D. Packwood and A.N. Gorban
_Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK_
Abstract: We describe how regularization of lattice Boltzmann methods can be
achieved by modifying dissipation. Classes of techniques used to try to
improve regularization of LBMs include flux limiters, enforcing the exact
correct production of entropy and manipulating non-hydrodynamic modes of the
system in relaxation. Each of these techniques corresponds to an additional
modification of dissipation compared with the standard LBGK model. Using some
standard 1D and 2D benchmarks including the shock tube and lid driven cavity,
we explore the effectiveness of these classes of methods.
Keywords: Lattice Boltzmann, Dissipation, Stability, Entropy, Filtering,
Multiple Relaxation Times (MRT), Shock Tube, Lid-Driven Cavity
1\. INTRODUCTION
Lattice Boltzmann methods (LBM) are a class of discrete computational schemes
which can be used to simulate hydrodynamics and more[33]. They have been
proposed as a discretization of Boltzmann’s kinetic equation. Instead of
fields of moments $M$, the LBM operates with fields of discrete distributions
$f$.
All computational methods for continuum dynamics meet some troubles with
stability when the gradients of the flows become too sharp. In Computational
Fluid Dynamics (CFD) such situations occur when the Mach number $Ma$ is not
small, or the Reynolds number $Re$ is too large. The possibility for using
grid refinement is bounded by computational time and memory restrictions.
Moreover, for nonlinear systems with shocks then grid refinement does not
guarantee convergence to the proper solution. Methods of choice to remedy this
are based on the modification of dissipation with limiters, additional
viscosity, and so on [36]. All these approaches combine high order methods in
relatively quiet regions with low order regularized methods in the regions
with large gradients. The areas of the high-slope flows are assumed to be
small but the loss of the order of accuracy in a small region may affect the
accuracy in the whole domain because of the phenomenon of error propagation.
Nevertheless, this loss of accuracy for systems with high gradients seems to
be unavoidable.
It is impossible to successfully struggle with some spurious effects without a
local decrease in the order of accuracy. In a more formal setting, this has
been proven. In 1959, Godunov [14] proved that a (linear) scheme for a partial
differintial equation could not, at the same time, be monotone and second-
order accurate. Hence, we should choose between spurious oscillations in high
order non-monotone schemes and additional dissipation in first-order schemes.
Lax [23] demonstrated that un-physical dispersive oscillations in areas with
high slopes are unavoidable due to discretization. For hydrodynamic
simulations using the standard LBGK model (Sec. 1) such oscillations become
prevalent especially at high $Re$ and non-small $Ma$. Levermore and Liu used
differential approximation to produce the “modulation equation" for the
dispersive oscillation in the simple initial-value Hopf problem [24] and
demonstrated directly how for a nonlinear problem a solution of the
discretized equation does not converge to the solution of the continuous model
with high slope when the step $h\to 0$.
Some authors expressed a hope that precisely keeping the entropy balance can
make the computation more “physical", and that thermodynamics can help to
suppress nonphysical effects. Tadmor and Zhong constructed a new family of
entropy stable difference schemes which retain the precise entropy decay of
the Navier–Stokes equations and demonstrated that this precise keeping of the
entropy balance does not help to avoid the nonphysical dispersive oscillations
[34].
To prevent nonphysical oscillations, most upwind schemes employ limiters that
reduce the spatial accuracy to first order through shock waves. A mixed-order
scheme may be defined as a numerical method where the formal order of the
truncation error varies either spatially, for example, at a shock wave, or for
different terms in the governing equations, for example, third-order
convection with second-order diffusion [31].
Several techniques have been proposed to help suppress these pollutive
oscillations in LBM, the three which we deal with in this work are entropic
lattice Boltzman (ELBM), entropic limiters and generalized lattice Boltzmann,
also known as multiple relaxation time lattice Boltzmann (MRT). Where
effective each of these techniques corresponds to an additional degree of
complexity in the dissipation to the system, above that which exists in the
LBGK model.
The Entropic lattice Boltzmann method (ELBM) was invented first in 1998 as a
tool for the construction of single relaxation time LBM which respect the
$H$-theorem [19]. For this purpose, instead of the mirror image with a local
equilibrium as the reflection center, the entropic involution was proposed,
which preserves the entropy value. Later, it was called the Karlin-Succi
involution [15]. In 2000, it was reported that exact implementation of the
Karlin-Succi involution (which keeps the entropy balance) significantly
regularizes the post-shock dispersive oscillations [1]. This regularization
seems very surprising, because the entropic lattice BGK (ELBGK) model gives a
second-order approximation to the Navier–Stokes equation similarly to the LBGK
model (different proofs of that degree of approximation were given in [33] and
[4]).
Entropic limiters [4] are an example of flux limiter schemes [4, 21, 29],
which are invented to combine high resolution schemes in areas with smooth
fields and first order schemes in areas with sharp gradients. The idea of flux
limiters can be illustrated by the computation of the flux $F_{0,1}$ of the
conserved quantity $u$ between a cell marked by 0 and one of two its neighbour
cells marked by $\pm 1$:
$F_{0,1}=(1-\phi(r))f^{\rm low}_{0,1}+\phi(r)f^{\rm high}_{0,1},$ (1)
where $f^{\rm low}_{0,\,1}$, $f^{\rm high}_{0,\,1}$ are low and high
resolution scheme fluxes, respectively, $r=(u_{0}-u_{-1})/(u_{1}-u_{0})$, and
$\phi(r)\geq 0$ is a flux limiter function. For $r$ close to 1, the flux
limiter function $\phi(r)$ should be also close to 1. Many flux limiter
schemes have been invented during the last two decades [36]. No particular
limiter works well for all problems, and a choice is usually made on a trial
and error basis. Particular examples of the limiters we use are introduced in
Section 3.3..
MRT has been developed as a true generalization of the collisions in the
lattice Bhatnagar–Gross–Krook (LBGK) scheme [11, 22] from a one parameter
diagonal relaxation matrix, to a general linear operation with more free
parameters, the number of which is dependent on the particular discrete
velocity set used and the number of conserved macroscopic variables. Different
variants of MRT have been shown to improve accuracy and stability, including
in our benchmark examples [26] in comparison with the standard LBGK systems.
The lattice Boltzmann paradigm is now mature, and explanations for some of its
successes are available. However, in its applications it approaches the
boundaries of the applicability and need special additional tools to extend
the area of applications. It is well-understood that near to shocks, for
instance, special and specific attention must be paid to avoid unphysical
effects. In this paper then, we will discuss a variety of add-ons for LBM and
apply them to a variety of standard 1D and 2D problems to test their
effectiveness. In particular we will describe a family of entropic filters and
show that we can use them to signficantly expand the effective range of
operation of the LBM.
2\. BACKGROUND
Lattice Boltzmann methods can be derived independently by a discretization
Boltzmann’s equation for kinetic transport or by naively creating a discrete
scheme which matches moments with the Maxwellian distribution up to some
finite order.
In each case the final discrete algorithm consists of two alternating steps,
advection and collision, which are applied to $m$ single particle distribution
functions $f_{i}\equiv f_{i}(\mathbf{x},t),(i=1\ldots m)$, each of which
corresponds with a discrete velocity vector $\mathbf{v}_{i},(i=1\ldots m)$.
The values $f_{i}$ are also sometimes known as _populations_ or _densities_ as
they can be thought of as representative of the densities of particles moving
in the direction of the corresponding discrete velocities.
The advection operation is simply free flight for the discrete time step
$\epsilon$ in the direction of the corresponding velocity vector,
$f_{i}(\mathbf{x},t+\epsilon)=f_{i}(\mathbf{x}-\epsilon\mathbf{v}_{i},t).$ (2)
The collision operation is instantaneous and can be different for each
distribution function but depends on every distribution function, this might
be written,
$f_{i}(\mathbf{x})\rightarrow F_{i}(\\{f_{i}(\mathbf{x})\\}).$ (3)
In order to have a slightly more compact notation we can write these
operations in vector form, in the below equation it should be inferred that
the $i$th distribution function is advecting along its corresponding discrete
velocity,
$\mathbf{f}(\mathbf{x},t+\epsilon)=\mathbf{f}(\mathbf{x}-\epsilon\mathbf{v}_{i},t),$
(4) $\mathbf{f}(\mathbf{x})\rightarrow F(\mathbf{f}(\mathbf{x})).$ (5)
To transform our vector of microscopic variables at a point in space
$\mathbf{f}(\mathbf{x})$ to a vector of macroscopic variables $M(\mathbf{x})$
we use a vector of linear functions $M(\mathbf{x})=m(\mathbf{f}(\mathbf{x}))$.
In the athermal hydrodynamic systems we consider in this work the momemts are
density $\rho$ and momentum density $\rho\mathbf{u}$,
$\\{\rho,\rho\mathbf{u}\\}(\mathbf{x})=\sum_{i}\\{1,\mathbf{v}_{i}\\}f_{i}(\mathbf{x})$.
These macroscopic moments are conserved by the collision operation,
$m(\mathbf{f})=m(F(\mathbf{f}))$.
The simplest and most common choice for the collision operation $F$ is the
Bhatnagar-Gross-Krook(BGK) [7, 9, 17, 33] operator with over-relaxation
$F(\mathbf{f})=\mathbf{f}+\alpha\beta(\mathbf{f}^{\mathrm{eq}}-\mathbf{f}).$
(6)
For the standard LBGK method $\alpha=2$ and $\beta\in[0,1]$ (usually,
$\beta\in[1/2,1]$) is the over-relaxation coefficient used to control
viscosity. For $\beta=1/2$ the collision operator returns the _local
equilibrium_ $\mathbf{f}^{\mathrm{eq}}$ and $\beta=1$ (the _mirror
reflection_) returns the collision for a liquid at the zero viscosity limit.
The definition of $\mathbf{f}^{\mathrm{eq}}$ defines the dynamics of the
system, often it chosen as an approximation to the continuous Maxwellian
distribution. An equilibrium can also be independently derived by constructing
a discrete system which matches moments of the Maxwellian up to some finite
order. For hydrodynamic systems often this finite order is chosen to be 2, as
this is sufficient to accurately replicate the Euler(non dissipative)
component of the Navier Stokes equations. For a dissipative fluid with
viscosity $\nu$ the parameter $\beta$ is chosen by
$\beta=\epsilon/(2\nu+\epsilon)$.
Each of the techniques we test in this paper can be introduced as developments
of the generic LBGK system and such a presentation follows in the next
sections.
3\. ENTROPIC LATTICE BOLTZMANN
## 3.1. LBM with $H$ theorem
In the continuous case the Maxwellian distribution maximizes entropy, as
measured by the Boltzmann $H$ function, and therefore also has zero entropy
production. In the context of lattice Boltzmann methods a discrete form of the
$H$-theorem has been suggested as a way to introduce thermodynamic control to
the system [18, 1].
From this perspective the goal is to find an equilibrium state equivalent to
the Maxwellian in the continuum which will similarly maximize entropy. Before
the equilibrium can be found an appropriate $H$ function must be known for a
given lattice. These functions have been constructed in a lattice dependent
fashion in [18], and $H=-S$ with $S$ from (7) is an example of a $H$ function
constructed in this way.
One way to implement an ELBM is as a variation on the LBGK, known as the ELBGK
[1]. In this case $\alpha$ is varied to ensure a constant entropy condition
according to the discrete $H$-theorem. In general the entropy function is
based upon the lattice and cannot always be found explicitly. However for some
examples such as the simple one dimensional lattice with velocities
$\mathbf{v}=(-c,0,c)$ and corresponding populations
$\mathbf{f}=(f_{-},f_{0},f_{+})$ an explicit Boltzmann style entropy function
is known [18]:
$S(\mathbf{f})=-f_{-}\log(f_{-})-f_{0}\log(f_{0}/4)-f_{+}\log(f_{+}).$ (7)
With knowledge of such a function $\alpha$ is found as the non-trivial root of
the equation
$S(\mathbf{f})=S(\mathbf{f}+\alpha(\mathbf{f}^{\ast}-\mathbf{f})).$ (8)
The trivial root $\alpha=0$ returns the entropy value of the original
populations. ELBGK then finds the non-trivial $\alpha$ such that (8) holds.
This version of the BGK collision one calls entropic BGK (or EBGK) collision.
A solution of (8) must be found at every time step and lattice site. Entropic
equilibria (also derived from the $H$-theorem) are always used for ELBGK.
## 3.2. ELBM algorithm and additional dissipation
The definition of ELBM for a given entropy equation (8) is incomplete. First
of all, it is possible that the non-trivial solution does not exist. Moreover,
for most of the known entropies (like the perfect entropy [18]) there always
exist such $f$ that the equation (8) for the ELBM collision has no non-trivial
solutions. These $f$ should be sufficiently far from equilibrium. For
completeness, every user of ELBM should define collisions when the non-trivial
root of (8) does not exist. We know and tried two rules for this situation:
1. 1.
The most radical approach gives the the Ehrenfest rule [16, 3]: ”if the
solution does not exist then go to equilibrium", i.e. if the solution does not
exists then take $\alpha=1$.
2. 2.
The most gentle solution gives the ”positivity rule" [3, 25, 35, 32]: to take
the maximal value of $\alpha$ that guarantees
$f_{i}+\alpha(f_{i}^{\ast}-f_{i})\geq 0$ for all $i$.
In general, the Ehrenfest rule prescribes to send the most non-equilibrium
sites to equilibrium and the positivity rule is applied for any LBM as a
recommendation to substitute the non-positive vectors $\mathbf{f}$ by the
closest non-negative on the interval of the straight line
$[\mathbf{f},\mathbf{f}^{\ast}]$ that connects $\mathbf{f}$ to equilibrium.
These rules give the examples of the pointwise LBM limiter and we discuss them
separately.
By its nature, the ELBM adds more dissipation than the positivity rule when
the non-trivial root of (8) does not exist. It does not always keep the
entropy balance but increases dissipation for highly nonequilibrium sites.
## 3.3. Numerical method for solution of the ELBM equation
Another source for additional dissipation in the ELBM may be the numerical
method used for the solution of (8). For the full description of ELBM we have
to select a numerical method for this equation. This method has to have an
uniform accuracy in the wide range of parameters, for all possible deviation
from equilibrium (distribution of these deviations has “heavy tails" [5]).
In order to investigate the stabilization properties of ELBGK it is necessary
to craft a numerical method capable of finding the non-trivial root in (8). In
this section we fix the population vectors $\mathbf{f}$ and $\mathbf{f}^{*}$,
and are concerned only with this root finding algorithm. We recast (8) as a
function of $\alpha$ only:
$S_{f}(\alpha)=S(\mathbf{f}+\alpha(\mathbf{f}^{\mathrm{eq}}-\mathbf{f}))-S(\mathbf{f}).$
(9)
In this setting we attempt to find the non-trivial root $r$ of (9) such that
$S_{f}(r)=0$. It should be noted that as we search for $r$ numerically we
should always take care that the approximation we use is less than $r$ itself.
An upper approximation could result in negative entropy production. A simple
algorithm for finding the roots of a concave function, based on local
quadratic approximations to the target function, has cubic convergence order.
Assume that we are operating in a neighbourhood $r\in N$, in which
$S_{f}^{\prime}$ is negative (as well of course $S_{f}^{\prime\prime}$ is
negative). At each iteration the new estimate for $r$ is the greater root of
the parabola $P$, the second order Taylor polynomial at the current estimate.
Analogously to the case for Newton iteration, the constant in the estimate is
the ratio of third and first derivatives in the interval of iteration:
$\displaystyle|(r-\alpha_{n+1})|\,\leq\,C|\alpha_{n}-r|^{3},$
$\displaystyle\mbox{where}\;\;C=\frac{1}{6}\sup_{a\in
N}|S_{f}^{\prime\prime\prime}(a)|\left/\inf_{b\in
N}|S_{f}^{\prime}(b)|\right.,$
where $\alpha_{n}$ is the evaluation of $r$ on the $n$th iteration.
We use a Newton step to estimate the accuracy of the method at each iteration:
because of the concavity of $S$
$|\alpha_{n}-r|\lesssim\left|S_{f}(\alpha_{n})/S_{f}^{\prime}(\alpha_{n})\right|.$
(10)
In fact we use a convergence criteria based not solely on $\alpha$ but on
$\alpha||\mathbf{f}^{*}-\mathbf{f}||$, this has the intuitive appeal that in
the case where the populations are close to the local equilibrium $\Delta
S=S(\mathbf{f}^{*})-S(\mathbf{f})$ will be small and a very precise estimate
of $\alpha$ is unnecessary. We have some freedom in the choice of the norm
used and we select between the standard $L_{1}$ norm and the entropic norm.
The entropic norm is defined as
$||\mathbf{f}^{\mathrm{eq}}-\mathbf{f}||_{\mathbf{f}^{\mathrm{eq}}}=-((\mathbf{f}^{\mathrm{eq}}-\mathbf{f}),\left.D^{2}S\right|_{\mathbf{f}^{\mathrm{eq}}}(\mathbf{f}^{\mathrm{eq}}-\mathbf{f})),$
where $\left.D^{2}S\right|_{\mathbf{f}^{\mathrm{eq}}}$ is the second
differential of entropy at point $\mathbf{f}^{\mathrm{eq}}$, and $(x,y)$ is
the standard scalar product.
The final root finding algorithm then is beginning with the LBGK estimate
$x_{0}=2$ to iterate using the roots of successive parabolas. We stop the
method at the point,
$|\alpha_{n}-r|\cdot||\mathbf{f}^{*}-\mathbf{f}||<\epsilon.$ (11)
To ensure that we use an estimate that is less than the root, at the point
where the method has converged we check the sign of $S_{f}(\alpha_{n})$. If
$S_{f}(\alpha_{n})>0$ then we have achieved a lower estimate, if
$S_{f}(\alpha_{n})<0$ we correct the estimate to the other side of the root
with a double length Newton step,
$\alpha_{n}=\alpha_{n}-2\frac{S_{f}(\alpha_{n})}{S_{f}^{\prime}(\alpha_{n})}.$
(12)
At each time step before we begin root finding we eliminate all sites with
$\Delta S<10^{-15}$. For these sites we make a simple LBGK step. At such sites
we find that round off error in the calculation of $S_{f}$ by solution of
equation (8) can result in the root of the parabola becoming imaginary. In
such cases a mirror image given by LBGK is effectively indistinct from the
exact ELBGK collision. In the numerical examples given in this work the case
where the non-trivial root of the entropy parabola does not exist was not
encountered.
4\. ENTROPIC FILTERING
All the specific LBM limiters [5, 29] are based on a representation of
distributions $f$ in the form:
$\mathbf{f}=\mathbf{f}^{\mathrm{eq}}+\|\mathbf{f}-\mathbf{f}^{\mathrm{eq}}\|\frac{\mathbf{f}-\mathbf{f}^{\mathrm{eq}}}{\|\mathbf{f}-\mathbf{f}^{\mathrm{eq}}\|},$
(13)
where $\mathbf{f}^{\mathrm{eq}}$ is the corresponding quasiequilibrium
(conditional equilibrium) for given moments $M$,
$\mathbf{f}-\mathbf{f}^{\mathrm{eq}}$ is the nonequilibrium “part" of the
distribution, which is represented in the form “norm$\times$direction” and
$\|f-f^{*}\|$ is the norm of that nonequilibrium component (usually this is
the entropic norm).
All limiters we use change the norm of the nonequilibrium component
$\mathbf{f}-\mathbf{f}^{\mathrm{eq}}$, but do not touch its direction or the
quasiequilibrium. In particular, limiters do not change the macroscopic
variables, because moments for $\mathbf{f}$ and $\mathbf{f}^{\mathrm{eq}}$
coincide. These limiters are transformations of the form
$\mathbf{f}\mapsto\mathbf{f}^{\mathrm{eq}}+\phi\times(\mathbf{f}-\mathbf{f}^{\mathrm{eq}})$
(14)
with $\phi>0$. If $\mathbf{f}-\mathbf{f}^{\mathrm{eq}}$ is too big, then the
limiter should decrease its norm.
For the first example of the realization of this pointwise filtering we use
the kinetic idea of the positivity rule, the prescription is simple [3, 25,
35, 32]: to substitute nonpositive $F(\mathbf{f})$ by the closest nonnegative
state that belongs to the straight line
$\Bigr{\\{}\lambda\mathbf{f}+(1-\lambda)\mathbf{f}^{\mathrm{eq}}|\>\lambda\in\mathbb{R}\Bigl{\\}}$
(15)
defined by the two points, $\mathbf{f}$ and the corresponding quasiequilibrium
(1). This operation is to be applied pointwise, at points of the lattice where
positivity is violated. This technique preserves the positivity of
populations, but can affect the accuracy of the approximation. This rule is
necessary for ELBM when the positive “mirror state” with the same entropy as
$\mathbf{f}$ does not exists on the straight line (15).
Fig. (1): Positivity rule in action. The motions stops at the positivity
boundary.
The positivity rule measures the deviation $f-f^{*}$ by a binary measure: if
all components of this vector $f-f^{*}$ are non-negative then it is not too
large. If some of them are negative that this deviation is too large and needs
corrections.
To construct pointwise flux limiters for LBM, based on dissipation, the
entropic approach remains very convenient. The local nonequilibrium entropy
for each site is defined
$\Delta S(\mathbf{f}):=S(\mathbf{f}^{\mathrm{eq}})-S(\mathbf{f}).$ (16)
The positivity limiter was targeted wherever population densities became
negative. Entropic limiters are targeted wherever non-equilibrium entropy
becomes large.
The first limiter is _Ehrenfests’ regularisation_ [4, 3], it provides “entropy
trimming”: we monitor local deviation of $\mathbf{f}$ from the corresponding
quasiequilibrium, and when $\Delta S(\mathbf{f})(\mathbf{x})$ exceeds a pre-
specified threshold value $\delta$, perform local Ehrenfests’ steps to the
corresponding equilibrium: $\mathbf{f}\mapsto\mathbf{f}^{\mathrm{eq}}$ at
those points.
Not all lattice Boltzmann models are entropic, and an important question
arises: “how can we create nonequilibrium entropy limiters for LBM with non-
entropic (quasi)equilibria?”. We propose a solution of this problem based on
the discrete Kullback entropy [20]:
$S_{\rm K}(\mathbf{f})=-\sum_{i}f_{i}\ln\biggl{(}\frac{f_{i}}{f_{i}^{\rm
eq}}\biggr{)}.$ (17)
For entropic quasiequilibria with perfect entropy the discrete Kullback
entropy gives the same $\Delta S$: $-S_{\rm K}(f)=\Delta S(f)$. Let the
discrete entropy have the standard form for an ideal (perfect) mixture [18]:
$S(\mathbf{f})=-\sum_{i}f_{i}\ln\biggl{(}\frac{f_{i}}{W_{i}}\biggr{)}.$
In quadratic approximation,
$-S_{\rm K}(\mathbf{f})=\sum_{i}f_{i}\ln\biggl{(}\frac{f_{i}}{f^{\rm
eq}_{i}}\biggr{)}\approx\sum_{i}\frac{(f_{i}-f^{\rm eq}_{i})^{2}}{f^{\rm
eq}_{i}}.$ (18)
If we define $\mathbf{f}$ as the conditional entropy maximum for given
$M_{j}=\sum_{k}m_{jk}f_{k}$, then
$\ln f^{\rm eq}_{k}=\sum_{j}\mu_{j}m_{jk},$
where $\mu_{j}(M)$ are the Lagrange multipliers (or “potentials”). For this
entropy and conditional equilibrium we find
$\Delta
S=S(\mathbf{f}^{\mathrm{eq}})-S(\mathbf{f})=\sum_{i}f_{i}\ln\biggl{(}\frac{f_{i}}{f^{\rm
eq}_{i}}\biggr{)}=-S_{\rm K}(\mathbf{f}),$ (19)
if $\mathbf{f}$ and $\mathbf{f}^{\mathrm{eq}}$ have the same moments,
$m(\mathbf{f})=m(\mathbf{f}^{\mathrm{eq}})$.
In what follows, $\Delta S$ is the Kullback distance $-S_{\rm
K}(\mathbf{f}^{\mathrm{eq}})$ (19) for general (positive) quasiequilibria
$\mathbf{f}^{\mathrm{eq}}$, or simply
$S(\mathbf{f}^{\mathrm{eq}})-S(\mathbf{f})$ for entropic quasiequilibria (or
second approximations for these quantities (18)).
So that the Ehrenfests’ steps are not allowed to degrade the accuracy of LBGK
it is pertinent to select the $k$ sites with highest $\Delta S>\delta$. An a
posteriori estimate of added dissipation could easily be performed by the
analysis of entropy production in the Ehrenfests’ steps. Numerical experiments
show (see, e.g., [4, 3]) that even a small number of such steps drastically
improves stability.
The positivity rule and Ehrenfests’ regularisation provide rare, intense and
localised corrections. Of course, it is easy and also computationally cheap to
organise more gentle transformations with a smooth shift of highly
nonequilibrium states to quasiequilibrium. The following regularisation
transformation with a smooths function $\phi$ distributes its action smoothly:
$\mathbf{f}\mapsto\mathbf{f}^{\mathrm{eq}}+\phi(\Delta
S(\mathbf{f}))(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}).$ (20)
The choice of function $\phi$ is highly ambiguous, for example,
$\phi=1/(1+\alpha\Delta S^{k})$ for some $\alpha>0$ and $k>0$. There are two
significantly different choices: (i) ensemble-independent $\phi$ (i.e., the
value of $\phi$ depends on local value of $\Delta S$ only) and (ii) ensemble-
dependent $\phi$, for example
$\phi(\Delta S)=\frac{1+(\Delta S/(\alpha{\rm E}(\Delta
S)))^{k-1/2}}{1+(\Delta S/(\alpha{\rm E}(\Delta S)))^{k}},$ (21)
where ${\rm E}(\Delta S)$ is the average value of $\Delta S$ in the
computational area, $k\geq 1$, and $\alpha\gtrsim 1$. For small $\Delta S$,
$\phi(\Delta S)\approx 1$ and for $\Delta S\gg\alpha{\rm E}(\Delta S)$ it
tends to $\sqrt{\alpha{\rm E}(\Delta S)/\Delta S}$. It is easy to select an
ensemble-dependent $\phi$ with control of total additional dissipation.
## 4.1. Monotonic and double monotonic limiters
Two monotonicity properties are important in the theory of nonequilibrium
entropy limiters:
1. 1.
A limiter should move the distribution to equilibrium: in all cases of (14)
$0\leq\phi\leq 1$. This is the _dissipativity_ condition which means that
limiters never produce negative entropy.
2. 2.
A limiter should not change the order of states on the line: if for two
distributions with the same moments, $\mathbf{f}$ and $\mathbf{f}^{\prime}$,
$\mathbf{f}^{\prime}-\mathbf{f}^{\mathrm{eq}}=x(\mathbf{f}-\mathbf{f}^{\mathrm{eq}})$
and $\Delta S(\mathbf{f})>\Delta S(\mathbf{f}^{\prime})$ before the limiter
transformation, then the same inequality should hold after the limiter
transformation too. For example, for the limiter (20) it means that $\Delta
S(\mathbf{f}^{\mathrm{eq}}+x\phi(\Delta
S(\mathbf{f}^{\mathrm{eq}}+x(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}))(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}))$
is a monotonically increasing function of $x>0$.
In quadratic approximation,
$\displaystyle\Delta
S(\mathbf{f}^{\mathrm{eq}}+x(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}))$
$\displaystyle=x^{2}\Delta S(\mathbf{f}),$ $\displaystyle\Delta
S(\mathbf{f}^{\mathrm{eq}}+x\phi(\Delta
S(\mathbf{f}^{\mathrm{eq}}+x(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}))(\mathbf{f}-\mathbf{f}^{\mathrm{eq}}))$
$\displaystyle=x^{2}\phi^{2}(x^{2}\Delta S(\mathbf{f})),$
and the second monotonicity condition transforms into the following
requirement: $y\phi(y^{2}s)$ is a monotonically increasing (not decreasing)
function of $y>0$ for any $s>0$.
If a limiter satisfies both monotonicity conditions, we call it “double
monotonic”. For example, Ehrenfests’ regularisation satisfies the first
monotonicity condition, but violates the second one. The limiter (21) violates
the first condition for small $\Delta S$, but is dissipative and satisfies the
second one in quadratic approximation for large $\Delta S$. The limiter with
$\phi=1/(1+\alpha\Delta S^{k})$ always satisfies the first monotonicity
condition, violates the second if $k>1/2$, and is double monotonic (in
quadratic approximation for the second condition), if $0<k\leq 1/2$. The
threshold limiter (26) is also double monotonic.
For smooth functions, the condition of double monotonicity (in quadratic
approximation) is equivalent to the system of differential inequalities:
$\begin{split}\phi(x)+2x\phi^{\prime}(x)&\geq 0;\\\ \phi^{\prime}(x)&\leq
0.\end{split}$
The initial condition $\phi(0)=1$ means that in the limit $\Delta S\to 0$
limiters do not affect the flow. Following these inequalities we can write:
$2x\phi^{\prime}(x)=-\eta(x)\phi(x)$, where $0\leq\eta(x)\leq 1$. The solution
of these inequalities with initial condition is
$\phi(x)=\exp\biggl{(}-\frac{1}{2}\int_{0}^{x}\frac{\eta(\chi)}{\chi}{\mathrm{d}}\chi\biggr{)},$
(22)
if the integral on the right-hand side exists. This is a general solution for
double monotonic limiters (in the second approximation for entropy). If
$\eta(x)$ is the Heaviside step function, $\eta(x)=H(x-\Delta S_{\rm t})$ with
threshold value $\Delta S_{\rm t}$, then the general solution (22) gives us
the threshold limiter. If, for example, $\eta(x)=x^{k}/(\Delta S_{\rm
t}^{k}+x^{k})$, then
$\phi(x)=\left(1+\frac{x^{k}}{\Delta S_{\rm t}^{k}}\right)^{-\frac{1}{2k}}.$
(23)
This special form of limiter function is attractive because for small $x$ it
gives
$\phi(x)=1-\frac{1}{2k}\frac{x^{k}}{\Delta S_{\rm t}^{k}}+o(x^{k}).$
Thus, the limiter does not affect the motion up to the $(k+1)$st order, and
the macroscopic equations coincide with the macroscopic equations for LBM
without limiters up to the $(k+1)$st order in powers of deviation from
quasiequilibrium. Furthermore, for large $x$ we get the $k$th order
approximation to the threshold limiter (26):
$\phi(x)=\sqrt{\frac{\Delta S_{\rm t}}{x}}+o(x^{-k}).$
Of course, it is not forbidden to use any type of limiters under the local and
global control of dissipation, but double monotonic limiters provide some
natural properties automatically, without additional care.
## 4.2. Monitoring total dissipation
For given $\beta$, the entropy production in one LBGK step in quadratic
approximation for $\Delta S$ is:
$\delta_{\rm LBGK}S\approx[1-(2\beta-1)^{2}]\sum_{x}\Delta S(\mathbf{x}),$
where $\mathbf{x}$ is the grid point, $\Delta S(\mathbf{x})$ is nonequilibrium
entropy (16) at point $\mathbf{x}$, $\delta_{\rm LBGK}S$ is the total entropy
production in a single LBGK step. It would be desirable if the total entropy
production for the limiter $\delta_{\rm lim}S$ was small relative to
$\delta_{\rm LBGK}S$:
$\delta_{\rm lim}S<\delta_{0}\delta_{\rm LBGK}S.$ (24)
A simple ensemble-dependent limiter (perhaps, the simplest one) for a given
$\delta_{0}$ operates as follows. Let us collect the histogram of the $\Delta
S(\mathbf{x})$ distribution, and estimate the distribution density, $p(\Delta
S)$. We have to estimate a value $\Delta S_{0}$ that satisfies the following
equation:
$\int_{\Delta S_{0}}^{\infty}p(\Delta S)(\Delta S-\Delta
S_{0})\,{\mathrm{d}}\Delta
S=\delta_{0}[1-(2\beta-1)^{2}]\int_{0}^{\infty}p(\Delta S)\Delta
S\,{\mathrm{d}}\Delta S.$ (25)
In order not to affect distributions with a small expectation of $\Delta S$,
we choose a threshold $\Delta S_{\rm t}=\max\\{\Delta S_{0},\delta\\}$, where
$\delta$ is some predefined value (as in the Ehrenfests’ regularization). For
states at sites with $\Delta S\geq\Delta S_{\rm t}$ we provide homothety with
equilibrium center $\mathbf{f}^{\mathrm{eq}}$ and coefficient $\sqrt{\Delta
S_{\rm t}/\Delta S}$ (in quadratic approximation for nonequilibrium entropy):
$\mathbf{f}(\mathbf{x})\mapsto\mathbf{f}^{\mathrm{eq}}(\mathbf{x})+\sqrt{\frac{\Delta
S_{\rm t}}{\Delta
S}}(\mathbf{f}(\mathbf{x})-\mathbf{f}^{\mathrm{eq}}(\mathbf{x})).$ (26)
To avoid the change of accuracy order “on average”, the number of sites with
this step should be $\leq\mathcal{O}(Nh/L)$ where $N$ is the total number of
sites, $h$ is the step of the space discretization and $L$ is the macroscopic
characteristic length. But this rough estimate of accuracy across the system
might be destroyed by a concentration of Ehrenfests’ steps in the most
nonequilibrium areas, for example, in boundary layers. In that case, instead
of the total number of sites $N$ in $\mathcal{O}(Nh/L)$ we should take the
number of sites in a specific region. The effects of such concentration could
be analysed a posteriori.
## 4.3. Median entropy filter
The Ehrenfest step described above provides pointwise correction of
nonequilibrium entropy at the “most nonequilibrium” points. Due to the
pointwise nature, the technique does not introduce any nonisotropic effects,
and provides some other benefits. But if we involve local structure, we can
correct local non-monotone irregularities without touching regular fragments.
For example, we can discuss monotone increase or decrease of nonequilibrium
entropy as regular fragments and concentrate our efforts on reduction of
“speckle noise” or “salt and pepper noise”. This approach allows us to use the
accessible resource of entropy change (24) more thriftily. Salt and pepper
noise is a form of noise typically observed in images. It represents itself as
randomly occurring white (maximal brightness) and black pixels. For this kind
of noise, conventional low-pass filtering, e.g., mean filtering or Gaussian
smoothing is unsuccessful because the perturbed pixel value can vary
significantly both from the original and mean value. For this type of noise,
_median filtering_ is a common and effective noise reduction method. Median
filtering is a common step in image processing [30] for the smoothing of
signals and the suppression of impulse noise with preservation of edges.
The median is a more robust average than the mean (or the weighted mean) and
so a single very unrepresentative value in a neighbourhood will not affect the
median value significantly. Hence, we suppose that the median entropy filter
will work better than entropy convolution filters.
For the nonequilibrium entropy field, the median filter considers each site in
turn and looks at its nearby neighbours. It replaces the nonequilibrium
entropy value $\Delta S$ at the point with the median of those values $\Delta
S_{\rm med}$, then updates $f$ by the transformation (26) with the homothety
coefficient $\sqrt{\Delta S_{\rm med}/{\Delta S}}$. The median, $\Delta S_{\rm
med}$, is calculated by first sorting all the values from the surrounding
neighbourhood into numerical order and then replacing that being considered
with the middle value. For example, if a point has 3 nearest neighbours
including itself, then after sorting we have 3 values $\Delta S$: $\Delta
S_{1}\leq\Delta S_{2}\leq\Delta S_{3}$. The median value is $\Delta S_{\rm
med}=\Delta S_{2}$. For 9 nearest neighbours (including itself) we have after
sorting $\Delta S_{\rm med}=\Delta S_{5}$. For 27 nearest neighbours $\Delta
S_{\rm med}=\Delta S_{14}$.
We accept only dissipative corrections (those resulting in a decrease of
$\Delta S$, $\Delta S_{\rm med}<\Delta S$) because of the second law of
thermodynamics. The analogue of (25) is also useful for the acceptance of the
most significant corrections. In “salt and pepper" terms, we correct the salt
(where $\Delta S$ exceeds the median value) and do not touch the pepper.
## 4.4. General nonlocal filters
The separation of $\mathbf{f}$ in equilibrium and nonequilibrium parts (13)
allows one to use any nonlocal filtering procedure. Let $\mathbf{f}^{\rm
neq}=\mathbf{f}-\mathbf{f}^{\mathrm{eq}}$. The values of moments for
$\mathbf{f}$ and $\mathbf{f}^{\mathrm{eq}}$ coincide, hence we can apply any
transformation of the form
$\mathbf{f}^{\rm neq}(\mathbf{x})\mapsto\sum_{y}d(\mathbf{y})\mathbf{f}^{\rm
neq}(\mathbf{x}+\mathbf{y})$
for any family of vectors $\mathbf{y}$ that shift the grid into itself and any
coefficients $d(\mathbf{y})$. If we apply this transformation, the macroscopic
variables do not change but their time derivatives may change. We can control
the values of some higher moments in order not to perturb significantly some
macroscopic parameters, the shear viscosity, for example [10]. Several local
(but not pointwise) filters of this type have been proposed and tested
recently [29].
5\. MULTIPLE RELAXATION TIMES
The MRT lattice Boltzmann system [22, 26, 11] generalizes the BGK collision
into a more general linear transformation of the population functions,
$F(\mathbf{f})=\mathbf{f}+A(\mathbf{f}^{\mathrm{eq}}-\mathbf{f}),$ (27)
where $A$ is a square matrix of size $m$. The use of this more general
operator allows more different parameters to be used within the collision, to
manipulate different physical properties, or for stability purposes.
To facilitate this a change of basis matrix can be used to switch the space of
the collision to the moment space. Since the moment space of the system may be
several dimensions smaller than the population space, to complete the basis
linear combinations of higher order polynomials of the discrete velocity
vectors may be used. For our later experiments we will use the D2Q9 system, we
should select a particular enumeration of the discrete velocity vectors for
the system, the zero velocity is numbered one and the positive x velocity is
numbered 2, the remainder are numbered clockwise from this system,
$\begin{array}[]{ccc}7&8&9\\\ 6&1&2\\\ 5&4&3\end{array}.$ (28)
The change of basis matrix given in some of the literature [22] is chosen to
represent specific macroscopic quantities and in our velocity system is as
follows,
$M_{1}=\left(\begin{array}[]{c c c c c c c c c}1&1&1&1&1&1&1&1&1\\\
-4&-1&2&-1&2&-1&2&-1&2\\\ 4&-2&1&-2&1&-2&1&-2&1\\\ 0&1&1&0&-1&-1&-1&0&1\\\
0&-2&1&0&-1&2&-1&0&1\\\ 0&0&-1&-1&-1&0&1&1&1\\\ 0&0&-1&2&-1&0&1&-2&1\\\
0&1&0&-1&0&1&0&-1&0\\\ 0&0&-1&0&1&0&-1&0&1\end{array}\right)$ (29)
As an alternative we could complete the basis simply using higher powers of
the velocity vectors, the basis would be
$1,v_{x},v_{y},v_{x}^{2}+v_{y}^{2},v_{x}^{2}-v_{y}^{2},v_{x}v_{y},v_{x}^{2}v_{y},v_{x}v_{y}^{2},v_{x}^{2}v_{y}^{2}$,
in our velocity system then this change of basis matrix is as follows,
$M_{2}=\left(\begin{array}[]{c c c c c c c c c}1&1&1&1&1&1&1&1&1\\\
0&1&0&-1&0&1&-1&-1&1\\\ 0&0&1&0&-1&1&1&-1&-1\\\ 0&1&1&1&1&2&2&2&2\\\
0&1&-1&1&-1&0&0&0&0\\\ 0&0&0&0&0&1&-1&1&-1\\\ 0&0&0&0&0&1&-1&-1&1\\\
0&0&0&0&0&1&1&-1&-1\\\ 0&0&0&0&0&1&1&1&1\end{array}\right)$ (30)
When utilized properly any basis should be equivalent (although with different
rates). In any case in this system there are 3 conserved moments and 9
population functions. Altogether then there are 6 degrees of freedom in
relaxation in this system. We need some of these degrees of freedom to
implement hydrodynamic rates such as shear viscosity or to force isotropy and
some are ‘spare’ and can be manipulated to improve accuracy or stability.
Typically these spare relaxation modes are sent closer to equilibrium than the
standard BGK relaxation rate. This is in effect an additional contraction in
the finite dimensional non-equilibrium population function space,
corresponding to an increase in dissipation.
Once in moment space we can apply a diagonal relaxation matrix $C_{1}$ to the
populations and then the inverse moment transformation matrix $M_{1}^{-1}$ to
switch back into population space, altogether $A_{1}=M_{1}^{-1}C_{1}M_{1}$. If
we use the standard athermal polynomial equilibria then three entries on the
diagonal of $C_{1}$ are actually not important as the moments will be
automatically conserved since $m(\mathbf{f})=m(\mathbf{f}^{\mathrm{eq}})$, for
simplicity we set them equal to 0 or 1 to reduce the complexity of the terms
in the collision matrix. There are 6 more parameters on the diagonal matrix
$C$ which we can set. Three of these correspond to second order moments, one
each is required for shear and bulk viscosity which are called $s_{e}$ and
$s_{\nu}$ respectively and one for isotropy. Two correspond to third order
moments, one gives a relaxation rate $s_{q}$ and again one is needed for
isotropy. Finally one is used to give a relaxation rate $s_{\epsilon}$ for the
single fourth order moment. We have then in total four relaxation parameters
which appear on the diagonal matrix in the following form:
$C_{1}=\left(\begin{array}[]{ c c c c c c c c c}1&0&0&0&0&0&0&0&0\\\
0&s_{e}&0&0&0&0&0&0&0\\\ 0&0&s_{\epsilon}&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\
0&0&0&0&s_{q}&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&s_{q}&0&0\\\
0&0&0&0&0&0&0&s_{\nu}&0\\\ 0&0&0&0&0&0&0&0&s_{\nu}\end{array}\right).$ (31)
Apart from the parameter $s_{\nu}$ which is used to control shear viscosity,
in an incompressible system the other properties can be varied to improve
accuracy and stability. In particular, there exists a variant of MRT known as
TRT (two relaxation time) [13] where the relaxation rates $s_{e},s_{\epsilon}$
are made equal to $s_{\nu}$. In a system with boundaries the final rate is
calculated $s_{q}=8(2=s_{\nu})/(8-s_{\nu})$, this is done, in particular, to
combat numerical slip on the boundaries of the system.
We should say that in some of the literature regarding MRT the equilibrium is
actually built in moment space, that is the collision operation would be
written,
$F(\mathbf{f})=\mathbf{f}+M_{1}^{-1}C_{1}(\mathbf{m}^{\mathrm{eq}}-M_{1}\mathbf{f}).$
(32)
This could be done to increase efficiency, depending on the implementation,
however each moment equilibrium $\mathbf{m}^{\mathrm{eq}}$ has an equivalent
population space equilibrium
$\mathbf{f}^{\mathrm{eq}}=M_{1}^{-1}\mathbf{m}^{\mathrm{eq}}$, so the results
of implementing either system should be the same up to rounding error.
We can also conceive of using an MRT type collision as a limiter, that is to
apply it only on a small number of points on the lattice where non equilibrium
entropy passes a certain threshold. This answers a criticism of the single
relaxation time limiters that they fail to preserve dissipation on physical
modes. As well as using the standard MRT form given above we can build an MRT
limiter using the alternative change of basis matrix $M_{2}$.
The limiter in this case is based on the idea of sending every mode except
shear viscosity directly to equilibrium again the complete relaxation matrix
is given by $A_{2}=M_{2}^{-1}C_{2}M_{2}$ where,
$C_{2}=\left(\begin{array}[]{ c c c c c c c c c}1&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\
0&0&0&0&s_{\nu}&0&0&0&0\\\ 0&0&0&0&0&s_{\nu}&0&0&0\\\ 0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&0&1\end{array}\right).$ (33)
This could be considered a very aggressive form of the MRT which maximizes
regularization on every mode except shear viscosity and would not be
appropriate for general use in a system, especially as most systems violate
the incompressibility assumption and hence bulk viscosity is not small. The
advantage of using the different change of basis matrix is that the complete
collision matrix $A_{2}$ is relatively sparse with just twelve off diagonal
elements and hence is easy to implement and not too expensive to compute with.
6\. 1D SHOCK TUBE
A standard experiment for the testing of LBMs is the one-dimensional shock
tube problem. The lattice velocities used are $\mathbf{v}=(-1,0,1)$, so that
space shifts of the velocities give lattice sites separated by the unit
distance. 800 lattice sites are used and are initialized with the density
distribution
$\rho(x)=\left\\{\begin{array}[]{ll}1,&\;\;\;\;1\leq x\leq 400,\\\
0.5,&\;\;\;\;401\leq x\leq 800.\end{array}\right.$
Initially all velocities are set to zero. We compare the ELBGK equipped with
the parabola based root finding algorithm using the entropic norm with the
standard LBGK method using both standard polynomial and entropic equilibria.
The polynomial equilibria are given in [7, 33]:
$\begin{split}&f_{-}^{*}=\frac{\rho}{6}\left(1-3u+3u^{2}\right),\;\;f_{0}^{*}=\frac{2\rho}{3}\left(1-\frac{3u^{2}}{2}\right),\\\
&f_{+}^{*}=\frac{\rho}{6}\left(1+3u+3u^{2}\right).\end{split}$
The entropic equilibria also used by the ELBGK are available explicitly as the
maximum of the entropy function (7),
$\begin{split}&f_{-}^{*}=\frac{\rho}{6}(-3u-1+2\sqrt{1\\!+\\!3u^{2}}),\;\;f_{0}^{*}=\frac{2\rho}{3}(2-\sqrt{1\\!+\\!3u^{2}}),\\\
&f_{+}^{*}=\frac{\rho}{6}(3u-1+2\sqrt{1\\!+\\!3u^{2}}).\end{split}$
Now following the prescription fromm Sec. 1 the governing equations for the
simulation are
$\begin{split}&f_{-}(x,t\\!+\\!1)\\!=\\!f_{-}(x\\!+\\!1,t)\\!+\\!\alpha\beta(f_{-}^{*}(x\\!+\\!1,t)\\!-\\!f_{-}(x\\!+\\!1,t)),\\\
&f_{0}(x,t\\!+\\!1)\\!=\\!f_{0}(x,t)\\!+\\!\alpha\beta(f_{0}^{*}(x,t)-f_{0}(x,t)),\\\
&f_{+}(x,t\\!+\\!1)\\!=\\!f_{+}(x\\!-\\!1,t)\\!+\\!\alpha\beta(f_{+}^{*}(x\\!-\\!1,t)\\!-\\!f_{+}(x\\!-\\!1,t)).\end{split}$
Fig. (2): Density profile of the simulation of the shock tube problem
following 400 time steps using (a) LBGK with polynomial equilibria
[$\nu=(1/3)\cdot 10^{-1}$]; (b) LBGK with entropic equilibria [$\nu=(1/3)\cdot
10^{-1}$]; (c) ELBGK [$\nu=(1/3)\cdot 10^{-1}$]; (d) LBGK with polynomial
equilibria [$\nu=10^{-9}$]; (e) LBGK with entropic equilibria [$\nu=10^{-9}$];
(f) ELBGK [$\nu=10^{-9}$].
From this experiment we observe no benefit in terms of regularization in using
the ELBGK rather than the standard LBGK method (Fig. (2)). In both the medium
and low viscosity regimes ELBGK does not supress the spurious oscillations
found in the standard LBGK method. The observation is in full agreement with
the Tadmor and Zhong [34] experiments for schemes with precise entropy
balance.
Entropy balance gives a nice additional possibility to monitor the accuracy
and the basic physics but does not give an omnipotent tool for regularization.
7\. 2D SHEAR DECAY
For the second test we use a simple test proposed to measure the observable
viscosity of a lattice Boltzmann simulation to validate the shear viscosity
production of the MRT models. We take the 2D isothermal nine-velocity model
with standard polynomial equilibria. Our computational domain will a square
which we discretize with $L+1\times L+1$ uniformly spaced points and periodic
boundary conditions. The initial condition is $\rho(x,y)=1,u_{x}(x,y)=0$ and
$u_{y}(x,y)=u_{0}sin(2\pi x/L)$, with $u_{0}=0.05$. The exact velocity
solution to this problem is an exponential decay of the initial condition:
$u_{x}(x,y,t)=0,u_{y}(x,y,t)=u_{0}\exp(-\lambda u_{0}t/{\rm Re}L)\sin(2\pi
x/L)$, where $\lambda$ is some constant and $Re=u_{0}L/\nu$ is the Reynolds
number of the flow. Here, $\nu$ is the theoretical shear viscosity of the
fluid due to the relaxation parameters of the collision operation.
Now, we simulate the flow over $L/u_{0}$ time steps and measure the constant
$\lambda$ from the numerical solution. We do this for LBGK, the ‘aggressive’
MRT with collision matrix $A_{2}$ and the MRT system with collision matrix
$A_{1}$ with additional parameters
$s_{e}=1.64,s_{\epsilon}=1.54,s_{q}=3(2-s_{\nu})/(3-s_{\nu})$ [22]. The shear
viscosity relaxation parameter $s_{\nu}$ is varied to give different
viscosities and therefore Reynolds numbers for $L=50$ and for $L=100$.
Fig. (3): Observed $\lambda$ in the shear decay experiment
From Figure (3) it can be seen that in all the systems, for increasing
theoretical Reynolds numbers (decreasing viscosity coefficient) following a
certain point numerical dissipation due to the lattice begins growing. The
most important observation from this system is that our MRT models do indeed
produce shear viscosity at almost the same rate as the BGK model.
In particular, the bulk viscosity in this system is zero, so all dissipation
is given by shear viscosity or higher order modes. Since the space derivatives
of the velocity modes are well bounded, as is the magnitude of the velocity
itself, the proper asymptotic decay of higher order modes is observed and the
varying higher order relaxation coefficients have only a very marginal effect.
We should reiterate that we while we use the ‘aggressive’ MRT across the
system, this is only appropriate as bulk viscosity is zero and in fact the
selection of the bulk viscosity coefficient makes no difference in this
example. This example is a special case in this regard.
8\. LID DRIVEN CAVITY
## 8.1. Stability
Our next 2D example is the benchmark 2D lid driven cavity. In this case this
is a square system of side length 129. Bounce back boundary conditions are
used and the top boundary imposes a constant velocity of $u_{\rm wall}=0.1$.
For a variety of Reynolds numbers we run experiments for up to 10000000 time
steps and check which methods have remained stable up until that time step.
The methods which we test are the standard BGK system, the BGK system equipped
with Ehrenfest steps, the BGK system equipped with the MRT limiter, the TRT
system, an MRT system with the TRT relaxation rate for the third order moment
and the other rates $s_{e}=1.64,s_{\epsilon}=1.54$ and finally an MRT system
which we call MRT1 with the rates $s_{q}=1.9s_{e}=1.64,s_{\epsilon}=1.54$
[26]. In each case of the system equipped with limiters the maximum number of
sites where the limiter is used is 9.
All methods are equipped with the standard 2nd order compressible quasi-
equilibrium, which is available as the product of the 1D equilibria 4.4..
When calculating the stream functions of the final states of these simulations
we use Simpson integration in first the x and then y directions.
Additionally we measure Enstrophy $\mathcal{E}$ in each system over time.
Enstrophy is calculated as the sum of vorticity squared across the system,
normalized by the number of lattice sites. This statistic is useful as
vorticity is theoretically only dissipated due to shear viscosity, at the same
time in the lid driven system vorticity is produced by the moving boundary.
For these systems $\mathcal{E}$ becomes constant as the vorticity field
becomes steady. The value of this constant indicates where the balance between
dissipation and production of vorticity is found. The lower the final value of
$\mathcal{E}$ the more dissipation produced in the system.
Fig. (4): Contour plots of stream functions of A: BGK, B: BGK + Ehrenfest
Steps, C: BGK + MRT Limiter, D: TRT, E: MRT, F: MRT1 following 10000000 time
steps at Re1000. Fig. (5): Enstrophy in the Re1000 systems during the final
$2\cdot 10^{5}$ time steps
All of these systems are stable for Re1000, the contour plots of the final
state are given in Figure (4) and there appears only small differences. We
calculate the average enstrophy in each system and plot it as a function of
time in Figure (5). We can see that in the different systems that enstrophy
and hence dissipation varies. Compared with the BGK system all the other
systems except MRT1 exhibit a lower level of enstrophy indicating a higher
rate of dissipation. For MRT1 the fixed relaxation rate of the third order
mode is actually less dissipative than the BGK relaxation rate for this
Reynolds number, hence the increased enstrophy. An artifact of using the
pointwise filtering techniques is that they introduce small scale local
oscillations in the modes that they regularize, therefore the system seems not
to be asymptotically stable. This might be remedied by increasing the
threshold of $\Delta S$ below which no regularization is performed.
Nevertheless in these experiments after sufficient time the enstrophy values
remain within a small enough boundary for the results to be useful.
Fig. (6): Contour plots of stream functions of A: BGK + Ehrenfest Steps, B:
BGK + MRT Limiter, C: TRT, D: MRT1,following 10000000 time steps at Re2500.
Fig. (7): Enstrophy in the Re2500 systems during the final $2\cdot 10^{5}$
time steps
The next Reynolds number we choose is Re2500. Only 4 of the original 6 systems
complete the full number of time steps for this Reynolds number, the contour
plots of the final stream functions are given in Figure (6). Of the systems
which did not complete the simulation it should be said that the MRT system
survived a few 10s of thousands of time steps while the BGK system diverged
almost immediately, indicating that it does provide stability benefits which
are not apparent at the coarse granularity of Reynolds numbers used in this
study. One feature to observe in the stream function plots is the absence of
an upper left vortex in the Ehrenfest limiter. This system selects the "most
non-equilibrium" sites to apply the filter. These typically occur near the
corners of the moving lid. It seems here that the local increase in shear
viscosity is enough to prevent this vortex forming. This problem does not seem
to affect the MRT limiter which preserves the correct production of shear
viscosity.
Again we check the enstrophy of the systems and give the results for the final
timesteps in Figure (7). Due to the failure of the BGK system to complete this
simulation there is no "standard" result to compare the improved methods with.
The surviving methods maintain their relative positions with respect to
enstrophy production.
Fig. (8): Contour plots of stream functions of A: BGK + Ehrenfest Steps, B:
MRT1,following 10000000 time steps at Re5000. Fig. (9): Enstrophy in the
Re5000 systems during the final $2\cdot 10^{5}$ time steps
For the theoretical Reynolds number of 5000 only two systems remain, their
streamfunction plots are given in Figure (8). At this Reynolds number the
upper left vortex has appeared in the Ehrenfest limited simulation, however a
new discrepancy has arisen. The lower right corner exhibits a very low level
of streaming.
In Figure (9) the enstrophy during the final parts of the simulation is given.
We note that for the first time the MRT1 system produces less enstrophy (is
more dissipative) than the BGK system with Ehrenfest limiter.
Fig. (10): Contour plots of stream functions of A: Re7500 and B: Re10000 BGK +
Ehrenfest systems,following 10000000 time steps at Re5000. Fig. (11):
Enstrophy in the Re7500 and Re10000 BGK + Ehrenfest systems during the final
$2\cdot 10^{5}$ time steps
For the final two Reynolds numbers we use, 7500 and 10000, only the BGK system
with the Ehrenfest limiter completes the simulation. The corresponding
streamfunction plots are given in Figure (10) and they exhibit multiple
vortices in the corners of the domain.
The enstrophy plots are given in Figure (11), as the theoretical Reynolds
number increases so does the level of enstrophy.
## 8.2. First Hopf Bifurcation
As Reynolds number increases the flow in the cavity is no longer steady and a
more complicated flow pattern emerges. On the way to a fully developed
turbulent flow, the lid-driven cavity flow is known to undergo a series of
period doubling Hopf bifurcations.
A survey of available literature reveals that the precise value of
$\mathrm{Re}$ at which the first Hopf bifurcation occurs is somewhat
contentious, with most current studies (all of which are for incompressible
flow) ranging from around $\mathrm{Re}=7400$–$8500$ [6, 27, 28]. Here, we do
not intend to give a precise value because it is a well observed grid effect
that the critical Reynolds number increases (shifts to the right) with
refinement (see, e.g., Fig. 3 in [28]). Rather, we will be content to localise
the first bifurcation and, in doing so, demonstrate that limiters are capable
of regularising without effecting fundamental flow features.
To localise the first bifurcation we take the following algorithmic approach.
Entropic equilibria are in use. The initial uniform fluid density profile is
$\rho=1.0$ and the velocity of the lid is $u_{0}=1/10$ (in lattice units). We
record the unsteady velocity data at a single control point with coordinates
$(L/16,13L/16)$ and run the simulation for $5000L/u_{0}$ time steps. Let us
denote the final 1% of this signal by $(u_{\mathrm{sig}},v_{\mathrm{sig}})$.
We then compute the _energy_ $E_{u}$ ($\ell_{2}$-norm normalised by non-
dimensional signal duration) of the deviation of $u_{\mathrm{sig}}$ from its
mean:
$E_{u}:=\biggl{\|}\sqrt{\frac{L}{u_{0}|u_{\mathrm{sig}}|}}(u_{\mathrm{sig}}-\overline{u_{\mathrm{sig}}})\biggr{\|}_{\ell_{2}},$
(34)
where $|u_{\mathrm{sig}}|$ and $\overline{u_{\mathrm{sig}}}$ denote the length
and mean of $u_{\mathrm{sig}}$, respectively. We choose this robust statistic
instead of attempting to measure signal amplitude because of numerical noise
in the LBM simulation. The source of noise in LBM is attributed to the
existence of an inherently unavoidable neutral stability direction in the
numerical scheme (see, e.g., [3]).
We opt not to employ the “bounce-back” boundary condition used in the previous
steady state study. Instead we will use the diffusive Maxwell boundary
condition (see, e.g., [8]), which was first applied to LBM in [2]. The essence
of the condition is that populations reaching a boundary are reflected,
proportional to equilibrium, such that mass-balance (in the bulk) and detail-
balance are achieved. The boundary condition coincides with “bounce-back” in
each corner of the cavity.
To illustrate, immediately following the advection of populations consider the
situation of a wall, aligned with the lattice, moving with velocity
$u_{\mathrm{wall}}$ and with outward pointing normal to the wall in the
negative $y$-direction (this is the situation on the lid of the cavity with
$u_{\mathrm{wall}}=u_{0}$). The implementation of the diffusive Maxwell
boundary condition at a boundary site $(x,y)$ on this wall consists of the
update
$f_{i}(x,y,t+1)=\gamma f^{*}_{i}(u_{\mathrm{wall}}),\qquad i=4,7,8,$
with
$\gamma=\frac{f_{2}(x,y,t)+f_{5}(x,y,t)+f_{6}(x,y,t)}{f^{*}_{4}(u_{\mathrm{wall}})+f^{*}_{7}(u_{\mathrm{wall}})+f^{*}_{8}(u_{\mathrm{wall}})}.$
Observe that, because density is a linear factor of the equilibria, the
density of the wall is inconsequential in the boundary condition and can
therefore be taken as unity for convenience. As is usual, only those
populations pointing in to the fluid at a boundary site are updated. Boundary
sites do not undergo the collisional step that the bulk of the sites are
subjected to.
We prefer the diffusive boundary condition over the often preferred “bounce-
back” boundary condition with constant lid profile. This is because we have
experienced difficulty in separating the aforementioned numerical noise from
the genuine signal at a single control point using “bounce-back”. We remark
that the diffusive boundary condition does not prevent unregularised LBGK from
failing at some critical Reynolds number.
Now, we conduct an experiment and record (34) over a range of Reynolds
numbers. In each case the median filter limiter is employed with parameter
$\delta=10^{-3}$. Since the transition between steady and periodic flow in the
lid-driven cavity is known to belong to the class of standard Hopf
bifurcations we are assured that $E_{u}^{2}\propto\mathrm{Re}$ [12]. Fitting a
line of best fit to the resulting data localises the first bifurcation in the
lid-driven cavity flow to $\mathrm{Re}=7135$ (Fig. (12)). This value is within
the tolerance of $\mathrm{Re}=7402\pm 4\%$ given in [28] for a $100\times 100$
grid. We also provide a (time averaged) phase space trajectory and Fourier
spectrum for $\mathrm{Re}=7375$ at the monitoring point (Fig. (13) and Fig.
(14)) which clearly indicate that the first bifurcation has been observed.
Fig. (12): Plot of energy squared, $E_{u}^{2}$ (34), as a function of Reynolds
number, $\mathrm{Re}$, using LBGK regularised with the median filter limiter
with $\delta=10^{-3}$ on a $100\times 100$ grid. Straight lines are lines of
best fit. The intersection of the sloping line with the $x$-axis occurs close
to $\mathrm{Re}=7135$. Fig. (13): A phase trajectory for velocity components
for the signal $(u_{\mathrm{sig}},v_{\mathrm{sig}})$ at the monitoring point
$(L/16,13L/16)$ using LBGK regularised with the median filter limiter with
$\delta=10^{-3}$ on a $100\times 100$ grid ($\mathrm{Re}=7375$). Dots
represent simulation results at various time moments and the solid line is a
$100$ step time average of the signal. Fig. (14): Relative amplitude spectrum
for the signal $u_{\mathrm{sig}}$ at the monitoring point $(L/16,13L/16)$
using LBGK regularised with the median filter limiter with $\delta=10^{-3}$ on
a $100\times 100$ grid ($\mathrm{Re}=7375$). We measure a dominant frequency
of $\omega=0.525$.
9\. CONCLUSION
In the 1D shock tube we do not find any evidence that maintaining the proper
balance of entropy (implementing ELBM) regularizes spurious oscillations in
the LBM. We note that entropy production controlled by $\alpha$ and viscosity
controlled by $\beta$ are composite in the collision integral (6). A weak
lower approximation to $\alpha$ would lead effectively to addition of
dissipation at the mostly far from equilibrium sites and therefore would
locally increase viscosity. Therefore the choice of the method to implement
the entropic involution is crucial in this scheme. Any method which is not
sufficiently accurate could give a misleading result.
In the 2D lid driven cavity test we observe that implementing TRT[13] or
MRT[22] with certain relaxation rates can improve stability. The increase in
stability from using TRT can be attributed to the correction of the numerical
slip on the boundary, as well as increasing dissipation. What is the best set
of parameters to choose for MRT is not a closed question. The parameters used
in this work originally proposed by Lallemand and Luo [22] are based on a
linear stability analysis. Certain choices of relaxation parameters may
improve stability while qualitatively changing the flow, so parameter choices
should be justified theoretically, or alternatively the results of simulations
should be somehow validated. Nevertheless the parameters used in this work
exhibit an improvement in stability over the standard BGK system.
Modifying the relaxation rates of the different modes changes the production
of dissipation of different components at different orders of the dynamics.
The higher order dynamics of latttice Boltzmann methods include higher order
space derivatives of the distribution functions. MRT could exhibit the very
nice property that where these derivatives are near to zero that MRT has
little effect, while where these derivatives are large (near shocks and
oscillations) that additional dissipation could be added, regularizing the
system.
Using entropic limiters explicitly adds dissipation locally[4]. The Ehrenfest
steps succeed to stabilize the system at Reynolds numbers where other tested
methods fail, at the cost of the smoothness of the flow. We also implemented
an entropic limiter using MRT technology. This also succeeded in stabilizing
the system to a degree, however the amount of dissipation added is less than
an Ehrenfest step and hence it is less effective. The particular advantage of
a limiter of this type over the Ehrenfest step is that it can preserve the
correct production of dissipation on physical modes across the system. Other
MRT type limiters can easily be invented by simply varying the relaxation
parameters.
As previously mentioned there have been more filtering operations
proposed[29]. These have a similar idea of local (but not pointwise) filtering
of lattice Boltzmann simulations. A greater variety of variables to filter
have been examined, for example the macroscopic field can be filtered rather
than the mesoscopic population functions.
We can use the Enstrophy statistic to measure effective dissipation in the
system. The results from the lid-driven cavity experiment indicate that
increased total dissipation does not necessarily increase stability. The
increase in dissipation needs to be targeted onto specific parts of the domain
or specific modes of the dynamics to be effective.
Using the global median filter in the lid-driven cavity we find that the
expected Reynolds number of the first Hopf bifurcation seems to be preserved,
despite the additional dissipation. This is an extremely positive result as it
indicates that if the addition of dissipation needed to stabilize the system
is added in an appropriate manner then qualitative features of the flow can be
preserved.
Finally we should note that the stability of lattice Boltzmann systems depends
on more than one parameter. In all these numerical tests the Reynolds number
was modified by altering the rate of production of shear viscosity. In
particular, for the lid driven cavity the Reynolds number could be varied by
altering the lid speed, which was fixed at 0.1 in all of these simulations.
Since the different modes of the dynamics include varying powers of velocity,
this would affect the stability of the system in a different manner to simply
changing the shear viscosity coefficient. In such systems the relative
improvements offered by these methods over the BGK system could be different.
The various LBMs all work well in regimes where the macroscopic fields are
smooth. Each method has its limitation as they attempt to simulate flows with,
for example, shocks and turbulence, and it is clear that something needs to be
done in order to simulate beyond these limitations. We have examined various
add-ons to well-known implementations of the LBM, and explored their
efficiency. We demonstrate that add-ons based on the gentle modification of
dissipation can significantly expand the stable boundary of operation of the
LBM.
REFERENCES
##
* [1] Ansumali S, Karlin IV. Stabilization of the lattice Boltzmann method by the H theorem: A numerical test. Phys Rev E 2000; 62(6): 7999–8003.
* [2] Ansumali S, Karlin IV. Kinetic boundary conditions in the lattice Boltzmann method. Phys Rev E 2002; 66: 026311.
* [3] Brownlee RA, Gorban AN, Levesley J. Stabilisation of the lattice-Boltzmann method using the Ehrenfests’ coarse-graining. Phys Rev E 2006; 74: 037703.
* [4] Brownlee RA, Gorban AN, Levesley J. Stability and stabilisation of the lattice Boltzmann method. Phys Rev E 2007; 75: 036711.
* [5] Brownlee RA, Gorban AN, Levesley J. Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A 2008; 387: 385–406.
* [6] Bruneau C-H, Saad M. The 2D lid-driven cavity problem revisited. Comput Fluids 2006; 35: 326–348.
* [7] Benzi R., Succi S, Vergassola M. The lattice Boltzmann-equation - theory and applications. Physics Reports 1992; 222(3): 145 197.
* [8] Cercignani C. Theory and application of the Boltzmann equation. Scottish Academic Press, Edinburgh 1975.
* [9] Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 1998; 30: 329 364.
* [10] Dellar PJ. Bulck and shear viscosities in lattice Boltzmann equation. Phys Rev E 2001; 64: 031203.
* [11] Dellar PJ. Incompressible limits of lattice Boltzmann equations using multiple relaxation times. J Comput Phys 2003; 190: 351–370.
* [12] Ghaddar NK, Korczak KZ, Mikic BB, Patera AT. Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations. J Fluid Mech 1986; 163: 99–127.
* [13] Ginzburg I. Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations. Adv Water Res 2005; 28: 1196 1216.
* [14] Godunov SK. A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Math Sbornik 1959; 47: 271–306.
* [15] Gorban AN. Basic type of coarse-graining. In: Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, ed. by AN Gorban, N Kazantzis, IG Kevrekidis, HC Öttinger, C. Theodoropoulos Springer, Berlin-Heidelberg, New York 2006: 117–176.
* [16] Gorban AN, Karlin IV, Öttinger HC, Tatarinova LL. Ehrenfests argument extended to a formalism of nonequilibrium thermodynamics, Physical Review E 2001; 63: 066124.
* [17] Higuera F, Succi S, Benzi R. Lattice gas dynamics with enhanced collisions. Europhys Lett 1989; 9: 345 349.
* [18] Karlin IV, Ferrante A, Öttinger HC. Perfect entropy functions of the Lattice Boltzmann method. Europhys Lett 1999; 47: 182–188.
* [19] Karlin IV, Gorban AN, Succi S, Boffi V. Maximum Entropy Principle for Lattice Kinetic Equations. Phys Rev Lett 1998; 81:6–9 .
* [20] Kullback S. Information theory and statistics. Wiley, New York 1959.
* [21] Kuzmin D, Lohner R, Turek S. Flux Corrected Transport, Springer 2005.
* [22] Lallemand P, Luo LS. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys Rev E 2000; 61(6): 6546-6562.
* [23] Lax PD. On dispersive difference schemes. Physica D 1986; 18: 250–254.
* [24] Levermore CD, Liu J-G. Oscillations arising in numerical experiments, Physica D 1996; 99: 191–216.
* [25] Li Y, Shock R, Zhang R, Chen H. Numerical study of flow past an impulsively started cylinder by the lattice-Boltzmann method. J Fluid Mech 2004; 519: 273–300.
* [26] Luo LS, Liao W, Chen X, Peng Y. and Zhang W. Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations. Phys Rev E 2011; 83: 056710.
* [27] Pan TW, Glowinksi R. A projection/wave-like equation method for the numerical simulation of incompressible viscous fluid flow modeled by the Navier–Stokes equations. Comp Fluid Dyn J 2000; 9: 28–42.
* [28] Peng Y-F, Shiau Y-H, Hwang RR. Transition in a 2-D lid-driven cavity flow. Comput Fluids 2003; 32: 337–352.
* [29] Ricot D, Marie S, Sagaut P, Bailly C. Lattice Boltzmann method with selective viscosity filter. J Comput Physics 2009; 228: 4478–4490.
* [30] Pratt WK. Digital Image Processing. Wiley, New York 1978.
* [31] Roy CJ. Grid convergence error analysis for mixed-order numerical schemes. AIAA J 2003; 41: 595–604
* [32] Servan-Camas B, Tsai FT-C. Non-negativity and stability analyses of lattice Boltzmann method for advection diffusion equation. J Comput Physics 2009; 228(1): 236-256
* [33] Succi S. The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press, New York 2001.
* [34] Tadmor E, Zhong W. Entropy stable approximations of Navier–Stokes equations with no artificial numerical viscosity. J Hyperbolic DEs 2006; 3: 529–559.
* [35] Tosi F, Ubertini S, Succi S, Chen H, Karlin IV. A comparison of single-time relaxation lattice Boltzmann schemes with enhanced stability. Math Comput Simulation 2006; 72: 227–231.
* [36] Wesseling P. Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics 29, Springer, Berlin 2001.
|
arxiv-papers
| 2011-10-03T05:38:41 |
2024-09-04T02:49:22.698862
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. A. Brownlee, J. Levesley, D. Packwood and A. N. Gorban",
"submitter": "David Packwood",
"url": "https://arxiv.org/abs/1110.0270"
}
|
1110.0327
|
# Internal Structure of Protocluster Galaxies: Accelerated Structural
Evolution in Overdense Environments?11affiliation: Based on observations with
the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-26555
Andrew W. Zirm22affiliation: Dark Cosmology Centre, Niels Bohr Institute,
University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark;
azirm@dark-cosmology.dk; sune@dark-cosmology.dk Sune Toft22affiliation: Dark
Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane
Maries Vej 30, DK-2100 Copenhagen, Denmark; azirm@dark-cosmology.dk;
sune@dark-cosmology.dk Masayuki Tanaka33affiliation: Institute for the Physics
and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8583, Japan; masayuki.tanaka@ipmu.jp
###### Abstract
We present a high spatial-resolution HST/NICMOS imaging survey in the field of
a known protocluster surrounding the powerful radio galaxy MRC1138-262 at
$z=2.16$. Previously, we have shown that this field exhibits a substantial
surface overdensity of red $J-H$ galaxies. Here we focus on the stellar masses
and galaxy effective radii in an effort to compare and contrast the properties
of likely protocluster galaxies with their field counterparts and to look for
correlations between galaxy structure and (projected) distance relative to the
radio galaxy.
We find a hint that quiescent, cluster galaxies are on average less dense than
quiescent field galaxies of similar stellar mass and redshift. In fact, we
find only two (of nine) quiescent protocluster galaxies are of simliar density
to the majority of the massive, quiescent compact galaxies (SEEDs) found in
several field surveys. Furthermore, there is some indication that the
structural Sersic $n$ parameter is higher ($n\sim 3-4$) on average for cluster
galaxies compared to the field SEEDs ($n\sim 1-2$) This result may imply that
the accelerated galaxy evolution expected (and observed) in overdense regions
also extends to structural evolution presuming that massive galaxies began as
dense (low $n$) SEEDs and have already evolved to be more in line with local
galaxies of the same stellar mass.
###### Subject headings:
galaxies: clusters: individual (MRC1138-262) – galaxies: evolution – galaxies:
high-redshift – galaxies: structure
## 1\. Introduction
The internal spatial and velocity distribution of stars is an indicator of the
manner in which galaxies have formed, assembled and evolved. In the local
universe, tidal streams, shells and kinematically distinct cores are examples
of archeological clues to past merging and formation events (e.g., Peng et
al., 2002b; Emsellem et al., 2007; van Dokkum, 2005; Blanton & Moustakas,
2009). Even coarse measures, such as the average stellar surface mass density
within the effective radius ($\Sigma_{50}$), correlate with the star formation
rate or the mean stellar age. Giant elliptical galaxies have high stellar mass
per unit area (or volume) and show negligible current star formation while
more diffuse stellar disks and dwarf irregulars are forming stars at sometimes
prodigious rates per unit stellar mass (specific star-formation rate; sSFR).
At higher redshift, analogous relations are already in place (e.g., Franx et
al., 2008). While observationally it remains difficult to separate high-
redshift galaxies into classical Hubble-types we can now photometrically
determine redshifts, stellar masses and galaxy sizes for large numbers of
galaxies at $z\sim 2$. Such studies (e.g., Zirm et al., 2007; Toft et al.,
2007; van Dokkum et al., 2008; Toft et al., 2009; Williams et al., 2010;
Mosleh et al., 2011) have found that quiescent galaxies are in general more
dense than their star-forming counterparts.
The origin of this bi-modal distribution of galaxy properties is unclear. It
is possible that the quiescent $z\sim 2$ galaxies are more compact because
they formed when the universe was smaller and more dense and mergers were more
gas-rich. It also seems plausible that the process which quenches star
formation may be linked to a morphological change. Or perhaps the dominant
formation processes differ for galaxies with different present-day stellar
masses (Khochfar & Silk, 2006, 2009, 2011; Dekel & Birnboim, 2008; Dekel et
al., 2009)
In general terms, dissipation should result in more compact stellar cores than
dissipationless assembly (e.g., Ciotti, Lanzoni, & Volonteri, 2007; Naab,
Johansson, & Ostriker, 2009; Oser et al., 2010). Dense stellar systems, once
formed, tend to persist through successive (minor) mergers. The relatively
recent discovery of quiescent, massive and compact galaxies at $z\sim 2$
(hereafter Semi-Evolved Elephantine Dense galaxies or “SEEDs”) implies that at
least some galaxies have their origin in high-redshift ($z\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 318\relax$}}4$), gas-rich mergers (Daddi et al., 2005;
Zirm et al., 2007; Toft et al., 2007; van Dokkum et al., 2008; Cimatti et al.,
2008). These mergers resulted in many stars being formed in a relatively small
volume. By $z\sim 2$ these systems have low star-formation rates and
relatively high stellar masses in addition to their small sizes
($r_{e}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}1$kpc). The SEEDs
therefore have extreme internal stellar mass densities They do not fall on the
size-mass relation defined by local galaxies. Their evolution from $z\sim 2$
to $z=0$ is therefore a puzzle.
The most massive galaxies in the present-day universe are located at the
centers of rich clusters. These galaxy overdensities were statistically the
first to separate from the Hubble flow and collapse and are therefore believed
to follow an accelerated timeline for the process of galaxy formation. There
is some observational evidence that galaxies in the progenitors of clusters,
protoclusters, do have significantly older stars and higher masses than
galaxies in the field at similar redshifts (Steidel et al., 2005; Tanaka et
al., 2010). Might cluster galaxies, having formed earlier, be even more dense
than field SEEDs? Or, alternatively, the ’fast-forward’ evolution of cluster
galaxies may lead to lower density galaxies in protoclusters compared to their
field counterparts. It is interesting, then, to look for dense SEED galaxies
in protoclusters at redshift $z\sim 2$.
We have undertaken a NICMOS imaging program to study the red galaxy population
in a protocluster at $z=2.16$. Broad and narrow-band imaging, both in the
optical and near-infrared, of the field surrounding the powerful radio galaxy
MRC 1138-262 ($z=2.16$) have identified more than 100 candidate companion
galaxies. There are surface-overdensities of both line-emitting candidates
(Lyman-$\alpha$ and H$\alpha$), X-ray point sources, sub-mm selected galaxies
and red optical–near-infrared galaxies (Pentericci et al., 2002; Kurk, 2003;
Kurk et al., 2004b; Stevens et al., 2003; Croft et al., 2005). Fifteen of the
Ly$\alpha$ and 9 of the H$\alpha$ emitters have been spectroscopically
confirmed to lie at the same redshift as the radio galaxy (Kurk et al.,
2004a). By obtaining deep images through the NICMOS $J_{110}$ and $H_{160}$
filters, which effectively span the 4000Å-break at $z=2.16$, we have
identified a large surface overdensity of red galaxies consistent with a
forming red sequence (Zirm et al., 2008). In this paper we present a more
detailed analysis of the masses and morphologies of galaxies in this field.
The article is organized as follows: in Section §2 we describe the data and
their reductions, in Section §3 we present the photometric redshifts, stellar
population models and morphological fits, in Section §4 we present the
internal stellar mass densities and other derived properties and finally in §5
we discuss these results in the context of galaxy evolution models. We use a
$(\Omega_{\Lambda},\Omega_{M})=(0.7,0.3)$, $H_{0}=73$ ${\rm km}$ ${\rm
s^{-1}}$ ${\rm Mpc^{-1}}$ cosmology throughout. At $z=2.16$ one arcsecond is
equivalent to 8.4 kpc. All magnitudes are referenced to the AB system (Oke,
1974) unless otherwise noted.
## 2\. Observations and Data Reductions
### 2.1. NICMOS Imaging
The NICMOS instrument on-board HST is capable of deep near-infrared imaging
over a relatively small field-of-view ($51\arcsec\times 51\arcsec$). In the
case of MRC 1138-262, we know that galaxies are overdense on the scale of a
few arcminutes (Kurk et al., 2004b; Croft et al., 2005) and are thus well-
suited for observations with NICMOS camera 3 on HST. We used 30 HST orbits to
image seven overlapping pointings in both filters and one additional pointing
in $H_{160}$ alone. These observations reach an AB limiting magnitude
($m_{10\sigma}$; 10$\sigma$, $0\farcs 5$ diameter circular aperture) of
$m_{10\sigma}=24.9$ mag in $J_{110}$ and $m_{10\sigma}=25.1$ mag in $H_{160}$.
The same field was imaged in the $g_{475}$ ($m_{10\sigma}=27.5$ mag) and
$I_{814}$ ($m_{10\sigma}=26.8$ mag) filters using the Wide-Field Channel of
the Advanced Camera for Surveys on HST as part of a Guaranteed Time program (#
10327; Miley et al. 2006).
The NICMOS images were reduced using the on-the-fly reductions from the HST
archive, the IRAF task pedsky and the dither/drizzle package to combine the
images in a mosaic. The dither offsets were calculated using image cross-
correlation and were refined iteratively. Alignment of the pointings relative
to each other was accomplished using a rebinned version of the ACS $I_{814}$
image as a reference. The final mosaic has a pixel scale of $0\farcs 1$.
Galaxies were selected using the $H_{160}$-band image for detection within
SExtractor (Bertin & Arnouts, 1996). We used a $2.2\sigma$ detection threshold
with a minimum connected area of 10 pixels. We also corrected the NICMOS data
for the count-rate dependent non-linearity (de Jong, 2006). Total galaxy
magnitudes were estimated by using the MAG_AUTO values from SExtractor. We
show the outline of the NICMOS mosaic in Figure 1 along with the positions of
the radio galaxy (yellow star) and star-forming (blue circles) and quiescent
(red circles) protocluster galaxies.
The $J_{110}-H_{160}$ colors were determined by running SExtractor (Bertin &
Arnouts, 1996) in two-image mode using the $H_{160}$ image for object
detection and isophotal apertures. The $J_{110}$ image was PSF-matched to the
$H_{160}$ band. We also incorporated the two ACS bands (Miley et al., 2006),
the Spitzer IRAC bands, $U_{n}$ (Zirm et al., 2008) and $V$ bands from
Keck/LRIS, $z$ and $R$ from VLT/FORS2 (Kurk et al., 2004b, a), $H$ band from
NTT/SOFI and $J$ and $Ks$ from Subaru/MOIRCS (Kodama et al., 2007). The
assembly of the merged multi-band catalog is detailed in Tanaka et al. 2010.
Figure 1.— Outline of the NICMOS mosaic. The red and blue points mark the
locations of the quiescent and star-forming cluster galaxies respectively. The
yellow star is the radio galaxy MRC1138-262.
### 2.2. FIREWORKS Survey Data and Literature Sample
The FIREWORKS data are described in detail in Wuyts et al. (2008). In brief,
the survey is $K_{S}$-band selected to $5\sigma$ depth of 24.3 (AB) over an
area of 113$\square\arcmin$. In addition to the deep $K_{S}$ band data there
is high-quality imaging in each of the $U$, $B$, $V$, $I$, $i$, $z$, $J$, $H$,
the four Spitzer/IRAC bands and the $24\mu$m Spitzer/MIPS band. The combined
multi-band catalog has been used to measure precise photometric redshifts,
galaxy sizes (Toft et al., 2009) and to model the spectral energy
distributions to derive stellar masses, ages and star-formation rates (Damen
et al., 2009). We have made three cuts to the FIREWORKS sample to ensure that
we are making appropriate field-to-cluster comparisons. First, since we are
comparing galaxy sizes (densities) we require that the galaxies are bright
enough to have a reliable size measurement in these ground-based data. Based
on the comparison of size measurements from VLT/ISAAC and HST/NICMOS for the
same galaxies, Toft et al. (2009) found that at $K\sim 21.5-22.0$ the scatter
between these two size determinations increases significantly. We therefore
select only $K<21.5$ galaxies from FIREWORKS. Next, we have made a photometric
redshift cut $1.9<z<2.6$ to select galaxies within the field at roughly the
same epoch as the protocluster galaxies. Finally, we select the quiescent
field population on the basis of the specific star-formation rate (log(sSFR)
$<-11$ yr-1). We note that after these cuts, the stellar mass distribution
remains similar to our protocluster galaxy sample.
For further comparison to our protocluster field data, we have compiled a
sample of $z\sim 2$ quiescent galaxies published in the literature. We used
four references for this sample: Cassata et al. (2010), van Dokkum et al.
(2008), Mancini et al. (2010) and Saracco, Longhetti, & Andreon (2009).
Saracco, Longhetti, & Andreon (2009) used HST/NICMOS Camera 3 as we have,
Mancini et al. (2010) used HST/ACS imaging, van Dokkum et al. (2008) studied
HST/NICMOS Camera 2 imaging while Cassata et al. (2010) use imaging from the
WFC3/IR channel on HST.
We have attempted to translate these published stellar mass estimates to the
same IMF (Salpeter) and to the same stellar population synthesis model set
(Maraston 2005). We have used the analyses of Salimbeni et al. (2009; see
their Fig. 1) to derive mean corrections between model sets. The adopted IMF
also affects the derived star-formation rates. The offsets in this quantity
are similar in magnitude to the systematic shift in derived stellar mass
(e.g., Erb et al., 2006; Nordon et al., 2010), so the specific star formation
rate (i.e., the ratio of star-formation rate to stellar mass) should be
effectively unchanged.
Figure 2.— Upper panels, left to right: Galaxy image, GALFIT best-fit model
and image residuals after subtraction of the model. Lower panel: Photometry
and best-fit SED model from FAST (red circles and solid black line Kriek et
al., 2009). The shaded regions represent the photometric redshift probability
distribution (upper scale) centered at rest-frame 4000Å.
## 3\. Analysis
Here we combine the multiband photometric catalog and the NICMOS high spatial-
resolution imaging to derive physical parameters for individual galaxies. We
pare down the total NICMOS galaxy sample to those which have a high quality-
of-fit for the photometric redshift (using EAZY, Brammer, van Dokkum, & Coppi
(2008)), the spectral energy distribution (using FAST, Kriek et al. (2009))
and 2D surface-brightness profile fit (using GALFIT, Peng et al. (2002a)).
This reduces the galaxy sample from the $H_{160}$-band detected total of 711.
We further restrict our attention to those galaxies which most likely lie
within the known protocluster (see Section §3.4).
Figure 3.— Fig. 1 cont.
### 3.1. Photometric Redshifts
The thirteen filter photometric catalog (Tanaka et al., 2010) was used to
determine galaxy photometric redshifts. We used the public code, EAZY, to fit
a set of model templates to each galaxy’s photometric data (Brammer, van
Dokkum, & Coppi, 2008). We required that each galaxy have at least 5 colors
measured for the photometric redshift fit. The set of SED templates we used
included both galaxy spectral energy distributions and a narrow emission line
spectrum. EAZY uses all linear combinations of the input templates to find the
best photometric redshift fit. For each fit, EAZY produces the full redshift
probability distribution (see Figs. 2 \- 4). We are most interested in the
galaxies detected in the relatively small ($\sim 5\square\arcmin$), but deep,
NICMOS $H_{160}$-band area. Therefore, we have only included sources detected
in the $H_{\rm 160}$ NICMOS images. For the target redshift of $z=2.2$, the
primary strong spectral feature covered by the photometric data is the 4000Å
break. We note that even with 13 bands of imaging, photo-$z$s are not
sufficiently precise to determine whether a galaxy is inside the cluster or
not. There are 12 spectroscopically confirmed (emission line) protocluster
members within the NICMOS mosaic. Of these, four $H\alpha$ and one Ly$\alpha$
emitters have well-determined photometric redshifts (the remaining members are
generally too faint to have detections in enough bands). Four of the five
photo-$z$s are around $z\sim 2.1$, ranging from $1.8$ to $2.1$. There is one
clear outlier, the $H\alpha$ emitter with $z_{\rm phot}=0.33$.
### 3.2. Stellar Population Modeling
Figure 4.— Fig. 1 cont.
Using the calculated best-fit photometric redshifts, we used FAST (Kriek et
al., 2009) to fit stellar population model templates to the rest-frame
photometry. These templates consist of a grid of models drawn from the
Maraston (2005) set. We chose to use the Salpeter IMF, exponentially declining
star-formation histories with $\tau$ varying between $10^{7}$ and $10^{10}$
years and $A_{V}$ between $0$ and $3$ magnitudes. FAST calculates the best-
fitting model template among the grid and thereby derives a luminosity-
weighted mean stellar age, stellar mass, star-formation rate and extinction
for each galaxy. FAST also outputs the 1$\sigma$ error estimates for each of
these fit parameters. We show the derived masses and their errors for the
protocluster galaxies in Table 1. We note that the star-formation rates from
SED fitting are equivalent to a dust-corrected rest-frame UV SFR and that none
of our quiescent protocluster galaxies have significant detections in the MIPS
24 $\mu{\rm m}$ image (20693, PI: Stanford). Since we are only concerned with
differentiating the quiescent and star-forming galaxies, our results are
sensitive only to catastrophic errors in these SFR determinations.
### 3.3. NICMOS Galaxy Sizes and Morphologies
NICMOS camera 3 provides good angular resolution over its field-of-view (PSF
FWHM $\approx 0\farcs 27$). To exploit this resolution we have used the GALFIT
code (Peng et al., 2002a) to fit analytic Sérsic surface-brightness profiles
(Sérsic, 1968) to all the $H_{160}\leq 26.5$ sources in our $H_{160}$-band
mosaic. We have used our own error map as input to GALFIT for properly
weighting the image pixels and have masked all neighboring objects. A model
point-spread function was created for each of these galaxies individually by
generating a TinyTim simulated PSF (Krist, 1993) at the galaxies’ positions in
each exposure and then drizzling these PSFs together in exactly the same
fashion as for the data themselves (see Zirm et al. 2007). We then executed
several different runs of GALFIT. We ran fits holding the Sérsic index
constant at $n=1$ and $4$, using a single model PSF for all galaxies, using a
stellar PSF instead of the model(s) and holding the sky value fixed at zero.
For all fits we restricted the Sérsic index, $n$, to be between 0.5 and 5. The
range of output fit values for all these different runs gives us an idea of
the variance of the derived parameters due to model assumptions. We choose the
best fit from these runs by applying the F-test to the resultant
$\chi^{2}_{\nu}$ values.
The primary source of systematic offsets in galaxy profile and size fitting is
the estimation of the local sky value. If the sky is underestimated the galaxy
size can be overestimated, particularly for small faint galaxies. Therefore,
we compare our fits where the sky level is a free parameter with those where
we explicitly fix the sky to zero. Many of the “zero-sky” fits fail to
converge, for those that do converge and have comparable chi-squared values to
the corresponding free fits, we can compare the output $r_{e}$ determinations.
It does not appear to be the case that equally good fits are obtained with and
without fitting the sky. In those five cases where an F-test shows the zero-
sky fit to be better, the sizes agree within the errors. Furthermore, none of
these where the zero-sky result is comparably good are for any or our
protocluster galaxies. We note that the GALFIT sky values while non-zero are
consistently several orders-of-magnitude smaller than the values corresponding
to galaxy pixels. This sensitivity of the fit parameters to even slight
variations in the sky highlights the importance of fitting the local sky along
with the galaxy parameters (even in sky-subtracted data).
We also note that additional scatter to the derived $r_{e}$ values introduced
by using a stellar rather than model PSF is about $10-20\%$ and therefore
comparable to the scatter on the single-fit measurements themselves.
### 3.4. Sample Selections
We use the surface-brightness profile and stellar population fits along with
the photometric redshifts to define three sub-samples of the NICMOS-detected
galaxies. We detail these in order of increasing restriction. The initial
sample is defined by a single $H_{160}$-band limit of $26.5$ (AB) and consists
of 711 galaxies.
#### 3.4.1 GALFIT Sample
We ran GALFIT on the full $H_{160}$-limited galaxy sample. Using the
distribution of GALFIT $\chi^{2}_{\nu}$ values for the fits we identify
galaxies with good quality-of-fit ($\chi^{2}_{\nu}<2$; 577 of the input 711).
We then calculate the circularized $r_{e}$ ($=\sqrt{ab}$) for these well-fit
galaxies. We use this GALFIT sample in our analysis of the dependence of the
Sersic index, $n$, on radial position within the cluster (see left panel of
Fig. 7). We note that we have re-normalized the input sigma (error) maps such
that the best fits have $\chi^{2}_{\nu}\sim 1$.
#### 3.4.2 Stellar Populations Sample
To select galaxies with good constraints on both the stellar mass and star-
formation rate we have selected another sub-sample for the initial 711
galaxies. For targets with both good photo-z fits and narrow redshift
probability distributions (ODDS $>0.90$, meaning 90% of the probability
distribution is contained within the $\Delta z=0.2$ around the peak value) the
sample comprises 190 galaxies out of the 711. The photometry for these targets
were fit using FAST. From this set we have identified 112 with reliably
derived parameters ($\chi^{2}_{\nu}<3.0$) based on the SED fits (for example,
see Figs. 2-4).
#### 3.4.3 Probable Protocluster Galaxies
Finally, we have identified a sample of candidate protocluster members using
photometric redshifts. In lieu of spectroscopic redshifts, which are difficult
to obtain for $z\sim 2$ red, quiescent galaxies, this selection should reject
most of the interlopers. We consider only galaxies with a robust photometric
redshift (as above) that have probability $P(z)>20$% at the protocluster
redshift ($z=2.156$). We further require that these galaxies also are members
of the “Stellar Populations” sample. These selections result in a sample of 11
galaxies, nine of which are quiescent (log10 sSFR $<-11$ yr-1). We show the
galaxy cutouts, the best-fit Sersic model, the model subtraction residuals,
the broad-band SED fit and photometric redshift probability distribution for
these galaxies in Figures 2-4.
### 3.5. Stellar Mass Density
Figure 5.— Surface stellar mass density ($\Sigma$) vs. total stellar mass for
individual galaxies. The blue and black squares are from the FIREWORKS survey
(Toft et al., 2009) and other literature (van Dokkum et al., 2008; Cassata et
al., 2010; Mancini et al., 2010; Saracco, Longhetti, & Andreon, 2009)
respectively. The large blue squares are the $K>21.5$ FIREWORKS galaxies
included in the KS test. The yellow circles with error bars are the 8 likely
quiescent protocluster members. The two yellow squares are the star-forming
protocluster galaxies (the third falls at $5\times 10^{9}M_{\odot}$). The
shaded regions are the local relations for early-type (light red) and late-
type galaxies (light blue). Note that most of the protocluster members have
lower densities than their field counterparts.
The surface (volume) mass density in individual galaxies is a fundamental
property which seems to correlate directly with the absence of star formation
(e.g., Kauffmann et al., 2003; Franx et al., 2008). To measure this quantity
requires accurate total mass estimates based on either stellar velocity
dispersion or, much more commonly, the stellar mass from SED fits to the
broad-band photometry. Along with the resolved surface-brightness profile to
determine the galaxy size we can calculate the mass density. We must make the
assumption that light traces mass and that there are no strong gradients in
the stellar mass-to-light ratio, i.e., we measure $M/L$ for the integrated
galaxy light and assume that value applies to the resolved profile in a single
broad-band image.
For our galaxies, we calculate the average surface mass density ($\Sigma_{\rm
50}$ in $M_{\odot}$ ${\rm kpc}^{-2}$) within the (circularized) effective
radius ($r_{e}$) as follows:
$\Sigma_{\rm 50}=\frac{M_{\star}/2}{\pi r_{e}^{2}}$ (1)
We present our measurements in Table 1.
## 4\. Results
### 4.1. Distribution of Internal Surface Mass Densities
We have used the combination of photometric redshifts, stellar population
modeling and surface-brightness profile fits to calculate internal surface
mass densities for our protocluster sample. We have also added data from the
literature and from FIREWORKS to construct a well-populated density versus
stellar mass diagram in Figure 5. If the published data had a measured star-
formation rate in addition to the stellar mass we have restricted the points
plotted to those with low sSFRs (quiescent; ${\rm log_{10}}$ ${\rm sSFR}<-11$
${\rm yr^{-1}}$). In cases where the star-formation rate was not quoted
explicitly, we only plot those galaxies which are described as “quiescent” by
the authors. This distribution for both our protocluster candidates (yellow
circles and squares for quiescent and star-forming) and the field sources from
the literature (black squares) and FIREWORKS (blue squares) is shown in Figure
5. For the comparison field sample we have restricted to FIREWORKS galaxies
brighter than $K=21.5$ and with photo-$z$ between 1.9 and 2.6. We have made
the same redshift cut for the literature sample. This redshift range
approximately corresponds to a 1 Gyr epoch centered on the protocluster
redshift.
The mean density of the protocluster sample (${\rm log}$ $<\Sigma_{50}>=9.9$)
is 0.5 dex lower than that for the field sample. For the stellar mass range of
our protocluster sample, $10^{10.5}M_{\odot}<M_{\star}<10^{11.4}M_{\odot}$,
where the mass distributions are similar, we can calculate the distribution of
the surface mass densities irrespective of total stellar mass. We have also
used the KS test to calculate the probability that the (quiescent)
protocluster and field densities are drawn from the same parent distribution.
A fiducial value of $P_{\rm KS}<5\%$ may be considered sufficient to reject
the null hypothesis that they are from the same parent. In this case $P_{\rm
KS}\sim 5\%$ and is therefore a relatively strong constraint. We have
perturbed our measured densities and re-calculated $P_{\rm KS}$ for 10000
trials. We show the distribution of $P_{\rm KS}$ in Figure 6. For the black
(yellow) histogram 30% (60%) of the realizations fall below $P_{\rm KS}=5\%$.
Both histograms have tails to higher probabilities.
Figure 6.— The KS probability distributions for the protocluster versus field
galaxy comparison. The black histogram are derived from 10000 runs with the
protocluster densities perturbed at random within the Gaussian errors. The
yellow histogram is the same but excluding the most dense protocluster galaxy.
The vertical dashed line marks $P_{\rm KS}=0.05$. About 55% of the
realizations fall below the 5% probability.
### 4.2. Radial Dependencies
In order to assess possible radial gradients in the galaxy properties within
the cluster, with respect to the radio galaxy, we have constructed the
histograms shown in Figure 7. For each of these four physical galaxy
parameters: Sérsic $n$ value, surface mass density ($\Sigma$), stellar mass
($M_{\star}$) and specific SFR, we have made a single cut in the galaxy sample
and plotted two radial histograms for above (orange) and below (blue) the
chosen cut value. The left panel of Fig. 7 shows the histograms for the full
galaxy sample appropriate to each parameter, i.e., the GALFIT sample for the
Sersic $n$ value. For the rest, the galaxies must also be in the stellar
population sample. The right panel shows the histograms for our protocluster
galaxy sample (both quiescent and star-forming). We have presented these two
analyses because while the presence of field galaxies in the larger samples
will dilute any result, the statistics are poor for the quiescent protocluster
galaxy sample. Furthermore, this field exhibits a factor of six surface
overdensity of red galaxies (Zirm et al., 2008), so the large sample may not
strongly dilute trends.
For each pair of histograms we have run the two-sample KS test to determine
whether the distributions are consistent with one another. However, for the
full sample (left) the Sersic $n$ distributions are more dissimilar than for
the other parameters with a low $P_{\rm KS}=5\%$. This low probability seems
to be the result of a relatively flat distribution of the $n<2.5$ galaxies
with radius contrasting with the structure in the radial distribution of the
higher $n$ galaxies. This hint may imply a scenario in which the galaxies are
deeper within the gravitational potential. We discuss this point further
below. In the future, with spectroscopic redshifts for red galaxies, it will
be possible to repeat this test with better interloper rejection to see if the
discrepancy between histograms is significant for bona fide protocluster
galaxies.
## 5\. Discussion
We have presented the combined analysis of 13 band photometry and high
spatial-resolution NIR imaging in the field of a known galaxy protocluster at
$z=2.16$. In cases like this where there is a known, confirmed overdensity and
strong statistical evidence for a dominant contribution from protocluster
members we can make progress despite the lack of spectrocopic redshifts. We
have identified a robust sample of likely protocluster galaxies. Our
conclusions are tentative and tempered by the following caveats. We do not
have spectroscopic redshifts for the quiescent cluster galaxies. While we have
done our best to isolate the most probable cluster members, they may still be
field galaxies. The cluster sample is also relatively small and the results
are therefore more suggestive rather than statistically robust. Finally, due
to the limited areal coverage of the protocluster (see Fig. 1), the very
massive galaxies may be underrepresented in the cluster sample. These may also
be more dense.
Possible stellar population model offsets have been minimized by converting
the literature points to the same models and IMF we used to fit the cluster
galaxies.
### 5.1. Evolution of Galaxy Structure and Stellar Mass Surface Density
From the initial discovery of SEEDs, in general field surveys, the primary
question has been what evolutionary processes affect the SEEDs between $z\sim
2-3$ and $z=0$ that bring them in line with the local mass-size relation. If
the dominant process is galaxy merging, then we might expect that most of the
full galaxy mergers, as opposed to tidal interactions and harassment, may have
already happened. While if the primary determinant of galaxy density is the
formation redshift, the young Universe being denser, we might expect that
cluster galaxies will be denser than their field counterparts having formed
earlier.
In our data we see some indication for a difference between the profile shapes
and density distributions for protocluster versus field galaxies (Fig. 5). For
our sample of likely quiescent protocluster galaxies their stellar densities
are lower and perhaps even the Sersic index is higher than for similarly
selected field galaxies. From other studies it appears that the majority of
field SEEDs have higher axial ratios (flattened) with $n\sim 2$ (van der Wel
et al., 2011).
### 5.2. Cosmic Merger Clocks
Several previous studies have shown evidence that protocluster galaxies tend
to be more massive and contain older stars than their field counterparts at
the same redshift (e.g., Steidel et al., 2005; Tanaka et al., 2010). This
advanced evolution in the cluster environment may also extend to the internal
structure and dynamics of the galaxies. At lower redshift, we observe a
morphology-density relation, and we may be seeing the beginnings of that
relation in the MRC 1138 protocluster. Furthermore, the lower densities of the
protocluster galaxies suggests that the necessary merging has also taken place
at a quicker pace than in the field. If we assume then that all galaxies begin
as dense SEEDs at high redshift, we can use the observed densities as a
measure of the “merger age” of the remnants. A similar idea was put forth by
Hopkins and Hernquist (2010) to derive the global star-formation history by
using the mass profiles of galaxies. They propose that the dense cores of
galaxies form early in starbursts and the outer parts form in a more quiescent
mode, perhaps in disks. Based on the results from this paper we suggest that
the resolved mass profiles can be used further to constrain the redshift of
formation and the subsequent merger history. The ratio of high-density stellar
mass to low-density stellar mass may tell us something about the merger age of
a galaxy while the absolute density of the highest density stellar components
may tell us about the formation redshift. Studies at low-redshift have already
found some correlation between galaxy density and the mean stellar age (e.g.,
van der Wel et al., 2009). With more precise stellar ages for $z\sim 2$
galaxies now becoming available we can extend this analysis to high-redshift
when the fractional age differences between galaxies are larger.
A. Zirm and S. Toft gratefully acknowledge support from the Lundbeck
Foundation. The authors thank the anonymous referee for their helpful
comments. This work is supported by World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan and also in part by Grant-In-
Aid for Young Scientists No. 23740144.
## References
* Bertin & Arnouts (1996) Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393
* Blanton & Moustakas (2009) Blanton, M. R. & Moustakas, J. 2009, ARA&A, 47, 159
* Brammer et al. (2008) Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503
* Cassata et al. (2010) Cassata, P., et al. 2010, 714, 79
* Cimatti et al. (2008) Cimatti, A., et al. 2008, A&A, 482, 21
* Ciotti et al. (2007) Ciotti, L., Lanzoni, B., & Volonteri, M. 2007, ApJ, 658, 65
* Croft et al. (2005) Croft, S., Kurk, J., van Breugel, W., Stanford, S. A., de Vries, W., Pentericci, L., & Röttgering, H. 2005, AJ, 130, 867
* Daddi et al. (2005) Daddi, E., et al. 2005, ApJ, 626, 680
* Damen et al. (2009) Damen, M., Förster Schreiber, N. M., Franx, M., Labbé, I., Toft, S., van Dokkum, P. G., & Wuyts, S. 2009, ApJ, 705, 617–623
* de Jong (2006) de Jong, R. 2006. Correcting the nicmos count-rate dependent non-linearity. Technical report, STScI
* Dekel & Birnboim (2008) Dekel, A. & Birnboim, Y. 2008, MNRAS, 383, 119
* Dekel et al. (2009) Dekel, A., et al. 2009, Nature, 457, 451
* Emsellem et al. (2007) Emsellem, E., et al. 2007, MNRAS, 379, 401
* Erb et al. (2006) Erb, D. K., Steidel, C. C., Shapley, A. E., Pettini, M., Reddy, N. A., & Adelberger, K. L. 2006, ApJ, 647, 128
* Franx et al. (2008) Franx, M., van Dokkum, P. G., Schreiber, N. M. F., Wuyts, S., Labbé, I., & Toft, S. 2008, ApJ, 688, 770
* Hopkins & Hernquist (2010) Hopkins, P. F. & Hernquist, L. 2010, MNRAS, 402, 985
* Kauffmann et al. (2003) Kauffmann, G., et al. 2003, MNRAS, 341, 54–69
* Khochfar & Silk (2006) Khochfar, S. & Silk, J. 2006, ApJ, 648, 21
* Khochfar & Silk (2009) Khochfar, S. & Silk, J. 2009, ApJ, 700, 21
* Khochfar & Silk (2011) Khochfar, S. & Silk, J. 2011, MNRAS, 410, L42
* Kodama et al. (2007) Kodama, T., Tanaka, I., Kajisawa, M., Kurk, J., Venemans, B., De Breuck, C., Vernet, J., & Lidman, C. 2007, MNRAS, 377, 1717
* Kriek et al. (2009) Kriek, M., van Dokkum, P. G., Labbé, I., Franx, M., Illingworth, G. D., Marchesini, D., & Quadri, R. F. 2009, ApJ, 700, 221
* Krist (1993) Krist, J. 1993, in ASP Conf. Ser. 52: Astronomical Data Analysis Software and Systems II 536
* Kurk (2003) Kurk, J. D. 2003. The cluster environments and gaseous halos of distant radio galaxies. Ph. D. thesis, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
* Kurk et al. (2004a) Kurk, J. D., Pentericci, L., Overzier, R. A., Röttgering, H. J. A., & Miley, G. K. 2004a, A&A, 428, 817
* Kurk et al. (2004b) Kurk, J. D., Pentericci, L., Röttgering, H. J. A., & Miley, G. K. 2004b, A&A, 428, 793
* Mancini et al. (2010) Mancini, C., et al. 2010, 401, 933
* Maraston (2005) Maraston, C. 2005, MNRAS, 362, 799
* Miley et al. (2006) Miley, G. K., et al. 2006, ApJ, 650, L29
* Mosleh et al. (2011) Mosleh, M., Williams, R. J., Franx, M., & Kriek, M. 2011, ApJ, 727, 5
* Naab et al. (2009) Naab, T., Johansson, P. H., & Ostriker, J. P. 2009, ApJ, 699, 178
* Nordon et al. (2010) Nordon, R., et al. 2010, A&A, 518, L24
* Oke (1974) Oke, J. B. 1974, ApJS, 27, 21
* Oser et al. (2010) Oser, L., Ostriker, J. P., Naab, T., Johansson, P. H., & Burkert, A. 2010, ApJ, 725, 2312
* Peng et al. (2002a) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002a, AJ, 124, 266
* Peng et al. (2002b) Peng, E. W., Ford, H. C., Freeman, K. C., & White, R. L. 2002b, AJ, 124, 3144
* Pentericci et al. (2002) Pentericci, L., Kurk, J. D., Carilli, C. L., Harris, D. E., Miley, G. K., & Röttgering, H. J. A. 2002, A&A, 396, 109
* Salimbeni et al. (2009) Salimbeni, S., Fontana, A., Giallongo, E., Grazian, A., Menci, N., Pentericci, L., & Santini, P. 2009, in American Institute of Physics Conference Series, ed. G. Giobbi, A. Tornambe, G. Raimondo, M. Limongi, L. A. Antonelli, N. Menci, & E. Brocato, Volume 1111 of American Institute of Physics Conference Series 207
* Saracco et al. (2009) Saracco, P., Longhetti, M., & Andreon, S. 2009, 392, 718
* Sérsic (1968) Sérsic, J. L. 1968, Atlas de galaxias australes Cordoba, Argentina: Observatorio Astronomico, 1968
* Steidel et al. (2005) Steidel, C. C., Adelberger, K. L., Shapley, A. E., Erb, D. K., Reddy, N. A., & Pettini, M. 2005, ApJ, 626, 44
* Stevens et al. (2003) Stevens, J. A., et al. 2003, Nature, 425, 264
* Tanaka et al. (2010) Tanaka, M., De Breuck, C., Venemans, B., & Kurk, J. 2010, A&A, 518, A18
* Toft et al. (2009) Toft, S., Franx, M., van Dokkum, P., Förster Schreiber, N. M., Labbe, I., Wuyts, S., & Marchesini, D. 2009, ApJ, 705, 255
* Toft et al. (2007) Toft, S., et al. 2007, ApJ, 671, 285
* van der Wel et al. (2009) van der Wel, A., Bell, E. F., van den Bosch, F. C., Gallazzi, A., & Rix, H.-W. 2009, ApJ, 698, 1232
* van der Wel et al. (2011) van der Wel, A., et al. 2011, ApJ, 730, 38
* van Dokkum (2005) van Dokkum, P. G. 2005, AJ, 130, 2647
* van Dokkum et al. (2008) Dokkum, van Pieter G., et al. 2008, 677, 5
* Williams et al. (2010) Williams, R. J., Quadri, R. F., Franx, M., van Dokkum, P., Toft, S., Kriek, M., & Labbé, I. 2010, ApJ, 713, 738
* Wuyts et al. (2008) Wuyts, S., Labbé, I., Schreiber, N. M. F., Franx, M., Rudnick, G., Brammer, G. B., & van Dokkum, P. G. 2008, ApJ, 682, 985
* Zirm et al. (2008) Zirm, A. W., et al. 2008, ApJ, 680, 224
* Zirm et al. (2007) Zirm, A. W., et al. 2007, ApJ, 656, 66
Table 1Protocluster Candidates Object | Photometric | Odds | $H_{160}$ | Line | Stellar | Specific | Effective | Effective | Mass
---|---|---|---|---|---|---|---|---|---
Object | Redshift | | (AB) | | Mass | SFR | Radius | Radius | Density
| | | | | (log($M_{\odot}$)) | (log(yr-1)) | ($\arcsec$) | (kpc) | ($10^{9}$ $M_{\odot}{\rm kpc}^{-2}$)
Quiescent Protocluster Galaxies
312 | 2.24 | 1.00 | 21.98 $\pm$ 0.01 | 0 | $11.33^{+0.04}_{-0.04}$ | -99.00 | 0.23 $\pm$ 0.09 | 1.89 | $9.67\pm 6.5$
394 | 2.23 | 1.00 | 24.07 $\pm$ 0.01 | 0 | $10.65^{+0.20}_{-0.20}$ | -11.24 | 0.14 $\pm$ 0.07 | 1.20 | $4.98\pm 2.6$
399 | 2.49 | 0.98 | 24.20 $\pm$ 0.01 | 0 | $10.59^{+0.20}_{-0.16}$ | -11.24 | 0.23 $\pm$ 0.09 | 1.91 | $1.78\pm 0.4$
493 | 2.12 | 1.00 | 22.73 $\pm$ 0.01 | 0 | $11.13^{+0.03}_{-0.04}$ | -99.00 | 0.19 $\pm$ 0.07 | 1.58 | $8.56\pm 6.2$
507 | 2.16 | 1.00 | 22.48 $\pm$ 0.01 | 0 | $11.04^{+0.05}_{-0.03}$ | -99.00 | 0.06 $\pm$ 0.06 | 0.47 | $80.14\pm 52.9$
547 | 2.10 | 1.00 | 22.68 $\pm$ 0.01 | 0 | $10.97^{+0.08}_{-0.03}$ | -11.72 | 0.09 $\pm$ 0.08 | 0.78 | $24.21\pm 14.2$
574 | 2.13 | 1.00 | 23.33 $\pm$ 0.01 | 0 | $10.65^{+0.04}_{-0.04}$ | -99.00 | 0.14 $\pm$ 0.09 | 1.17 | $5.15\pm 3.6$
599 | 2.54 | 1.00 | 23.89 $\pm$ 0.01 | 0 | $10.75^{+0.21}_{-0.05}$ | -12.20 | 0.15 $\pm$ 0.08 | 1.22 | $6.41\pm 1.0$
Star-forming Protocluster Galaxies
347 | 2.08 | 0.92 | 23.69 $\pm$ 0.01 | 0 | 9.69 | -8.79 | 0.19 $\pm$ 0.05 | 1.57 | 0.31
456 | 2.06 | 0.95 | 23.57 $\pm$ 0.07 | 0 | 10.49 | -9.27 | 0.63 $\pm$ 0.06 | 5.19 | 0.18
595 | 2.06 | 0.91 | 23.47 $\pm$ 0.01 | 0 | 10.24 | -8.79 | 0.21 $\pm$ 0.07 | 1.72 | 0.92
Line-emitting CandidatesaaNarrow-band selected objects that are not yet
spectroscopically confirmed.
700 | 2.16 | $\cdots$ | 24.02 $\pm$ 0.02 | 1 | $\cdots$ | $\cdots$ | 0.23 $\pm$ 0.06 | 1.88 | $\cdots$
463 | 2.16 | $\cdots$ | 24.51 $\pm$ 0.03 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
1078 | 2.16 | $\cdots$ | 22.94 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.17 $\pm$ 0.06 | 1.38 | $\cdots$
1070 | 2.16 | $\cdots$ | 23.44 $\pm$ 0.02 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
511 | 2.16 | $\cdots$ | 24.50 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
516 | 2.16 | $\cdots$ | 24.07 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.14 $\pm$ 0.08 | 1.15 | $\cdots$
575 | 2.16 | $\cdots$ | 24.62 $\pm$ 0.02 | 1 | $\cdots$ | $\cdots$ | 0.01 $\pm$ 0.03 | 0.09 | $\cdots$
988 | 2.16 | $\cdots$ | 23.41 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.32 $\pm$ 0.06 | 2.67 | $\cdots$
536 | 2.16 | $\cdots$ | 23.90 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.13 $\pm$ 0.08 | 1.11 | $\cdots$
1069 | 2.16 | $\cdots$ | 23.91 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.00 $\pm$ 0.54 | 0.01 | $\cdots$
457 | 2.16 | $\cdots$ | 23.84 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
451 | 2.16 | $\cdots$ | 23.74 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
275 | 2.16 | $\cdots$ | 23.62 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.12 $\pm$ 0.06 | 0.96 | $\cdots$
897 | 2.16 | $\cdots$ | 24.09 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.11 $\pm$ 0.05 | 0.92 | $\cdots$
300 | 2.16 | $\cdots$ | 22.50 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
311 | 2.16 | $\cdots$ | 23.94 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 6.77 $\pm$ 4.36 | 56.15 | $\cdots$
361 | 2.16 | $\cdots$ | 24.50 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.03 $\pm$ 0.04 | 0.27 | $\cdots$
365 | 2.16 | $\cdots$ | 26.16 $\pm$ 0.06 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
215 | 2.16 | $\cdots$ | 24.09 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | 0.14 $\pm$ 0.08 | 1.18 | $\cdots$
448 | 2.16 | $\cdots$ | 21.93 $\pm$ 0.02 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
435 | 2.16 | $\cdots$ | 23.74 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
431 | 2.16 | $\cdots$ | 22.61 $\pm$ 0.01 | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Confirmed Line-emitting Galaxies
53 | 2.16 | $\cdots$ | 24.00 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | 0.02 $\pm$ 0.04 | 0.19 | $\cdots$
263 | 2.16 | $\cdots$ | 24.01 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | 0.18 $\pm$ 0.08 | 1.52 | $\cdots$
270 | 2.16 | $\cdots$ | 21.11 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
450 | 2.16 | $\cdots$ | 22.75 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
945 | 2.16 | $\cdots$ | 23.79 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | 0.03 $\pm$ 0.02 | 0.22 | $\cdots$
289 | 2.16 | $\cdots$ | 22.05 $\pm$ 0.01 | 2 | 11.52 | -11.24 | 0.43 $\pm$ 0.09 | 3.53 | 4.22
757 | 2.16 | $\cdots$ | $\cdots$ | 2 | 10.54 | -9.27 | $\cdots$ | $\cdots$ | $\cdots$
561 | 2.16 | $\cdots$ | 21.99 $\pm$ 0.01 | 2 | 11.23 | -12.20 | 0.86 $\pm$ 0.09 | 7.09 | 0.54
296 | 2.16 | $\cdots$ | 22.27 $\pm$ 0.01 | 2 | 10.50 | -8.18 | 0.10 $\pm$ 0.07 | 0.84 | 7.10
648 | 2.16 | $\cdots$ | 23.38 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | 0.14 $\pm$ 0.06 | 1.18 | $\cdots$
535 | 2.16 | $\cdots$ | 23.90 $\pm$ 0.01 | 2 | $\cdots$ | $\cdots$ | 0.14 $\pm$ 0.05 | 1.19 | $\cdots$
387 | 2.16 | $\cdots$ | 24.70 $\pm$ 0.02 | 2 | $\cdots$ | $\cdots$ | 0.06 $\pm$ 0.04 | 0.47 | $\cdots$
Figure 7.— Radial (measured from the radio galaxy) distributions of galaxies
as a function of four derived physical parameters. Top to bottom: Sérsic $n$
value, surface mass density ($\Sigma$), stellar mass ($M_{\star}$) and
specific star-formation rate. For each parameter we have made a single cut of
the total sample into two bins and plotted those two histograms separately for
those above (orange) and below (blue) this cut. The value of the cut is shown
below the parameter name and the number of galaxies in each bin to the left of
the name. Finally, the KS probability, that the two histograms are drawn from
the same parent distribution is also shown.
|
arxiv-papers
| 2011-10-03T11:49:44 |
2024-09-04T02:49:22.713213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.W. Zirm, S. Toft, M. Tanaka",
"submitter": "Andrew Zirm",
"url": "https://arxiv.org/abs/1110.0327"
}
|
1110.0463
|
# A binary noisy channel to model errors in printing process
V.N. Gorbachev, E.S. Yakovleva 111email: 2305lena@mail.ru
( _North-Western Institute of Printing,
St.-Petersburg State University of Technology and Design,
13, Djambula, St-Petersburg, 191180, Russia_ )
###### Abstract
To model printing noise a binary noisy channel and a set of controlled gates
are introduced. The channel input is an image created by a halftoning
algorithm and its output is the printed picture. Using this channel robustness
to noise between halftoning algorithms can be studied. We introduced relative
entropy to describe immunity of the algorithm to noise and tested several
halftoning algorithms.
###### Contents
1. 1 Introduction
2. 2 Binary channel for printing
3. 3 Binary noise
4. 4 Robustness to noise
## 1 Introduction
Printing is a complicated process that can be considered from the numerous
points of view. In literature it is payed more attention to halftoning, a
process that creates a binary image to reproduce it as a continuous-tone
picture by a binary printing device [1]. The goal of all halftoning techniques
is to generate an image that is perceptually similar to the original. To
estimate the original and printed image, a large number of distortion measures
and criteria based on the models of the human vision system have been proposed
[2]. Absence of the universal solutions stimulates progress in these fields.
In this paper we consider a printing process as a communication channel that
transforms a halftoned image into a printed image. We are interested in the
main question can the noisy channel be suitable to describe errors in printing
process? We introduced a binary channel, well known in theory of information
[3], and a set of controlled-gates to model printing noise. We found that such
simple noise of the binary channel as bit-flip and erasing noise introduce a
visible distortion. The introduced noisy channel allows us to consider
robustness of halftone images against to noise. By this way robustness between
halftoning algorithms can be studied. Immunity to noise is important because
it provides the quality of the printed images. Noise has a physical nature and
results in errors in printing always. However one can reduce errors if one
chooses a halftoning algorithm such that its immunity to noise is large.
## 2 Binary channel for printing
Any printing device has binary output, if a dot is printed or not. From the
point of view of the theory of information the printing process can be
considered as transmitting information trough a binary channel with noise that
can model errors of printing.
To introduce the binary channel for printing, we begin with the general scheme
of Shannon [3] for transmitting and storing information. The scheme has a
source of information whose output, a message, is transformed by a coder into
a signal acceptable for sending through the channel or storing it in a memory
device. The channel has a noise source that describes errors due to its
physical nature. In the output of the channel a decoder transforms the message
into the signal suitable for the receiver. Considering printing one can find
the similar organization, shown in Fig. 1 . Here the source of information is
a digital grayscale image. By halftoning the image is transformed into a
binary one, that is input of the printing binary channel. This channel
consists of "ink" and "paper". Output is a printed image addressed for the
receiver. The receiver plays an important role, his features are very
complicated and we will ignore him. So the introduced scheme is not perfect
but we think this scheme may be sufficient to model printing process.
The binary channel transmits strings of 0 and 1 generated by the printing
device whose main elements are "ink" and "paper". The terms "ink" and "paper"
are conventional. They can be a usual ink and paper, when the image is
presented by an usual way. Also, one can find electronic paper or inks. Anyway
the methods of reproducing halftone or colored image have the similar
features.
## 3 Binary noise
The binary channel has a binary noise. To model numerous sources of noise in
printing process by the binary channel a set of controlled gates can be
suitable.
Because of the noise zeros and ones can be transmitted incorrectly from input
to output. It results in information loss and decreasing capacity of the
communication channel. For printing it means a degradation of the image
quality. Two main types of noise are known as bit-flip $0\to 1,$ $1\to 0$, and
erasing $\\{0;1\\}\to 1$. They occur with some probability $p$. For example,
if the bit-flip error has $p=1/2$ then the channel capacity is zero and
information can’t be transmitted. There are numerous noise sources in printing
process. For example, consider a printing head, or more generally the printing
devices. They have to produce an ink dot of a fixed size in a given position.
Let introduce a simple vibration, it affects on the printing head and the head
makes errors. Then one can find an unnecessary dot, a dot in the unwanted
position also the dot size may be increased and so on.
The observations tell that in printing process the noise can be described
mathematically by conditional operations C-U, known in the computer literature
[4] . The C-U gate or controlled gate, has two inputs and two outputs named as
control and target. The gate works as follows. If and only if the control bit
is 1, then operation $\mathbf{U}$ is applied to target. Let $x\in\\{0;1\\}$ be
a control bit and $y$ be a target input, then C-U maps $x\otimes y\to
x\otimes\mathbf{U}^{x}y.$ By this way one can describe any noise, particular
the vibration of the printing head, assuming $x$ be a random string of 0 and 1
and $\mathbf{U}$ be an appropriate operation. In more general case both the
control and target signal can be _n-d_ arrays. If we wish to describe
processing image then $n=2$ and it needs 2d-binary noise.
We generated binary noise by a simple way with the help of threshold
halftoning. Let $\mathbf{r}$ be a random matrix, a 8-bit grayscale image. Let
the probability distribution function of lightness
$\mathbf{r}[m,n]=0,1,..,255$ be uniform. Then binary matrix is obtained like
that
$\displaystyle\mathbf{v}[m,n]=\begin{cases}1,&\text{if
$\mathbf{r}[m,n]<T$;}\\\ 0,&\text{if $\mathbf{r}[m,n]\geq T$,}\end{cases}$
where $T$ is a threshold. It follows from this equation that in matrix
$\mathbf{v}$ the number of ones is proportional to threshold $T$ so we will
name $T$ power of binary noise. Fig. 2 shows some results on binary noise and
its entropy, presented as functions of $T$. Perceptually noise refers to
lightness of the image. The image looks more black when power increases. Due
to random matrix $\mathbf{r}$ entropy of binary noise is a random function of
$T$ and it needs averaging. In Fig. 2 (c) entropy has been calculated by
averaging over 1024 realizations of $\mathbf{r}$. Note, entropy achieves its
maximum at $T=0.5$ because this threshold results in matrix $\mathbf{v}$ that
has zeros and ones generated uniformly with probability close to 1/2.
Erasing noise $\\{0,1\\}\to 1$ can be modeled by a C-U gate with 2d input. Let
the control signal be the binary noise $\mathbf{v}$ and let input target
signal be a halftoned image $\mathbf{g}$. Then $x=\mathbf{v}[m,n]$,
$y=\mathbf{g}[m,n]$ and the C-U gate accomplishes the task, if
$\mathbf{U}^{x}y=x\vee y$, where $\vee$ is disjunction. This noise increases
number of the black dots at printed image. An example of erasing is shown in 3
(a,b,c), noise results in distortion of contour. A more exact erasing noise
can process a block $K_{g}$ of binary image. Let one of the pixels,
$\mathbf{g}[m,n]\in K_{g}$, be transformed into 1 by erasing noise, if and
only if the center pixel of the block $\mathbf{g}[m_{c},n_{c}]=1$. For this
case input of the C-U gate is
$x=\mathbf{v}[m,n]\otimes\mathbf{g}[m_{c},n_{c}]$ and $y=\mathbf{g}[m,n]$. The
gate accomplishes the task, if
$\mathbf{U}^{x}y=\mathbf{v}[m,n]\wedge\mathbf{g}[m_{c},n_{c}]\vee y$, where
$\wedge$ is conjunction. This noise may lead to blooming and smearing of image
as it is shown in Fig. 3(d,e,f).
## 4 Robustness to noise
By considering distortion of a halftoned image due to noise one can learn
about its robustness to noise. If the image is created by a halftoning
algorithm, then one can study robustness of the algorithm.
Let a halftoned image $\mathbf{g}$ be transmitted trough a channel with binary
noise $\mathbf{v}$ of power $T$ and $\mathbf{g}^{\prime}$ be the output noisy
image. To describe distinction between $\mathbf{g,v,g^{\prime}}$ we introduce
two measures of distortion, euclidian distance and relative entropy. Both
measures maps pairs of images given by matrices $\mathbf{A}$ and $\mathbf{B}$
into $[0,\infty)$.
Euclidian distance between two matrices is denoted by
$e(\mathbf{A,B})=\\{(1/n)\sum_{m,n}(\mathbf{A}[m,n]-\mathbf{B}[m,n])^{2}\\}^{1/2}$,
where $n$ is the number of elements of matrices. This is an average distance
between matrices. It describes how much images $\mathbf{A}$ and $\mathbf{B}$
are close, the more euclidian distance the more discrepancy between images.
But this is a mathematical measure and it may be not in accordance with visual
perception. To examine a halftoned image $g$ obtained by D-algorithm
introduced in our work [5] has been chosen. We calculated euclidian distance
$e(\mathbf{v,g})$ between binary noise and halftone image and found that it
increases monotonically with power noise $T$, Fig. 4 (a). It means that the
binary noise of maximally power is maximally closer to image $\mathbf{g}$. It
doesn’t correspond to visual perception and this is one of the reasons to
consider relative entropy.
In statistics the relative entropy known also as Kulback-Leibler entropy is
used to establish the difference between two distributions of a random
variable. For images $\mathbf{A}$ and $\mathbf{B}$ it is defined as
$Q(p_{A}||p_{B})=\sum_{i}p_{A}[i](\log p_{A}[i]-\log p_{B}[i])$, where
$p_{A}[i]$, $p_{B}[i])$ are histograms of lightness $i=0,1,..$. The relative
entropy is not negative and $Q(p_{A}||p_{B})=0$ if and only if $p_{A}=p_{B}$.
The relative entropy is not a metric because it is in general not symmetric
$Q(p_{A}||p_{B})\neq Q(p_{B}||p_{A})$ and does’t satisfy the triangle
inequality. It is nevertheless often used in image processing, particular in
steganography and digital watermarking [6].
We calculated the relative entropy $Q(\mathbf{g}||\mathbf{v})$ between the
considered above halftone image $g$ and binary noise. It is shown in Fig. 4
(b) versus the noise power $T$. As it is seen the function
$Q(\mathbf{g}||\mathbf{v})$ has its minimum at $T=0.7$ and increases if power
noise moves off 0.7. The minimum tells that histograms of $\mathbf{g}$ and
$\mathbf{v}$ are very close, if $T=0.7$. This is a mathematical fact and it is
not accordance with the human vision. We think the reason is that these images
have a quite different semantic information that can’t properly be described
by relative entropy. Clear, that from the semantic point of view two images
$\mathbf{g}$ and $\mathbf{g^{\prime}}$ are more ”closer”, than $\mathbf{g}$
and noise $\mathbf{v}$. Indeed, we calculated the relative entropy
$Q(\mathbf{g}||\mathbf{g}^{\prime})$ that increases monotonically with the
noise power $T$ (Fig. 4 (c)). This is an evident fact because noise introduces
a difference in histograms and what is more important this is in accordance
with human vision.
Taking into account observation about semantic information we will use the
relative entropy for comparative studies of robustness to noise between
halftoning algorithms. This choice is also based on the next statement. _The
relative entropy can describe an expert estimation of the quality of the
image._
Let $\mathbf{g}$ and $\mathbf{g}^{\prime}$ be input and output image of the
binary noisy channel and $\mathbf{g}$ be obtained by a halftoning algorithm
$\mathbf{a}$. We will name the algorithm $\mathbf{a}$ $\epsilon$-robust to
noise, if relative entropy $Q(\mathbf{g}||\mathbf{g^{\prime}})\leq\epsilon$.
The algorithm is perfect, if $Q(\mathbf{g}||\mathbf{g^{\prime}})=0$. The
introduced definition tells that the smaller $\epsilon$ is, or the closer the
histograms of $\mathbf{g}$ and $\mathbf{g^{\prime}}$ are, the greater
robustness of algorithm is. To better understand what is meant by
$\epsilon$-robustness consider that an expert examines the quality by viewing
images $\mathbf{g}$ and $\mathbf{g}^{\prime}$. Let the expert have to answer
the question, is the given image $\mathbf{g}$ or $\mathbf{g}^{\prime}$. When
$\epsilon$ is small he can make two types of errors. So the expert can decide
that the given image is $\mathbf{g}$, when it fact it is $\mathbf{g^{\prime}}$
and vice versa. This detection scheme is well known in statistical signal
detection and estimation [7]. Let probability of the expert errors be $\alpha$
and $\beta$. Then, if only $\mathbf{g}$ and, if only $\mathbf{g^{\prime}}$ is
shown to expert, the relative entropy between his decisions is
$Q(\alpha||\beta)$. Because the expert with his decisions is a type of
processing and processing can’t increase relative entropy for
$\epsilon$-robust algorithm one finds
$\displaystyle Q(\alpha||\beta)\leq
Q(\mathbf{g}||\mathbf{g^{\prime}})\leq\epsilon.$
As it follows from this equation $\epsilon$-robustness of algorithm refers to
errors of the expert, that tests the image quality.
The relative entropy can be used to compare robustness to noise between
halftoning algorithms. Let $\mathbf{g}_{k}$ be an image obtained by a
halftoning algorithm $\mathbf{a}_{k},k=1,2,\dots$ and let
$\mathbf{g^{\prime}}_{k}$ be its noisy version. One can calculate the relative
entropies $q_{k}=Q(\mathbf{g_{k}}||\mathbf{g^{\prime}_{k}})$. Algorithm
$\mathbf{a}_{k}$ is more robust than $\mathbf{a}_{t}$, if $q_{k}\leq q_{t}$.
In practice this inequality may hold in some region of parameters and for
certain types of images because there are not ideal halftoning algorithms.
As example we studied some algorithms. First is DITHER, it is based on the the
Floyd-Steinberg error diffusion method and can be evaluated as dither(I) in
MATLAB. Second is D-algorithm, it has an input parameter $h$ that denotes a
grayscale $h\times h$ block processed [5]. We calculated the relative entropy
between DITHER and D-algorithm, $q_{1}$ and $q_{2}$ for bit-flip noise. Fig. 5
shows difference $q_{1}-q_{2}$ as function of the power noise $T$ and size
block $h$. It follows that there is a range, $h\geq 25$, where $q_{2}<q_{1}$.
It this range D-algorithm is more robust against noise. Fig. 6 illustrates
distortions introduced by noise in halftone images obtained by D and DITHER
techniques for $T=0.2$ and $h=19,q_{2}<q_{1}$. One can see that distortions
are more noticeable in the DITHER techniques, that is in agreement with
mathematical results. The results presented at Fig. 5 and Fig. 6 are valid for
an image only and it needs a large set of images for investigation. Clear,
that a given halftoning algorithm can’t processes properly all possible
images. Then there is a problem to find a suitable class of images. One of the
solutions is presented at Fig. 7. It shows relative entropy of eight
halftining algorithms versus power of noise. We used seven original halhtoning
algorithms DOTDIF, ERRDIF, LAU, ULICHNEY, CDOT1, CDOT2, BAYER from [1] and D
algorithm. Each point at Fig. 7 is averaged over 80 images from the famous
collection Caprichos by Goya. Indeed, there is a monotonic behavior that tells
that relative entropy of D algorithm is the least. It means, for example, that
to print Caprichos D algorithm may by more preferable.
## References
* [1] Ulichney R.A. Digital Halftoning. MIT Press, Cambridge, MA, 1987.
Lau D.L., Arce G.R. Modern Digital Halftoning. Second Edition, CRC Press,
2008.
* [2] Keelan B.W. Handbook of Image Quality. 2002.
Wincler S. Digital Video Quality. Vision Models and Metrics. John Willey $\&$
Sons, Ltd, 2005.
* [3] Shannon C.E. A Mathematical Theory of Communication, Bell System Technical Journal, 27, pp. 379423, 623656, 1948.
* [4] Deutsch D. Quantum computational networks, Proc. Roy. Soc. Lond., V. A 425, p.73, 1985.
* [5] Gorbachev V.N., Drozdov V.N., Yakovleva E.S. An algorithm for halftoning grayscale images. Dizain, Material Tehnologiya No 2, p. 104, 2009.
* [6] Cox I.J., Miller M.L., Bloom J.F.,Fridrich J., Kaler T. Digital Watermarking and Steganography. ELSEVIER, Morgan Kaufmann Publishers, 2008.
* [7] Helstrom C.W. Element of Signal Detection and Estimation. Engle-wood Cliffs, N.J: Prentice-Hall, 1995.
Figure 1: Binary channel for printing. Input halftoning image is transmitted
trough the channel, a printing device, which output is a printed image.
W
Figure 2: Binary noise and its properties. Random binary pattern obtained by a
threshold halftoning vs threshold T, that is power of noise (a) and (b),
entropy of binary noise (c).
W
Figure 3: Erasing noise.
W
Figure 4: Euclidian distance and relative entropy vs power of noise. Euclidian
distance between halftone image and binary noise (a), relative entropy between
halftone image and binary noise (b), relative entropy between halftone image
and noisy image (c).
W
Figure 5: Difference between relative entropy of DITHER and D-algorithm. If
the difference is positive immunity of D-algorithm to noise is greater and
vice versa.
W
Figure 6: Distortion introduced by bit-flip noise with power $T=0.2$. Halftone
images created by DITHER (left) and D-algorithm (right), $h=19$, their noisy
versions at the bottom.
W
Figure 7: Relative entropy overaged over 80 images. Seven halftoning
algorithms DOTDIF, ERRDIF, LAU, ULICHNEY, CDOT1, CDOT2, BAYER show very close
results, blue curves. The eighth algorithm D is distinguished, red curve, its
relative entropy is least.
W
|
arxiv-papers
| 2011-10-02T15:45:47 |
2024-09-04T02:49:22.729248
|
{
"license": "Public Domain",
"authors": "V.N. Gorbachev, E.S. Yakovleva",
"submitter": "Valery Gorbachev",
"url": "https://arxiv.org/abs/1110.0463"
}
|
1110.0627
|
ROM2F/2011/13
Electrified branes
Luca Martucci
I.N.F.N. Sezione di Roma “TorVergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”,
Via della Ricerca Scienti ca, 00133 Roma, Italy
Abstract
A geometrical form of the supersymmetry conditions for D-branes on arbitrary
type II supersymmetric backgrounds is derived, as well as the associated BPS
bounds. The treatment is general and allows to consider, for instance, non-
static configurations or D-branes supporting a non-vanishing electric flux,
hence completing previous partial results. In particular, our discussion
clarifies how the notion of calibration can be extended in order to be
applicable to the most general supersymmetric configurations. As an
exemplifying preliminary step, the procedure followed is first applied to
fundamental strings.
e-mail: luca.martucci@roma2.infn.it,
###### Contents
1. 1 Introduction
2. 2 Background structures
3. 3 Fundamental strings
1. 3.1 Geometric supersymmetry conditions
2. 3.2 Conserved current and charge
3. 3.3 Supersymmetry and BPS bound
4. 3.4 BPS bound and energetics
4. 4 D-branes
1. 4.1 Geometric supersymmetry conditions
2. 4.2 Conserved current and charge
3. 4.3 Supersymmetry and BPS bound
4. 4.4 BPS bound and energetics
5. 4.5 Gauss law, Bianchi identity and equations of motion
5. 5 Some examples
1. 5.1 Magnetized D-branes
1. 5.1.1 Static magnetized D-branes
2. 5.1.2 Compactifications to ${\rm Mink}_{4}$ and non-static branes
2. 5.2 Simple D-branes with non-vanishing electric field
1. 5.2.1 BIons
2. 5.2.2 Supertubes
6. 6 Conclusion
## 1 Introduction
String theory is populated by branes of various kinds and their properties are
directly related to the supersymmetric structure of the theory. This deep
interplay has one of its most explicit manifestations in the appearance of
integrable background geometrical structures which are directly related to the
geometry of branes. This relation constitutes an important ingredient in
several applications. For instance, the presence of an integrable complex
structure is naturally associated with holomorphically embedded branes. One
can then often avoid facing the detailed supersymmetric structure, working
directly with its physically relevant geometrical manifestation.
The relation between branes and background geometry can be concretely realized
in terms of calibrations, background forms with particular algebraic and
differential properties – see for instance [1] for a review emphasizing some
of the aspects that are important for applications to string theory. In their
original mathematical definition [2], calibrations are closed forms which
identify volume minimizing submanifolds with special properties. They appeared
in string theory when physicists started studying string theory vacua on
Ricci-flat spaces with special holonomy (i.e. with reduced supersymmetry), and
the associated brane configurations – see for instance [3]. Physically, brane
energy is expected to be subject to a BPS lower bound saturated by
supersymmetric configurations and, in string theory vacua on special holonomy
spaces, the energy of static branes is just given by the volume of the cycle
wrapped by the brane. On the other hand, special holonomy spaces are naturally
equipped with calibrations and indeed these provide a nice geometrical
realization of the expected BPS energetic bound, see e.g. [4].
Clearly, more complicated backgrounds require a modification of the notion of
calibration given in [2]. Still focusing on static settings, the inclusion of
background fluxes which couple minimally to branes has been considered in [5]
and leads to the general receipt that the associated background calibrations
are no-longer closed but rather their exterior derivative is given by the
background flux itself. After this work, several papers followed developing
this idea at various levels of generality, see for instance [6, 7, 8, 9, 10].
On the other hand various branes, and in particular D-branes, can support non-
trivial world-volume fluxes, which contribute to the brane energetics and need
to be taken into account in a proper analysis. The way to incorporate magnetic
world-volume fluxes for static D-branes on static backgrounds has been
identified in [11, 12, 13, 14] and exhibits the relevance of generalized
geometry [15, 16]111For a review for physicists, see for instance [17]. as the
proper framework for describing not only the D-brane geometry but also their
dynamics [18, 19]. Not surprisingly, the results on the D-brane side uniform
well with results the on the closed string sector side [20, 21, 22, 23, 24,
25, 26]. Importantly, it has also become evident that the calibration
structures play a crucial role in proving the integrability properties of
supergravity backgrounds fully coupled to backreacting D-branes and
orientifolds, not only in supersymmetric settings [27] but also in non-
supersymmetric backgrounds which preserve some BPS-like properties [28].
This paper studies the geometrical properties of general supersymmetric
D-branes on general supersymmetric type II backgrounds. In particular,
backgrounds as well as D-branes can be non-static and the latter can support
an electric flux in addition to the magnetic one.222The analogous problem for
M5-branes has been addressed in [29]. The analysis explicitly confirms
generalized geometry as the natural language to be used and indeed we will see
how the background supersymmetry conditions found in [30] within this same
framework are the ones which naturally enter in the geometric characterization
of supersymmetric D-branes and the associated BPS bounds. In particular, our
discussion shows how various notions of calibrations should be extended in
order to uncover non-static configurations and world-volume electric fluxes.
Furthermore, supersymmetry may be broken still preserving the calibrations and
the associated BPS structures, along the lines of what was discussed for
static settings in [28]. In this paper we will not consider this interesting
possibility and the adjectives “BPS” and “supersymmetric” can be considered as
synonymous.
This paper is structured as follows. After a brief summary in section 2 of the
background properties, in section 3 we start by considering the simpler case
of fundamental strings. This will allow us to explain without too many
technical complications the basic conceptual strategy which we will follow
when we will turn to D-branes. These are discussed in section 4 along the same
lines followed in section 3 for fundamental strings. In section 5 we provide a
few simple explicit examples in which the we apply the general results of the
previous sections.
## 2 Background structures
This section briefly reviews the geometric structures characterizing the
background.
Take a supersymmetric type II vacuum, hence characterized by a pair of ten-
dimensional Majorana-Weyl spinors ($\epsilon_{1},\epsilon_{2}$) which satisfy
the supersymmetric Killing spinor equations. In our conventions $\epsilon_{1}$
has positive chirality, $\Gamma_{11}\epsilon_{1}=\epsilon_{1}$, while
$\Gamma_{11}\epsilon_{2}=\mp\epsilon_{2}$ in IIA/IIB respectively. These can
be grouped into the Killing doubled-spinor
$\epsilon:=\left(\begin{array}[]{c}\epsilon_{1}\\\
\epsilon_{2}\end{array}\right)$ (2.1)
The supersymmetric Killing spinor conditions have been analyzed in several
papers, see for instance [6, 7, 9, 31, 30]. For our purposes, the most natural
approach is the one adopted [30], which we follow.
By using $\epsilon$ one can construct several tensorial quantities, which have
special differential properties deriving from the Killing spinor equations.
First of all, one can construct a vector field
$K:=-\frac{1}{2}\bar{\epsilon}\,\Gamma^{M}\epsilon\,\partial_{M}\equiv-\frac{1}{2}(\bar{\epsilon}_{1}\Gamma^{M}\epsilon_{1}+\bar{\epsilon}_{2}\Gamma^{M}\epsilon_{2})\,\partial_{M}$
(2.2)
where we use capital letters from the middle of the alphabet as ten-
dimensional curved indices. By using the Killing spinor equations, $K$ turns
out to be a Killing vector
${\cal L}_{K}g=0$ (2.3)
and actually generates a symmetry of the entire background, hence including
all bosonic supergravity fields and supersymmetric structures. This will be
important for us, since $K$ will identify the naturally selected ‘energy’
which is minimized by the supersymmetric branes. In general $K$ can be either
time-like or null and the overall minus-sign in (2.2) has been introduced in
order for $K$ to be future-pointing.
In addition to the vector $K$, one can introduce a one-form $\omega$ defined
by
$\omega:=-\frac{1}{2}\bar{\epsilon}\,\Gamma_{M}\sigma_{3}\epsilon\,\text{d}x^{M}\equiv-\frac{1}{2}(\bar{\epsilon}_{1}\Gamma_{M}\epsilon_{1}-\bar{\epsilon}_{2}\Gamma_{M}\epsilon_{2})\,\text{d}x^{M}$
(2.4)
Notice that $g(K,K)=-g^{-1}(\omega,\omega)\geq 0$, which follows from the fact
that the vectors $\bar{\epsilon}_{1}\Gamma^{M}\epsilon_{1}$ and
$\bar{\epsilon}_{2}\Gamma^{M}\epsilon_{2}$ are null, so that either $K$ and
$\omega$ are both null or $K$ is time-like and $\omega$ space-like.
Furthermore, $\omega$ is related to $K$ by the differential condition
$\text{d}\omega=-\iota_{K}H$ (2.5)
where $H$ is the (closed) NS-NS three-form which is locally given by the
exterior derivative of the $B$-field, $H=\text{d}B$.
There is one further set of tensors which can be constructed by using
$\epsilon$. These can be organized into a ‘polyform’
$\Psi=\sum_{k}\Psi_{\it(k)}$ (2.6)
with ${\it k}=$even/odd in IIA/IIB and
$\Psi_{\it(k)}=-\frac{1}{{\it k}!}(\bar{\epsilon}_{1}\Gamma_{M_{1}\ldots
M_{k}}\epsilon_{2})\,\text{d}x^{M_{1}}\wedge\ldots\text{d}x^{M_{k}}$ (2.7)
Alternatively, $\Psi$ can be defined by the bi-spinor
$\not{\Psi}=32\epsilon_{1}\otimes\bar{\epsilon}_{2}\Gamma_{11}$.333We use the
supergravity and D-brane conventions described in appendix of the first paper
of Ref. [28] and in [32]. Our conventions and definitions slightly differ from
those of [30]. The dictionary is the following: $H_{\rm here}=-H_{\rm there}$,
$F^{\rm IIA}_{\rm here}=-F^{\rm IIA}_{\rm there}$, $F^{\rm IIB}_{\rm
here}=F^{\rm IIB}_{\rm there}$, $K_{\rm here}=-32K_{\rm there}$, $\omega_{\rm
here}=-32\tilde{K}_{\rm there}$, $\Psi^{\rm IIB}_{\rm here}=-32\Phi^{\rm
IIB}_{\rm there}$, $\Psi^{\rm IIA}_{\rm here}=32\Phi^{\rm IIA}_{\rm there}$.
Notice that $\Psi$ obeys the self-duality relation
$\Psi=*\lambda(\Psi)$ (2.8)
where $\lambda(\Psi_{(k)})=(-)^{\frac{k(k-1)}{2}}\Psi_{(k)}$. Furthermore
$\iota_{K}\Psi+\omega\wedge\Psi=0$ (2.9)
The Killing spinor equations imply that $\Psi$ is linked with the R-R fluxes
by the following differential equations
$\text{d}_{H}(e^{-\phi}\,\Psi)=-\iota_{K}F-\omega\wedge F$ (2.10)
where
$F:=\sum_{\it k}F_{\it(k)}$ (2.11)
is the sum of (even/odd) R-R field strengths (in IIA/IIB) and we have
introduced the $H$-twisted exterior derivative
$\text{d}_{H}:=\text{d}+H\wedge$ (2.12)
which acts on polyforms. Notice that $\text{d}_{H}^{2}=0$ by the Bianchi
identity $\text{d}H=0$. The R-R field strengths are subject to the same self-
duality constraint of $\Psi$: $F=*\lambda(F)$.
Not only the supergravity bosonic fields are invariant under the
diffeomorphism generated by $K$, but also the supersymmetric spinorial
structure. This implies in particular that
${\cal L}_{K}\omega=0\,,\quad{\cal L}_{K}\Psi=0$ (2.13)
as can be derived from (2.5) and (2.10) by using the Bianchi identities.
Notice that the differential conditions reviewed above are necessary but not
generically sufficient for guaranteeing the background to be supersymmetry.
Rather, they must be supplemented by other conditions [30]. However, the
differential conditions above are the only ones that we will need when
discussing fundamental strings and D-branes.
In discussing D-branes, it will be much more natural to use the language of
generalized geometry [15, 16], see [17] for a review. In particular, the
combination
${\cal K}=K+\omega$ (2.14)
can be seen as a generalized vector in $TM\oplus T^{*}M$, known as generalized
tangent bundle of the ten-dimensional manifold $M$. On this there exists a
natural inner product defined by
${\cal I}(X,X)=\iota_{v}\eta\qquad\text{for}\qquad X=v+\eta\in TM\oplus
T^{*}M$ (2.15)
which induces a canonical $O(10,10)$ structure on $TM\oplus T^{*}M$. Polyforms
can be seen as $O(10,10)$ spinors and the Clifford action of a generalized
vector $X=v+\eta$ on them is given by the operator
$X\cdot:=\iota_{v}+\eta\wedge$ (2.16)
Furthermore, by using the ordinary metric $g$, one can introduce the
generalized metric444We are using the SO(10,10) frame in which there are no
off-diagonal elements in ${\cal G}$. They can be obtained by a general
$b$-twist which would however shift $H\rightarrow H-\text{d}b$ – see for
instance [17].
${\cal G}=\left(\begin{array}[]{cc}g&0\\\ 0&g^{-1}\end{array}\right)$ (2.17)
which further reduces the generalized structure group to $SO(1,9)\times
SO(1,9)$.
${\cal K}$ is null in both ${\cal I}$ and ${\cal G}$:
${\cal I}({\cal K},{\cal K})={\cal G}({\cal K},{\cal K})=0$ (2.18)
Furthermore, $\Psi$ can be seen as a spinor in $TM\oplus T^{*}M$ and (2.9) can
be translated by saying that $\Psi$ is annihilated by the Clifford action of
${\cal K}$:
${\cal K}\cdot\Psi=0$ (2.19)
By adopting this viewpoint, one can write (2.10) as
$\text{d}_{H}(e^{-\phi}\Psi)=-{\cal K}\cdot F$ (2.20)
and check that ${\cal K}$ generates a generalized isometry which preserves the
bosonic background as well as the spinorial structure. For instance
$\displaystyle{\cal L}^{H}_{\cal K}(e^{-\phi}\Psi)=0$ (2.21)
$\displaystyle{\cal L}^{H}_{\cal K}F={\cal K}\cdot\text{d}_{H}F=0$ (2.22)
where we have used the Bianchi identity $\text{d}_{H}F=0$ – we will later come
back on the effect of localized sources – and the generalized Lie derivative
on forms is defined by
${\cal L}^{H}_{\cal K}:=\\{\text{d}_{H},{\cal K}\\}$ (2.23)
In conclusion, ${\cal K}$ can be considered as a generalized Killing vector.
${\cal K}$ and $\Psi$ will play an important role when discussing the
energetics of D-branes.
## 3 Fundamental strings
In this section we consider fundamental (F1) strings in general supersymmetric
backgrounds, by using the framework described in section 2. We study the
supersymmetry conditions and we derive the associated BPS bound and its
relation to background calibrations. This section provides also a useful warm-
up for the discussion on D-branes presented in section 4, which will follow a
path which is conceptually identical although slightly more difficult from the
technical point of view. However, this section and section 4 are almost
completely independent and the reader more interested on D-branes may safely
jump directly to section 4.
Let us briefly review some basic facts about the F1 world-volume theory,
basically to fix our notation and conventions. The action is given by
$S_{\rm F1}=-\mu_{\rm F1}\int_{\cal S}\text{d}^{2}\xi\sqrt{-\det g|_{\cal
S}}\,+\mu_{\rm F1}\int_{\cal S}B$ (3.1)
where $\xi^{\alpha}$ are world-sheet coordinates, $\mu_{\rm
F1}=1/(2\pi\alpha^{\prime})$, ${\cal S}$ is the two-dimensional submanifold
wrapped by the F1-string and $g|_{\cal S}$ refers to the pull-back of bulk
metric to ${\cal S}$. In the following we will sometimes use an economic
notation for pulled-back tensors in components, leaving implicit the pull-back
operation $|_{\cal S}$. For instance, $B_{\alpha\beta}\equiv
B_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\equiv(B|_{\cal
S})_{\alpha\beta}$.
It will result useful to explicitly split the action in terms of the Nambu-
Goto and Wess-Zumino terms: $S=\int\text{d}^{2}\xi\,{\cal
L}=\int\text{d}^{2}\xi({\cal L}_{\rm NG}+{\cal L}_{\rm WZ})$, with
${\cal L}_{\rm NG}=-\mu_{\rm F1}\sqrt{-\det g|_{\cal S}}\,,\qquad{\cal L}_{\rm
WZ}=\frac{\mu_{\rm F1}}{2}\,\epsilon^{\alpha\beta}B_{\alpha\beta}$ (3.2)
### 3.1 Geometric supersymmetry conditions
We now study the conditions that need to be imposed on supersymmetric
fundamental strings and how they can be written in terms of the background
tensors characterizing the bulk supersymmetry.
Let us start from the supersymmetry condition in spinorial form. This reads
$\Gamma_{\rm F1}\epsilon=\epsilon$ (3.3)
where $\epsilon$ is the doubled-spinor introduced in (2.1) and $\Gamma_{\rm
F1}$ is the $\kappa$-symmetry operator [33]
$\Gamma_{\rm F1}=\frac{1}{2\sqrt{-\det g|_{\cal
S}}}\,\epsilon^{\alpha\beta}\Gamma_{\alpha\beta}\otimes\sigma_{3}$ (3.4)
where, according to our general conventions,
$\Gamma_{\alpha\beta}=\Gamma_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}$.
Our aim is to rewrite (3.3) in a more geometrical way. This is readily
obtained just by observing that (3.3) implies that
$\bar{\epsilon}\,\Gamma^{M}\Gamma_{\rm
F1}\epsilon=\bar{\epsilon}\,\Gamma^{M}\epsilon$. In turn, by using the
definitions (2.2) and (2.4), it is easy to see that this latter condition can
be written as
$(\text{d}X^{M}\wedge\omega)|_{\cal S}=K^{M}\sqrt{-\det g|_{\cal
S}}\,\text{d}^{2}\xi$ (3.5)
We then find that (3.5) is a necessary condition for the F1-string to be
supersymmetric. On the other hand, we will show in the following sections that
(3.5) is in fact also sufficient for it. Hence, (3.5) is completely equivalent
to (3.3) and can be taken as defining supersymmetric F1-strings.555Actually,
as we will see, (3.5) is redundant and only one particular ‘component’
provides the minimal necessary and sufficient condition for the F1 to be BPS.
We can write (3.5) in an alternative way. Consider the vector field
$V=V^{M}\partial_{M}$ defined by
$V^{M}=\frac{\mu_{{\rm F}1}}{2\sqrt{-\det g|_{\cal
S}}}\,\epsilon^{\alpha\beta}(\text{d}X^{M}\wedge\omega)_{\alpha\beta}$ (3.6)
$V$ is tangent to the F1 world-sheet, $V\in T{\cal S}\subset(TM)|_{\cal S}$.
Hence the supersymmetry condition (3.5) states that $V$ must be equal to the
Killing vector:
$V=K\qquad\text{(on ${\cal S}$)}$ (3.7)
Cleary, (3.7) implies that $K$ is tangent to the world-sheet ${\cal S}$ of
supersymmetric F1-strings. This means that ${\cal S}$ does not break the
symmetry generated by $K$, as expected. This condition can be alternatively
formulated by saying that the embedding ${\cal S}$ is a stationary
configuration with respect to the ‘time’ $t$ (which can actually be either
time-like or light-like) generated by $K=\partial_{t}$. In particular, (3.5)
implies that for supersymmetric strings
$(\text{d}t\wedge\omega)|_{\cal S}=\sqrt{-\det g|_{\cal S}}\,\text{d}^{2}\xi$
(3.8)
Hence, for supersymmetric F1-strings we can write
$S_{\rm F1}=-\mu_{\rm F1}\int_{\cal S}(\text{d}t\wedge\omega-B)$ (3.9)
On general grounds, supersymmetric configurations are expected to minimize
some BPS-like lower bound on some appropriately defined energy. This is indeed
true, as discussed in the following sections.
### 3.2 Conserved current and charge
We first recall some well known results about symmetries and conserved
currents.
Noether’s theorem states that one can construct a conserved current associated
with any symmetry under which the action is invariant. More explicitly, take a
generic Lagrangian ${\cal L}(\phi^{I},\partial\phi^{I})$ and suppose that it
is invariant, up to a total derivative $\partial_{\alpha}\Lambda^{\alpha}$,
under an infinitesimal deformation $\delta\phi^{I}$:
$\delta{\cal L}=\partial_{\alpha}\Lambda^{\alpha}$ (3.10)
Notice that $\Lambda^{\alpha}$ is defined up to divergence-less additional
terms.
Then, one can construct the current
$j^{\alpha}=-\delta\phi^{I}\,\frac{\partial{\cal
L}}{\partial(\partial_{\alpha}\phi^{I})}+\Lambda^{\alpha}$ (3.11)
which is conserved, $\partial_{\alpha}j^{\alpha}=0$ (on-shell).
Let us now apply Noether’s theorem to our problem. Take the transformation
$\delta_{K}x^{M}=K^{M}$ (3.12)
which corresponds to shifting the fundamental string in the direction of the
Killing vector $K$ defined in (2.2). Consider first ${\cal L}_{\rm NG}$, see
(3.2). Being ${\cal L}_{K}g=0$, we obviously have $\delta_{K}{\cal L}_{\rm
NG}=0$. Hence, we can construct the following associated current
$j^{\alpha}_{\rm NG}=\mu_{\rm F1}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\beta}K_{\beta}$ (3.13)
Consider now ${\cal L}_{\rm WZ}$. First notice that, by (2.5), we have
$\displaystyle{\cal L}_{K}B$ $\displaystyle=$
$\displaystyle\iota_{K}H+\text{d}(\iota_{K}B)$ (3.14) $\displaystyle=$
$\displaystyle\text{d}(\iota_{K}B-\omega)$ (3.15)
Hence, we can still apply Noether’s procedure but now we have to be careful
since ${\cal L}_{\rm CS}$ is only invariant up to a total derivative.
Comparing with (3.10), we can take
$\Lambda^{\alpha}_{\rm WZ}=\mu_{\rm
F1}\,\epsilon^{\alpha\beta}(\iota_{K}B-\omega)_{\beta}$ (3.16)
Hence, the corresponding current is given by
$\displaystyle j^{\alpha}_{\rm WZ}$ $\displaystyle=$ $\displaystyle-\mu_{\rm
F1}\,\epsilon^{\alpha\beta}(\iota_{K}B)_{\beta}+\Delta^{\alpha}_{\rm WZ}$
(3.17) $\displaystyle=$ $\displaystyle-\mu_{\rm
F1}\,\epsilon^{\alpha\beta}\omega_{\beta}$ (3.18)
Let us stress that, in deriving $j^{\alpha}_{\rm WZ}$, we have crucially used
the background supersymmetry condition (2.5).
We can then conclude that the conserved current associated with the world-
volume symmetry induced by the bulk Killing vector $K$ is given by the sum of
$j^{\alpha}_{\rm NG}$ and $j^{\alpha}_{\rm WZ}$:
$j_{\rm F1}^{\alpha}=j^{\alpha}_{\rm NG}+j^{\alpha}_{\rm WZ}=\mu_{\rm
F1}\big{(}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\beta}K_{\beta}-\epsilon^{\alpha\beta}\omega_{\beta}\big{)}$
(3.19)
Suppose now to split the world-sheet coordinates as
$\xi^{\alpha}=(\tau,\sigma)$. More precisely, consider world-sheet embeddings
which have structure ${\cal S}=\mathbb{R}\times\Sigma$, where $\mathbb{R}$
parametrizes a time-like world-sheet coordinate $\tau$ whereas $\Sigma$ is a
space-like one-dimensional submanifold parametrized by the space-like
coordinate $\sigma$. Hence, we can use the conserved current $j^{\alpha}_{\rm
F1}$ to define the conserved charge
$\displaystyle\Delta_{\rm F1}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\text{d}\Sigma_{\alpha}\,j^{\alpha}_{\rm
F1}\equiv\int_{\Sigma}\text{d}\sigma\,j^{\tau}_{\rm F1}$ (3.20)
$\displaystyle=$
$\displaystyle-\int_{\Sigma}\big{[}\text{d}\sigma\,g(P,K)+\mu_{\rm
F1}\,\omega\big{]}$ (3.21)
where we have introduced the vector
$P^{M}=-\mu_{\rm F1}\sqrt{-\det g|_{\cal
S}}\,g^{\alpha\tau}\partial_{\alpha}x^{M}\qquad\Rightarrow\qquad j^{\tau}_{\rm
NG}=-g(P,K)$ (3.22)
$P^{M}$ is a vector tangent to ${\cal S}$ and can be identified with the
‘kinematical’ gauge-invariant momentum conjugated to the embedding field
$x^{M}(\xi)$.
### 3.3 Supersymmetry and BPS bound
Experience with supersymmetric theories leads us to expect that the energy
obeys a BPS lower bound which is saturated exactly when the F1-string is
supersymmetric. Standard arguments are usually applied to static situations
and lead to the identification of various kinds of background calibrations,
which naturally realize the BPS bound.
On the other hand, in our analysis we are considering the most generic
supersymmetric background and F1-string configuration. Hence, we need to
extend the usual approach to calibrations. As we now show, such a
generalization is actually very natural. Furthermore, our procedure
straightforwardly generalizes to other branes, as we will explicitly see for
the case of D-branes.
Let us start again from the supersymmetry condition (3.3). By applying the
split ${\cal S}=\mathbb{R}\times\Sigma$ we can write
$\Gamma_{\rm F1}=\frac{1}{\mu_{\rm F1}\sqrt{g|_{\Sigma}}}\not{P}\gamma_{\rm
F1}=-\frac{1}{\mu_{\rm F1}\sqrt{g|_{\Sigma}}}\gamma_{\rm F1}\not{P}$ (3.23)
where $\not{P}=P^{M}\Gamma_{M}$, with $P^{M}$ defined in (3.22), and
$\gamma_{\rm F1}=\frac{1}{\mu_{\rm
F1}\sqrt{g|_{\Sigma}}}\,\Gamma_{\sigma}\otimes\sigma_{3}$ (3.24)
Furthermore, notice that if the string is not collapsed then $C\not{P}$ is a
positive definite matrix, in the sense that
$\chi^{T}C\not{P}\chi\equiv\bar{\chi}\not{P}\chi>0$ (3.25)
for any (non-vanishing) doubled Majorana spinor $\chi$. Indeed, from the
definition (3.22) it is easy to see that $P$ is time-like, $g(P,P)=-\mu_{\rm
F1}^{2}g_{\sigma\sigma}$, whenever the string is not collapsed. In the real
representation that we are using, the charge-conjugation matrix is simply
given by $C=\Gamma^{\underline{0}}$. Being $P$ future pointing by
construction, we can locally choose a frame $e_{\underline{M}}$ such that
$e_{\underline{0}}\propto P$, so that
$C\not{P}=P^{\underline{0}}\mathbbm{1}>0$, which immediately leads to (3.25).
Clearly, such a conclusion cannot depend on the choice of frame and then is
valid in general.
We can now apply the general inequality (3.25) to the spinor
$\chi=(\mathbbm{1}-\Gamma_{\rm F1})\epsilon$ (3.26)
Using $\Gamma^{T}_{\rm F1}=-C^{-1}\Gamma_{\rm F1}C$, a few manipulations allow
to rewrite (3.25) with (3.26) as
$\bar{\epsilon}\not{P}\epsilon\,\geq\,\bar{\epsilon}\not{P}\Gamma_{\rm
F1}\epsilon$ (3.27)
In turn, by taking into account (3.23) and the definitions (2.2) and (2.4),
this can be rewritten in more geometric way as
$-g(P,K)\,\text{d}\sigma\geq\mu_{\rm F1}\,\omega|_{\Sigma}$ (3.28)
which should be read $-g(P,K)\geq\mu_{\rm F1}\omega_{\sigma}$. Recalling
(3.20), we see that (3.28) provides exactly the expected local bound
$\Delta_{\rm F1}\geq 0$ (3.29)
On the other hand, it is clear by construction that (3.29) is saturated if and
only if the F1-string satisfies the supersymmetry condition (3.3). Hence, the
supersymmetry condition can be alternatively written as
$\text{SUSY
F1-string}\qquad\Leftrightarrow\qquad-g(P,K)\,\text{d}\sigma=\mu_{\rm
F1}\,\omega|_{\Sigma}$ (3.30)
We then conclude that $\omega$ can be seen as the proper generalization of the
more standard calibrations for static backgrounds and static world-sheet
configurations. In particular, (3.30) substitutes the usual calibration bound
for minimally coupled static branes [5], where the the l.h.s. simply reduces
to the volume density.
Notice that in order to interpret the algebraic bound (3.29) in terms of a
conserved charge, a key role is played by the construction of the currents
$j^{\alpha}_{\rm NG}$ and $j^{\alpha}_{\rm WZ}$, which in turn uses
differential background conditions. In particular for $j^{\alpha}_{\rm WZ}$ we
have crucially used the bulk supersymmetry condition (2.5). Without such
condition $S_{\rm WZ}$ would not be invariant and no current could be
constructed. In a sense, consistency of the F1 world-sheet theory with
supersymmetry demands that the background fields $\omega$ and $K$ must be
related by the differential condition (2.5).
The discussion followed in this section makes explicit use of the time-plus-
space split, but can be easily related to the covariant condition presented in
(3.5). Indeed, (3.30) can be obtained as the equation resulting from the
contraction of (3.5) with $P$. In terms of the vector $V$ introduced in (3.6),
the BPS condition can be written as $g(P,K)=g(P,V)$ and the BPS bound (3.29)
can be rephrased as
$|g(P,K)|\geq|g(P,V)|$ (3.31)
We then see that, as already anticipated, the conditions encoded in (3.5) are
not just necessary but also sufficient for characterizing BPS F1-strings.
Actually, just the projection of $(\ref{F1form2})$ along $P$ is sufficient and
the remaining components in the orthogonal directions follow from this.
### 3.4 BPS bound and energetics
In order to relate the above analysis to a more standard Hamiltonian
treatment, pick-up a field $B$ such that $\text{d}\iota_{K}B=-\iota_{K}H=0$
or, in other words, ${\cal L}_{K}B=0$. This is consistent with the fact that
${\cal L}_{K}H=0$ and $\iota_{K}H$ is exact, as required by the supersymmetry
condition (2.5). In this case $\delta_{K}{\cal L}_{\rm F1}$ is exactly
invariant and we can apply Noether’s procedure with $\Lambda^{\alpha}_{\rm
CS}\equiv 0$.666This particular choice of $B$ is equivalent to correcting
$\Lambda^{\alpha}_{\rm F1}$ given in (3.16) by a divergence-less term which
make it vanish – see comment after (3.10). It is then easy to see that the
conserved charge is simply given by $-g(\hat{P},K)$, the component along $K$
of the canonical momentum
$\hat{P}_{M}=\frac{\partial{\cal L}_{\rm
F1}}{\partial(\partial_{\tau}x^{M})}=P_{M}+\mu_{\rm F1}(\iota_{M}B)|_{\Sigma}$
(3.32)
which, compared to the gauge-invariant momentum $P_{M}$, incorporates the
effect of the WZ-term too. Hence, we get the conserved energy
$E_{\rm F1}:=-\int_{\Sigma}\text{d}\sigma\,g(\hat{P},K)$ (3.33)
The local bound (3.28) can now be translated into a BPS bound on the energy
$E_{\rm F1}$
$E_{\rm F1}\geq E^{\rm BPS}_{\rm F1}$ (3.34)
where
$E^{\rm BPS}_{\rm F1}:=\int_{\Sigma}(\omega-\iota_{K}B)$ (3.35)
Notice that, thanks to the background supersymmetry condition (2.5), $E^{\rm
BPS}_{\rm F1}$ is a topological quantity which does not change under
continuous deformations of $\Sigma$. Indeed, suppose that $\Sigma$ can be
deformed to $\Sigma^{\prime}=\Sigma+\partial{\cal C}$, for some chain ${\cal
C}$. Hence, by using (2.5) and the fact that $\text{d}\iota_{K}B=-\iota_{K}H$,
it is easy to see that
$\int_{\Sigma^{\prime}}(\omega-\iota_{K}B)=\int_{{\cal
C}}[\text{d}\omega-\text{d}(\iota_{K}B)]+\int_{\Sigma}(\omega-\iota_{K}B)=\int_{\Sigma}(\omega-\iota_{K}B)$
(3.36)
In this sense, $E^{\rm BPS}_{\rm F1}$ can be seen as the central charge which
extremizes the energy canonically associated with the Killing vector $K$. In
fact, by (2.5) we may pick-up $B$ such that $\iota_{K}B=\omega$. With this
choice, $E_{\rm F1}\equiv\Delta_{\rm F1}$ and $E^{\rm BPS}_{\rm F1}\equiv 0$.
By imposing a partial static gauge in which $\tau$ is identified with the
(time- or light-like) coordinate $t$ generated by $K$, $-g(\hat{P},K)$
coincides with the canonical Hamiltonian density. In this case $E_{\rm F1}$
would indeed coincide with the canonical energy, along the lines of what
happens for (generalized) calibrations [5] for static backgrounds and branes.
See also [7, 8, 9] for analogous conclusions obtained by starting from the
local type II supersymmetry algebra.
## 4 D-branes
We now turn to D-branes. The line we will follow is practically identical to
one for F1-strings, up to some technical complication due to the more
sophisticated D-brane world-volume theory.
The effective theory governing the bosonic sector of a D$p$-brane wrapping a
$(p+1)$-cycle ${\cal S}$ is given by Dirac-Born-Infeld (DBI) and Chern-Simons
(CS) terms
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int_{\cal
S}\text{d}^{p+1}\xi\,{\cal L}_{\rm DBI}+\int_{\cal S}\text{d}^{p+1}\xi\,{\cal
L}_{\rm CS}$ (4.1) $\displaystyle=$ $\displaystyle-\mu_{{\rm D}p}\int_{\cal
S}\text{d}^{p+1}\xi e^{-\phi}\sqrt{-\det(g|_{\cal S}+{\cal F})}+\mu_{{\rm
D}p}\int_{\cal S}C\wedge e^{{\cal F}}$ (4.2)
where $\mu_{{\rm D}p}=(2\pi)^{p}(\alpha^{\prime})^{-\frac{p+1}{2}}$ and ${\cal
F}$ is the gauge-invariant world-volume field-strength, which satisfies the
Bianchi identity $\text{d}{\cal F}=H|_{\cal S}$.
As we will see, generalized geometry will naturally enter the description and
will be useful to make more manifest the formal analogy between D-branes and
F1-strings.
### 4.1 Geometric supersymmetry conditions
As in the discussion for fundamental strings, let us start from the
supersymmetry conditions for a D$p$-brane in its spinorial form (see e.g. [34,
32])
$\Gamma_{{\rm D}p}\epsilon=\epsilon$ (4.3)
where $\Gamma_{{\rm D}p}$ acts on type II doubled-spinors as
$\Gamma_{{\rm D}p}=\left(\begin{array}[]{cc}0&\hat{\Gamma}_{{\rm D}p}\\\
\hat{\Gamma}^{-1}_{{\rm D}p}&0\end{array}\right)$ (4.4)
with
$\hat{\Gamma}_{\text{D}p}=\frac{1}{\sqrt{-\det(g|_{\cal S}+{\cal
F})}}\sum_{2l+s=p+1}\frac{\epsilon^{\alpha_{1}\ldots\alpha_{2l}\beta_{1}\ldots\beta_{s}}}{l!s!2^{l}}{\cal
F}_{\alpha_{1}\alpha_{2}}\cdots{\cal
F}_{\alpha_{2l-1}\alpha_{2l}}\Gamma_{\beta_{1}\ldots\beta_{s}}\ .$ (4.5)
and $\hat{\Gamma}^{-1}_{{\rm D}p}=(-)^{p}C^{-1}\hat{\Gamma}^{T}_{{\rm D}p}C$.
As for fundamental strings, we would like now to re-express (4.3) in a more
geometrical way, which highlights its physical implications.
These can be obtained by noticing that the spinorial supersymmetry condition
(4.3) implies in particular $\bar{\epsilon}(\Gamma^{M}\Gamma_{{\rm
D}p})\epsilon=\bar{\epsilon}\,\Gamma^{M}\epsilon$ and
$\bar{\epsilon}(\Gamma_{M}\sigma_{3}\Gamma_{{\rm
D}p})\epsilon=\bar{\epsilon}(\Gamma_{M}\sigma_{3})\epsilon$, which can be
translated as follows in terms of the bulk structures $K$, $\omega$ and
$\Psi$:
$\displaystyle[(\text{d}X^{M}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}$ $\displaystyle=$ $\displaystyle K^{M}\sqrt{-\det(g|_{\cal S}+{\cal
F})}\,\text{d}^{p+1}\xi$ (4.6) $\displaystyle[(\iota_{M}\Psi)|_{\cal S}\wedge
e^{\cal F}]_{\rm top}$ $\displaystyle=$
$\displaystyle\omega_{M}\sqrt{-\det(g|_{\cal S}+{\cal F})}\,\text{d}^{p+1}\xi$
(4.7)
We then obtain that (4.6) are necessary conditions for the D-brane to be
supersymmetric. In fact, as it will be clear from the discussion of the
following sections, (4.6) are also sufficient and then (4.6) provides a
complete, actually redundant, set of conditions which a supersymmetric D-brane
must satisfy.
With the help of generalized geometry one can rewrite (4.6) in more inspiring
way. First take the following generalized vector in $(TM\oplus T^{*}M)|_{\cal
S}$:
${\cal V}=\frac{\epsilon^{\alpha_{0}\ldots\alpha_{p}}}{\sqrt{-\det{\cal
M}}}\big{\\{}[(\text{d}x^{M}\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{\alpha_{0}\ldots\alpha_{p}}\partial_{M}+[(\iota_{M}\Psi)|_{\cal S}\wedge
e^{\cal F}]_{\alpha_{0}\ldots\alpha_{p}}\text{d}x^{M}\big{\\}}$ (4.8)
where we have introduced the shorthand notation
${\cal M}:=g|_{\cal S}+{\cal F}$ (4.9)
This is the D-brane counterpart of the vector $V$ introduced in (3.6) for
F1-strings. It is easy to see that ${\cal V}$ is tangent to $({\cal S},{\cal
F})$ in the generalized sense, i.e. it takes values in the D-brane generalized
tangent bundle $T_{({\cal S},{\cal F})}$, which is defined as [16]
$T_{({\cal S},{\cal F})}=\\{v+\eta\in T{\cal S}\oplus T^{*}M|_{\cal
S}\,:\,\eta|_{\cal S}=\iota_{v}{\cal F}\\}$ (4.10)
Then, it is easy to see that the supersymmetry condition (4.6) is equivalent
to
${\cal V}={\cal K}$ (4.11)
where ${\cal K}$ is given in (2.14). It is clear that (4.11) necessary
requires that
${\cal K}\in T_{({\cal S},{\cal F})}$ (4.12)
This condition can be interpreted as the condition that the D-brane is
stationary with respect to ${\cal K}$. More in detail, this incorporates the
following two conditions. First, $K$ is tangent to ${\cal S}$, $K\in T{\cal
S}$, and then ${\cal S}$ is invariant under the symmetry generated by $K$.
Second, $K$ identifies an naturally selected electric flux component ${\cal
E}=\iota_{K}{\cal F}$, which is completely fixed in terms of background one-
form $\omega$:
${\cal E}=\omega|_{\cal S}$ (4.13)
In particular, this implies that there cannot be supersymmetric electric field
${\cal E}=\iota_{K}{\cal F}\neq 0$ if the background supersymmetry has
$\omega=0$.777As an aside remark, let us mention that, although so far we have
treated D-branes as probes, the same conditions must hold in the case of
backreacting D-branes too. Notice in particular that the condition (4.12) for
all localized RR sources in the background is equivalent to the requirement
that ${\cal K}\cdot j_{\rm loc}=0$, where $j_{\rm loc}\simeq\delta({\cal
S})\wedge e^{-{\cal F}}$ is the current associated with backreacting D-branes.
The same current appears on the r.h.s of the R-R Bianchi identity
$\text{d}_{H}F=j_{\rm loc}$, implying (2.22) even in presence of
supersymmetric back-reacting D-branes. From the Bianchi identity
$\text{d}{\cal F}=H|_{\cal S}$ and the bulk equation (2.5) it immediately
follows that ${\cal F}$ is stationary too: ${\cal L}_{K}{\cal F}=0$.
Finally, notice that by introducing the ‘time’ $t$ (which, actually, can be
light-like) defined by $K=\partial_{t}$, the first of (4.6) implies the
following value for the D-brane action evaluated on the BPS configuration
$S^{\rm BPS}_{{\rm D}p}=-\mu_{{\rm D}p}\int_{\cal S}(\text{d}t\wedge
e^{-\phi}\Psi-C)\wedge e^{{\cal F}}$ (4.14)
In the next sections we will discuss the bounds associated with the BPS
condition.
### 4.2 Conserved current and charge
In order to discuss the the BPS bound associated with the supersymmetry
conditions we must identify the proper conserved current and charge that enter
the bound. Of course, these quantities are associated with the Killing vector
$K$.
The bulk symmetry generated by $K$ induces a symmetry of would-volume theory.
The action of such a symmetry on the embedding world-volume fields
$x^{M}(\xi)$ is given by
$\delta_{K}x^{M}=K^{M}$ (4.15)
Being ${\cal L}_{K}g=0$, the pulled-back metric $g|_{\cal S}$ is clearly
invariant under (4.15). On the other hand, $B$ is not in general invariant
under the symmetry generated by $K$, but rather we have (3.14). Hence ${\cal
F}$, which can be locally written as $\text{d}A+B|_{\cal S}$, is generically
not automatically invariant under (4.15). On the other hand, it should be
clear from (3.14) that we can compensate (4.15) by a transformation of the
world-volume gauge-field
$\delta_{K}A=(\omega-\iota_{K}B)|_{\cal S}$ (4.16)
so that $\delta_{K}{\cal F}=0$.
With this definition of $\delta_{K}A$, the DBI lagrangian ${\cal L}_{\rm DBI}$
in (4.1) is manifestly invariant under the symmetry generated by $K$:
$\delta_{K}{\cal L}_{\rm DBI}=0$. By following Noether’s procedure, recalled
in section 3.2, we can then construct the following current
$j^{\alpha}_{\rm DBI}=\mu_{{\rm D}p}\,e^{-\phi}\sqrt{-\det{\cal
M}}\,\big{(}{\cal M}^{(\beta\alpha)}K_{\beta}+{\cal
M}^{[\beta\alpha]}\omega_{\beta}\big{)}$ (4.17)
Take now the CS Lagrangian ${\cal L}_{\rm CS}$ in (4.1). By using (2.10), it
is possible to see that ${\cal L}_{\rm CS}$ tranforms into a total derivative
$\delta_{K}{\cal L}_{\rm CS}=\partial_{\alpha}\Lambda^{\alpha}_{\rm CS}$ under
(4.15) and (4.16), with
$\Lambda^{\alpha}_{\rm CS}=\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}[({\cal K}\cdot
C-e^{-\phi}\Psi)|_{\cal S}\wedge e^{{\cal
F}}\big{]}_{\beta_{1}\ldots\beta_{p}}$ (4.18)
where ${\cal K}$ is the generalized Killing vector defined in (2.14) and
${\cal K}\cdot$ acts as in (2.16). Then, Noether’s procedure gives the current
$j^{\alpha}_{\rm CS}=-\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}[(e^{-\phi}\Psi)|_{\cal
S}\wedge e^{{\cal F}}\big{]}_{\beta_{1}\ldots\beta_{p}}$ (4.19)
Notice that the bulk differential condition (2.10) has been crucial in
deriving this expression.
The total conserved current is then given by
$j^{\alpha}_{{\rm D}p}=j^{\alpha}_{\rm DBI}+j^{\alpha}_{\rm CS}$ (4.20)
Consider now a time-plus-space split ${\cal S}=\mathbb{R}\times\Sigma$, with
adapted coordinate $(\tau,\sigma^{a})$, $a=1,\ldots,p$, where $\Sigma$ is the
space-like surface (alias space-volume) spanning ${\cal S}$, and let us define
the split
${\cal F}=\text{d}\tau\wedge{\cal E}+{\cal F}_{\rm
mg}\quad\Rightarrow\quad{\cal F}_{\rm mg}:={\cal F}|_{\Sigma}$ (4.21)
in terms of the electric field ${\cal E}$ and the magnetic component ${\cal
F}_{\rm mg}$.
We can then construct the conserved charge
$\displaystyle\Delta_{{\rm D}p}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\text{d}\Sigma_{\alpha}\,j^{\alpha}_{{\rm
D}p}\equiv\int_{\Sigma}\text{d}^{p}\sigma\,j^{\tau}_{{\rm D}p}$ (4.22)
$\displaystyle=$ $\displaystyle-\int_{\Sigma}\text{d}^{p}\sigma\,{\cal
G}({\cal P},{\cal K})-\mu_{{\rm
D}p}\int_{\Sigma}(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}$ (4.23)
Here we have used the generalized metric defined in (2.17). Furthermore, in
the last line we have introduced the ‘generalized momentum’, which is a
generalized vector defined by
${\cal
P}:=P^{M}\partial_{M}+g_{MN}\partial_{\alpha}x^{N}\Pi^{\alpha}\text{d}x^{M}$
(4.24)
where
$\displaystyle P^{M}$ $\displaystyle:=$ $\displaystyle-\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal
M}^{-1})^{(\alpha\tau)}\partial_{\alpha}X^{M}$ (4.25)
$\displaystyle\Pi^{\alpha}$ $\displaystyle:=$ $\displaystyle-\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal M}^{-1})^{[\alpha\tau]}$ (4.26)
can be considered as the kinematical gauge-invariant momenta associated with
$x^{M}$ and $A_{\alpha}$. It is easy to check that ${\cal P}$ belongs to the
generalized tangent space $T_{({\cal S},{\cal F})}$ defined in (4.10) and,
moreover, for non collapsed D-branes it is always ‘time-like’ with respect to
the generalized metric ${\cal G}$:
${\cal G}({\cal P},{\cal
P})=P^{M}P_{M}+h_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}=-\mu^{2}_{p}\,e^{-2\phi}\det
m$ (4.27)
where
$m:=g|_{\Sigma}+{\cal F}_{\rm mg}$ (4.28)
The identity (4.27) corresponds to the superhamiltonian constraint originating
from the world-volume time-like diffeomorphism invariance.888The supermomentum
constraint, which is associated with the space-like diffeomorphism invariance,
can be equally elegantly written as ${\cal P}\in T^{\perp}_{(\Sigma,{\cal
F}_{\rm mg})}$, where $T^{\perp}_{(\Sigma,{\cal F}_{\rm mg})}$ is the
orthogonal complement of the generalized tangent bundle $T_{(\Sigma,{\cal
F}_{\rm mg})}$ in $(TM\oplus T^{*}M)|_{\cal S}$ with respect to the
generalized metric ${\cal G}$. Furthermore,
${\cal I}({\cal P},{\cal P})=P_{\alpha}\Pi^{\alpha}=0$ (4.29)
consistently the fact that ${\cal P}$ belongs to $T_{({\cal S},{\cal F})}$,
which is maximally isotropic with respect to ${\cal I}$.
We see that by using the language of generalized geometry the conserved charge
$\Delta_{{\rm D}p}$ has a structure very similar to the corresponding
conserved charge for F1-strings found in (3.20). As explained in section 3.3,
$\omega$ can be interpreted as a calibration for fundamental strings. Indeed,
a similar interpretation holds for $e^{-\phi}\Psi$ too, as we will presently
explain.
### 4.3 Supersymmetry and BPS bound
We start by considering the following operator acting on bi-spinors
$\not{{\cal
P}}:=P^{M}\Gamma_{M}+\Pi^{\alpha}\Gamma_{\alpha}\sigma_{3}=\left(\begin{array}[]{cc}P^{M}\Gamma_{M}+\Pi^{\alpha}\Gamma_{\alpha}&0\\\
0&P^{M}\Gamma_{M}-\Pi^{\alpha}\Gamma_{\alpha}\end{array}\right)$ (4.30)
The notation has been chosen on purpose, in order to highlight the analogy
with the analysis for F1-strings. By using (4.27) and (4.29), one can
immediately conclude that the vector fields
$P^{M}\pm\Pi^{\alpha}\partial_{\alpha}x^{M}$ are time-like and future-ponting
which, by the same reasoning presented in section 3.3, implies that
$\chi(C\not{{\cal P}})\chi>0$ (4.31)
for any non-vanishing doubled-spinor $\chi$. We can now apply this inequality
to
$\chi=(1-\Gamma_{\rm Dp})\epsilon$ (4.32)
in order to obtain a bound which is saturated exactly by supersymmetric
D-branes, i.e. when (4.3) is satisfied.
The resulting bound can be written in physically more meaningful form by
considering a time-plus-space split ${\cal S}=\mathbb{R}\times\Sigma$, with
adapted coordinates $\xi^{\alpha}=(\tau,\sigma^{a})$, $a=1,\ldots,p$. Observe
that
$\Gamma_{{\rm D}p}=\frac{1}{\mu_{{\rm D}p}\,e^{-\phi}\sqrt{\det
m}}\,\not{{\cal P}}\,\gamma_{{\rm D}p}=\frac{(-)^{p}}{\mu_{{\rm
D}p}\,e^{-\phi}\sqrt{\det m}}\,\gamma_{{\rm D}p}\,\not{{\cal P}}$ (4.33)
where
$\displaystyle\gamma_{\text{D}p}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&\hat{\gamma}_{{\rm D}p}\\\
(-)^{p+1}\hat{\gamma}^{-1}_{{\rm D}p}&0\end{array}\right)$ (4.36) with
$\displaystyle\hat{\gamma}_{\text{D}p}=\frac{1}{\sqrt{\det
m}}\sum_{2l+s=p}\frac{\epsilon^{a_{1}\ldots a_{2l}b_{1}\ldots
b_{s}}}{l!s!2^{l}}{\cal F}_{a_{1}a_{2}}\cdots{\cal
F}_{a_{2l-1}a_{2l}}\Gamma_{b_{1}\ldots b_{s}}\ .$ (4.37)
One can then rewrite the inequality (4.31) with $\chi$ given in (4.32) as
$\bar{\epsilon}\,\not{{\cal P}}\epsilon\geq\bar{\epsilon}\,\not{{\cal
P}}\Gamma_{{\rm D}p}\epsilon$ (4.38)
which, by (4.33), in turn translates into the following bound involving the
world-volume fields and the background polyform $\Psi$
$-{\cal G}({\cal P},{\cal K})\geq\frac{\mu_{{\rm D}p}}{p!}\epsilon^{\tau
a_{1}\ldots a_{p}}[(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{a_{1}\ldots a_{p}}$ (4.39)
or, in more intrinsic notation,
$-{\cal G}({\cal P},{\cal K})\,\text{d}^{p}\sigma\geq\mu_{{\rm
D}p}[(e^{-\phi}\Psi)|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}]_{(p)}$ (4.40)
This bound is saturated if and only if the D-brane is supersymmetric
$\text{SUSY D-brane}\quad\Leftrightarrow\quad-{\cal G}({\cal P},{\cal
K})\,\text{d}^{p}\sigma=\mu_{{\rm D}p}[(e^{-\phi}\Psi)|_{\Sigma}\wedge
e^{{\cal F}_{\rm mg}}]_{(p)}$ (4.41)
Recalling (4.22) we see that (4.39) translates into the lower bound on the
conserved charge $\Delta_{{\rm D}p}$ associated with the symmetry $K$:
$\Delta_{{\rm D}p}\geq 0$ (4.42)
This is exactly what we expected from the discussion of section 4.2! D-branes
preserving the supersymmetry $\epsilon$ are exactly those that minimize the
conserved charge $\Delta_{{\rm D}p}$. This is in line with what we expect from
our experience on static backgrounds. In particular $e^{-\phi}\Psi$ can be
seen as an extension of the generalized calibrations of [11, 12, 13, 14] to
the most general supersymmetric setting.
Let us stress once again that the bulk differential condition (2.10) has been
crucial for deriving this current. As it happened for fundamental strings, a
consistent D-brane world-volume theory seems to ‘know’ about (at least part
of) the bulk differential conditions.
Finally, notice that the supersymmetry condition (4.41) is obtained by
projecting (4.6) along ${\cal P}$ by using the metric ${\cal G}$. This proves
the statement anticipated in section 4.1 that the conditions (4.6) are not
only necessary but actually also sufficient for a D-brane to be
supersymmetric. Moreover, the reformulation in terms of the generalized vector
${\cal V}$ defined in (4.8) allows us to write the local bound (4.41) as
$|{\cal G}({\cal P},{\cal K})|\geq|{\cal G}({\cal P},{\cal V})|$ (4.43)
whose saturation reproduces the supersymmetry condition in the form (4.11).
### 4.4 BPS bound and energetics
As done for fundamental strings, in order to give a Hamiltonian interpretation
of the above results, let us pick-up potential forms $B$ and $C$ that preserve
the symmetry generated by $K$, namely ${\cal L}_{K}B=0$ and ${\cal L}_{K}C=0$.
In this case the complete Lagrangian ${\cal L}_{{\rm D}p}$ is exactly
invariant under the symmetry generated by $K$ and the conserved charge density
is $-K^{M}\hat{P}_{M}$, with
$\displaystyle\hat{P}_{M}:=\frac{\partial{\cal L}_{{\rm
D}p}}{\partial(\partial_{\tau}x^{M})}$ $\displaystyle=$ $\displaystyle
P_{M}-\mu_{{\rm D}p}e^{-\phi}\sqrt{-\det{\cal M}}\,({\cal
M}^{-1})^{[\alpha\tau]}B_{MN}\partial_{\alpha}X^{N}$ (4.45)
$\displaystyle+\frac{\mu_{{\rm
D}p}}{p!}\,\epsilon^{\tau\alpha_{1}\ldots\alpha_{p}}\big{[}\iota_{M}(C\wedge
e^{B})\wedge e^{{\cal F}-B}\big{]}_{\alpha_{1}\ldots\alpha_{p}}$
The conserved charge is then given by the energy
$E_{{\rm D}p}=-\int_{\Sigma}\text{d}^{p}\sigma\,g(\hat{P},K)$ (4.46)
and the bound (4.42) translates into
$E_{{\rm D}p}\geq E^{\rm BPS}_{{\rm D}p}$ (4.47)
with
$E^{\rm BPS}_{{\rm
D}p}=\int_{\Sigma}\text{d}^{p}\sigma\,\hat{\Pi}^{a}(\omega-\iota_{K}B)_{a}+\mu_{{\rm
D}p}\,\int_{\Sigma}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge e^{{\cal F}_{\rm
mg}}$ (4.48)
where we have introduced the momenta conjugated to the gauge field
$\hat{\Pi}^{\alpha}:=\frac{\partial{\cal L}_{{\rm
D}p}}{\partial(\partial_{\tau}A_{\alpha})}=\Pi^{\alpha}+\frac{\mu_{{\rm
D}p}}{(p-1)!}\,\epsilon^{\tau\alpha\beta_{1}\ldots\beta_{p-1}}\big{[}C\wedge
e^{{\cal F}_{\rm mg}}\big{]}_{\beta_{1}\ldots\beta_{p-1}}$ (4.49)
These satisy the constraint $\hat{\Pi}^{\tau}\equiv 0$ and the Gauss law
constraint
$\partial_{\alpha}\hat{\Pi}^{\alpha}\equiv\partial_{a}\hat{\Pi}^{a}=0$.
Remarkably, $E^{\rm BPS}_{{\rm D}p}$ is a topological quantity, which is
invariant under continuous deformations of the D-brane configuration. Let us
discuss this point in some detail.
First consider the first term on the r.h.s. of (4.48) and rewrite it as
$\mu_{\rm F1}\int_{\Sigma}(\omega-\iota_{K}B)|_{\Sigma}\wedge\rho_{\rm F1}$
(4.50)
where we have introduced the $(p-1)$-form $\rho_{\rm F1}$ defined by
$(\rho_{\rm F1})_{a_{1}\ldots
a_{p-1}}=2\pi\alpha^{\prime}\,\hat{\Pi}^{b}\,\epsilon_{ba_{1}\ldots a_{p-1}}$
(4.51)
As the notation suggests, $\rho_{\rm F1}$ corresponds to the density of
fundamental strings ‘dissolved’ on the world-space $\Sigma$. By the Gauss law
constraint $\partial_{a}\hat{\Pi}^{a}=0$, $\rho_{\rm F1}$ is closed and
defines a cohomology class on $\Sigma$ which, once integrally quantized, is
Poicaré dual to the space-like one-cycle $\Sigma_{\rm F1}\subset\Sigma$
wrapped by the dissolved fundamental strings. Hence, we allow for continuous
deformations of $\hat{\Pi}^{a}$ which does not change the homology class of
the dual strings, i.e. which correspond to deformations of $\rho_{\rm F1}$ by
an exact piece. By (2.5) and ${\cal L}_{K}B=0$, the one-form
$\omega-\iota_{K}B$ is closed and then we can write (4.51) as
$\mu_{\rm F1}\int_{\Sigma_{\rm F1}}(\omega-\iota_{K}B)$ (4.52)
Comparing with (3.35), we see that it clearly reproduces the contribution of a
BPS F1-string wrapping $\Sigma_{\rm F1}$. Repeating the arguments following
(3.35), (4.52) is clearly invariant under continuous deformations of
$\Sigma_{\rm F1}$ (as well as of its hosting cycle $\Sigma$) within the same
homology class.
Consider now the second term on the r.h.s. of (4.48) and notice that by ${\cal
L}_{k}C=0$ and $\text{d}_{H}C=F$ one can easily check that $\text{d}_{H}({\cal
K}\cdot C)=-{\cal K}\cdot F$ and then, by (2.10), we also have
$\text{d}_{H}(e^{-\phi}\Psi-{\cal K}\cdot C)=0$. This immediately implies that
by continuously deforming $(\Sigma,{\cal F}_{\rm mg})$ the second term on the
r.h.s. of (4.48) does not change. Indeed, consider a $(p+1)$-dimensional chain
$\tilde{\Sigma}$ supporting $\tilde{\cal F}$ and interpolating between two
configurations $(\Sigma,{\cal F}_{\rm mg})$ and $(\Sigma^{\prime},{\cal
F}^{\prime}_{\rm mg})$, i.e. such that $\partial{\cal
C}=\Sigma-\Sigma^{\prime}$, $\tilde{\cal F}|_{\Sigma}={\cal F}_{\rm mg}$ and
$\tilde{\cal F}|_{\Sigma^{\prime}}={\cal F}^{\prime}_{\rm mg}$. Then
$\displaystyle\int_{\Sigma}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge e^{{\cal
F}_{\rm mg}}$ $\displaystyle=$
$\displaystyle\int_{\Sigma^{\prime}}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge
e^{{\cal F}^{\prime}_{\rm mg}}+\int_{{\cal C}}\text{d}[(e^{-\phi}\Psi-{\cal
K}\cdot C)\wedge e^{\tilde{\cal F}}]$ (4.53) $\displaystyle=$
$\displaystyle\int_{\Sigma^{\prime}}(e^{-\phi}\Psi-{\cal K}\cdot C)\wedge
e^{{\cal F}^{\prime}_{\rm mg}}$ (4.54)
where in the last step we have used the Bianchi identity $\text{d}\tilde{\cal
F}=H|_{\cal S}$ and the bulk equation $\text{d}_{H}(e^{-\phi}\Psi-{\cal
K}\cdot C)=0$. Hence, the r.h.s. of (4.48) depends just on the generalized
homology class [13] of $(\Sigma,{\cal F}_{\rm mg})$ and is, in this sense,
topological.
In conclusion, we find that the energy of D-branes is (globally) minimized by
the supersymmetric configurations.
### 4.5 Gauss law, Bianchi identity and equations of motion
So far we have focused on the supersymmetry conditions for D-branes. On the
other hand, general on-shell D-brane configurations must satisfy a certain set
of equations of motion and Bianchi identities. Here we would like to discuss
which equations are automatic consequences of the supersymmetry condition and
which ones must be independently added.
The equations of motion associated with the embedding scalar fields are
automatically satisfied thanks to the energetics arguments implied by the BPS
bound discussed above. On the other hand, the story is not that simple when we
turn to the equations associated with the world-volume field-stregth ${\cal
F}$. These are the Bianchi identity999The delta-like $k$-form
$\delta^{(k)}(M)$ on a $n$-dimensional manifold $M$ associated with a
$(n-k)$-dimensional submanifold $N\subset M$ is defined by the identity
$\int_{M}\alpha\wedge\delta^{(k)}(M)\equiv\int_{N}\alpha$ for any $(n-k)$
smooth differential form $\alpha$ on $M$.
$\text{d}{\cal F}=H|_{\cal S}+\delta^{(3)}(\partial\tilde{\cal S})$ (4.55)
where $\partial\tilde{\cal S}\subset{\cal S}$ is the boundary of the world-
volume $\tilde{\cal S}$ of D$(p-2)$-branes ending on ${\cal S}$, and the
equations of motion
$\partial_{\beta}\left(\frac{\partial{\cal L}_{{\rm D}p}}{\partial{\cal
F}_{\alpha\beta}}\right)=-\frac{\mu_{\rm
F1}}{p!}\,\epsilon^{\alpha\beta_{1}\ldots\beta_{p}}\,\delta^{(p)}(\partial{\cal
S}_{\rm F1})_{\beta_{1}\ldots\beta_{p}}$ (4.56)
where now $\partial{\cal S}_{\rm F1}\subset{\cal S}$ is the world-line which
is the boundary of the world-sheet ${\cal S}_{\rm F1}$ of F1-strings ending on
${\cal S}$.
For completeness, we have indicated the possible delta-like contributions from
localized sources, neglected so far. The localized sources contributing to the
the Bianchi identity are given by D$(p-2)$-branes ending on a codimension-
three submanifold of ${\cal S}$. The localized sources contributing to (4.56)
are given by the end-points of fundamental strings attached to the D-brane.
Notice that the overall sign of the localized contributions depends on the
orientation of open branes and strings ending on the hosting D-brane.
Let us now use the natural time-plus-space split introduced by the Killing
vector $K=\partial_{t}$, by choosing the partial static gauge $\tau=t$. The
split (4.21) in this partial static gauge reads
${\cal F}=\text{d}t\wedge{\cal E}+{\cal F}_{\rm mg}$ (4.57)
Analogously, we can split $H=\text{d}t\wedge\iota_{K}H+H_{\rm mg}$. Remember
now the supersymmetry condition (4.13). By using (2.5), it directly implies
that the Bianchi identity for the electric field ${\cal E}$ is automatically
implied by the supersymmetry condition. Hence, it cannot receive any
contribution from localized sources, which is consistent with the fact that
the possible D$(p-2)$-branes wrapping $\tilde{\cal S}$ must be supersymmetric
and then tangent to $K$ as well, hence giving a vanishing localized
contribution $\iota_{K}\delta^{(3)}(\partial\tilde{\cal S})\equiv 0$.
On the other hand, it remains to independently impose by hand the Bianchi
identity for ${\cal F}_{\rm mg}$. By splitting $\tilde{\cal
S}=\mathbb{R}\times\tilde{\Sigma}$, with
$\partial\tilde{\Sigma}\subset\Sigma$, we have
$\text{d}{\cal F}_{\rm mg}=H_{\rm
mg}|_{\Sigma}+\delta^{(3)}(\partial\tilde{\Sigma})$ (4.58)
Consistency requires that the r.h.s. of (4.58) is exact. Hence, if $H_{\rm
mg}|_{\Sigma}$ is cohomologically non-trivial then $\Sigma$ must host the
boundary of an appropriate number of D$(p-2)$-branes.
The situation is reversed for the equations of motion (4.56). By splitting
${\cal S}_{\rm F1}=\mathbb{R}\times\Sigma_{\rm F1}$, the time component of
(4.56) gives rise to the Gauss law constraint
$\partial_{a}\hat{\Pi}^{a}=-\frac{\mu_{\rm F1}}{p!}\,\epsilon^{b_{1}\ldots
b_{p}}\,\delta^{(p)}(\partial\Sigma_{\rm F1})_{b_{1}\ldots b_{p}}$ (4.59)
which must be independently imposed, in addition to the BPS conditions. It is
easier to describe the localized source terms entering (4.59) by using the
dual description of $\hat{\Pi}^{a}$ provided by the $(p-1)$-form $\rho_{\rm
F1}$ defined in (4.51). $\rho_{\rm F1}$ is closed up to localized
contributions coming from the end-points of F1-strings
$\text{d}\rho_{\rm F1}=-\delta^{(p)}(\partial\Sigma_{\rm F1})$ (4.60)
Notice that, by (4.49), $\rho_{\rm F1}$ receives a contribution from the R-R
potentials, which could be not well defined globally. However, we can split
$\rho_{\rm F1}=\tilde{\rho}_{\rm
F1}-\frac{1}{(2\pi\ell_{s})^{p}}\,[C|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p-1)}$, with $\ell_{s}:=\sqrt{\alpha^{\prime}}$, where now
$\tilde{\rho}_{\rm F1}$ is globally defined and is given by the same formula
(4.51) with $\Pi^{a}$ instead of $\hat{\Pi}^{a}$. Hence, (4.60) can be written
as
$\text{d}\tilde{\rho}_{\rm
F1}=\frac{1}{(2\pi\ell_{s})^{p}}\,[F|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}-\delta^{(p)}(\partial\Sigma_{\rm F1})$ (4.61)
Being $\tilde{\rho}_{\rm F1}$ globally defined on $\Sigma$, consistency
requires the r.h.s. of this equation to be exact. For instance, this is the
reason why a D0-brane in a massive IIA background with $F_{(0)}=k$ requires
$k$ attached F1-strings.
The fact that Bianchi identity and Gauss law must be imposed by hand in
addition to the supersymmetry conditions should be also clear from the
argument leading to the BPS energy bound, which uses them as external
ingredients. On the other hand, the BPS energy bound itself guarantees that
the remaining equations are automatically fulfilled.
Finally notice that, by using the source-modified equations (4.58) and (4.60),
one can extend the description of D-brane networks given in [13] to our more
general framework while incorporating fundamental strings too.
## 5 Some examples
We now discuss a few examples of applications of the above general results.
### 5.1 Magnetized D-branes
In this section we consider examples of D-branes with at most pure magnetic
flux ${\cal F}_{\rm mg}$ on static backgrounds. In the following two
subsections, we focus on two subcases.
In the first subcase, discussed in section 5.1.1, $K$ is timelike and we
identify it with the generator of the time characterizing the static
background $K\equiv\partial_{t}$. In this case all supersymmetric branes,
being tangent to $K$, must be static themselves and the energetics governing
these configurations can be characterized in terms of the generalized
calibrations described in [11, 12, 13, 14]. These are a direct extension of
the calibrations of [2] and [5] which allows to incorporate the effect of
${\cal F}_{\rm mg}$. We then show how to recover these cases from our general
analysis.
In the second subcase $K$ is null and then, in principle, supersymmetric
D-branes can be either static (filling the 1+1 directions spanned by
$\partial_{t}$ and $K$) or travel at the speed of light along $K$. In
particular, $K$ is null in the interesting case of ${\cal N}=1$ (static)
compactifications to four dimensions and, in section 5.1.2, we focus for
concreteness on this important class of backgrounds.
#### 5.1.1 Static magnetized D-branes
Here we consider the case in which $K$ is time-like and the space is static.
Hence, introducing the time $t$ defined by $K=\partial_{t}$, we can split the
ten-dimensional space as $\mathbb{R}\times M_{9}$, with metric
$\text{d}s^{2}_{(10)}=-e^{2A(y)}\text{d}t^{2}+g_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.1)
Furthermore, since we want to focus on magnetized branes, we impose the
condition $\omega=0$, which implies that $H$ has only internal legs,
$\iota_{t}H=0$. We can then split
$\Psi=\psi+\text{d}t\wedge\tilde{\psi}$ (5.2)
where $\iota_{t}\psi=0$ and $\text{d}t\wedge\tilde{\psi}=*\lambda(\psi)$ by
(2.8). The supersymmetry condition (2.10) splits into
$\text{d}_{H}\psi=-\iota_{t}F\,,\qquad\text{d}_{H}\tilde{\psi}=0$ (5.3)
Consider now a D$p$-brane wrapping ${\cal S}=\mathbb{R}\times\Sigma$, with
$\Sigma\subset M_{9}$, possibly supporting a non-trivial magnetic flux ${\cal
F}_{\rm mg}$, such that $\text{d}{\cal F}_{\rm mg}=H|_{\Sigma}$. By defining
$m=g|_{\Sigma}+{\cal F}_{\rm mg}$, we have $\sqrt{-\det{\cal
M}}=e^{A}\sqrt{\det m}$ and
$P^{M}=\mu_{{\rm D}p}\,e^{-A-\phi}\sqrt{\det
m}\,\delta^{M}_{t}\,,\qquad\Pi^{\alpha}\equiv 0$ (5.4)
Hence, $-{\cal G}(P,{\cal K})=\mu_{{\rm D}p}\,e^{A-\phi}\sqrt{\det m}$ and the
BPS bound $\Delta_{{\rm D}p}\geq 0$ given by (4.40) becomes
$e^{A}\sqrt{\det m}\,\text{d}^{p}\sigma\,\geq\,[\psi|_{\Sigma}\wedge e^{{\cal
F}_{\rm mg}}]_{(p)}$ (5.5)
This is saturated, i.e. the D$p$-brane is supersymmetric, exactly when
$(\Sigma,{\cal F}_{\rm mg})$ is calibrated by $\psi$. This reproduces the
result of the discussion for static configurations presented in [14].
Of course, in the limit in which one turns off all background and world-volume
fluxes, $e^{A}\sqrt{\det m}\,\text{d}^{p}\sigma$ is just the volume form,
$\psi$ reduces to a calibration of [2] and (5.5) gives the ordinary
calibration bound on the volume of the cycle $\Sigma$.
#### 5.1.2 Compactifications to ${\rm Mink}_{4}$ and non-static branes
A particularly important class of static backgrounds is provided by the ${\cal
N}=1$ compactifications to four-dimensional Minkowski space with
SU(3)$\times$SU(3) structure considered in [20], whose supersymmetry
conditions can be completely interpreted in terms of (static) calibrations
[12]. Concerning the background structures, the relation between the general
ten-dimensional approach and this particular sub-case is discussed in detail
in [30].
In this case the space has structure $\mathbb{R}^{1,3}\times M_{6}$, the
general metric has the form
$\text{d}s^{2}=e^{2A(y)}\text{d}x^{\mu}\text{d}x_{\mu}+\hat{g}_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.6)
and all fields preserve the four-dimensional Poincaré invariance. We can split
the ten-dimensional gamma-matrices as follows
$\Gamma_{\mu}=e^{A}\gamma_{\mu}\otimes\mathbbm{1}\,,\qquad\Gamma_{m}=\gamma_{5}\otimes\hat{\gamma}_{m}$
(5.7)
where $\gamma_{\mu}$ are the four-dimensional gamma matrices for Minkowski
space. By using the real representation in which $C=\Gamma^{\underline{0}}$,
the four independent Killing spinors (2.1) have the form
$\epsilon_{1}=\zeta\otimes\eta_{1}+\text{c.c.}\,,\quad\epsilon_{2}=\zeta\otimes\eta_{2}+\text{c.c.}$
(5.8)
where $\zeta$ is an arbitrary four-dimensional spinor of positive chirality
$\gamma_{5}\zeta=\zeta$, while the internal spinors have chiralities
$\hat{\gamma}_{7}\eta_{1}=\eta_{1}$ and $\hat{\gamma}_{7}\eta_{2}=\mp\eta_{2}$
in IIA/IIB. We denote the norms of the internal spinors by
$|a|^{2}=\eta^{\dagger}_{1}\eta_{1}$ and $|b|^{2}=\eta^{\dagger}_{2}\eta_{2}$.
One can readily compute that $K^{m}=\omega_{m}=0$ while
$K^{\mu}=-e^{-A}\,\bar{\zeta}\gamma^{\mu}\zeta\,(|a|^{2}+|b|^{2})\,,\qquad\omega_{\mu}=-e^{A}\,\bar{\zeta}\gamma_{\mu}\zeta\,(|a|^{2}-|b|^{2})$
(5.9)
Furthermore, (2.3) implies that $\partial_{m}[e^{-A}(|a|^{2}+|b|^{2})]=0$,
while (2.5) implies that $\partial_{m}[e^{A}(|a|^{2}-|b|^{2})]=0$ since by
Poincaré invariance $H$ has only internal legs.
By (4.13) the background is compatible with an electric flux (along $K$) only
if $|a|^{2}\neq|b|^{2}$. On the other hand, in ordinary flux compactifications
O-planes are usually necessary and they preserve the same supersymmetry of
non-fluxed D-branes, hence requiring $|a|^{2}=|b|^{2}$. Focusing on this
latter case, we have $\omega=0$ and we can choose $|a|^{2}=|b|^{2}=e^{A}$, so
that
$K^{\mu}=-2\,\bar{\zeta}\gamma^{\mu}\zeta$ (5.10)
We can use the real representation of the four-dimensional gamma matrices in
which
$\gamma_{0}={\rm
i}\sigma_{2}\otimes\mathbbm{1}_{2}\,,\quad\gamma_{1}=\sigma_{1}\otimes\mathbbm{1}_{2}\,,\quad\gamma_{2}=\sigma_{3}\otimes\sigma_{3}\,,\quad\gamma_{3}=\sigma_{3}\otimes\sigma_{1}$
(5.11)
and the four-dimensional chirality operator is $\gamma_{5}:={\rm
i}\gamma^{0123}=\mathbbm{1}\otimes\sigma_{2}$, and pick up a particular
$\zeta$
$\zeta=\frac{1}{2}\left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\left(\begin{array}[]{c}1\\\ {\rm
i}\end{array}\right)$ (5.12)
We can then easily compute
$K=\frac{\partial}{\partial t}+\frac{\partial}{\partial x^{1}}$ (5.13)
We have used the notation $(t,x^{i})\equiv x^{\mu}$ in order to emphasize the
distinguished role of time. In turn, the polyform $\Psi$ becomes
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(\text{d}x^{1}-\text{d}t)\wedge\,e^{2A}{\rm Im\hskip
1.00006pt}\Phi_{1}+(\text{d}x^{1}-\text{d}t)\wedge\text{d}x^{2}\wedge\text{d}x^{3}\wedge\,e^{4A}{\rm
Re\hskip 1.00006pt}\Phi_{1}$ (5.15)
$\displaystyle+(\text{d}x^{1}-\text{d}t)\wedge\,e^{3A}{\rm Im\hskip
1.00006pt}\big{[}(\text{d}x^{2}+{\rm i}\text{d}x^{3})\wedge\Phi_{2}\big{]}$
where we have introduced the internal pure spinors defined, through the
Clifford map, by
$\Phi_{1}:=-8{\rm
i}e^{-A}\,\eta_{1}\otimes\eta_{2}^{\dagger}\,,\qquad\Phi_{2}:=\mp\ 8{\rm
i}e^{-A}\,\eta_{1}\otimes\eta_{2}^{T}$ (5.16)
the upper/lower sign in $\Phi_{2}$ is for IIB/IIA respectively.
On the right-hand side of (5.15) one can recognize the calibrations ${\rm
Re\hskip 1.00006pt}\Phi_{1}$, ${\rm Im\hskip 1.00006pt}\Phi_{1}$ and ${\rm
Re\hskip 1.00006pt}(e^{{\rm i}\theta}\Phi_{2})$ (for some constant $e^{{\rm
i}\theta}$) on $M_{6}$ which were introduced in [12] to describe static space-
filling, string-like and domain-wall D-branes respectively.
For instance, a static D$p$-brane which appears as a string in
$\mathbb{R}^{1,3}$ and preserves (5.12) must be stretched along $x^{1}$ and
wrap an internal magnetized $(p-1)$-cycle $(\hat{\Sigma},{\cal F}_{\rm mg})$
in $M_{6}$ calibrated by ${\rm Im\hskip 1.00006pt}\Phi_{1}$:
$[{\rm Im\hskip 1.00006pt}\Phi_{1}|_{\hat{\Sigma}}\wedge e^{{\cal F}_{\rm
mg}}]_{(p-1)}=\sqrt{\det m}\,\text{d}^{p-1}\sigma$ (5.17)
where $m:=\hat{g}|_{\hat{\Sigma}}+{\cal F}_{\rm mg}$.
More generically, if we want to consider more complicated static D-branes or
D-brane networks, analogously to the case of time-like $K$ considered in
section 5.1.1 we could actually write
$\Psi=\psi+*\lambda(\psi)$ (5.18)
where
$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\text{d}x^{1}\wedge\,e^{2A}{\rm Im\hskip
1.00006pt}\Phi_{1}+\text{d}x^{1}\wedge\text{d}x^{2}\wedge\text{d}x^{3}\wedge\,e^{4A}{\rm
Re\hskip 1.00006pt}\Phi_{1}$ (5.20)
$\displaystyle+\text{d}x^{1}\wedge\,e^{3A}{\rm Im\hskip
1.00006pt}\big{[}(\text{d}x^{2}+{\rm i}\text{d}x^{3})\wedge\Phi_{2}\big{]}$
is the calibration for static (possibly magnetized) branes on
$\mathbb{R}^{3}\times M_{6}$ which was already considered in [13,
14].101010See in particular [13] for a discussion on how D-brane networks can
be described in this framework.
On the other hand, differently from the assumptions of section 5.1.1, here $K$
is null and this opens the possibility of having non-static branes. A simple
example of non-static supersymmetric D-branes is obtained as follows. Consider
a D-brane which appears as a (two-dimensional) string in $\mathbb{R}^{1,3}$
and wraps a internal magnetized cycle $(\hat{\Sigma},{\cal F}_{\rm mg})$
calibrated by ${\rm Im\hskip 1.00006pt}\Phi_{1}$ as in (5.17). If $(\tau,\xi)$
are the two world-sheet coordinates along the string in $\mathbb{R}^{1,3}$,
take an embedding of the form
$t=\tau\,,\quad x^{1}=\tau+f^{1}(\xi)\,,\quad x^{2}=f^{2}(\xi)\,,\quad
x^{3}=f^{3}(\xi)$ (5.21)
where $f^{1,2,3}$ are arbitrary functions. Such an embedding describes a
string of arbitrary shape which travels at the speed of light in the $x^{1}$
direction. One can readily verify that $\sqrt{-\det{\cal
M}}=|\partial_{\xi}f^{1}|\sqrt{\det m}$ and then, by using the formula (4.6),
one can easily check that all these configurations are supersymmetric.
Alternatively, one can compute ${\cal P}$ from (4.25) and obtain that
$-{\cal G}({\cal P},{\cal K})=-g(P,K)=\mu_{{\rm
D}p}\,e^{-\phi}\,|\partial_{\xi}f^{1}|\sqrt{\det m}$ (5.22)
On the other hand, we have that the $p$-dimensional world-space is given by
$\Sigma=\gamma\times\hat{\Sigma}$, where $\gamma$ is the curve in
$\mathbb{R}^{3}$ described by $x^{i}=f^{i}(\xi)$. Hence
$[\Psi|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}=(\partial_{\xi}f^{1})\text{d}\xi\wedge[{\rm Im\hskip
1.00006pt}\Phi_{1}|_{\hat{\Sigma}}\wedge e^{{\cal F}_{\rm mg}}]_{(p-1)}$
(5.23)
and by (5.17) the supersymmetry condition in the form (4.41) is satisfied by
appropriately choosing the orientation of $\xi$.
Similar supersymmetric configurations are known in the literature and the
peculiar mechanism which allows them to be supersymmetric is somewhat
analogous to the one behind the existence of giant gravitons [35], as
discussed in [36]. Indeed, giant gravitons can be equally well described in
our general formalism, although we refrain from explicitly discussing them in
the present paper.111111A proposal of calibration for giant gravitons appears
in [7], which is consistent with the results of our approach.
### 5.2 Simple D-branes with non-vanishing electric field
In the literature, there are several examples of BPS D-branes supporting an
electric field. In this section we revisit a few simple known examples by
exploiting the perspective offered by our general approach. We will consider
the BIon solution [37] and the supertube solutions of [38, 39, 40, 41]. Our
discussion will be purposely slightly more general than in the original works,
in order to highlight the power of our framework.
These examples can be seen as describing threshold bound states of D-branes
and fundamental strings. Hence, this class of examples can be obtained by
selcting backgrounds whose supersymmetry is compatible with supersymmetric
F1-strings. A simple, still quite general class of backgrounds of this kind
have the form $\mathbb{R}^{1,1}\times M_{8}$ with $M_{8}$ any eight-
dimensional (Ricci-flat) special holonomy space. We can write the ten-
dimensional metric as
$\text{d}s^{2}_{M_{10}}=-\text{d}t^{2}+\text{d}x^{2}+\hat{g}_{mn}(y)\text{d}y^{m}\text{d}y^{n}$
(5.24)
These backgrounds preserve at least two supersymmetries. We are interested in
the particular supersymmetry which is mutually supersymmetric with a
fundamental string stretching along $\mathbb{R}^{1,1}$. We can describe it
quite explicitly, by using a two-plus-eight split $x^{M}=(t,x,y^{m})$ and
accordingly decomposing the gamma matrices (in the real representation) as
follows
$\Gamma_{t}={\rm
i}\sigma_{2}\otimes\mathbbm{1}\,,\quad\Gamma_{x}=\sigma_{1}\otimes\mathbbm{1}\,,\quad\Gamma_{m}=\sigma_{3}\otimes\hat{\gamma}_{m}$
(5.25)
and $\hat{\gamma}_{(8)}=\hat{\gamma}_{1\ldots 8}$ as eight-dimensional chiral
operator. Then, it is easy to see that the preserved supersymmetry has the
form
$\epsilon_{1}=\left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\eta_{1}\,\quad\epsilon_{2}=\left(\begin{array}[]{c}0\\\
1\end{array}\right)\otimes\eta_{2}$ (5.26)
for $\eta_{1}$ and $\eta_{2}$ two Killing spinors on $M_{8}$, with
$\gamma_{(8)}\eta_{1}=\eta_{1}$ and $\gamma_{(8)}\eta_{2}=\pm\eta_{2}$ in
IIA/IIB respectively. We can then easily compute the tensors characterizing
this supersymmetry:
$K=\frac{\partial}{\partial
t}\,,\quad\omega=\text{d}x\,,\quad\Psi=(1-\text{d}t\wedge\text{d}x)\wedge\psi$
(5.27)
where $\psi$ is the calibration for static magnetized D-branes (in the sense
of [11, 12, 13, 14] recalled in section 5.1) for D-branes filling the time
direction $t$ and wrapping a cycle along $M_{8}$. Take for instance IIA and
$M_{8}$ to be a Spin(7) space, with Majorana-Weyl Killing spinor $\eta$,
$\eta^{T}\eta=1$. Hence we can take $\eta_{1}=\eta_{2}=\eta$ and
$\psi=1+\Omega_{4}+\rm{vol}_{8}$, where $\Omega_{4}$ is the Cayley four-form,
which is the calibration for the so-called Cayley cycles.
#### 5.2.1 BIons
In the space $\mathbb{R}^{1,1}\times M_{8}$ described above, consider a
D$p$-brane with the following profile
$t=\tau\,,\quad x=X(\sigma)\,,\quad y^{m}=Y^{m}(\sigma)$ (5.28)
More precisely, we take $\sigma^{a}$ to parametrize an internal cycle
$\Sigma\subset M_{8}$, defined by the embedding $Y^{m}(\sigma)$, and allow for
a possible displacement of the embedding in the flat $x$ direction.
From our general result (4.13) we can immediately conclude that the electric
field is fixed to be
${\cal E}=\partial_{a}X\,\text{d}\sigma^{a}$ (5.29)
We then have
${\cal M}_{\alpha\beta}:=(g|_{\cal S}+{\cal
F})_{\alpha\beta}=\left(\begin{array}[]{cc}-1&\nabla_{b}X\\\
-\nabla_{a}X&\hat{m}_{ab}+\nabla_{a}X\nabla_{b}X\end{array}\right)$ (5.30)
where $\hat{m}=\hat{g}|_{\Sigma}+{\cal F}_{\rm mg}$. One then easily gets
$\det{\cal M}=-\det\hat{m}$. On the other hand, by using (5.27) it is also
easy to see that
$\displaystyle[(\text{d}t\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+1)}=[(\iota_{x}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+1)}=\text{d}t\wedge[\psi|_{\Sigma}\wedge e^{{\cal F}_{\rm mg}}]_{(p)}$
(5.31) $\displaystyle[(\text{d}x\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}=[(\text{d}y^{m}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm top}$ (5.32)
$\displaystyle=[(\iota_{t}\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm
top}=[(\iota_{m}\Psi)|_{\cal S}\wedge e^{\cal F}]_{\rm top}=0$ (5.33)
Take now $\Sigma\subset M_{8}$ to be calibrated by $\psi$:
$[\psi|_{\Sigma}\wedge e^{{\cal F}_{\rm
mg}}]_{(p)}=\sqrt{\det\hat{m}}\,\text{d}^{p}\sigma$ (5.34)
With this prescription on $\Sigma$, one can readily verify that the
supersymmety conditions in the form (4.6) are satisfied. In IIA with $M_{8}$
Spin(7) – see comment after (5.27) – we have that $\Sigma$ can be a point
(D2-brane), or a Cayley four-cycle (D6-brane), or a $M_{8}$-filling eight-
cycle (D8-brane), or a more complicated configuration including a possible
world-volume flux ${\cal F}_{\rm mg}$ too [42].
What remains is to impose the space-like Bianchi identity $\text{d}{\cal
F}_{\rm mg}=0$ and the Gauss law constraint $\partial_{a}\hat{\Pi}^{a}=0$. The
canonical momentum turns out to be given by
$\hat{\Pi}^{a}=\frac{\mu_{{\rm
D}p}}{g_{s}}\,\sqrt{\det\hat{m}}\,\hat{m}^{(ab)}\partial_{b}X$ (5.35)
where $g_{s}\equiv e^{\phi}$ is constant and $\hat{m}^{(ab)}$ is the inverse
of $\hat{m}_{(ab)}$. Now, the Gauss law constraint (4.59) can be rewritten as
the condition
$\mu_{{\rm
D}p}\partial_{a}(\sqrt{\det\hat{m}}\,\hat{m}^{(ab)}\partial_{b}X)=\sqrt{\det\tilde{g}}\,\tilde{\nabla}^{2}X=-\mu_{\rm
F1}\delta^{(p)}(\partial\Sigma_{\rm F1})_{1\ldots p}$ (5.36)
where $\tilde{\nabla}_{a}$ is computed by using the world-volume metric
$\tilde{g}_{ab}=e^{2B}(\hat{g}_{ab}-{\cal F}_{ac}\hat{g}^{cd}{\cal F}_{db})$
(5.37)
with $e^{2B}=[(\det\hat{m})/(\det\hat{g}|_{\Sigma})]^{\frac{2}{2-p}}$. Hence,
$X$ must be harmonic with respect to the metric $\tilde{g}$, up to localized
sources. If we specialize to a flat internal space $M_{8}=\mathbb{R}^{8}$,
with the D$p$-brane just spanning $p$ flat directions, we obtain the funnel-
shaped solution $X\sim{\ell^{p-1}_{s}}/r^{p-2}$, which describes a fundamental
string ending on the D-brane. This is the BIon of [37] and the solution
described above is just a generalization of this kind of configuration.
Recall that, in terms of the $(p-1)$ form $\rho_{\rm F1}$ introduced in
(4.51), the first term on the r.h.s. of (4.48) can be written as
$\int_{\Sigma}\text{d}^{p}\sigma\,\hat{\Pi}^{a}\omega_{a}=\mu_{\rm
F1}\int_{\Sigma}\omega\wedge\rho_{\rm F1}$ (5.38)
On the other hand, $\omega|_{\Sigma}=\text{d}X(\sigma)$. Let us now impose
boundary conditions corresponding to $N$ F1 endpoints $\hat{\sigma}_{(i)}$ on
$\Sigma$, $\partial\Sigma_{\rm F1}=\sum^{N}_{i=1}\hat{\sigma}_{(i)}$. Then
$\text{d}\rho_{\rm F1}=-\sum^{N}_{i=1}\delta^{p}(\hat{\sigma}_{(i)})$ and by
integrating by part we obtain that (5.38) becomes
$\mu_{\rm
F1}\sum_{i}\int_{\Sigma}X(\sigma)\,\delta^{p}(\hat{\sigma}_{(i)})-\mu_{\rm
F1}\int_{\partial\Sigma}X\rho_{\rm F1}$ (5.39)
If we can approximate $X$ on $\partial\Sigma$ to a constant $X_{\infty}$, by
observing that $\int_{\partial\Sigma}\rho_{\rm F1}=N$, we see that (5.39)
reduces to
$\mu_{\rm F1}\sum^{N}_{i=1}\Delta X_{(i)}\quad\quad\text{with }\quad\Delta
X_{(i)}\equiv X(\hat{\sigma}_{(i)})-X_{\infty}$ (5.40)
which is just the energy of the $N$ strings ending on the D$p$-brane and
stretching along the $x$-direction with lengths $\Delta X_{(i)}$.
In conclusion, the formula (4.48) for the energy reduces to
$E^{\rm BPS}_{\rm BIon}=\mu_{\rm F1}\sum^{N}_{i=1}\Delta X_{(i)}+{\mu_{{\rm
D}p}}\int_{\Sigma}e^{-\phi}\psi\wedge e^{{\cal F}_{\rm mg}}$ (5.41)
which, as expected, is just the sum of the F1 energies and the energy
associated with a purely magnetized D-brane.
#### 5.2.2 Supertubes
In the same background $\mathbb{R}^{1,1}\times M_{8}$ described above, take
now a D$(p+2)$-brane of the form ${\cal S}=\mathbb{R}^{1,1}\times\Sigma$.
Moreover, we take $\Sigma$ to be of the form $I\times\hat{\Sigma}$, where $I$
is an interval parametrized by the coordinate $\chi$ and
$\hat{\Sigma}=\hat{\Sigma}(\chi)$ is a one-parameter family of $p$-cycles
calibrated by $\psi$. Clearly, this is possible only if $\hat{\Sigma}$ is
deformable in $M_{8}$. If there are different possible deformations, we can
take an arbitrary one.121212The simplest possibility [38, 39, 40, 41] is to
take $p=2$, i.e. a D2-brane, which implies that $\hat{\Sigma}$ is just a
point. If for instance $M_{8}$ is Spin(7) – see comment after (5.27) – a point
is obviously calibrated by $\psi$ and then always deformable to any other
point.
We work in partial static gauge by choosing world-volume coordinates
$\xi^{\alpha}=(t,x,\chi,\sigma^{a})$, $a=1,\ldots,p$. Hence the embedding
$\hat{\Sigma}$ is described by
$y^{m}=Y^{m}(\chi,\sigma)$ (5.42)
and the calibration condition on $\hat{\Sigma}$ translates into
$\psi|_{\hat{\Sigma}}=\sqrt{\det\hat{g}|_{\hat{\Sigma}}}\,\text{d}^{p}\sigma$
(5.43)
Take then a world-volume flux of the form
${\cal F}=\text{d}t\wedge\text{d}x+b(\chi)\text{d}x\wedge\text{d}\chi$ (5.44)
The first term on the r.h.s. corresponds to the electric field ${\cal
E}=\text{d}x$ required by (4.13).
It is then easy to see that $\det{\cal
M}=-|b|^{2}{\det\hat{g}|_{\hat{\Sigma}}}$ while by using (5.27) it is
immediate to check that
$\displaystyle[(\text{d}t\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\iota_{x}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=b\,\text{d}t\wedge\text{d}x\wedge\text{d}\chi\wedge\psi|_{\hat{\Sigma}}$
(5.45) $\displaystyle[(\text{d}x\wedge\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\text{d}y^{m}\wedge\Psi)|_{\cal S}\wedge e^{\cal F}]_{(p+3)}=0$
(5.46) $\displaystyle[(\iota_{t}\Psi)|_{\cal S}\wedge e^{\cal
F}]_{(p+3)}=[(\iota_{m}\Psi)|_{\cal S}\wedge e^{\cal F}]_{(p+3)}=0$ (5.47)
Hence, by combining these simple results, one can readily verify that the
supersymmetry conditions (4.6) are satisfied for any $b(\chi)$ and for any
choice any family of calibrated submanifolds $\hat{\Sigma}(\chi)$.
What remains is to impose the space-like Bianchi identity and Gauss law. Both
are automatically satisfied. The former because $b$ depends just on $\chi$.
The latter because $\hat{\Pi}^{a}$ has as only non-trivial component
$\hat{\Pi}^{x}$, which manifestly fulfills $\partial_{x}\hat{\Pi}^{x}=0$ since
$\hat{\Pi}^{x}\equiv\hat{\Pi}^{x}(\chi,\sigma)$ does not depend on $x$.
By using $\rho_{\rm F1}$ as defined in (4.51), the total energy is given by
$E_{\text{STube}}=\mu_{\rm F1}\int_{\Sigma}\text{d}x\wedge\rho_{\rm
F1}+\mu_{{\rm D}(p+2)}\int_{\Sigma}{\cal F}_{\rm mg}\wedge e^{-\phi}\psi$
(5.48)
The first term provides the contribution coming from a number of dissolved
F1-strings wrapping a curve $\Sigma_{\rm F1}$ stretched along the
$x$-direction. The second term corresponds to the energy associated with a
number of dissolved D$p$-branes wrapping an internal cycle $\Sigma_{{\rm D}p}$
belonging to the family $\hat{\Sigma}(\chi)$, hence calibrated by $\psi$, and
appearing as points in $\mathbb{R}^{1,1}$. In other words, $\rho_{\rm F1}$ and
$(2\pi\ell_{s})^{-2}{\cal F}_{\rm mg}$ are cohomology classes Poincarè dual to
$\Sigma_{\rm F1}\subset\Sigma$ and $\Sigma_{{\rm D}p}\subset\Sigma$ and then
$E_{\text{STube}}=\mu_{\rm F1}\int_{\Sigma_{\rm F1}}\text{d}x+\mu_{{\rm
D}p}\int_{\Sigma_{{\rm D}p}}e^{-\phi}\psi$ (5.49)
On the other hand, there is no contribution associated with the supporting
D$(p+2)$-brane, which appears as ‘effectively tensionless’, consistently with
the fact that $\hat{\Sigma}(\chi)$ is arbitrary – see [38, 39, 40, 41] for
more discussions.
## 6 Conclusion
In this paper we have obtained a geometric characterization of the
supersymmetry conditions for fundamental strings and D-branes. The analysis is
completely general concerning both background and brane configurations. In
particular, D-branes can support the most general world-volume flux compatible
with supersymmetry.
The various equations for F1-strings and D-branes look very similar and come
in pairs, with the difference that D-branes ‘see’ a generalized geometry
rather than an ordinary one: the covariant supersymmetry conditions are
provided by (3.5) and (4.6) [or equivalently (3.7) and (4.11)] respectively;
the non-covariant minimal BPS conditions are (3.30) and (4.41) respectively;
the local BPS bounds saturated by supersymmetric configurations are (3.29) and
(4.42) [or equivalently (3.31) and (4.43)] respectively.
Furthermore, we have shown that the total energy satisfies lower bounds
saturated by its BPS-saturating values $E_{\rm F1}^{\rm BPS}$ and $E_{{\rm
D}p}^{\rm BPS}$ respectively – see (3.35) and (4.48). These are defined by
integrals which are invariant under continuous deformations of the brane
configuration. Hence $E_{\rm F1}^{\rm BPS}$ and $E_{{\rm D}p}^{\rm BPS}$ can
be considered as ‘topological’ and provide the natural central charge
associated with the preserved supersymmetry in presence of F1-strings and
D-branes. In particular, our formalism takes into account the effect of the
most general world-volume flux on D-branes and can be applied to study
D-branes and F1-strings networks extending the approach of [13].
The results obtained here can be useful in several contexts, for instance in
the study of black holes or more general black brane configurations. In
particular, it would be important to study their implications to fully coupled
bulk-plus-branes systems. With regard to this point, the relation between bulk
structures and brane energetics highlighted in this paper should play a key
role as it happens for static settings [27, 28]. Furthermore, it would be
interesting to understand whether one can break the bulk and brane
supersymmetry while preserving (part of) the brane BPS bounds. This could
allow better control over the fully coupled system in non-supersymmetric
settings as well, as for instance it is realized in the non-supersymmetric
flux compactifications considered in [28].
Acknowledgments
I would like to thank M. Bianchi, F. Fucito, P. Koerber and specially A.
Tomasiello for useful discussions. This work is partially supported by the ERC
Advanced Grant n.226455 “Superfields”, by the Italian MIUR-PRIN contract
20075ATT78 and by the NATO grant PST.CLG.978785.
## References
* [1] J. P. Gauntlett, “Branes, calibrations and supergravity,” [hep-th/0305074].
* [2] R. Harvey, H. B. Lawson, Jr., “Calibrated geometries,” Acta Math. 148 (1982) 47.
* [3] D. Joyce, “Compact manifolds with special holonomy,” Oxford University Press (2000).
* [4] K. Becker, M. Becker, A. Strominger, “Five-branes, membranes and nonperturbative string theory,” Nucl. Phys. B456 (1995) 130-152. [hep-th/9507158].
* [5] J. Gutowski, G. Papadopoulos, “AdS calibrations,” Phys. Lett. B462 (1999) 81-88. [hep-th/9902034]. J. Gutowski, G. Papadopoulos, P. K. Townsend, “Supersymmetry and generalized calibrations,” Phys. Rev. D60 (1999) 106006. [hep-th/9905156].
* [6] J. P. Gauntlett, N. w. Kim, D. Martelli and D. Waldram, “Fivebranes wrapped on SLAG three-cycles and related geometry,” JHEP 0111 (2001) 018 [arXiv:hep-th/0110034]; J. P. Gauntlett, D. Martelli, S. Pakis and D. Waldram, “G-structures and wrapped NS5-branes,” Commun. Math. Phys. 247 (2004) 421 [arXiv:hep-th/0205050]; J. P. Gauntlett, D. Martelli and D. Waldram, “Superstrings with intrinsic torsion,” Phys. Rev. D 69 (2004) 086002 [arXiv:hep-th/0302158]; D. Martelli and J. Sparks, “G-structures, fluxes and calibrations in M-theory,” Phys. Rev. D 68 (2003) 085014 [arXiv:hep-th/0306225].
* [7] E. J. Hackett-Jones, D. J. Smith, “Type IIB Killing spinors and calibrations,” JHEP 0411 (2004) 029. [hep-th/0405098].
* [8] J. F. G. Cascales and A. M. Uranga, “Branes on generalised calibrated sub-manifolds,” JHEP 0411 (2004) 083 [arXiv:hep-th/0407132].
* [9] P. M. Saffin, “Type IIA Killing spinors and calibrations,” Phys. Rev. D71 (2005) 025018. [hep-th/0407156].
* [10] G. Papadopoulos, P. Sloane, “Kappa symmetry, generalized calibrations and spinorial geometry,” JHEP 0605 (2006) 050. [hep-th/0601164].
* [11] P. Koerber, “Stable D-branes, calibrations and generalized Calabi-Yau geometry,” JHEP 0508 (2005) 099. [hep-th/0506154].
* [12] L. Martucci, P. Smyth, “Supersymmetric D-branes and calibrations on general N=1 backgrounds,” JHEP 0511 (2005) 048. [hep-th/0507099].
* [13] J. Evslin, L. Martucci, “D-brane networks in flux vacua, generalized cycles and calibrations,” JHEP 0707 (2007) 040. [hep-th/0703129 [HEP-TH]].
* [14] P. Koerber, L. Martucci, “D-branes on AdS flux compactifications,” JHEP 0801 (2008) 047. [arXiv:0710.5530 [hep-th]].
* [15] N. Hitchin, “Generalized Calabi-Yau manifolds,” Quart. J. Math. Oxford Ser. 54 (2003) 281-308. [math/0209099 [math-dg]].
* [16] M. Gualtieri, “Generalized complex geometry,” [math/0401221 [math-dg]].
* [17] P. Koerber, “Lectures on Generalized Complex Geometry for Physicists,” Fortsch. Phys. 59 (2011) 169-242. [arXiv:1006.1536 [hep-th]].
* [18] L. Martucci, “D-branes on general N=1 backgrounds: Superpotentials and D-terms,” JHEP 0606 (2006) 033. [arXiv:hep-th/0602129 [hep-th]].
* [19] P. Koerber, L. Martucci, “Deformations of calibrated D-branes in flux generalized complex manifolds,” JHEP 0612 (2006) 062. [hep-th/0610044].
* [20] M. Grana, R. Minasian, M. Petrini, A. Tomasiello, “Generalized structures of N=1 vacua,” JHEP 0511 (2005) 020. [hep-th/0505212].
* [21] M. Grana, J. Louis, D. Waldram, “Hitchin functionals in N=2 supergravity,” JHEP 0601 (2006) 008. [hep-th/0505264]; “SU(3) x SU(3) compactification and mirror duals of magnetic fluxes,” JHEP 0704 (2007) 101. [hep-th/0612237].
* [22] I. Benmachiche, T. W. Grimm, “Generalized N=1 orientifold compactifications and the Hitchin functionals,” Nucl. Phys. B748 (2006) 200-252. [hep-th/0602241].
* [23] P. Koerber, L. Martucci, “From ten to four and back again: How to generalize the geometry,” JHEP 0708 (2007) 059. [arXiv:0707.1038 [hep-th]].
* [24] L. Martucci, “On moduli and effective theory of N=1 warped flux compactifications,” JHEP 0905 (2009) 027. [arXiv:0902.4031 [hep-th]].
* [25] M. Haack, D. Lust, L. Martucci, A. Tomasiello, “Domain walls from ten dimensions,” JHEP 0910 (2009) 089. [arXiv:0905.1582 [hep-th]].
* [26] D. Lüst, P. Patalong, D. Tsimpis, “Generalized geometry, calibrations and supersymmetry in diverse dimensions,” JHEP 1101 (2011) 063. [arXiv:1010.5789 [hep-th]].
* [27] P. Koerber, D. Tsimpis, “Supersymmetric sources, integrability and generalized-structure compactifications,” JHEP 0708 (2007) 082. [arXiv:0706.1244 [hep-th]].
* [28] D. Lüst, F. Marchesano, L. Martucci, D. Tsimpis, “Generalized non-supersymmetric flux vacua,” JHEP 0811 (2008) 021. [arXiv:0807.4540 [hep-th]]. J. Held, D. Lüst, F. Marchesano, L. Martucci, “DWSB in heterotic flux compactifications,” JHEP 1006 (2010) 090. [arXiv:1004.0867 [hep-th]].
* [29] O. Barwald, N. D. Lambert, P. C. West, “A Calibration bound for the M theory five-brane,” Phys. Lett. B463 (1999) 33-40. [hep-th/9907170].
* [30] A. Tomasiello, “Generalized structures of ten-dimensional supersymmetric solutions,” [arXiv:1109.2603 [hep-th]].
* [31] J. Gillard, U. Gran, G. Papadopoulos, “The Spinorial geometry of supersymmetric backgrounds,” Class. Quant. Grav. 22 (2005) 1033-1076. [hep-th/0410155]. U. Gran, J. Gutowski, G. Papadopoulos, “The Spinorial geometry of supersymmetric IIb backgrounds,” Class. Quant. Grav. 22 (2005) 2453-2492. [hep-th/0501177].
* [32] L. Martucci, J. Rosseel, D. Van den Bleeken, A. Van Proeyen, “Dirac actions for D-branes on backgrounds with fluxes,” Class. Quant. Grav. 22 (2005) 2745-2764. [hep-th/0504041].
* [33] M. T. Grisaru, P. S. Howe, L. Mezincescu, B. Nilsson, P. K. Townsend, “N=2 Superstrings in a Supergravity Background,” Phys. Lett. B162 (1985) 116.
* [34] E. Bergshoeff, R. Kallosh, T. Ortin, G. Papadopoulos, “Kappa symmetry, supersymmetry and intersecting branes,” Nucl. Phys. B502 (1997) 149-169. [hep-th/9705040].
* [35] J. McGreevy, L. Susskind, N. Toumbas, “Invasion of the giant gravitons from Anti-de Sitter space,” JHEP 0006 (2000) 008. [hep-th/0003075]. M. T. Grisaru, R. C. Myers, O. Tafjord, “SUSY and goliath,” JHEP 0008 (2000) 040. [hep-th/0008015]. A. Hashimoto, S. Hirano, N. Itzhaki, “Large branes in AdS and their field theory dual,” JHEP 0008 (2000) 051. [hep-th/0008016].
* [36] A. Mikhailov, “Giant gravitons from holomorphic surfaces,” JHEP 0011 (2000) 027. [hep-th/0010206].
* [37] C. G. Callan, J. M. Maldacena, “Brane death and dynamics from the Born-Infeld action,” Nucl. Phys. B513 (1998) 198-212. [hep-th/9708147]. G. W. Gibbons, “Born-Infeld particles and Dirichlet p-branes,” Nucl. Phys. B514 (1998) 603-639. [hep-th/9709027].
* [38] D. Mateos, P. K. Townsend, “Supertubes,” Phys. Rev. Lett. 87, 011602 (2001). [hep-th/0103030].
* [39] R. Emparan, D. Mateos, P. K. Townsend, “Supergravity supertubes,” JHEP 0107 (2001) 011. [hep-th/0106012].
* [40] D. Mateos, S. Ng, P. K. Townsend, “Tachyons, supertubes and brane / anti-brane systems,” JHEP 0203 (2002) 016. [hep-th/0112054].
* [41] J. Gomis, T. Mateos, P. J. Silva, A. Van Proeyen, “Supertubes in reduced holonomy manifolds,” Class. Quant. Grav. 20 (2003) 3113-3128. [hep-th/0304210].
* [42] M. Marino, R. Minasian, G. W. Moore, A. Strominger, “Nonlinear instantons from supersymmetric p-branes,” JHEP 0001 (2000) 005. [hep-th/9911206].
|
arxiv-papers
| 2011-10-04T10:18:59 |
2024-09-04T02:49:22.744193
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luca Martucci",
"submitter": "Luca Martucci",
"url": "https://arxiv.org/abs/1110.0627"
}
|
1110.0645
|
# Strategies and payoffs in quantum minority games
Puya Sharif and Hoshang Heydari
Physics Department, Stockholm university 10691 Stockholm Sweden
Email: ps@puyasharif.net
###### Abstract
Game theory is the mathematical framework for analyzing strategic interactions
in conflict and competition situations. In recent years quantum game theory
has earned the attention of physicists, and has emerged as a branch of quantum
information theory [1]. With the aid of entanglement and linear superposition
of strategies, quantum games are shown to yield significant advantage over
their classical counterparts. In this paper we explore optimal and equilibrium
solutions to quantum minority games. Initial states with different level of
entanglement are investigated. Focus will be on 4 and 6 player games with some
$N$-player generalizations.
## 1 Introduction
Game theory is the systematic study of decision-making in strategic
situations. Its models are widely used in economics, political science,
biology and computer science to capture the behavior of individual
participants in conflict and competition situations. The field attempts to
describe how decision makers do and should interact within a well-defined
system of rules to maximize their payoff. The kind of games we will be
considering here is called minority games and arises in situations when a
group of non communicating agents has to independently choose between two
different choices $\left|0\right\rangle$ and $\left|1\right\rangle$. Payoff $
of one unit goes to those agents that makes the minority choice. If agents are
evenly distributed between the two choices, everybody loses. In the classical
case, the game has a mixed-strategy solution, where agents chooses randomly
between $\left|0\right\rangle$ and $\left|1\right\rangle$. This yields an
expected payoff $<\$>$ which is given by the number of combinations that
results in some player being in the minority group divided by the total number
of possible combinations. For a four player game there are 16 possible
combinations, with two minority states for each player. This gives an expected
payoff ¡$¿ of 1/8 to each.
Generally a game is defined as a set
$\Gamma=\Gamma(N,\,\\{s_{i}\\},\,\\{\$_{i}\\})$, where $N$ denotes the number
of players, $\\{s_{i}\\}$ the set of available strategies of player $i$, and
$\\{\$_{i}\\}$ the payoffs of different game outcomes. For quantum games, we
add the associated Hilbert space $\mathcal{H}$, generally of dim $2^{N}$, and
the initial state $\rho$. In a quantum game, the choice of strategy $s_{i}$
translates to choosing a unitary operator $M_{i}$, which is applied locally on
the qubit held by the player. The games will be analyzed with regard to two of
the most important solution concepts in game theory is the Nash equilibrium
and Pareto optimality. Nash equilibrium is defined as the combination of
strategies $s_{i}$ for which no player gains by unilaterally changing their
strategy. Pareto optimality occurs when no player can rise its payoff without
lowering the payoff of others.
## 2 Quantum minority games
Following the scheme presented in [2], in the quantum version of the minority
game, each player is provided with a qubit from an entangled set. Strategy
$s_{i}$ of player $i$ is played by doing a local unitary operation on the
players own qubit, by applying its strategy operator $M\in$ SU(2). $M$ will be
parameterized in the following way:
$M\left(\theta,\alpha,\beta\right)=\left(\begin{array}[]{cc}e^{i\alpha}\cos\left(\theta/2\right)&ie^{i\beta}\sin\left(\theta/2\right)\\\
ie^{-i\beta}\sin\left(\theta/2\right)&e^{-i\alpha}\cos\left(\theta/2\right)\end{array}\right),$
(2.0.1)
with $\theta\in[0,\pi]$ and $\alpha,\beta\in[-\pi,\pi]$. The game starts out
in an entangled initial state $\rho_{in}$.
$\rho_{in}=\left|\psi\right\rangle\langle\psi|,$ (2.0.2)
where$\left|\psi\right\rangle$ is usually taken to be a $N$ qubit GHZ-state,
from which each player is provided with one qubit [2][3]. The final state
$\rho_{\mathrm{fin}}$ of the game becomes
$\rho_{\mathrm{fin}}=(\bigotimes_{i=1}^{N}M_{i})\rho_{\mathrm{in}}(\bigotimes_{i=1}^{N}M_{i}){}^{\dagger}.$
(2.0.3)
To calculate the expected payoff of player $i$ we take the trace of the final
state $\rho_{\mathrm{fin}}$ multiplied with the projection operator $P_{i}$ of
the player. The projection operator projects the final state onto the desired
states of player $i$.
$P_{i}=\sum_{j=1}^{k}\left|\xi_{i}^{j}\right\rangle\left\langle\xi_{i}^{j}\right|.$
(2.0.4)
The sum is over all the $k$ different states $\left|\xi_{i}^{j}\right\rangle$,
for which player $i$ is in the minority. For $N=4$, we have the following
projection operator $P_{1}$ for player 1.
$P_{1}=\left|1000\right\rangle\left\langle
1000\right|+\left|0111\right\rangle\left\langle 0111\right|$. In the 6-player
game, each player has a sum of 12 such states. The expected payoff $<\$>$ is
finally given by:
$\mathrm{<\$_{i}>=Tr[}\rho_{\mathrm{fin}}P_{i}].$ (2.0.5)
The local unitary operations of the players eliminates the possibility for the
system to end up in most states where nobody wins, and therefore yields higher
than classical payoff.
### 2.1 Solutions with different initial states
As a generalization of the broadly used GHZ-state as the initial state
$\left|\psi_{in}\right\rangle$ we consider a superposition with products of
symmetric bell pairs. A four qubit version of this state was used in a
experimental implementation of a quantum minority game by C. Schmid and A.P.
Flitney in [3].
$\left|\Psi(x)\right\rangle=\frac{x}{\sqrt{2}}\left|\mathrm{GHZ}_{N}\right\rangle+\sqrt{\frac{1-x^{2}}{2^{N/2}}}(\left|01\right\rangle+\left|10\right\rangle)^{\otimes
N/2}.$ (2.1.1)
The parameter $x\in[0,1]$ denotes the level of mixture. $x=1$ just gives back
the GHZ-state and $x=0$ product of the Bell-pairs. To account for loss in
fidelity in the creation of the initial state, we form a density matrix
$\rho_{in}$ out of $\mid\Psi_{in}\rangle$ and add noise that can be controlled
by the parameter $f$. We get:
$\rho_{in}=f\mid\Psi_{in}\rangle\langle\Psi_{in}\mid+\frac{1-f}{64}\mathbb{I_{\mathrm{64}}},$
(2.1.2)
where $\mathbb{I_{\mathrm{64}}}$ is the $64\times 64$ identity matrix. By
adjusting $f\rightarrow 0$, the initial state gets mixed with an even
distribution of all basis states in $\mathcal{H}=(\mathbb{C}^{2})^{\otimes 6}$
For the GHZ-state alone i.e $x=1$ and $f=1$ it has been shown that the Nash
equilibrium solution $s_{NE}=M(\theta,\alpha,-\alpha)$ for the 4-player game
is $M(\frac{\pi}{2},-\frac{\pi}{8},\frac{\pi}{8})$, and for the 6-player game,
$M(\frac{\pi}{2},-\frac{\pi}{12},\frac{\pi}{12})$. For the state above, Schmid
and Flitney showed that when starting with only the product of Bell-pairs i.e
$x=0$, no advantage is achieved over the classical counterpart. For
$x\leq\sqrt{\frac{2}{3}}$, a new set of Nash equilibria occurs, where the
payoff is a function of $x$. This Bell-dominated region has a new Pareto
optimal strategy: $M(\frac{\pi}{4},0,0)$ compared to
$M(\frac{\pi}{2},-\frac{\pi}{8},\frac{\pi}{8})$ in the GHZ-dominated region
$x>\sqrt{\frac{2}{3}}$.
Figure 1: Payoffs for $\hat{M}(\theta,\alpha,-\alpha).$ When $N=4$. Top left:
$x=1$. Top right: $x=\sqrt{2/3}+0.1$, Bottom left: $x=\sqrt{2/3}$. Bottom
right: $x=\sqrt{2/3}-0.1$
The case is different for $N=6$, here the equilibrium strategy remains the
same throughout any change of $x$. For $f=1$, the payoff is given by
$<\$>=\frac{1}{4}+\frac{x^{2}}{16}.$ (2.1.3)
For the pure GHZ-state this gives an equilibrium payoff of 5/16. When
$x\rightarrow 0$ the payoff approaches 1/4, which is still better than the
classical payoff of 3/16. This shows that even the initial state containing
only the products of Bell-pairs yields an advantage compared to the classical
expected payoff. This is not the case for general $N$. When noise is taken
into account the payoff function becomes
$<\$>=\frac{1}{16}(3+f+fx^{2}).$ (2.1.4)
When the noise reaches maximum: $f\rightarrow 0$, the classical payoff of 3/16
returns. It can be demonstrated that
$M_{NE}=M(\frac{\pi}{2},-\frac{\pi}{12},\frac{\pi}{12})$ is a Nash equilibrium
solution for all $x\in[0,1]$, by letting one player deviate from the NE
solution, by playing $M_{D}(\theta^{*},-\alpha^{*},\alpha^{*})$. The following
inequality holds for a Nash equilibrium:
$\$_{i}(M_{NE}^{\otimes 6})\geq\$_{i}(M_{D}\otimes M_{NE}^{\otimes 5}).$
(2.1.5)
Figure 2: Payoff of a player that plays
$M_{D}(\theta^{*},-\frac{\pi}{12},\frac{\pi}{12})$, when the rest plays
$M_{NE}$
.
Figure 3: Payoff of a player that plays
$M_{D}(\frac{\pi}{2},-\alpha^{*},\alpha^{*})$, when the rest plays $M_{NE}$
.
#### 2.1.1 exponential entangler
A GHZ-state can be created by acting with an entanglement operator $J(\gamma)$
on a product state
$\left|0\right\rangle\otimes\left|0\right\rangle\otimes\left|0\right\rangle\otimes\cdots\otimes\left|0\right\rangle$
, where
$J(\gamma)=\mathrm{exp\left(i\frac{\gamma}{2}\sigma_{x}^{\otimes\mathit{N}}\right)}.$
(2.1.6)
We then have
$\left|\Psi(\gamma)\right\rangle=J\left|00\cdots 0\right\rangle,$ (2.1.7)
where $\gamma\in[0,\frac{\pi}{2}]$ is a parameter that controls the level of
entanglement. This gives an output state of the following form
$\left|\Psi(\gamma)\right\rangle=\cos(\frac{\gamma}{2})\left|00\cdots
0\right\rangle+i\sin(\frac{\gamma}{2})\left|11\cdots 1\right\rangle.$ (2.1.8)
Maximum is reached for $\gamma=\frac{\pi}{2}.$ If
$\left|\Psi(\gamma)\right\rangle$ is used as initial state for a quantum
minority game, the Nash equilibrium payoffs will depend on the parameter
$\gamma$. For $\gamma=0$ the classical payoffs are obtained, since the game
starts out in an unentangled initial state [6]. A $N$-player generalization
has been conjectured:
$<\$>_{N}=\left(<\$>_{C}-\frac{1}{2}<\$>_{Q}\right)\left(\cos\frac{\gamma}{2}-\sin\frac{\gamma}{2}\right)^{2}+\frac{1}{2}<\$>_{Q}\left(\cos\frac{\gamma}{2}+\sin\frac{\gamma}{2}\right)^{2},$
(2.1.9)
where $<\$>_{C}$ is the classically obtainable payoffs for for classical NE
strategies, and $<\$>_{C}$ for the quantum versions [5].
#### 2.1.2 Products of W-states
A six player game could use a product of two three qubit W-states as its
initial state $\left|\psi_{in}\right\rangle$.
$\left|\psi_{in}\right\rangle=\left|W_{3}\right\rangle\otimes\left|W_{3}\right\rangle,$
(2.1.10)
where
$\left|W_{3}\right\rangle=\frac{1}{\sqrt{3}}(\left|001\right\rangle+\left|010\right\rangle+\left|100\right\rangle).$
(2.1.11)
$\left|\psi_{in}\right\rangle$ is a symmetric superposition of nine states
with four qubits in the $\left|0\right\rangle$ -state and two in the
$\left|1\right\rangle$ -state, compactly written as
$\left|\psi_{in}\right\rangle=\left|4,2\right\rangle$. This state therefore
has tree minority combinations for each player, and no undesired states! The
game simply starts out in the best possible configuration, and the only thing
the players should do is to apply the identity operator $I$, to obtain an
expected payoff 1/3, the theoretical maximum for a six-player game. This
solution Pareto optimal, compared to the six-player game starting with an GHZ-
state, which is not.
## References
* [1] E. W. Piotrowski, J. Sladkowski, An invitation to quantum game theory, International Journal of Theoretical Physics Springer, 42 (5): 10891099 1, (2003).
* [2] S. C. Benjamin, P.M. Hayden, Phys. Rev. Lett 87, 069801 (2001).
* [3] C. Schmid, A.P. Flitney, Experimental implementation of a four-player quantum game, New J. Phys. 12, 063031, (2010).
* [4] A. P. Flitney, A. Greentree, Coalitions in the minority game: classical cheats and quantum bullies, Elsiever Science, (2008)
* [5] Q. Chen, Y. Wang, N-player quantum minority game, Physics Letters A, A 327, 98,102, (2004).
* [6] A. P. Flitney, L. C. L. Hollenberg, Multiplayer quantum Minority game with decoherence, Quant. Inform. Comput. 7, 111-126,(2007).
|
arxiv-papers
| 2011-10-04T11:25:23 |
2024-09-04T02:49:22.757786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Puya Sharif and Hoshang Heydari",
"submitter": "Puya Sharif",
"url": "https://arxiv.org/abs/1110.0645"
}
|
1110.0730
|
# Search for the lepton number violating decays
$\bm{B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}}$ and $\bm{B^{+}\rightarrow
K^{-}\mu^{+}\mu^{+}}$
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39,
M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h,
P.M. Bjørnstad50, T. Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52,
A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S.
Borghi47, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den
Brand24, J. Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T.
Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A.
Bursche39, J. Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7,
M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14,
G. Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho
Akiba23, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N.
Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M.
Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P.
Collins37, A. Comerma-Montells35, F. Constantin28, G. Conti38, A. Contu51, A.
Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C.
D’Ambrosio37, P. David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C.
Deplano15, O. Deschamps5, F. Dettori15,d, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, A. Dosil Suárez36, D.
Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D.G. d’Enterria35,o,
D. Esperante Pereira36, L. Estève43, A. Falabella16,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S.
Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph.
Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K.
Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E.
Hicks48, W. Hofmann10, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51,
T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P.
Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E.
Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D.
Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40, M. Karacson37, T.M.
Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M.
Kim46, M. Knecht38, S. Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L.
Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9,
K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25, R. Kumar14,37, T.
Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15,
D. Lambert46, R.W. Lambert37, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R.
Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L.
Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J. Luisier38, F. Machefert7,
I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51,
R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14,
R. Märki38, J. Marks11, G. Martellotti22, A. Martens7, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P.
Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R.
Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T.
Nakada38, R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Oblakowska-Mucha26, V. Obraztsov34, S.
Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M.
Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C. Parkes47,37,
C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49,
G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36,
A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, E. Picatoste
Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du
Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H.
Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24,
S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36,
G.J. Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T.
Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P.
Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S.
Schleich9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P.
Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y.
Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A.
Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37,
N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F.
Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B.
Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51,45, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, M.T. Tran38, A.
Tsaregorodtsev6, N. Tuning23, A. Ukleja27, P. Urquijo52, U. Uwer11, V.
Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S.
Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7,
X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A.
Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z.
Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
###### Abstract
A search is performed for the lepton number violating decay
$B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$, where $h^{-}$ represents a $K^{-}$
or a $\pi^{-}$, using data from the LHCb detector corresponding to an
integrated luminosity of $36\mbox{\,pb}^{-1}$. The decay is forbidden in the
Standard Model but allowed in models with a Majorana neutrino. No signal is
observed in either channel and limits of ${\cal B}(B^{+}\rightarrow
K^{-}\mu^{+}\mu^{+})<5.4\times 10^{-8}$ and ${\cal
B}(B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})<5.8\times 10^{-8}$ are set at the
95% confidence level. These improve the previous best limits by factors of 40
and 30, respectively.
###### pacs:
11.30.Fs, 13.20.He, 13.35.Hb
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | LHCb-PAPER-2011-009
| | CERN-PH-EP-2011-156
Lepton number is conserved in the Standard Model but can be violated in a
range of new physics models such as those with Majorana neutrinos
Majorana:1937vz or left-right symmetric models with a doubly charged Higgs
boson Pati:1974yy . In this letter a search for lepton number violating decays
of the type $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$, where $h^{-}$
represents a $K^{-}$ or a $\pi^{-}$, is presented. The inclusion of charge
conjugated modes is implied throughout. A search for any lepton number
violating process that mediates the $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$
decay is made. A specific search for $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$
decays mediated by an on-shell Majorana neutrino (Fig. 1) is also performed.
Such decays would give rise to a narrow peak in the invariant mass spectrum of
the hadron and one of the muons Pascoli:2007qh , $m_{\nu}=m_{h\mu}$, if the
mass of the neutrino is between $m_{K(\pi)}+m_{\mu}$ and $m_{B}-m_{\mu}$. The
previous best experimental limit on the $B^{+}\\!\rightarrow
K^{-}(\pi^{-})\mu^{+}\mu^{+}$ branching fraction is $1.8(1.2)\times 10^{-6}$
at 90% confidence level (CL) Edwards:2002kq .
Figure 1: s-channel diagram for $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$
($B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$) where the decay is mediated by an
on-shell Majorana neutrino.
The search for $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$ is carried out with
data from the LHCb experiment at the Large Hadron Collider at CERN. The data
corresponds to 36$\mbox{\,pb}^{-1}$ of integrated luminosity of proton-proton
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ collected in 2010. The
LHCb detector is a single-arm spectrometer designed to study $b$-hadron decays
with an acceptance for charged tracks with pseudorapidity between 2 and 5.
Primary proton-proton vertices (PVs), and secondary $B$ vertices are
identified in a silicon strip vertex detector. Tracks from charged particles
are reconstructed by the vertex detector and a set of tracking stations. The
curvature of the tracks in a dipole magnetic field allows momenta to be
determined with a precision of $\delta p/p=0.35$–$0.5\%$. Two Ring Imaging
CHerenkov (RICH) detectors allow kaons to be separated from pions/muons over a
momentum range $2<p<100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons with
momentum above 3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are identified on the
basis of the number of hits left in detectors interleaved with an iron muon
filter. Further details about the LHCb detector can be found in Ref.
Alves:2008zz .
The search for $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$ decays is based on
the selection of $B^{+}\\!\rightarrow h^{\pm}\mu^{+}\mu^{\mp}$ candidates. The
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ is included in the same selection. It is
subsequently used as a normalisation mode when setting a limit on the
branching fraction of the $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$ decays.
The selection is designed to minimise and control the difference between
decays with same- and opposite-sign muons and thus cancel most of the
systematic uncertainty from the normalisation. The only differences in
efficiency between the signal and normalisation channels are due to the decay
kinematics and the presence of a same-sign muon pair, rather than an opposite-
sign pair, in the final state.
In the trigger, the $B^{+}\\!\rightarrow h^{\pm}\mu^{+}\mu^{\mp}$ candidates
are required to pass the initial hardware trigger based on the $p_{\rm T}$ of
one of the muons. In the subsequent software trigger, one of the muons is
required to have a large impact parameter (IP) with respect to all the PVs in
the event and to pass requirements on the quality of the track fit and the
compatibility of the candidate with the muon hypothesis. Finally, the muon
candidate combined with another track is required to form a vertex displaced
from the PVs.
Further event selection is applied offline on fully reconstructed $B$ decay
candidates. The selection is designed to reduce combinatorial backgrounds,
where not all the selected tracks come from the same decay vertex; and peaking
backgrounds, where a single decay is selected but with some of the particle
types misidentified. The combinatorial background is smoothly distributed in
the reconstructed $B$-candidate mass and the level of background is assessed
from the sidebands around the signal window. Peaking backgrounds from $B$
decays to hadronic final states, final states with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and semileptonic final states
are also considered.
Proxies are used in the optimisation of the selection for both the signal and
the background to avoid a selection bias. The
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
is used as a proxy for the signal. The background proxy comprises opposite-
sign $B^{+}\\!\rightarrow h^{+}\mu^{+}\mu^{-}$ candidates with an invariant
mass in the upper mass sideband and with muon pairs incompatible with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or a $\psi{(2S)}$ hypothesis.
The bias introduced by using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ for
both optimisation and as a normalisation mode is insignificant due to the
large number of candidates.
The combinatorial background is reduced by requiring that the decay products
of the $B$ have $\mbox{$p_{\rm T}$}>800{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Tracks are selected which are incompatible with originating from a PV based on
the $\chi^{2}$ of the tracks’ impact parameters ($\chi^{2}_{\rm IP}>45$). The
direction of the candidate $B^{+}$ momentum is required to be within
8$\rm\,mrad$ of the reconstructed $B^{+}$ line of flight. The $B^{+}$ vertex
is also required to be of good quality ($\chi^{2}<12$ for three degrees of
freedom) and significantly displaced from the PV ($\chi^{2}$ of vertex
separation larger than 144).
The selection uses a range of particle identification (PID) criteria, based on
information from the RICH and muon detectors, to ensure the hadron and the
muons are correctly identified. For example, $\mathrm{DLL}_{K\pi}$ is the
difference in log-likelihoods between the $K$ and $\pi$ hypotheses. For the
$B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ final state, $\mathrm{DLL}_{K\pi}>1$
is required to select kaon candidates. For the
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ final state the selection criterion
is mirrored to select pions with $\mathrm{DLL}_{K\pi}<-1$. The
$B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ and
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ selections are otherwise identical.
To reject background events where two tracks that are close together in the
tracking system share hits in the muon detector, a requirement is made on the
maximum number of muon system hits that two candidate muons may have in
common. This requirement can introduce a bias in the relative efficiency
between signal and normalisation channels, as both tracks from the same-sign
muon pair for $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$ will curve in the same
direction in the dipole field. Simulated events give an estimate of 0.3% for
the effect on the relative efficiency between the signal and normalisation
channel. In order to avoid selecting a muon as the pion or kaon, the candidate
hadron is also required to be within the acceptance of the muon system but not
have a track segment there.
After the application of the above criteria the combinatorial background is
completely dominated by candidates with two real muons, rather than by hadrons
mis-identified as muons.
The invariant mass distribution and the relevant mis-identification rates are
required in order to evaluate the peaking background. These are evaluated,
respectively, from a full simulation using Pythia Sjostrand:2006za followed
by Geant4 Agostinelli:2002hh , and from control channels which provide an
unambiguous and pure source of particles of known type. The control channel
events are selected to have the same kinematics as the signal decay, without
the application of any PID criteria. $D^{*}\\!\rightarrow D^{0}\pi$,
$D^{0}\\!\rightarrow K\pi$ decays give pure sources of pions and kaons. A pure
source of muons is selected by using a tag-and-probe approach with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
decays Aaij:2011rj .
Under the $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ hypothesis, any crossfeed
from $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
decays would peak strongly in the signal mass region. The $K\rightarrow\mu$
mis-id rate is evaluated from the above $D^{*}$ sample and the
$\mu\\!\rightarrow K$ mis-id rate from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample. The later mis-id rate
is consistent with zero. The number of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
expected in the signal region is therefore also zero but with a large
uncertainty which dominates the error on the total exclusive background
expected in the signal region. The $B^{+}\\!\rightarrow\pi^{-}\pi^{+}K^{+}$
decay contributes the most to the peaking background with an expected $(1.7\pm
0.1)\times 10^{-3}$ candidates, followed by the $B^{+}\\!\rightarrow
K^{-}\pi^{+}K^{+}$ decay with $(6.1\pm 0.8)\times 10^{-4}$ candidates. The
total peaking background expected in the $B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+}$ signal region is $(3.4^{+14.0}_{-0.2})\times 10^{-3}$
events with the asymmetric error caused by the zero expectation from the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay.
Under the $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ hypothesis,
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
are reconstructed with invariant masses below the nominal $B^{+}$ mass, in the
lower mass sideband. The dominant background decay in this case is
$B^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$, where the two same-sign pions are
misidentified as muons. The $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ peaking
background level is $(2.9\pm 0.6)\times 10^{-2}$ events.
Figure 2: Invariant mass distribution of $K^{+}\mu^{+}\mu^{-}$ events after
the application of the selection criteria. In (a) requiring the muon pair to
be compatible with coming from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ decay and in (b) excluding invariant mass windows around the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ for the muon
pair. The curve is the fit to data as described in the text.
In Fig. 2(a), the $m_{K^{+}\mu^{+}\mu^{-}}$ invariant mass distribution for
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ events with
$|m_{\mu^{+}\mu^{-}}-m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}|<50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is shown, after the
application of the selection. In the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ sample,
there are no events containing more than one candidate. An unbinned maximum
likelihood fit to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ mass
peak is made with a Crystal Ball Skwarnicki:1986xj function which accounts
for the radiative tail. The combinatorial background is assumed to be flat,
and the partially reconstructed events in the lower mass sideband are fitted
with a Gaussian distribution. The signal peak has a Gaussian component of
width 20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and a signal mass window
of $5280\pm 40{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is chosen. The
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ peak
contains $3407\pm 59$ signal events within the signal window.
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
candidates were also examined and, accounting for a shoulder in the mass
distribution from $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$, the yield observed agrees with the expectation.
The $m_{K^{+}\mu^{+}\mu^{-}}$ invariant mass distribution for events with
$|m_{\mu^{+}\mu^{-}}-m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},\psi{(2S)}}|>70{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is shown in
Fig. 2(b). Using the same fit model, with all shape parameters fixed to those
from the above fit, the signal peak was determined to contain $27\pm 5$ events
from the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay. The ratio of
branching fractions between
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays Nakamura:2010zzi and the
trigger efficiency ratio predicted by the simulation, give an expectation of
$29\pm 4$ $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays. The observed yield
is consistent with the expectation showing that the selection does not favour
candidates with a dimuon mass close to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.
The difference in efficiency between the signal and normalisation channels was
evaluated using Monte Carlo simulation samples. The relative selection
efficiency across the phase space is shown for $B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+}$ in Fig. 3. The efficiency of the signal selection in a
given phase space bin is divided by the average efficiency of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, to
yield the relative efficiency for that bin. The $D^{*}$ control channel is
used to determine the PID efficiencies required to normalise
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ to
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$.
Figure 3: Relative efficiency between the $B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+}$ signal and the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
normalisation channel. The plot has been symmetrised over the diagonal.
Assuming a signal that is uniformly distributed in phase space, the relative
efficiency of $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ was
calculated to be $89.1\pm 0.4\,\mathrm{(stat)}\pm 0.3\,\mathrm{(syst)}\%$. The
relative efficiency of $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ was
calculated to be $82.7\pm 0.6\,\mathrm{(stat)}\pm 0.8\,\mathrm{(syst)}\%$. The
systematic uncertainties associated with these estimates are detailed below.
These relative efficiencies together with the number of events observed in the
normalisation channel, give single event sensitivities of $2.0\times 10^{-8}$
($2.1\times 10^{-8}$) in the $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$
($B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$) case.
In order to compute the efficiency under a given Majorana neutrino mass
hypothesis, a model for the variation of efficiency with $m_{h\mu}$ is
required. For a given value of $m_{h\mu}$ this is obtained by varying the
polarisation of the Majorana neutrino in the decay and taking the lowest (most
conservative) value of the efficiency.
The dominant systematic uncertainty (under the assumption of a flat phase-
space distribution) for the single event sensitivity is the 3.4% uncertainty
on the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
branching fraction. The statistical uncertainty on the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield
gives an additional systematic uncertainty of 1.7% and the uncertainty from
the model used to fit the data is 1.6%. The latter is evaluated by changing
the Crystal Ball signal function used in the fit to a Gaussian and the
polynomial background function to an exponential.
There are several sources of uncertainty associated with the calculation of
the relative efficiency between the signal and normalisation channels. In
addition to the statistical uncertainty of the simulation samples, there are
systematic uncertainties from: the differences in the effect of the IP
selection criteria between the simulation and data; the statistical
uncertainty on the measured PID efficiencies; the uncertainties associated
with the simulation of the trigger; and the uncertainty in the tracking
efficiency. In each case the systematic uncertainty is estimated by varying
the relevant criteria at the level of the expected effect and re-evaluating
the relative efficiency. For the $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$
decay, there is an additional uncertainty from the correction for the relative
kaon- and pion-identification efficiencies. The systematic uncertainties
averaged over the three-body phase space are given in Table 1.
Table 1: Sources of systematic error and their fractional uncertainty on the relative efficiency. Source | $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ | $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$
---|---|---
$\mathcal{B}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$ | 3.4% | 3.4%
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield | 1.7% | 1.7%
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ fit models | 1.6% | 1.6%
Simulation statistics | 0.4% | 0.6%
IP modelling | 0.2% | 0.2%
PID modelling | 0.1% | 0.8%
Trigger efficiency | 0.1% | 0.1%
Tracking efficiency | 0.1% | 0.1%
A limit on the branching fraction of each of the $B^{+}\\!\rightarrow
h^{-}\mu^{+}\mu^{+}$ decays is set by counting the number of observed events
in the mass windows, and using the single event sensitivity. The probability
is modelled with a Poisson distribution where the mean has contributions from
a potential signal, the combinatorial and peaking backgrounds. The
combinatorial background is unconstrained by measurements from the simulation
or the opposite-sign data. The number of events in the upper mass sideband is
therefore used to constrain the contribution of the combinatorial background
to the Poisson mean. The upper mass sideband is restricted to masses above
$m_{h\mu\mu}>5.4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ such that any
peaking background component can be ignored. In both the $B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+}$ and $B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ cases no
events are found in either the upper or lower mass sidebands. This is
consistent with the observation of three opposite-sign candidates seen in the
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ upper mass sideband (Fig. 2) and two
candidates in the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ upper mass
sideband. The peaking background estimates are explicitly split into two
components, the contribution from $B^{+}\rightarrow h^{-}h^{+}h^{+}$ decays
and that from $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ decays. The latter has a large uncertainty. The central values
for both peaking background components are taken from the estimates described
above.
Systematic uncertainties on the peaking background, single event sensitivity
and signal-to-sideband scale factor are included in the limit-setting
procedure using a Bayesian approach. The unknown parameter is integrated over
and included in the probability to observe a given number of events in the
signal and upper mass window.
In the signal mass windows of $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ and
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ no events are observed. This
corresponds to limits on the $B^{+}\\!\rightarrow h^{-}\mu^{+}\mu^{+}$
branching fractions of
$\begin{split}\mathcal{B}(B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+})&<5.4\,(4.1)\times 10^{-8}~{}\text{at 95\%\,(90\%) CL},\\\
\mathcal{B}(B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+})&<5.8\,(4.4)\times
10^{-8}~{}\text{at 95\%\,(90\%) CL}.\\\ \end{split}$
Figure 4: The 95 % CL branching fraction limits for $B^{+}\\!\rightarrow
K^{-}\mu^{+}\mu^{+}$ (light-coloured line) and
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ (dark-coloured line) as a function
of the Majorana neutrino mass $m_{\nu}=m_{h\mu}$.
The observation of no candidates in the sidebands as well as the signal region
is compatible with a background-only hypothesis. The $m_{h\mu}$ dependence of
the limit in models where the Majorana neutrino can be produced on mass shell
is shown in Fig. 4. The shapes of the limits arise from the changing
efficiency as a function of mass.
In summary, a search for the $B^{+}\\!\rightarrow K^{-}\mu^{+}\mu^{+}$ and
$B^{+}\\!\rightarrow\pi^{-}\mu^{+}\mu^{+}$ decays modes has been performed
with $36\mbox{\,pb}^{-1}$ of integrated luminosity collected with the LHCb
detector in 2010. No signal is observed in either decay and, using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ as a
normalisation channel, the present best limits on ${\cal B}(B^{+}\rightarrow
K^{-}\mu^{+}\mu^{+})$ and ${\cal B}(B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})$
are improved by factors of 40 and 30, respectively Edwards:2002kq .
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Région Auvergne.
## References
* (1) E. Majorana, Teoria simmetrica dell’elettrone e del positrone, Nuovo Cim. 14 (1937) 171–184.
* (2) J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D10 (1974) 275–289. Erratum-ibid. D11 (1975) 703.
* (3) S. Pascoli and S. Petcov, Majorana Neutrinos, Neutrino Mass Spectrum and the $|\langle m\rangle|\sim 10^{-3}$ eV Frontier in Neutrinoless Double Beta Decay, Phys. Rev. D77 (2008) 113003, [arXiv:0711.4993].
* (4) CLEO collaboration, K. W. Edwards et al., Search for lepton flavor violating decays of B mesons, Phys. Rev. D65 (2002) 111102, [arXiv:hep-ex/0204017].
* (5) LHCb collaboration, A. A. Alves et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005.
* (6) T. Sjöstrand, S. Mrenna, and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026, [arXiv:hep-ph/0603175].
* (7) GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250–303.
* (8) LHCb collaboration, R. Aaij et al., Search for the rare decays $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$, Phys. Lett. B699 (2011) 330–340, [arXiv:1103.2465].
* (9) T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances. PhD thesis, Institute of Nuclear Physics, Krakow, 1986. DESY-F31-86-02.
* (10) Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021.
|
arxiv-papers
| 2011-10-04T15:45:45 |
2024-09-04T02:49:22.769737
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y.\n Amhis, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L.\n Arrabito, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann,\n J.J. Back, D.S. Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S.\n Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, S. Brisbane, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, J.M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P.\n Collins, A. Comerma-Montells, F. Constantin, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L.\n Est\\'eve, A. Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli,\n S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M.\n Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C.\n Gaspar, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T.\n Gys, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R.\n Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V. Heijne, K. Hennessy,\n P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, W. Hofmann, K.\n Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F.\n Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei, M. Karacson,\n T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M.\n Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, S. Kukulak, R. Kumar, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li\n Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.H. Lopes, E.\n Lopez Asamar, N. Lopez-March, J. Luisier, F. Machefert, I.V. Machikhiliyan,\n F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel,\n R. Messi, S. Miglioranzi, D.A. Milanes, M.-N. Minard, S. Monteil, D. Moran,\n P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, J.\n Nardulli, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M.\n Nicol, S. Nies, V. Niess, N. Nikitin, A. Nomerotski, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M.\n Otalora Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J.\n Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N. Patrick, C. Patrignani,\n C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrella, A.\n Petrolini, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pilar, D.\n Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S. Redford, M.M. Reid,\n A.C. dos Reis, S. Ricciardi, K. Rinnert, D.A. Roa Romero, P. Robbe, E.\n Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V.\n Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, B. Shao, M. Shapkin,\n I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, H.P. Skottowe, T.\n Skwarnicki, A.C. Smith, N.A. Smith, E. Smith, K. Sobczak, F.J.P. Soler, A.\n Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, N. Styles, V.K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, M.T. Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja,\n P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, K. Vervink, B. Viaud, I.\n Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Voong, A.\n Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D.R. Ward, A.D. Webber,\n D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P.\n Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A.\n Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O.\n Yushchenko, M. Zavertyaev, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L.\n Zhong, E. Zverev, A. Zvyagin",
"submitter": "Mitesh Patel",
"url": "https://arxiv.org/abs/1110.0730"
}
|
1110.0744
|
# Electronic states in heterostructures formed by ultranarrow layers
F. T. Vasko fedirvas@buffalo.edu V. V. Mitin Department of Electrical
Engineering, University at Buffalo, Buffalo, NY 14260-1920, USA
###### Abstract
Low-energy electronic states in heterosrtuctures formed by ultranarrow layers
(single or several monolayers thickness) are studied theoretically. The host
material is described within the effective mass approximation and effect of
ultranarrow layers is taken into account within the framework of the transfer
matrix approach. Using the current conservation requirement and the inversion
symmetry of ultranarrow layer, the transfer matrix is evaluated through two
phenomenological parameters. The binding energy of localized state, the
reflection (transmission) coefficient for the single ultranarrow layer case,
and the energy spectrum of superlattice are determined by these parameters.
Spectral dependency of absorption due to photoexcitation of electrons from
localized states into minibands of superlattice is determined by the
ultranarrow layers characteristics. Such a dependency can be used for
verification of the transfer matrix and should modify characteristics of
optoelactronic devices with ultranarrow layers. Comparision with experimental
data shows that the effective mass approach is not valid for description of
ultranarrow layer.
###### pacs:
73.21.-b, 78.67.Pt
## I Introduction
Multi-layer heterostructures are widely used in different devices, such as bi-
or monopolar heterosrtucture lasers, photodetectors and solar cells, see
reviews 1 or 1a , 2 , and 3 , respectively. Thicknesses of layers in such
structures are varied widely, starting from an ultranarrow layer (UNL) of
thickness $a$ or $Na$ ($a$ is the monolayer thickness and $N=1,2,\ldots$ is a
number of monolayers) for a single or several monolayer structures, e.g. the
case of the quantum dot sheets with ultranarrow wetting layers, see Refs. 5.
The case of $N\gg 1$ corresponds to a wide well or a barrier which are
described in the framework of the effective mass approximation (EMA) or the
$\bf kp$-method supplied by appropriate boundary conditions at
heterointerfaces. 5 Early in the eighties, similar approaches were used for
description of abrupt heterojunctions between different bulk semiconductors,
see 6 ; 7 and references in Ref. 8. The electronic properties of short-period
superlattices (SL), i. e. heterostructures formed by periodical UNLs, were
studied experimentally and numerically based on different approximations, see
Refs. 9 and 10, respectively. These approaches determine the energy spectra of
SL only but they are not suitable for consideration of transport and optical
phenomena. To the best of our knowledge, an effect of UNLs on these phenomena
was not considered in details because the effective mass approximation is not
valid over scales $\sim a$. If a heterostructure includes UNL, one should
consider it as a new object which is described by boundary conditions added to
the EMA equations at UNL’s positions, $z=z_{0}$. Because heterostructures with
UNLs are routinely used in different optoelectronic devices without an
investigation of the peculiarities mentioned, it is important and timely to
develop an adequate theory of electronic states at UNL and to perform a
verification of UNL’s parameters.
Figure 1: Band diagrams of UNLs enclosed by host semiconductor for: (a) well-
like case with $|\tau|<1$, (b) barrier-like case with $\tau>1$ , and (c)
combined case with $\tau<-1$, and (d) non-symmetric case. Here $\tau$ is the
diagonal element of transfer matrix, see Eq. (5).
In this paper we consider an UNL placed at $\\{z_{0}\\}$, by applying the
effective mass approximation (or the $\bf kp$-method) for the host material,
at $z\neq z_{0}$, and by using boundary conditions at $z\to z_{0}$ which
describe modifications of envelope functions at UNL. In order to describe low-
energy electron states, with energies in the vicinity of the conduction band
extremum, in heterosrtuctures formed by UNLs we employ the EMA-approximation
in the host material and the boundary conditions at UNLs written through the
transfer matrix. The parameters of such a matrix are restricted by the current
conservation condition and the boundary conditions are written below for a
symmetric UNL, see Figs. 1a - 1c (the case of non-symmetric UNL, shown in Fig.
1d will be considered elsewhere). Depending on the diagonal element of
transfer matrix $\tau$ introduced in Eq. (5), one should separate of UNLs into
three categories: (i) a well-like UNL with $|\tau|<1$, (ii) a barrier-like UNL
with $\tau>1$, and (iii) a non-conventional UNL with $\tau<-1$. These cases
are sketched in Figs. 1a - 1c, respectively, where thicknesses of layers
(filled in by grey) are comparable to $a$. While the cases (i) or (ii)
correspond to EMA description of a narrow well or a barrier at $Na\to 0$, 10
a non-conventional UNL has no simple EMA analogy. In general, it is convenient
to characterize the transfer matrix by a dimensionless phase, $\Theta$, and a
characteristic wave vector, $K$, see Eqs. (6) and (7) below. These
phenomenological parameters determine a character of low-energy states
together with the effective mass (or parameters of $\bf kp$-method, if energy
is comparable to gap of host semiconductor) and they should be determined from
experimental data.
Here we have evaluated the transfer matrix which is written through the
parameters $\Theta$ and $K$ (or through characteristic energy $E_{K}$) and
have considered electronic properties of different types of single UNLs as
well as SL formed by UNLs. We calculated the binding energy of localized state
at a single UNL, which appears to be allowed state for the cases (i) and (iii)
and to be forbidden state for the barrier-like case (ii). The energy-dependent
reflection (transmission) coefficient is found when considering the scattering
of electron on UNL. For the SL formed by UNLs, we analyze the energy spectrum,
the density of states, and the absorption coefficient due to photoexcitation
of localized electrons under THz or mid-IR excitation. In addition, we
demonstrate that these properties are modified essentially under variation of
the UNL’s characteristics and we discuss possibilities for verification of the
transfer matrix parameters.
The consideration below is organized as follows. In Sect. II we describe an
UNL in the framework of the transfer matrix approach. The single-layer case is
considered in Sect. III and the properties of SL formed by UNL are considered
in Sect. IV. Discussion on experimental verification of UNL’s parameters is
performed in Sect. V. Concluding remarks and list of assumptions used are
given in the last section.
## II Transfer matrix approach
Within the one-band effective mass approximation, the electronic states in
bulk semiconductor with an UNL placed at $z=z_{0}$ is described by the
Schrodinger equation:
$\frac{{\hat{p}_{z}^{2}}}{{2m}}\Psi_{z}=E\Psi_{z},~{}~{}~{}~{}z\neq z_{0}.$
(1)
Here $m$ is the effective mass and $\hat{p}_{z}$ is the momentum operator. The
second-order differential equation (1) should be solved with the boundary
conditions which connect the wave functions and their first derivatives at
$z_{0}-0$ and $z_{0}+0$. Similarly to the consideration of an abrupt
heterojunction case, 6 ; 7 we write the connection rules for the column
$\Phi_{z}$ in the form:
$\Phi_{z_{0}+0}=\hat{T}\Phi_{z_{0}-0},~{}~{}~{}~{}\Phi_{z}\equiv\left|{\begin{array}[]{*{20}c}{\Psi_{z}}\\\
{d\Psi_{z}/dz}\\\ \end{array}}\right|.$ (2)
The $2\times 2$ transfer matrix $\hat{T}$ is determined by the current
conservation requirement and by the symmetry properties of layer.
From Eq. (1) it follows that
$\widetilde{\Phi}_{z}^{+}\hat{\sigma}_{y}\Phi_{z}$ does not dependent on $z$
if $z\neq z_{0}$ (here and below $\hat{}\mbox{\boldmath$\sigma$}$ is the Pauli
matrix which connects $\Psi_{z}$ and its derivative). We impose the current
conservation requirement at UNL as
$\widetilde{\Phi}_{z}^{+}\hat{\sigma}_{y}\Phi_{z}|_{z_{0}-0}^{z_{0}+0}=0$.
Since two arbitrary states $\widetilde{\Phi}_{z}$ and $\Phi_{z}$ are
considered here, the current conservation gives the condition for the transfer
matrix:
$\hat{T}^{+}\hat{\sigma}_{y}\hat{T}=\hat{\sigma}_{y},$ (3)
see similar evaluation for heterojunction in Ref. 8. Further, we restrict
ourselves by the case of symmetric UNL, when $\Phi_{z}$ and
$\hat{\sigma}_{z}\Phi_{-z}$ should be determined by the same equations. As a
result one obtains the additional condition:
$\hat{\sigma}_{z}\hat{T}\hat{\sigma}_{z}=\hat{T}^{-1}$ (4)
and 4 complex parameters of $\hat{T}$ should be determined from 8 conditions
given by Eqs. (3) and (4). Straightforward calculations give us the transfer
matrix
$\hat{T}=\left|{\begin{array}[]{*{20}c}\tau&{\tau_{12}}\\\ {\tau_{21}}&\tau\\\
\end{array}}\right|,~{}~{}~{}~{}\det\hat{T}=1$ (5)
written through real parameters $\tau$, $\tau_{12}$ and $\tau_{21}$. Since the
determinant is fixed here, the matrix $\hat{T}$ depends on two real parameters
which should be considered as phenomenological characteristics of UNL.
At $\left|\tau\right|<1$, it is convenient to introduce the phase, $\Theta$,
and wave vector, $K$, so that the transfer matrix takes the form:
$\hat{T}=\left|{\begin{array}[]{*{20}c}{\cos\Theta}&{K^{-1}\sin\Theta}\\\
{-K\sin\Theta}&{\cos\Theta}\\\ \end{array}}\right|$ (6)
which depends on two phenomenological parameters, $K>0$ and $0<\Theta<2\pi$.
If $\left|\tau\right|>1$, one can use similar expressions for $\hat{T}_{\pm}$
which is written through the hyperbolic functions:
$\hat{T}_{\pm}=\left|{\begin{array}[]{*{20}c}{\pm\cosh\Theta}&{K^{-1}\sinh\Theta}\\\
{K\sinh\Theta}&{\pm\cosh\Theta}\\\ \end{array}}\right|$ (7)
Here the signs $+$ and $-$ correspond to $\tau>1$ and $\tau<-1$, respectively.
Once again, we suppose $K>0$ in Eq. (7) because $\Theta$ can have any sign.
The explicit expressions for $\Theta$ and $K$ can be written for $N\gg 1$,
when $\hat{T}$ can be evaluated within the EMA approximation, for the wide
layer cases (i) and (ii). For the $N$-monolayer well or barrier of the
thickness $Na$ with the band offset $\Delta U$ and the effective mass $m_{l}$,
one obtains the transfer matrix $\hat{T}$ or $\hat{T}_{+}$ given by Eqs. (6)
or (7), respectively. The parameters $\Theta$ and $K$ are determined by
$\Theta=N\sqrt{\frac{m_{l}}{m}}Ka,~{}~{}~{}~{}~{}E_{K}=\frac{(\hbar
K)^{2}}{2m}=\frac{m}{m_{l}}\Delta U,$ (8)
where the characteristic energy $E_{K}$ is introduced. For the case of InAs
well placed in GaAs 11 one obtains that the energy $E_{K}$ varies between 1 -
2.8 eV depending on the mismatch stress contributions. The phase $\Theta$
varies between $0.32N$ \- $0.42$N, where $N\gg 1$ [for the case (i), $\Theta$
should be reduced to the interval (0, $2\pi$)]. Similar estimates can be
obtained for other heterostructures with well- or barrier-like UNL. Notice,
that the non-conventional case $\tau<-1$ described by matrix $\hat{T}_{-}$ has
no analogy with the results for well or barrier described by Eq. (8) within
the EMA approximation.
## III Single-layer case
Here we consider the single UNL placed at $z_{0}=0$. The localized state, with
energy $-E_{0}<0$, is described by the wave function
$\Psi_{z}=\Psi_{0}\left[{\theta(z)e^{-\kappa z}+\theta(-z)e^{\kappa
z}}\right],~{}~{}~{}E_{0}=\frac{(\hbar\kappa)^{2}}{2m},$ (9)
where $\theta(z)$ is the Heaviside step function, $E_{0}$ is the binding
energy written through the characteristic size of localization, $\kappa^{-1}$,
and $\Psi_{0}=\sqrt{\kappa}$ is the normalization coefficient. The boundary
condition (2) gives us the dispersion relation which determines $\kappa>0$
through the transfer matrix parameters, $\Theta$ and $E_{K}$. As a result, at
$\tau<1$ the binding energy of localized state $E_{0}$ is given by:
$E_{0}=E_{K}\left\\{\begin{array}[]{*{20}c}\left(\frac{\sqrt{1+\tan^{2}\Theta}-1}{\tan\Theta}\right)^{2},&\left|\tau\right|<1\\\
\left(\frac{\sqrt{1-\tanh^{2}\Theta}+1}{\tanh\Theta}\right)^{2},&\tau<-1\\\
\end{array}\right.$ (10)
and there is no localized state for the barrier-like case $\tau>1$.
Figure 2: (Color online) Dimensionless energy of localized state $E_{0}/E_{K}$
versus phase $\Theta$ for the cases of $|\tau|<1$ (a) and $\tau<-1$ (b).
Dashed curves in panel (b) correspond to asymptotes at $\Theta\ll 1$ and
$\Theta\gg 1$.
In Fig. 2 we plot the dependencies of $E_{0}/E_{K}$ versus $\Theta$. If
$|\tau|<1$ then $E_{0}/E_{K}<1$, moreover a deep level with $E_{0}\to E_{K}$
is realized at $\Theta\to\pi/2$ and such a dependency is plotted over the
interval (0, $\pi$) because it is the periodical function. At
$\Theta\leq\pi/4$ or $\pi-\Theta\leq\pi/4$ a shallow state at UNL is realized.
For a non-conventional UNL with $\tau\leq-1$, a deep level with
$E_{0}/E_{K}>1$ is realized and $E_{0}\to E_{K}$ if $|\Theta|\gg 1$ while in
the opposite case $|\Theta|\to 0$ one obtains the divergent asymptote
$E_{0}/E_{K}\approx(2/\Theta)^{2}$. These asymptotes are also shown in Fig.
2b.
Figure 3: (Color online) (a) Reflection coefficient $R_{E}$ versus
dimensionless energy $E/E_{K}$ for the case $|\tau|<1$ at $\Theta=\pi/16$ (1),
$\pi/8$ (2), $\pi/4$ (3), and $\pi/2$ (4). (b) The same for the case
$|\tau|>1$ at $\Theta=0.1$ (1), 0.4 (2), 0.8 (3), and 1.2 (4).
The scattering problem, for an electron with energy $E>0$ propagating from the
left, is described by the wave function
$\Psi_{z}=\theta(-z)\left(\Psi_{i}e^{ikz}+\Psi_{r}e^{-ikz}\right)+\theta(z)\Psi_{t}e^{ikz},$
(11)
where $\Psi_{i}$, $\Psi_{r}$, and $\Psi_{t}$ are the amplitudes of incident,
reflected, and transmitted waves. The wave vector $k$ is connected with the
energy $E$ by the standard relation $E=(\hbar k)^{2}/2m$. The reflected and
transmitted amplitudes, $\Psi_{r}$, and $\Psi_{t}$, are expressed through
$\Psi_{i}$ from the boundary condition (2). Further, we introduce the flows
$J_{\gamma}=\left|{\Psi_{\gamma}}\right|^{2}\hbar k/m$, which correspond to
the incident ($\gamma=i$), reflected ($\gamma=r$), and transmitted
($\gamma=t$) waves, and calculate the reflection and transmission coefficients
according to $R_{E}=J_{r}/J_{i}$ and $T_{E}=J_{t}/J_{i}$. Since the particle
conservation law, $R_{E}+T_{E}=1$, only the reflection coefficient is
considered below:
$\displaystyle R_{E}=\left[{1+F(\Theta,k/K)}\right]^{-1},$ $\displaystyle
F(\Theta,z)=\left\\{{\begin{array}[]{*{20}c}{\left[{\frac{{2z}}{{\sin\Theta(1-z^{2})}}}\right]^{2},}&{\left|\tau\right|<1}\\\
{\left[{\frac{{2z}}{{\sinh\Theta(1+z^{2})}}}\right]^{2},}&{|\tau|>1}\\\
\end{array}}\right..$ (14)
Here $F(\Theta,z)=F(-\Theta,z)$ and $R_{E}$ is determined by positive
$\Theta$; the interval $0<\Theta<\pi/4$ is enough in order to describe the
case of UNL with $|\tau|<1$.
Fig. 3 shows the reflection coefficient $R_{E}$ versus $E/E_{K}=(k/K)^{2}$.
Since $F(\Theta,0)=0$, there is no tunneling of low-energy electrons through
any UNL because $R_{E\to 0}\to 1$. If energy increases, $R_{E}$ decrease up to
$E=E_{K}$ and, once again, $R_{E}\to 1$ if $E\gg E_{K}$. But in the region
$E\sim E_{K}$ behavior of $R_{E}$ is different for $|\tau|<1$ and $|\tau|>1$:
the well-like case $R_{E_{K}}=0$, while for $|\tau|<1$ the reflection
coefficient approaches constant, compare Figs. 3a and 3b [there is no
difference between the cases (ii) and (iii) in panel (b)].
## IV Superlattice
We turn now to the consideration of electronic states in SL of the period $l$
formed by the UNLs placed at $z_{0}\to\left\\{rl\right\\},~{}r=0,\pm
1,\cdots$. Based on the boundary conditions (2) at $z=rl$, below we consider
the miniband energy spectrum and calculate the absorption coefficient of doped
SL under photoexcitation of localized states.
### IV.1 Miniband spectrum
The eigenvalue problem for the periodic system of UNLs is solved introducing a
quasimomentum $0<p_{\bot}<2\pi\hbar/l$ and using the Bloch’s theorem,
$\Psi_{p_{\bot}z}=\exp(ip_{\bot}l/\hbar)\Psi_{p_{\bot}z-l}$. The wave function
takes form
$\Psi_{p_{\bot}z}=\frac{N_{p_{\bot}}}{\sqrt{l}}\left[e^{ikz}+R(p_{\bot},k)e^{-ikz}\right],~{}~{}~{}~{}0<z<l,$
(15)
where $N_{p_{\bot}}$ is the normalization factor. By analogy with the general
consideration, 10 the coefficient $R(p_{\bot},k)$ and the dispersion relation
between $p_{\bot}$ and $k$ are determined from the periodicity requrement and
the boundary conditions (2) at $z=0,~{}l$. For the UNL with $|\tau|<1$ we use
the transfer matrix (6) and the factor $R(p_{\bot},k)$ takes form
$R(p_{\bot},k)=\frac{e^{ip_{\bot}l/\hbar}-\left[\cos\Theta+(ik/K)\sin\Theta
e^{ikl}\right]}{e^{ip_{\bot}l/\hbar}-\left[\cos\Theta-(ik/K)\sin\Theta
e^{-ikl}\right]},$ (16)
while the dispersion relation is written as follows:
$\cos\frac{p_{\bot}l}{\hbar}=\cos\Theta\cos
kl-\frac{1}{2}\left(\frac{K}{k}+\frac{k}{K}\right)\sin\Theta\sin kl.$ (17)
If $|\tau|>1$ one uses the transfer matrix (7) and the factors
$R_{\pm}(p_{\bot},k)$ should be written similarly to Eq. (14) but through the
hyperbolic functions, $\pm\cosh\Theta$ and $\sinh\Theta$ instead of
$\cos\Theta$ and $\sin\Theta$. The corresponding dispersion relation is given
by
$\displaystyle\cos\frac{p_{\bot}l}{\hbar}=\pm\cosh\Theta\cos kl$
$\displaystyle+\frac{1}{2}\left(\frac{K}{k}-\frac{k}{K}\right)\sinh\Theta\sin
kl.$ (18)
As in Eq. (7), here and in $R_{\pm}(p_{\bot},k)$ the signs $+$ and $-$
correspond to $\tau>1$ and $\tau<-1$, respectively. The last case has no
analogy with the standard dispersion relations obtained in the EMA
approximation, while Eq. (15) and Eq. (16) with $+\cosh\Theta$ are similar to
the results for well or barrier of finite width. 10
The contour plots of the right-hand sides of Eqs. (15) and (16) versus
$\Theta$ and dimensionless energy $E/\sqrt{E_{K}\varepsilon_{l}}$, [here
$\varepsilon_{l}=(\hbar/l)^{2}/2m$ is a small energy corresponding to SL of
period $l$] are presented in Fig. 4. Here the gap regions, which are above +1
and below $-1$, are shaded by grey and the dashed curves are plotted at the
middle of miniband energies, when $\cos p_{\bot}l/\hbar=0$. For the well-like
case, $|\tau|<1$, the weakly-coupled SL is realized if $\Theta\to
0,~{}\pi,~{}2\pi$ and far from these points the tight-binding coupling regime
takes place, see Figs. 4a and 4b. With increasing of the dimensionless
parameter $\sqrt[4]{E_{K}/\varepsilon_{l}}$, a number of minibands increases
and a weakly-coupled regime is under more rigid conditions, compare Figs. 4a
and 4b. If $|\tau|>1$ (i.e. SL is formed by barrier-like or non-conventional
UNLs), the tight-binding regime of coupling takes place if $\Theta>0.5$ and a
weakly-coupled SL is realized at $\Theta\to 0$. Transformations between these
regimes are different if $\tau>1$ or $\tau<-1$, compare Figs. 4c and 4d.
Figure 4: (Color online) (a) Contour plot of the right-hand side of Eq. (15)
versus $\Theta$ and dimensionless energy $E/\sqrt{E_{K}\varepsilon_{l}}$ at
$\sqrt[4]{E_{K}/\varepsilon_{l}}=$10\. (b) The same as in panel (a) at
$\sqrt[4]{E_{K}/\varepsilon_{l}}=$20\. (c) The same as in panel (a) for
dispersion equation (16) at $\tau>1$ and
$\sqrt[4]{E_{K}/\varepsilon_{l}}=$15\. (d) The same as in panel (c) at
$\tau<-1$. Gap regions are shaded, dashed curves correspond to zero level, and
dotted lines correspond to cross-sections shown in Fig. 5 below.
The dispersion laws $\varepsilon_{rp_{\bot}}=(\hbar k_{rp_{\bot}})^{2}/2m$,
which are written through $k_{rp_{\bot}}$ determined by the dispersion
equations (15) or (16) depending on UNL parameters, are shown in Fig. 5 at
$\Theta$ marked in each panel of Fig. 4. Energy scale of gaps and allowed
bands is determined by the characteristic energy $\sqrt{E_{K}\varepsilon_{l}}$
which is between 70 - 25 meV for $l=$10 nm - 50 nm if $E_{K}$ is taken as 1
eV. The dispersion laws plotted in Fig. 5 are close to cosine or sine
dependencies.
Figure 5: Miniband energy spectra $\varepsilon_{rp_{\bot}}$ for the parameters
used in Figs. 4a-4d at $\Theta=\pi/8$ (a, b) and 0.2 (c,d).
Farther, we consider the density of states which is introduced by the standard
formula:
$\rho_{E}=(2/L^{3})\sum\nolimits_{\delta}\delta(E-\varepsilon_{\delta})$ where
$\varepsilon_{\delta}\to\varepsilon_{rp_{\bot}}+\varepsilon_{p}$. Here we
separated the in-plane kinetic energy, $\varepsilon_{p}=p^{2}/2m$, which
corresponds to the 2D momentum $\bf p$, and the superlattice contribution,
$\varepsilon_{rp_{\bot}}$ plotted in Fig. 5. Integration over $\bf p$ gives
the 2D density of states, $\rho_{2D}$, and $\rho_{E}$ appears to be written
through the integrals taken over step function, $\theta(z)$:
$\rho_{E}=\rho_{2D}\sum\limits_{r}\int\limits_{0}^{2\pi\hbar/l}\frac{dp_{\bot}}{2\pi\hbar}\theta\left(E-\varepsilon_{rp_{\bot}}\right).$
(19)
If $E$ belongs to $\bar{r}$th gap, the $\theta$-functions for $r\leq\bar{r}$
should be replaced by unit and $\rho_{E}$ is constant. In $\bar{r}$th miniband
(below $r$th gap), the integral over $p_{\bot}$ should be taken over the
interval $(0,p_{E})$ where $p_{E}$ is found as a root of the equation
$E=\varepsilon_{\bar{r}p_{E}}$. As a result, the density of states takes the
form:
$\rho_{E}=\frac{\rho_{2D}}{l}\left\\{\begin{array}[]{*{20}c}\bar{r},&E\subset\bar{r}{\rm
th~{}gap}\\\ \bar{r}-1+\frac{p_{E}l}{2\pi\hbar},&E\subset\bar{r}{\rm
th~{}miniband}\\\ \end{array}\right.$ (20)
and a ladder-like shape of $\rho_{E}$ is determined by the gap-induced steps
with miniband contributions between them.
In Fig. 6 we plot the density of states for the same parameters as in Fig. 5.
Here the thick straight lines correspond to gap contributions with the
miniband contributions between them which are similar to arccosine
dependencies. One can see that a number of minibands increases with the
parameter $\sqrt[4]{E_{K}/\varepsilon_{l}}$. An approach to the square-root
dependence, corresponding to the bulk density of states, takes place at
$E/\sqrt{E_{K}\varepsilon_{l}}>$3\. Since $\rho_{E}$ is connected directly to
the shape of interband optical spectra, see Ref. 6b, the step-like
dependencies over the interval $E/\sqrt{E_{K}\varepsilon_{l}}\leq$3 permit one
to extract UNL parameters which determine the bandstructure of SL. Effect of
the above-barrier states on PLE spectra in wide InGaAs/GaAs structure was
measured and calculated within the EMA approach in Ref. 12.
Figure 6: Normalized density of states for the parameters used in panels (a-d)
of Figs. 4 and 5.
### IV.2 Absorption coefficient
Using the solutions obtained, we consider below the process of photoexcitation
of electrons localized at levels of energy $-E_{0}$ into minibands caused by
the radiation polarized along SL axis. Such absorption takes place in heavily
doped SL formed by the well-like or non-conventional UNLs; we do not consider
here the case (ii) which is similar to an ordinary SL, see Ref. 13. The
absorption coefficient $\alpha_{\omega}$ is determined by the general Kubo
formula as follows:
$\displaystyle\alpha_{\omega}=\frac{8(\pi e)^{2}}{\sqrt{\epsilon}c\omega
L^{3}}\sum\limits_{\delta\delta^{\prime}}\left[f(\varepsilon_{\delta})-f(\varepsilon_{\delta}+\hbar\omega)\right]$
$\displaystyle\times\left|(\delta|\hat{v}_{z}|\delta^{\prime})\right|^{2}\delta\left(\varepsilon_{\delta}-\varepsilon_{\delta^{\prime}}+\hbar\omega\right),$
(21)
where $\epsilon$ is the dielectric permittivity of the host semiconductor,
$L^{3}$ stands for the normalization volume, and the matrix element
$\left|(\delta|\hat{v}_{z}|\delta^{\prime})\right|^{2}$ corresponds to
transitions between $\delta$\- and $\delta^{\prime}$-states of energies
$\varepsilon_{\delta}$ and $\varepsilon_{\delta^{\prime}}$. We use the
equilibrium distribution $f(\varepsilon_{\delta})$ and take into account
$\varepsilon_{\delta}\to\varepsilon_{p}-E_{0}$ because only localized states
are populated. Since transitions are vertical, the energy conservation law and
the matrix element do not depend on $\bf p$ and the 2D concentration
$n_{2D}=(2/L^{2})\sum\nolimits_{\bf p}f(\varepsilon_{p}-E_{0})$ appears in
(19) if $\hbar\omega>E_{0}$. In addition, the matrix element
$M_{rp_{\bot}}=\left|(0|\hat{v}_{z}|rp_{\bot})\right|^{2}$ is the same for any
UNL and (19) may be transformed into the sum over different minibands
$\alpha_{\omega}=\frac{2\pi
e^{2}n_{2D}}{\sqrt{\epsilon}c\hbar\omega}\sum\limits_{r}\int\limits_{0}^{2\pi\hbar/l}dp_{\bot}M_{rp_{\bot}}\delta\left({\hbar\Delta\omega-\varepsilon_{rp_{\bot}}}\right)$
(22)
written through the frequency detuning, $\Delta\omega=\omega-E_{0}/\hbar$.
The transparency regions take place if $\hbar\Delta\omega$ belongs to any gap.
If $\hbar\Delta\omega$ is brought into the $\bar{r}$th miniband, the
corresponding absorption coefficient takes the form
$\alpha_{\omega}^{(\bar{r})}=\frac{{2\pi
e^{2}n_{2D}}}{{\sqrt{\epsilon}c\hbar\omega}}\frac{M_{\bar{r}p_{\Delta\omega}}}{|d\varepsilon_{\bar{r}p_{\bot}}/dp_{\bot}|_{p_{\Delta\omega}}},$
(23)
where $p_{\Delta\omega}$ should be determined from the equation
$\hbar\Delta\omega=\varepsilon_{rp_{\Delta\omega}}$. Under the condition
$\kappa l\gg 1$, the matrix element $M_{rp_{\bot}}$ is written through the
wave functions (9) and (14) in the form:
$\displaystyle
M_{rp_{\bot}}=\left({\frac{\hbar}{m}}\right)^{2}\frac{\kappa}{l}\left|{N_{p_{\bot}}}\right|^{2}~{}~{}~{}$
(24)
$\displaystyle\times\left|{1+e^{ikl}+R\left({p_{\bot},k}\right)\left({1+e^{-ikl}}\right)}\right|^{2}$
for the UNL with $|\tau|<1$. If $|\tau|>1$ one should use
$R_{\pm}\left({p_{\bot},k}\right)$ in Eq. (22). Notice, that $M_{rp_{\bot}}$
and the velocity $d\varepsilon_{\bar{r}p_{\bot}}/dp_{\bot}$ vanish at the
edges of minibands, so that jumps of absorption are possible at edges of
absorption regions. Thus, spectral dependencies of
$\alpha_{\omega}^{(\bar{r})}$ appear to be strongly dependent on the transfer
matrix parameters and they can be extracted from these data.
Figure 7: (a) Spectral dependencies of dimensionless absorption coefficient
$\alpha_{\Delta\omega}/\alpha_{0}$ for $\Theta=\pi/12$ at the same conditions
as in Figs. 4a - 6a. (b) The same as in panel (a) for the conditions of Figs.
4b - 6b. (c) The same as in panels (a,b) for $\Theta=$0.1 and the conditions
of Figs. 4d - 6d.
The shapes of absorption peaks are determined by Eqs. (21) and (22) and by the
dispersion relations $\varepsilon_{\bar{r}p_{\bot}}$, as it is shown in Fig. 7
for lower peaks. Here we plotted the dimensionless spectral dependencies
$\alpha_{\Delta\omega}/\alpha_{0}$ where
$\alpha_{0}=\frac{8\pi e^{2}}{\sqrt{\epsilon}c}n_{2D}\sqrt{\frac{2}{mE_{0}}}$
(25)
is the characteristic absorption. For the case of IR absorption of SL with
concentration $n_{2D}=5\times 10^{11}$ cm-2 and $E_{0}\sim$0.1 eV,12a one
obtains $\alpha_{0}\simeq 1.25\times 10^{4}$ cm-1 if
$\sqrt{\epsilon}\simeq$3.3. The positions and widths of absorption peaks
correspond to the miniband energy spectra shown in Figs. 4 and 5. There is an
essential difference between odd and even peaks due to the inversion symmetry
of the eigenstate problem at edges of minibands, when
$p_{\bot}l/\hbar=0,~{}2\pi$. The odd peaks are non-symmetric ones, they are
higher (see the scaling factors in Fig. 7) and more narrow in comparison to
the even peaks which are more symmetric and wide. Since the condition $\kappa
l>1$ is satisfied at $l\geq$10 nm, a visible ($>$30%) absorption takes place
for 20-layer periodic structure and the mid-IR measurements permit a direct
verification of the SL parameters.
## V Experimental verification
Next, we briefly discuss possibilities for experimental verification or for ab
initio calculations of the UNL’s parameters. It is not a simple task because
the UNL’s parameters affect on response of heterostructure in a complicated
way and one needs to design an appropriate structure and to perform special
measurements. Also, any first-principle calculations of the UNL’s parameters
should include both UNL and host material, so that it is necessary to perform
calculations on atomic level for a long period SL in order to compare with the
results of Sect. IV written through the UNL’s parameters. Any method used for
short-period SL, see 8 and Refs. therein, can be applied providing a
generalization for the case of a long host material. Similar calculations for
InAs UNL embedded by GaAs together with the measurements of capacitance-
voltage characteristics were performed in Ref. 15 and the binding energy of
localized state $E_{0}$ appears to be $\sim$60 meV. This value is about 3
times smaller than EMA estimate of $E_{0}$ given by Eq. (8) and Fig. 2a. Thus,
the EMA-approach is not valid for description of the InAs UNL in GaAs host
matrix but a complete determination of parameters $\Theta$ and $E_{K}$ is not
possible due to luck of data in Ref. 15.
Below we list a set of papers, where UNLs based on different heteropairs were
realized, and discuss possible measurement schemes in order to verify UNL’s
parameters. Different optoelectronic devices contain the quantum dot sheets
with an ultranarrow wetting layer 1 ; 1a ; 2 ; 3 and influence of these
layers on electronic states is described by the results obtained here if one
neglects the quantum dots effect. UNLs with $N\sim$5 are widely used in the
quantum cascade lasers, 12c where each period contains about 10 layers of
different thickness, so that a verification of UNL’s parameters is difficult.
At least several measurements of InAs UNLs placed into GaAs or AlGaAs matrix
13 as well as results for UNLs in AIVBVI 14 and in AIIBVI 15
heterostructures were published. But a verification of UNL’s parameters was
not performed in these papers and a special investigation for each system is
necessary.
All the results obtained in Sect. IV depend strongly on $\Theta$ and $E_{K}$,
which determine the transfer matrices (6) or (7), and these parameters can be
verified from the spectroscopical measurements. Indeed, the results of Sect.
IVB demonstrate that mid-IR (or THz, depending on parameters in question)
absorption is strongly varied depending on transfer matrix parameters. The
near-IR interband transitions in SL depend essentially on the density of
states considered in Sect. IVA for different UNL’s parameters. Variations of
the near-IR spectra from the bulk case permit one to verify $\Theta$ and
$E_{K}$ using the electromodulation spectroscopy or the PLE measurements.
Beside this, the transfer matrix parameters can be extracted from the
transport data but in addition to UNL’s parameters, some other
characteristics, e. g. scattering rates, should be involved under
interpretation of these data.
## VI Conclusions
In summary, we have re-examined the theory of electronic states in the
heterostructures formed by ultranarrow layers taking into account that the EMA
approximation can not be applied for description of UNLs. Our consideration is
based on the transfer matrix method which was generalized for the case of UNL.
It is found that the three types of UNL are possible depending on a value of
diagonal element in the transfer matrix (5). The cases $|\tau|<1$ or $\tau>1$
are similar to narrow well or barrier while there is no simple analogy with a
wide layer for the non-conventional UNL with $\tau<-1$. The localized level
appears at single UNL with $\tau<1$. The energy-dependent reflection
coefficient of a single UNL as well as the energy spectrum of SL formed by a
periodical array of UNLs are different for the cases $|\tau|<1$ and $\tau>1$.
The spectral dependencies of absorption are analyzed for the case of
photoexcitation of localized electrons into SL’s minibands. The EMA approach
fails to explain the results of Ref. 15 but there is a luck of experimental
data for verification of UNL’s parameters.
Let us discuss now the main assumptions applied to the consideration
performed. Using the single-band Hamiltonian in Eq. (1) we suppose that
electron energy $E$ is small in comparison with the gap of the host
semiconductor. In order to consider the high-energy states, one needs to use
kp-Hamiltonian and more complicated boundary conditions, see similar
consideration of abrupt heterojunction in Refs. 8 and 21. In addition, the
transfer matrix approach is valid for description of UNL of width $Na$ if $Na$
is less than electron wavelength, i. e. $(\hbar/Na)^{2}/2m\gg E$. We restrict
ourselves by the model of symmetric UNL based on the condition (4). It should
be mentioned that the EMA-based estimates given by Eq. (8) are not valid for
$N\sim 1$ at least for InAs UNLs, see discussion in Ref. 20. A more
complicated consideration is necessary in order to take into account a non-
symmetry of UNL; this case will be considered elsewhere. Also, we consider
ideal UNL neglecting an in-plane scattering processes or inhomogeneous
broadening; this approximation is valid if $E$ exceeds a typical broadening
energy. A possible segregation of UNLs (some experimental data for InAs see in
17 ) will not be essential because a short-scale lateral inhomogeneities is
realized.
To conclude, we believe that the results obtained will stimulate a
verification of phenomenological parameters describing the electronic
properties of UNLs using the mid-IR spectroscopy when the valence band states
are not essential. These data should be important for applications of
heterostructures with UNLs in different devices.
## References
* (1) Quantum Well Lasers Ed. by P. S. Zory, (Academic Press, San Diego, 1993).
* (2) C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Rep. on Progr. in Phys., 64, 1533 (2001).
* (3) A. Rogalski, J. Appl. Phys. 93, 4355 (2003); J. N. Pan and C. G. Fonstad, Material Sci. and Engineering 28, 65 (2000).
* (4) V. Avrutin, N. Izyumskaya, and H. Morkoc, Superlattices and Microstructures, 49, 337 (2011); G. F. Brown and J. Q. Wu, Laser and Photonics Reviews, 3, 394 (2009).
* (5) D. Bimberg, M. Grundmann, N. N. Ledentsov, Quantum Dot Heterostructures (J. Wiley and Sons, New York, 1999); I. L. Krestnikov, N. N. Ledentsov, A. Hoffmann, and D. Bimberg, Physica Status Solidi A, 183, 207 (2001).
* (6) G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Editions de Physique, Paris, 1988); F. T. Vasko and A. Kuznetsov, Electronic States and Optical Transitions in Semiconductor Heterostructures (Springer, New York, 1998).
* (7) T. Ando and S. Mori, Surf. Sci. 113, 124 (1982); I.M. Sokolov, Zh. Eksp. Teor. Fiz. 89, 556 (1985) [Sov. Phys. JETP 62, 317 (1985)]; B. Laikhtman, Phys. Rev. B 46, 4769 (1992).
* (8) I. V. Tokatly, A. G. Tsibizov, and A. A. Gorbatsevich, Phys. Rev. B, 65, 165328 (2002).
* (9) D. Donetsky, S. P. Svensson, L. E. Vorobjev, and G. Belenky, Appl. Phys. Lett. 95, 212104 (2009); H. Li, S. Katz, G. Boehm, and M.-C. Amann, Appl. Phys. Lett. 98, 131113 (2011).
* (10) H. Schmidt, R. Pickenhain, and G. Bohm, Phys. Rev. B, 65, 045323 (2002).
* (11) M. Herman, Semiconductor Superlattices (Academie-Verlag, Berlin, 1986); V. Mitin, D. Sementsov and N. Vagidov, Quantum Mechanics for Nanostructures (Cambridge University Press, Cambridge 2010).
* (12) T. Worren, K. B. Ozanyan, O. Hunderi, and F. Martelli, Phys. Rev. B 58, 3977 (1998).
* (13) A. Ya. Shik, Sov. Phys. Semiconductors, 8, 1841 (1974); ibid. 6, 1110 (1972); A. Kastalsky, T. Duffield, S. J. Allen, and J. Harbison, Appl. Phys. Lett. 52, 1320 (1988).
* (14) This value is taken in agreement with experimental data of Ref. 15.
* (15) R. Pickenhain, H. Schmidt, and V. Gottschalch, J. Appl. Phys. 88, 948 (2000).
* (16) O. Cathabard, R. Teissier, J. Devenson, J. C. Moreno, and A. N. Baranov, Appl. Phys. Lett., 96, 141110 (2010); D. G. Revin, J. P. Commin, S. Y. Zhang, A. B. Krysa, K. Kennedy, and J. W. Cockburn, IEEE J. of Quant. Electr. 17, 1417 (2011).
* (17) D. Guimard, R. Morihara, D. Bordel, K. Tanabe, Y. Wakayama, M. Nishioka, and Y. Arakawa, Appl. Phys. Lett. 96, 203507 (2010); M. Di Ventra and K. A. Mader, Phys. Rev. B, 55, 13148 (1997)
* (18) J. D. Jeffers, K. Namjou, Z. Cai, P. J. McCann, and L. Olona, Appl. Phys. Lett. 99, 041903 (2011).
* (19) S. V. Ivanov, A. A. Toropov, T. V. Shubina, S. V. Sorokin, A. V. Lebedev, I. V. Sedova, P. S. Kopev, G. R. Pozina, J. P. Bergman, and B. Monemar, J. Appl. Phys. 83, 3168 (1998).
* (20) P. Paki, R. Leonellia, L. Isnard, and R. A. Masut, Appl. Phys. Lett. 74, 1445 (1999); V. Albe and L. J. Lewis, Physica B 301, 233 (2001).
* (21) M. V. Kisin, B. L. Gelmont, S. Luryi, Phys. Rev. B, 58, 4605 (1998).
* (22) S. Martini, J. E. Manzoli, and A. A. Quivy, J. Vac. Sci. Technol. B 28, 277 (2010); R. Ares, C. A. Tran, and S. P. Watkins, Appl. Phys. Lett. 67, 1576 (1995).
|
arxiv-papers
| 2011-10-04T16:26:09 |
2024-09-04T02:49:22.779629
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. T. Vasko and V. V. Mitin",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/1110.0744"
}
|
1110.0828
|
# Observation of non-linear magnetic vortex gyration by X-ray microscopy and
micromagnetic simulations
A. Vansteenkiste1, B. Van de Wiele2, M. Weigand3, H. Stoll3, K.W. Chou4, T.
Tyliszczak4, G. Woltersdorf5, C. H. Back5, G. Schütz3, B. Van Waeyenberge 1 1
Department of Solid State Sciences, Ghent University, Kortrijksesteenweg
281-S1, 9000 Gent, Belgium.
2 Department of Electrical Energy, Systems and Automation, Ghent University,
Sint Pietersnieuwstraat 41, B-9000 Ghent, Belgium.
3 Max-Planck-Institut für Intelligente Systeme (ehemals Max-Planck-Institut
für Metallforschung), Heisenbergstr. 3, 70569 Stuttgart
4 Advanced Light Source, LBNL, 94720 Berkeley, CA, USA.
5 Institut für Experimentelle und Angewandte Physik, Universität Regensburg,
93040 Regensburg, Germany.
###### Abstract
Magnetic vortex gyration in a 500 nm wide, 50 nm thick Permalloy square
platelet was investigated using time-resolved scanning transmission x-ray
microscopy and micromagnetic simulations. The response of the vortex core on
an in-plane oscillating magnetic field was studied as a function of the
excitation frequency and amplitude. Non-linear behavior was observed in the
form of a _redshift_ of the gyrotropic resonance. I.e., when the excitation
amplitude was increased, the resonance frequency was found to decrease. The
nonlinearity is in agreement with extensive micromagnetic simulations. This
work complements previous reports on thinner nanostructures where a frequency
_blueshift_ was observed.
###### pacs:
75.75.+a, 76.50.+g, 07.85.Tt
## 1 Introduction
A magnetic vortex configuration occurs in, e.g., disk- and square-shaped soft
magnetic nanostructures of suited size [1, 2]. This configuration exhibits an
in-plane curling magnetization which forms a flux-closure (see Fig. 7). In the
case of a square platelet structure, four domains separated by 90∘ Néel walls
are formed, while no domain walls occur in disk-shaped structures. At the
center of the vortex, an out-of-plane magnetized vortex core appears. The
magnetization of the core has two possible directions (“up” or “down”), which
could enable vortices to be used as magnetic memory bits [3].
The vortex has a characteristic excitation mode: the vortex gyration mode [4].
This mode corresponds to a circular motion of the vortex core around the
center of the structure. It can be excited with, e.g., an in-plane alternating
[5] or rotating [6] magnetic field, by magnetic field pulses [7] or directly
by an electrical current [8]. The gyrotropic mode currently receives
considerable attention, since sufficient excitation causes the vortex core
polarization to switch its direction [3]. This happens whenever the motion of
the vortex core is accelerated beyond a threshold velocity in the order of 300
m/s [9, 10], independent of the excitation type. Alternating magnetic fields
with a frequency close to the gyrotropic frequency require only modest
amplitudes of the order of one mT to cause vortex core switching [3, 6]. A
precise determination of this frequency is thus a prerequisite to low-power
vortex core memories.
Subjected to a small excitation, the vortex gyration behaves linearly: the
dynamics can be well described by a two-dimensional harmonic oscillator model
[11, 12, 13, 14]. Such theoretical models describe the vortex core as a rigid
quasi particle that moves in a harmonic potential. The resulting gyration
radius as a function of the excitation frequency is a typical damped harmonic
oscillator curve, showing a single peak at the eigenfrequency which is
independent of the excitation amplitude.
At large excitations, on the other hand, these harmonic models break down and
the gyration dynamics show nonlinear behavior. Argyle was the first to
investigate the gyrotropic resonance and already noticed such nonlinearity in
1984 [4]. His group investigated a 15 $\mu$m wide garnet structure. The
optical Cotton-Mouton effect was used to image the displacement of the domain
walls while the vortex was excited by in-plane alternating fields. When the
excitation frequency was scanned, a peak response was found at the gyrotropic
eigenfrequency. When the same scan was repeated at a higher excitation
amplitude however, the resonance peak had moved towards a lower frequency and
also became broader. This frequency “redshift” was attributed to nonlinear
dynamics at large excitation amplitudes, without further investigation. More
recently, Ferromagnetic Resonance (FMR) techniques have been employed [15]. In
such studies, an ensemble of presumably identical nanostructures is positioned
on top of an electrical strip line through which an alternating current is
sent. This induces an in-plane alternating magnetic field in the radio
frequency range, which excites the gyrotropic mode. The resonance is then
studied by measuring the microwave absorption as a function of the excitation
frequency and amplitude. However, such experiments can not determine whether
vortex core switching occurs, making the interpretation difficult.
In this work, a more direct study of the frequency- and field dependence of
the gyrotropic resonance in Permalloy nanostructures is presented.
Experimental investigation with time-resolved magnetic Scanning Transmission
X-ray Microscopy (STXM) allowed for a single nanostructure, rather than an
array, to be investigated. The dynamics of the vortex core was imaged
directly, allowing its position and velocity to be determined. As vortex core
switching could be detected, the effects of switching could clearly be
separated from the non-linearities in the dynamics.
Additionally, micromagnetic simulations were carried out on Graphical
Processing Units using the in-house developed MuMax software [16]. The speedup
delivered by this hardware allowed an elaborate scan of the
frequency/amplitude parameter space. Previous micromagnetic studies like [17]
included only a relatively limited number of simulations.
Finally, previous works [17, 18] focused on relatively thin nanostructures, in
which case the vortex core can easily approach the nanostructure boundaries.
This was found to lead to a frequency _blueshift_ at high excitation
amplitudes — in contrast to the _redshift_ found in the thicker nanostructures
studied here.
## 2 Experiment
### 2.1 Experimental method
One element of an array of nominally 500 nm$\times$ 500 nm $\times$ 50 nm
square nanostructures, patterned in an evaporated Permalloy film was
investigated with the Scanning Transmission X-ray Microscope at the Advanced
Light Source (ALS, beamline 11.0.2) [19]. By orienting the sample plane
perpendicular to the photon beam and using the X-ray magnetic circular
dichroism (XMCD) effect [20], the out-of-plane component of the magnetization
was measured. This allowed for the vortex core to be directly imaged and its
polarization to be determined [5].
The nanostructures were positioned on top of a 2.5 $\mu$m wide, 150 nm thick
Cu strip line. To excite the gyrotropic mode, an in-plane alternating magnetic
field $B(t)=B_{0}\sin\left(2\pi ft\right)$, was induced by sending an
alternating current through the strip line. By taking advantage of the short,
intense X-ray flashes provided by the synchrotron, stroboscopic images of the
vortex core during a steady-state gyration could be recorded. In this
experiment, the synchronization requirements between the acquisition and the
synchrotron ring allowed only excitation frequencies $f=n\times$ 20 MHz with
$n$ no multiple of 5. Per period of the excitation, 25 images were acquired.
In order to collect sufficient photon counts, the responses of many gyration
periods were summed up, and consequently only a steady-state motion could be
recorded. Non-magnetic and non-dynamic contributions to the contrast were
removed by subtracting the average of all frames. The resulting images show
the gyrating vortex core as a white or black spot, depending on its
polarization (see Fig. 1).
Since the vortex core was imaged directly, vortex core switching could be
detected. When the magnetic excitation amplitude is sufficiently increased,
the vortex core starts toggling back and forth between the two possible
polarizations which causes the recorded contrast in the experiment to vanish
entirely (as only steady-state dynamics can be recorded). In such a case, the
switching could be confirmed by again lowering the excitation amplitude and
determining the core polarization. In about half of the cases, the
polarization was found to have switched, which can be expected when the core
has been constantly toggling back and forth for a long time [9]. Also, each
time the vortex core had switched, a short magnetic pulse was employed to
switch it back to its original polarization [7] before the experiment was
continued. In this way, all data was recorded for a vortex core pointing up,
and possible effects due to the small asymmetry between the core polarizations
[5, 6, 21] were excluded.
We have excluded the possibility that the observed changes in gyrotropic
frequency (see further) could be caused by sample heating due to the
electrical excitation. Sufficient heating could cause a frequency redshift due
to a corresponding reduction of the saturation magnetization. E.g., Kamionka
et al. found about a 5% decrease of resonance frequency when heating the
sample by 100∘C [22]. Therefore, we have determined the temperature of the
copper stripline by accurately measuring its electrical resistance while
sending different currents through. Even at the highest currents used in the
experiment, the temperature increase remained below 1∘C, so it can not be
responsible for the observed non-linearities. Furthermore, the micromagnetic
simulations presented in section 3 were carried out at $T=0$ and reproduce the
non-linearities.
### 2.2 Experimental results
Figure 1: Time-resolved STXM images of the vortex core gyration shown at five
phases of alternating magnetic fields. The excitation frequencies of 560 MHz
and 580 MHz are near the gyrotropic resonance frequency. The respective
gyration radii can easily be compared by from the rightmost images, which are
Fourier-filtered at a harmonic of the excitation frequency. Under the low-
field excitation of 0.48 mT, the gyration radius is larger for the 580 MHz
excitation. On the other hand, at the higher excitation of 1.19 mT, the
resonance has shifted downwards and the gyration radius is now larger at 560
MHz. The gyrotropic mode is thus anharmonic, as the resonance frequency
depends on the excitation strength.
First of all, the experiment shows qualitatively that the resonance frequency
decreases when the excitation amplitude is increased. This effect is
illustrated in Fig. 1. Five of the 25 recorded frames of the gyrating vortex
core are shown for different combinations of excitation frequency and
amplitude. The radius of gyration is highlighted in the Fourier-filtered
images. These show the magnitude of the image obtained by Fourier-filtering
the 25 time-dependent images at a harmonic of the excitation frequency. This
technique allows to easily observe the gyration radius with the bare eye, but
is not used in the further analysis. It can be observed that at the lowest
excitation amplitude the gyration radius is larger at 580 MHz, while at the
highest excitation amplitude the radius is larger at 540 MHz. The resonance
frequency thus shifts down while the field is increased. This frequency
“redshift” is a nonlinear effect that does appear in a perfect harmonic
oscillator.
Figure 2: Experimentally measured vortex core gyration velocity as a function
of excitation frequency and amplitude in a small region around the gyrotropic
resonance. The frequencies of maximal gyration are marked with a box,
indicating the resonance frequency for each amplitude. This shows that the the
gyrotropic resonance frequency shifts down as the amplitude of the excitation
field is increased. Above 1.5 mT, vortex core switching occurs (marked in
grey) at the resonance frequency. At even higher fields, switching also occurs
at frequencies further away from resonance.
For a more quantitative analysis, the vortex core position in each set of 25
recorded frames was determined by fitting two-dimensional Gaussian profiles to
the images. The subtraction of the average image was hereby taken into
account. From those positions, the average gyration radius and velocity during
one period was deduced. This procedure was repeated at four excitation
frequencies near the gyrotropic resonance frequency, as well as at several
different excitation amplitudes.
A full scan of excitation frequencies and amplitudes is shown in Fig. 2. Here,
the average vortex core velocity is shown. It was deduced by multiplying the
average gyration radius by the angular velocity of the core, which is fixed by
the excitation frequency $f_{0}$. For each magnetic excitation amplitude, the
frequency which lies closest to the resonance, and where the response is thus
maximal, has been marked with a box. It can again be observed that the
resonance shifts down as the field is increased.
At 1.5 mT and 540 MHz, the vortex core started toggling back and forth,
indicating that the critical switching velocity (about 300 m/s [23, 9]) was
reached. When the excitation amplitude was even further increased, the core
also switched at frequencies further away form the resonance. This results in
a _switching region_ with a typical triangle-like shape, marked in grey.
Although it was not yet identified as such, this kind of region is also
visible in the FMR spectra in [15].
## 3 Micromagnetic Simulations
### 3.1 Simulation Method
The experimental results were compared to micromagnetic simulations of a 500
nm$\times$ 500 nm $\times$ 50 nm Permalloy square structure (saturation
magnetization $M_{s}$=736 $\times$ 103 A/m (taken from [24]), exchange
constant $A$=13 $\times$ 10-12 J/m, damping parameter $\alpha$=0.01,
anisotropy constant $K_{1}$=0 and gyromagnetic ratio $\gamma$=2.211 $\times$
105 m/As). A 2D discretization with 3.9 $\times$ 3.9 nm2 finite difference
cells was used. The simulations were carried out with our in-house developed
MuMax micromagnetic simulator [16]. This program runs on Graphical Processing
Units (GPUs) rather than on standard CPUs, providing a computational speedup
of up to two orders of magnitude. This allowed a large parameter range to be
scanned with fine resolution. In total, 8000 separate field/amplitude
combinations were simulated. Each simulation started from the relaxed ground
state. The field was first applied for 20 periods to allow the system to
absorb energy and approach a stationary state. The next 20 periods of the
field were used for data-taking. The vortex core position and average velocity
were output during those 20 periods. We choose to start each simulation from
the ground state and not to, e.g., slowly scan the frequency or field. This
avoids possible effects due to a too high scan speed.
### 3.2 Simulation Results
Figure 3: Simulated frequency and field dependence of the vortex core gyration
velocity for a larger range than the experimental data in Fig. 2. The
simulated data qualitatively show the same behavior as the experimental data.
The simulations also reveal the triangular region in parameter space where
vortex core switching occurs. In this region the average core velocity is
lowered due to the continuous switching [9]. Experimentally, the velocity in
this region could not be determined.
The simulated gyration velocity as a function of excitation frequency and
amplitude is shown in Fig. 3. This simulation qualitatively agrees with the
experimental results: the frequency redshift at high excitations and the
typical shape of the switching region can here also be observed. The small
quantitative discrepancies with the experimental results are attributed to
uncertainties on the nanostructure size, thickness and material parameters. A
more detailed analyses is presented in the next section.
## 4 Detailed Analysis
### 4.1 Frequency redshift
First of all, we analyse frequency scans at a few fixed excitation amplitudes.
Fig. 1 already illustrated qualitatively that the resonance frequency drops as
the excitation amplitude is increased. This redshift is shown in more detail
in Fig. 4. Here, experimental and simulated frequency scans at two
representative excitation amplitudes are presented. One amplitude is chosen
well below the vortex core switching threshold, while the second amplitude
closely approaches it. It is clear that in both cases high amplitudes cause
the resonance peak to become broader and to shift to lower frequencies. The
simulated data also reveal that the resonance peak becomes asymmetric — a
typical property of anharmonic oscillators [15]. Fig. 5 shows the resonance
peaks for a wider range of excitation amplitudes. Here, the gradual
deformation and redshift of the resonance curve at increasingly higher
amplitudes can be clearly seen.
Figure 4: Experimental (left) and simulated (right) vortex core velocity $v$
as a function of the excitation frequency $f$ for a low and a high excitation
amplitude. The high excitation lies just below the vortex core switching
threshold. At high excitation, the peak of the response is clearly redshifted
towards a lower frequency. The detailed simulated curve also reveals that the
resonance peak shape becomes asymmetric. Figure 5: Simulated vortex core
velocity as a function of the excitation frequency $f$ at various excitation
amplitudes until just below the vortex core switching threshold. At high
excitations, the resonance peak shifts towards lower frequencies and develops
an asymmetric shape.
### 4.2 Amplitude jumps
Figure 6: Experimental (left) and simulated (right) vortex core velocity $v$
as a function of the excitation amplitude $B$, for a frequency just below and
a frequency just above the respective resonances. At the lowest frequency, the
velocity increases suddenly around 1mT: while increasing the field amplitude,
the resonance redshifts towards the excitation frequency until the resonance
frequency suddenly is reached. The higher frequency lies already above the
resonance so the resonance peak redshifts away from the excitation frequency
as the amplitude increases. Hence, no sharp jump of the response occurs.
Secondly, we analyse amplitude scans at a few fixed frequencies. For a perfect
harmonic oscillator, the response (either velocity or displacement) should be
proportional to the excitation amplitude. Fig. 6 presents experimental and
simulated amplitude scans at two fixed frequencies, one slightly below and one
slightly above the resonance. At the lowest of these two frequencies, the
response exhibits a sharp jump around 1mT, which is clearly non-linear
behavior. This behavior can be explained by the redshift as follows: Let us
consider a frequency just below the resonance frequency at a relatively small
amplitude. By increasing the amplitude, not only the response naturally
becomes larger, but also the resonance frequency redshifts towards the
excitation frequency, causing an extra increase of the response. In contrary,
this does not happen when a frequency is considered above the (small-
amplitude) resonance frequency. In that case, increasing the amplitude causes
the resonance to shift away from the considered excitation and thus
counteracts the resonance increase.
## 5 Discussion
Until now, theoretical studies on thinner nanostructures predicted a
_blueshift_ of the gyrotropic frequency at high excitation amplitudes [17,
18]. In these studies, relatively thin nanostructures were considered, which
allows the vortex core to approach the edges of the magnet. A steepening of
the potential well in which the vortex core moves was attributed to the
expulsion of the core by the edge, hence the blueshift. In our case however,
the vortex core did not approach the edges of the system. The maximum gyration
radius was about 150 nm (see, e.g., Fig 1) — much smaller than the lateral
size of the magnet.
The redshift in this case can be explained by domain wall bending during the
vortex movement. Fig. 7 shows a simulated magnetization snapshot where such
bending occurs. It was taken at an excitation just below the threshold for
vortex core swichting — where there is a strong redshift. It can be seen that
the domain walls slightly bend so that the total dipole moment is reduced with
respect to straight domain walls. Hence the magnetostatic energy of the
potential well and thus the resonance frequency is reduced [14].
Nevertheless, this effect is not specific to square-shaped nanostructures.
Micromagnetic simulations on disks (not presented here) also revealed a
redshift. Although domain walls are absent in that case, the magnetization
adapted in a similar way, reducing the global dipole moment.
Figure 7: (a) Simulated magnetization snapshot at an excitation that leads to
gyrotropic redshift. The domain walls have slightly bended, reducing the
global dipole moment. As this lowers the dipolar energy, it weakens the
potential well in which the vortex core moves —hence decreasing the
eigenfrequency. (b) Exaggerated sketch for clarity.
## 6 Conclusion
We have performed a detailed analysis of the non-linearity of the magnetic
vortex gyration in a 50 nm thick, 500 nm wide Permalloy nanostructure. The
study has shown that a frequency _redshift_ occurs at high excitation
amplitudes. By using time-resolved magnetic STXM microscopy the gyrotropic
motion could be imaged directly, allowing for a straightforward interpretation
of the results. STXM also allows to detect the vortex core polarization and
switching. In this way, the region in frequency/amplitude space where the
vortex core switches could be determined. This was not possible in earlier FMR
studies like [15]. Extensive micromagnetic simulations were carried out on
Graphical Processing Units and support the experimental results.
## Acknowledgments
Financial support by The Institute for the promotion of Innovation by Science
and Technology in Flanders (IWT-Flanders) and by the Research Foundation
Flanders (FWO-Flanders) through the research grant 60170.06 are gratefully
acknowledged. The Advanced Light Source is supported by the Director, Office
of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy.
We would also like to cordially thank André Drews for the fruitful
discussions, including the explanation of the redshift by domain wall bending.
## References
## References
* [1] J. Raabe, R. Pulwey, R. Sattler, T. Schweinbock, J. Zweck, and D. Weiss. Magnetization pattern of ferromagnetic nanodisks. J. Appl. Phys., 88:4437–4439, 2000.
* [2] X. B. Zhu, Z. G. Liu, V. Metlushko, P. Grutter, and M. R. Freeman. Broadband spin dynamics of the magnetic vortex state: Effect of the pulsed field direction. Phys. Rev. B, 71:180408, 2005.
* [3] B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Bruckl, K. Rott, G. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Schütz. Magnetic vortex core reversal by excitation with short bursts of an alternating field. Nature, 444:461–464, 2006.
* [4] B. E. Argyle, E. Terrenzio, and J. C. Slonczewski. Magnetic vortex dynamics using the optical Cotton-Mouton effect. 1984 Digests Of Intermag ’84. International Magnetics Conference (cat. No. 84ch1918-2), pages 350–350, 1984.
* [5] K. W. Chou, A. Puzic, H. Stoll, D. Dolgos, G. Schütz, B. Van Waeyenberge, A. Vansteenkiste, T. Tyliszczak, G. Woltersdorf, and C. H. Back. Direct observation of the vortex core magnetization and its dynamics. Appl. Phys. Lett., 90:202505, 2007.
* [6] M. Curcic, B. Van Waeyenberge, A. Vansteenkiste, M. Weigand, V. Sackmann, H. Stoll, H. Fahnle, T. Tyliszczak, G. Woltersdorf, C. H. Back, and G. Schütz. Polarization selective magnetic vortex dynamics and core reversal in rotating magnetic fields. Physical Review Letters, 101(19):197204, 2008.
* [7] M. Weigand, B. Van Waeyenberge, A. Vansteenkiste, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, K. Kaznatcheev, D. Bertwistle, G. Woltersdorf, C. H. Back, and G. Schütz. Vortex core switching by coherent excitation with single in-plane magnetic field pulses. Physical Review Letters, 102(7):077201, February 2009.
* [8] M. Bolte, G. Meier, B. Kruger, A. Drews, R. Eiselt, L. Bocklage, S. Bohlens, T. Tyliszczak, A. Vansteenkiste, B. Van Waeyenberge, K. W. Chou, A. Puzic, and H. Stoll. Time-resolved x-ray microscopy of spin-torque-induced magnetic vortex gyration. Phys. Rev. Lett., 100:176601, 2008.
* [9] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono. Electrical switching of the vortex core in a magnetic disk. Nat. Mater., 6:269–273, 2007.
* [10] A. Vansteenkiste, K.W. Chou, M. Weigand, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, G. Woltersdorf, C.H. Back, G. Schütz, and B. Van Waeyenberge. X-ray imaging of the dynamic magnetic vortex core deformation. Nature Physics, 5:332–334, 2009.
* [11] A. A. Thiele. Steady-state motion of magnetic domains. Phys. Rev. Lett., 30:230–233, 1973.
* [12] D. L. Huber. Equation of motion of a spin vortex in a two-dimensional planar magnet. J. Appl. Phys., 53:1899–1900, 1982.
* [13] V. Novosad, F. Y. Fradin, P. E. Roy, K. S. Buchanan, K. Y. Guslienko, and S. D. Bader. Magnetic vortex resonance in patterned ferromagnetic dots. Phys. Rev. B, 72:024455, 2005.
* [14] B. Kruger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G. Meier. Harmonic oscillator model for current- and field-driven magnetic vortices. Phys. Rev. B, 76:224426, 2007.
* [15] K. S. Buchanan, M. Grimsditch, F. Y. Fradin, S. D. Bader, and V. Novosad. Driven dynamic mode splitting of the magnetic vortex translational resonance. Phys. Rev. Lett. (USA), 99:267201–1–267201–267201–4, 2007.
* [16] A. Vansteenkiste and B. Van de Wiele. Mumax: A new high-performance micromagnetic simulation tool. Journal of Magnetism and Magnetic Materials, 323(21):2585–2591, November 2011.
* [17] K. S. Lee and S. K. Kim. Gyrotropic linear and nonlinear motions of a magnetic vortex in soft magnetic nanodots. Applied Physics Letters, 91(13):132511, September 2007.
* [18] K. Y. Guslienko, R. H. Heredero, and O. Chubykalo-Fesenko. Nonlinear gyrotropic vortex dynamics in ferromagnetic dots. Physical Review B, 82(1):014402, July 2010.
* [19] ALD Kilcoyne, T Tyliszczak, WF Steele, S Fakra, P Hitchcock, K Franck, E Anderson, B Harteneck, EG Rightor, GE Mitchell, AP Hitchcock, L Yang, T Warwick, and H Ade. Interferometer-controlled scanning transmission x-ray microscopes at the advanced light source. Journal of Synchrotron Radiation, 10:125–136, 2003.
* [20] G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, and G. Materlik. Absorption of circularly polarized x-rays in iron. Phys. Rev. Lett., 58:737–740, 1987.
* [21] A. Vansteenkiste, M. Weigand, M. Curcic, H. Stoll, G. Schütz, and B. Van Waeyenberge. Chiral symmetry breaking of magnetic vortices by sample roughness. New Journal Of Physics, 11:063006, 2009.
* [22] Thomas Kamionka, Michael Martens, Andre Drews, Benjamin Krueger, Ole Albrecht, and Guido Meier. Influence of temperature on the gyrotropic eigenmode of magnetic vortices. Physical Review B, 83(22):224424, June 2011.
* [23] K. Y. Guslienko, K. S. Lee, and S. K. Kim. Dynamic origin of vortex core switching in soft magnetic nanodots. Phys. Rev. Lett., 100:027203, 2008.
* [24] A. Vansteenkiste, J. De Baerdemaeker, K. W. Chou, H. Stoll, M. Curcic, T. Tyliszczak, G. Woltersdorf, C. H. Back, G. Schütz, and B. Van Waeyenberge. Influence of domain wall pinning on the dynamic behavior of magnetic vortex structures: Time-resolved scanning x-ray transmission microscopy in NiFe thin film structures. Phys. Rev. B, 77:144420, 2008.
|
arxiv-papers
| 2011-10-04T08:12:43 |
2024-09-04T02:49:22.790069
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arne Vansteenkiste, Ben Van de Wiele, Markus Weigand, Hermann Stoll,\n Kang W. Chou, Tolek Tyliszczak, Georg Woltersdorf, Christian H. Back, Gisela\n Sch\\\"utz, Bartel Van Waeyenberge",
"submitter": "Arne Vansteenkiste",
"url": "https://arxiv.org/abs/1110.0828"
}
|
1110.0904
|
On partitions avoiding right crossings
Sherry H. F. Yan, Yuexiao Xu
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R.
China
huifangyan@hotmail.com
Abstract. Recently, Chen et al. derived the generating function for partitions
avoiding right nestings and posed the problem of finding the generating
function for partitions avoiding right crossings. In this paper, we derive the
generating function for partitions avoiding right crossings via an
intermediate structure of partial matchings avoiding $2$-right crossings and
right nestings. We show that there is a bijection between partial matchings
avoiding 2-right crossing and right nestings and partitions avoiding right
crossings.
Key words: Partial matchings, partitions, 2-right crossings, right nestings,
right crossings.
AMS Mathematical Subject Classifications: 05A05, 05C30.
## 1 Introduction
This paper is concerned with the enumeration of set partitions that avoid
right crossings. Recall that a partition $\pi$ of $[n]=\\{1,2,\ldots,n\\}$ can
be represented as a diagram with vertices drawn on a horizontal line in
increasing order. For a block $B$ of $\pi$, we write the elements of $B$ in
increasing order. Suppose that $B=\\{i_{1},i_{2},\ldots,i_{k}\\}$. Then we
draw an arc from $i_{1}$ to $i_{2}$, an arc from $i_{2}$ to $i_{3}$, and so
on. Such a diagram is called the linear representation of $\pi$. If $(i,j)$ is
an arc in the diagram of $\pi$ with $i<j$, we call vertex $i$ an opener, and
call vertex $j$ a transient if there is an arc $(j,k)$ for some integer $k$
such that $i<j<k$; otherwise, we call vertex $j$ a closer. A partial matching
is a partition for which each block contains at most two elements. A partial
matching without singletons is called a perfect matching.
A nesting of a partition is a pair of arcs $(i_{1},j_{1})$ and $(i_{2},j_{2})$
with $i_{1}<i_{2}<j_{2}<j_{1}$. We call such a nesting a left nesting if
$i_{2}=i_{1}+1$. Similarly, we call it a right nesting if $j_{1}=j_{2}+1$. A
crossing of a partition is a pair of arcs $(i_{1},j_{1})$ and $(i_{2},j_{2})$
with $i_{1}<i_{2}<j_{1}<j_{2}$ and we can define left crossing and right
crossing analogously to how it was defined for nesting arcs. A neighbor
alignment of a partition is a pair of arcs $(i_{1},j_{1})$ and $(i_{2},j_{2})$
such that $i_{2}=j_{1}+1$.
Zagier [7] derived the generating function for perfect matchings avoiding left
nestings and right nestings, which was introduced by Stoimenow [6]. Recently,
Bousquet-Mélou et al. [1] constructed bijections between perfect matchings
avoiding left nestings and right nestings and three other classes of
combinatorial objects, that is, unlabeled $(2+2)$-free posets, permutations
avoiding a certain pattern and ascent sequences. Dukes and Parviainen [4]
presented a bijection between ascent sequences and upper triangular matrices
whose non-negative entries are such that all rows and columns contain at least
one non-zero entry. Claesson and Linusson [3] established a bijection between
perfect matchings avoiding left nestings and permutations and showed that
perfect matchings avoiding left crossings are equinumerous with perfect
matchings avoiding left nestings. They also introduced the notion of k-left
nesting and conjectured that the number of perfect matchings of $[2n]$ without
2-left nestings is equal to the number of perfect matchings of $[2n]$ without
left nestings and right nestings. This conjecture was confirmed by Levande
[5]. Recall that two arcs $(i_{1},j_{1})$ and $(i_{2},j_{2})$ form a $k$-left
nesting if $i_{1}<i_{2}<j_{2}<j_{1}$ and $i_{2}-i_{1}\leq k$. Similarly, we
call it a $k$-right nesting if $i_{1}<i_{2}<j_{2}<j_{1}$ and $j_{1}-j_{2}\leq
k$. Note that 1-left nesting is exactly a left nesting and 1-right nesting is
exactly a right nesting. Similarly, we can define k-left crossing and k-right
crossing analogously to how it was defined for nesting arcs. The left
nestings, left crossings, right crossings, right crossings, neighbor
alignments are called neighbor patterns.
Recently, Chen et al. [2] derived the generating functions for partial
matchings avoiding neighbor alignments and for partial matchings avoiding
neighbor alignments and left nestings. They obtained the generating function
for partitions avoiding right nestings by presenting a bijection between
partial matchings avoiding three neighbor patterns ( left nestings, right
nestings and neighbor alignments) and partitions avoiding right nestings. In
general, the number of partitions of $[n]$ avoiding right crossings is not
equal to the number of partitions of $[n]$ avoiding right nestings. In this
paper, we derive the generating functions for partitions avoiding right
crossings via an intermediate structure of partial matchings avoiding
$2$-right crossings and right nestings.
Denote by $\mathcal{M}(n,k)$ the set of partial matchings of $[n]$ with $k$
arcs. The set of partial matchings in $\mathcal{M}(n,k)$ with no $2$-right
crossings and right nestings is denoted by $\mathcal{P}(n,k)$. Let
$\mathcal{T}(n,k)$ be the set of partitions of $[n]$ with $k$ arcs. Denote by
$\mathcal{CT}(n,k)$ the set of partition in $\mathcal{T}(n,k)$ with no right
crossings.
Denote by $P(n,k)$ and $CT(n,k)$ the cardinalities of the sets
$\mathcal{P}(n,k)$ and $\mathcal{CT}(n,k)$, respectively.
We obtain the generating function formula for the numbers $P(n+k-1,k)$ by
establishing a bijection between partial matchings avoiding 2-right crossings
and right nestings and a certain class of integer sequence. Moreover, we show
that there is a correspondence between $\mathcal{P}(n+k-1,k)$ and
$\mathcal{CT}(n,k)$.
###### Theorem 1.1
We have
$\sum_{n\geq 1}\sum_{k=0}^{n-1}P(n+k-1,k)x^{n}y^{k}=\sum_{n\geq
1}{{x^{n}(1+xy)^{n\choose 2}}\over\prod_{k=0}^{n-1}(1-(1+xy)^{k}xy)}.$ (1.1)
###### Theorem 1.2
We have
$\sum_{n\geq 1}\sum_{k=0}^{n-1}CT(n,k)x^{n}y^{k}=\sum_{n\geq
1}{{x^{n}(1+xy)^{n\choose 2}}\over\prod_{k=0}^{n-1}(1-(1+xy)^{k}xy)}.$ (1.2)
## 2 Partial matchings avoiding 2-right crossings and right nestings
In this section, we construct a bijection between partial matchings avoiding
2-right crossings and right nestings and a certain class of integer sequences.
As a consequence, we obtain the bivariate generating function for the number
of partial matchings of $[n+k-1]$ with $k$ arcs and containing no 2-right
crossing and right nestings.
Let $x=x_{1}x_{2}\ldots x_{n}$ be an integer sequence. Denote by $max(x)$ the
maximum element of $x$. For $1<i\leq n$, an element $x_{i}$ of $x$ is said to
be a left-to-right maximum if $x_{i}>max(x_{1}x_{2}\ldots x_{i-1})$. For
$n\geq 1$, let $\mathcal{S}(n,k)$ be the set of integer sequences
$x=x_{0}x_{1}\ldots x_{n-1}$ with $n-k$ left-to-right maxima satisfying that
* •
$x_{0}=0$;
* •
for all $1\leq i\leq n-1$, $0\leq x_{i}\leq max(x_{0}x_{1}\ldots x_{i-1})+1$;
* •
for all $0\leq i\leq n-1$, if $0\leq x_{i}<max(x_{0}x_{1}\ldots x_{i-1})$,
then $x_{i}<x_{i-1}$.
Denote by $\mathcal{S}(n)$ the set of such integer sequences with $n$ left-to-
right maxima. Denote by $S(n,k)$ the cardinality of the set
$\mathcal{S}(n,k)$. Let $f(x,y)$ be the generating function for the numbers
$S(n,k)$. We derive the following generating function formula of $f(x,y)$.
###### Theorem 2.1
$f(x,y)=\sum_{n\geq 1}\sum_{k=0}^{n-1}S(n,k)x^{n}y^{k}=\sum_{n\geq
1}{{x^{n}(1+xy)^{n\choose 2}}\over\prod_{k=0}^{n-1}(1-(1+xy)^{k}xy)}.$ (2.1)
Proof. Let $x$ be a sequence in $\mathcal{S}(n)$ with $n$ left-to-right
maxima. It can be uniquely decomposed as
$0w_{0}1w_{1}2w_{2}\ldots(n-1)w_{n-1}$. For all $0\leq i\leq n-1$, each
$w_{i}$ is a (possibly empty) integer sequence whose elements are nonnegative
integers less than or equal to $i$. For all $1\leq i\leq n-1$, suppose that
there are $i_{l}$ occurrences of $i$’s in $w_{i}$, then $w_{i}$ reads as
$m_{0}im_{1}im_{2}\ldots im_{i_{l}}$, where for all $0\leq j\leq i_{l}$, the
sequence $m_{j}$ is a (possibly empty) decreasing sequence whose elements are
nonnegative integers less than $i$. Since $w_{0}$ is a (possibly empty)
sequence of $0$’s. Hence, $w_{0}$ contributes $1\over{1-xy}$ to the generating
function $f(x,y)$, while for all $1\leq i\leq n-1$, $w_{i}$ contributes
$(1+xy)^{i}\over{1-(1+xy)^{i}xy}$ to the generating function $f(x,y)$.
Furthermore, each left-to-right maximum contributes $x$ to the generating
function $f(x,y)$. Summing over all $n\geq 1$, we derive the generating
function formula (2.1). This completes the proof.
Now we proceed to construct a bijection between the set $\mathcal{P}(n+k-1,k)$
and the set $\mathcal{S}(n,k)$.
###### Theorem 2.2
There exists a bijection between the set $\mathcal{P}(n+k-1,k)$ and the set
$\mathcal{S}(n,k)$.
Proof. Let $M$ be a partial matching in $\mathcal{P}(n+k-1,k)$ of $[n+k-1]$
with $k$ arcs and containing no 2-right crossings and right nestings. We wish
to generate a sequence $\alpha(M)=x_{0}x_{1}x_{2}\ldots x_{n-1}$ from $M$
recursively. First, we remove the labels of the closers and relabel the
vertices of $M$ in the natural order. Denote by $\overline{M}$ the obtained
diagram. For all $1\leq i\leq n-1$, let $\mathcal{O}(i)$ be the set of openers
of arcs whose closers are left to vertex $i$ of $\overline{M}$. Denote by
$\mathcal{O}(n)$ the set of openers of all arcs of $\overline{M}$. Set
$x_{0}=0$ and assume that we have obtained $x_{i-1}$. Now we proceed to
generate $x_{i}$ in the following manner.
* Case 1.
If there is no closer immediately after vertex $i$, then let
$x_{i}=max(x_{0}x_{1}\ldots x_{i-1})+1$.
* Case 2.
If there is an arc whose closer is immediately after vertex $i$ and opener is
labelled with $j$ and $j$ is the $m$-th minimum element of the set
$[i]\setminus\mathcal{O}(i)$, then let $x_{i}=m-1$.
We claim that for all $0\leq i\leq n-1$, $max(x_{0}x_{1}x_{2}\ldots
x_{i})=i-|\mathcal{O}(i+1)|$. We prove this claim by induction on $i$.
Obviously, the claim holds for $i=0$. So let $i\geq 1$ and assume that the
claim holds for $i-1$. In Case 1, we have $max(x_{0}x_{1}x_{2}\ldots
x_{i})=max(x_{0}x_{1}\ldots x_{i-1})+1=i-1-|\mathcal{O}(i)|+1$. Since there
exist no closers immediately after vertex $i$ in $\overline{M}$, we are led to
$i-|\mathcal{O}(i+1)|=i-1-|\mathcal{O}(i)|+1=max(x_{0}x_{1}x_{2}\ldots
x_{i})$. In Case 2, we have $0\leq x_{i}\leq
i-|\mathcal{O}(i)|-1=max(x_{0}x_{1}\ldots x_{i-1})$, which yields that
$max(x_{0}x_{1}\ldots x_{i})=max(x_{0}x_{1}\ldots
x_{i-1})=i-1-|\mathcal{O}(i)|$. Since there is exactly one closer immediately
after vertex $i$, we have
$i-|\mathcal{O}(i+1)|=i-1-|\mathcal{O}(i)|=max(x_{0}x_{1}x_{2}\ldots x_{i})$.
Therefore, the claim holds for all $0\leq i\leq n-1$. This yields that $0\leq
x_{i}\leq max(x_{0}x_{1}x_{2}\ldots x_{i-1})+1$ for all $1\leq i\leq n-1$ and
$max(x_{0}x_{1}\ldots x_{n-1})=n-1-|\mathcal{O}(n)|=n-1-k$, which implies that
there are $n-k$ left-to-right maxima in $\alpha(M)$.
In order to prove that the obtained sequence $\alpha(M)\in\mathcal{S}(n,k)$,
it remains to show that in the sequence $\alpha(M)$ if $0\leq
x_{i}<max(x_{1}x_{2}\ldots x_{i-1})$, then $x_{i}<x_{i-1}$. According to the
construction of the map $\alpha$, there exists an arc whose closer is
immediately after vertex $i$. Suppose that the opener of this arc is labelled
with $p$ in $\overline{M}$. Since $x_{i}<max(x_{1}x_{2}\ldots
x_{i-1})=i-1-|\mathcal{O}(i)|$, we are led to $p<i$. Obviously, when
$x_{i-1}=max(x_{1}x_{2}\ldots x_{i-1})$, we have $x_{i}<x_{i-1}$. Now suppose
that $x_{i-1}<max(x_{1}x_{2}\ldots x_{i-1})$. According to the construction of
the map $\alpha$, there exists an arc whose closer is immediately after vertex
$i-1$. Suppose that the opener of this arc is labelled with $q$ in
$\overline{M}$. Since there are no 2-right crossings in $M$, we have $p<q$. It
is easy to check that $\mathcal{O}(i)=\mathcal{O}(i-1)\cup\\{q\\}$. From the
construction of the map $\alpha$, we have $x_{i}<x_{i-1}$. Hence, the obtained
sequence $\alpha(M)\in\mathcal{S}(n,k)$.
Conversely, given a sequence $x=x_{0}x_{1}x_{2}\ldots
x_{n-1}\in\mathcal{S}(n,k)$, we wish to construct a partial matching of
$[n+k-1]$ with $k$ arcs. First, we arrange $n-1$ vertices on a horizontal line
and label them in the natural order. For $i=1,2,\ldots,n-1$, at step $i$, we
insert at most one closer after vertex $i$ described as follows:
* (i)
if $x_{i}=max(x_{0}x_{1}x_{2}\ldots x_{i-1})+1$, then we do nothing for vertex
$i$;
* (ii)
if $0\leq x_{i}\leq max(x_{0}x_{1}x_{2}\ldots x_{i-1})$, then we insert an arc
whose opener is the $(x_{i}+1)$-th vacant vertex (from left to right) of the
set $[i]$ and closer is immediately after vertex $i$, where a vacant vertex is
a vertex which is not joined to any arc.
Finally, we relabel the vertices in the natural order to get a partial
matching.
By induction on $i$ $(0\leq i\leq n-1)$, it is easy to see that after step
$i$, the number of vacant vertices of the set $[i]$ is equal to
$max(x_{0}x_{1}x_{2}\ldots x_{i})$. So (ii) is valid and the construction of
the inverse map is well defined. Since there are exactly $n-k$ left-to-right
maxima in $x$, the obtained partial matching is a matching of $[n+k-1]$ with
$k$ arcs. Note that we add at most one closer immediately after each vertex
$i$ for all $1\leq i\leq n-1$. Thus there is no two consecutive closers in the
resulting partial matchings, which implies that there are no two right
crossings and right nestings. Since when $0\leq x_{i}<max(x_{1}x_{2}\ldots
x_{i-1})$, we have $x_{i}<x_{i-1}$, there are no 2-right crossings. Hence, the
resulting partial matching is in the set $\mathcal{P}(n+k-1,k)$. This implies
that the above map $\alpha$ is a bijection.
For example, let
$M=\\{\\{1,6\\},\\{2,3\\},\\{4,12\\},\\{5,10\\},\\{7,8\\},\\{9\\},\\{11\\}\\}\in\mathcal{P}(12,5)$.
We obtain a diagram $\overline{M}$ from $M$ by removing the labels of the
closers and relabeling the vertex of $M$ in the natural order, see Figure 1.
We have $\mathcal{O}(1)=\emptyset$, $\mathcal{O}(2)=\emptyset$,
$\mathcal{O}(3)=\\{2\\}$,
$\mathcal{O}(4)=\\{2\\},\mathcal{O}(5)=\\{1,2\\},\mathcal{O}(6)=\\{1,2,5\\}$,
$\mathcal{O}(7)=\\{1,2,4,5\\}$ and $\mathcal{O}(8)=\\{1,2,3,4,5\\}$. So the
corresponding sequence is $\alpha(M)=01120210$.
$1$$2$$3$$4$$5$$6$$7$ Figure 1: The diagram $\overline{M}$.
Conversely, given the integer sequence
$x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}=01120210\in\mathcal{S}(8,5)$, the
construction of the corresponding partial matching is illustrated in Figure 2.
$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$\downarrow$$x_{7}=0$$1$$2$$3$$4$$5$$6$$7$$\downarrow$$x_{6}=1$$1$$2$$3$$4$$5$$6$$7$$\downarrow$$x_{5}=2$$1$$2$$3$$4$$5$$6$$7$$\downarrow$$x_{4}=0$$1$$2$$3$$4$$5$$6$$7$$\downarrow$$x_{2}=1$$1$$2$$3$$4$$5$$6$$7$$\downarrow$$1$$2$$3$$4$$5$$6$$7$
Figure 2: The bijection $\alpha$.
Combining Theorems 2.1 and 2.2, we are led to the generating function formula
(1.1).
## 3 Partitions with no right crossings
In this section, we present a bijection between the set $\mathcal{P}(n+k-1,k)$
and the set $\mathcal{CT}(n,k)$. As a result, we derive the generating
functions for partitions avoiding right crossings. Before we present the
bijection, we should recall the notion of $2$-paths defined by Chen et al.
[2]. Recall that a pair of two arcs $(i,j)$ and $(j,k)$ with $i<j<k$ in the
diagram of a partition is said to be a $2$-path.
$i$$j$$k$$\Longrightarrow$$i$$j$$j+1$$k$ Figure 3: change a neighbor alignment
to a $2$-path.
###### Theorem 3.1
There is a bijection between the set $\mathcal{P}(n+k-1,k)$ and the set
$\mathcal{CT}(n,k)$. Moreover, this bijection transforms the number of
neighbor alignments of a partial matching to the number of transients of a
partition.
Proof. Let $M$ be a partial matching in $\mathcal{P}(n+k-1,k)$. we may reduce
it to a partition by the following procedure.
* •
Change neighbor alignments to 2-paths from left to right until there are no
more neighbor alignments, see Figure 3 for an illustration. More precisely,
suppose that there is a neighbor alignment consisting of two arcs $(i,j)$ and
$(j+1,k)$. We change the arc $(j+1,k)$ to $(j,k)$ and delete the vertex $j+1$.
* •
Delete the singleton immediately after each closer, except for the last
closer.
* •
Relabel the vertices in the natural order.
Denote by $P$ the obtained partition. We claim that there are no right
crossings in the resulting partition $P$. Suppose that there are two crossing
arcs $(i,j)$ and $(k,j+1)$ with $i<k<j$ in the resulting partition. Since we
have either deleted an opener or a singleton immediately after each closer of
$M$, except for the last closer, the two crossing arcs forms a 2-right
crossing in the partial matching $M$, a contradiction. Hence, the claim is
proved.
Conversely, given a partition $P^{\prime}\in\mathcal{CT}(n,k)$, we wish to
construct a partial matching $M^{\prime}\in\mathcal{R}(n+k-1,k)$. First, we
add a vertex after each closer, except for the last closer. Then, we change
each pair of arcs $(i,j)$ and $(j,k)$ with $i<j<k$ into a neighbor alignment.
Finally, we relabel the vertices in the natural order to get the partial
matching $M^{\prime}$.
From the construction of the inverse map, we know that we have either added a
singleton or an opener after each closer of the partition $P^{\prime}$. Hence,
there are no right crossings and right nestings in the resulting partial
matching. We claim that there are no 2-right crossings in the resulting
partial matching $M^{\prime}$. Suppose that there are two crossing arcs
$(i,j)$ and $(k,j+2)$ with $i<k<j$ in the resulting partial matching
$M^{\prime}$. Then $j+1$ is either a singleton or an opener. From the
construction of the reverse map, the vertex $j+1$ is an added vertex. This
implies that the two crossing arcs is a right crossing in the partition
$P^{\prime}$, a contradiction. Hence, the claim is proved. This implies that
there is a bijection between the set $\mathcal{P}(n+k-1,k)$ and the set
$\mathcal{CT}(n,k)$. Obviously, this bijection transforms the number of
neighbor alignments of a partial matching to the number of transients of a
partition. This completes the proof.
Combining Theorems 1.1 and 3.1, we derive the generating function Formula
(1.2).
Figure 4 gives an example of a partial matching $M\in\mathcal{P}(12,5)$ and
its corresponding partition $P\in\mathcal{CT}(8,5)$.
$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$\Longleftrightarrow$$1$$2$$3$$4$$5$$6$$7$$8$
Figure 4: The partial matching $M$ and its corresponding partition $P$.
Acknowledgments. This work was supported by the National Natural Science
Foundation of China (No. 10901141).
## References
* [1] M. Bousquet-Mélou, A. Claesson, M. Dukes, S. Kitaev, $(2+2)$-free posets, ascent sequences and pattern avoiding permutations, J. Combin. Theory Ser. A 117 (2010), 884–909.
* [2] W.Y.C. Chen, N.J.Y. Fan and A.F.Y.Zhao, Partitions and partial matchings avoiding neighbor patterns, Europ. J. Combin., to appear.
* [3] A. Claesson, S. Linusson, $n!$ matchings, $n!$ posets, Proc. Amer. Math. Soc. 139 2011, 435–449.
* [4] M. Dukes, R. Parviainen, Ascent sequences and upper triangular matrices containing non-negative integers, Electron. J. combin. 17 (2010), R53.
* [5] P. Levande, Two new interpretations of the Fishburn numbers and their refined generating functions, arXiv: math.CO /1006.3013.
* [6] A. Stoimenow, Enumumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications 7 (1998), 93–114.
* [7] D. Zagier, Vassiliev invariants and a stange identity related to the Dedeking eta-function, Topology 40 (2001), 945–960.
|
arxiv-papers
| 2011-10-05T06:43:32 |
2024-09-04T02:49:22.802897
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sherry H.F. Yan and Yuexiao Xu",
"submitter": "Sherry H.F. Yan",
"url": "https://arxiv.org/abs/1110.0904"
}
|
1110.0922
|
# Light quark correlators in a mixed-action setup
Fabio Bernardonia, Nicolas Garron 111N.G. acknowledges supported from STFC
grant ST/G000522/1 and EU grant 238353 (STRONGnet). b, Pilar Hernándezc, d,
Carlos Penae
a NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
b SUPA, School of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, UK
c Instituto de Física Corpuscular, CSIC-Universitat de València, Apartado de
Correos 22085, E-46071 Valencia, Spain
d CERN, Physics Departement, 1211 Geneva 23, Switzerland
e Dpto. de Física Téorica and Instituto de Física Téorica UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco E-28049 Madrid, Spain E-mail
###### Abstract:
We report our progress in simulating Neuberger valence fermions on $N_{f}=2$
Wilson $O(a)$-improved sea quarks. We compute correlators with valence quark
masses both in the $p$\- and in the $\epsilon$-regime, and we match the
results with the predictions of the Chiral Effective Theory in the mixed
regime. This allows us to extract the Low Energy Couplings (LECs) of the
$N_{f}=2$ theory and to test the validity of the approach.
CERN-PH-TH/2011-229
IFT-UAM/CSIC-11-68
Edinburgh 2011/28
FTUAM-11-55
DESY 11-171
IFIC/11-54
## 1 Introduction
Lattice QCD simulations with different discretizations in the valence and in
the sea sector have been performed by several groups and with different setups
in the last years [1, 2, 3, 4, 5, 6, 7]. This is a viable approach to preserve
exact chiral symmetry for the valence quarks with a “moderate” numerical cost,
which could be particularly advantageous for the computation of weak matrix
elements.
In our work we apply the Neuberger operator on $N_{f}=2$ non-perturbatively
$O(a)$-improved Wilson fermions configurations generated within the CLS common
effort. 222https://twiki.cern.ch/twiki/bin/view/CLS/WebHome In a recent
publication [8] we have reported our results on the topological susceptibility
and the low-lying eigenvalues of the massless Dirac operator, which we used to
test the so-called mixed Chiral Effective Theory approach and finally to
extract the chiral condensate. We now extend our analysis to two- and three-
point functions.
## 2 Left current correlator in Chiral Effective Theory
We consider partially quenched QCD with $N_{s}$ sea flavors with mass $m_{s}$
and $N_{v}$ valence flavors with mass $m_{v}$, and define the two-point
correlators of the left-handed current
${\rm Tr}[T^{a}T^{b}]C(x_{0})\equiv\int d^{3}x\langle
J^{a}_{0}(x)J^{b}_{0}(0)\rangle,\;\;\;\;\;J^{a}_{\mu}\equiv\bar{\psi}T^{a}\gamma_{\mu}P_{-}\psi,$
(1)
where $T^{a}$ is a traceless generator of $SU(N_{v})$ and
$P_{-}=(1-\gamma_{5})/2$ is the left projector. Low energy properties can be
described by the corresponding $SU(N_{s}+N_{v})$ Partially Quenched Chiral
Perturbation Theory (PQChPT); in particular, in a finite volume $V=L^{3}T$ we
consider the following two situations [9]:
* •
$m_{v/s}\Sigma V\gg 1$: all quarks are in the $p$-regime. In this case the
correlator $C(x_{0})$ at NLO in PQChPT is given by
$\mathcal{C}(x_{0})=\frac{F^{2}_{\rm{eff}}}{4}\frac{M_{vv;\rm{eff}}\cosh\left[M_{vv;\rm{eff}}(T/2-x_{0})\right]}{\sinh\left[M_{vv;\rm{eff}}T/2\right]}-\frac{N_{s}}{2}\frac{dG^{1}(0;M^{2}_{vs})}{dT},$
(2)
with
$\displaystyle F^{2}_{\rm eff}=F^{2}(1+\Delta_{F})$ $\displaystyle=$
$\displaystyle
F^{2}\left\\{1-\frac{N_{s}}{F^{2}}G^{1}(0;M^{2}_{vs})+\frac{8}{F^{2}}(N_{s}M^{2}_{ss}L_{4}+M^{2}_{vv}L_{5})\right\\},$
(3) $\displaystyle M^{2}_{vv;\rm{eff}}=M^{2}_{vv}(1+\Delta_{M})$
$\displaystyle=$ $\displaystyle
M^{2}_{vv}\Bigg{\\{}1+\frac{1}{N_{s}F^{2}}\left[G^{1}(0;M^{2}_{vv})+(M^{2}_{ss}-M^{2}_{vv})G^{2}(0;M^{2}_{vv})\right]$
(4) $\displaystyle-$
$\displaystyle\frac{8}{F^{2}}\left[N_{s}M^{2}_{ss}(L_{4}-2L_{6})+M^{2}_{vv}(L_{5}-2L_{8})\right]\Bigg{\\}},$
with the LO pseudoscalar masses $M_{ij}=(m_{i}+m_{j})B$ and the pseudoscalar
propagators in finite volume
$G^{r}(x;M^{2})=\frac{1}{V}\sum_{p}\frac{e^{ipx}}{(p^{2}+M^{2})^{r}},\;\;\;r\geq
1.$ (5)
* •
$m_{v}\Sigma V\lesssim 1$, $m_{s}\Sigma V\gg 1$: valence quarks are in the
$\epsilon$-regime, sea quarks are in the $p$-regime. In this case the NLO left
correlator at fixed topology is given by
$\mathcal{C}^{\nu}(x_{0})=\frac{F^{2}_{\epsilon;\rm{eff}}}{2T}\left\\{1+\frac{2T^{2}}{F^{2}V}\mu_{v}\sigma_{\nu}(\mu_{\nu})h_{1}\left(\frac{x_{0}}{T}\right)\right\\}-\frac{N_{s}}{2}\frac{dG^{1}(0;M^{2}_{ss}/2)}{dT},$
(6)
with $\mu_{v}=m_{v}\Sigma V$ and
$\displaystyle F^{2}_{\epsilon;\rm{eff}}$ $\displaystyle=$ $\displaystyle
F^{2}\left(1+\Delta_{F}|_{M_{vv}=0}\right)=F^{2}\left\\{1-\frac{N_{s}}{F^{2}}G^{1}(0;M^{2}_{ss}/2)+\frac{8}{F^{2}}(N_{s}M^{2}_{ss}L_{4})\right\\},$
(7) $\displaystyle h_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\tau-\frac{1}{2}\right)^{2}-\frac{1}{12}\right],$
(8) $\displaystyle\sigma_{\nu}(x)$ $\displaystyle=$ $\displaystyle
x\left[I_{\nu}(x)K_{\nu}(x)+I_{\nu+1}(x)K_{\nu-1}(x)\right]+\frac{\nu}{x},$
(9)
where $I_{\nu}$, $K_{\nu}$ are modified Bessel functions.
A well-established strategy to extract the LO ($F,B$) and the NLO ($L_{i}$)
Low Energy Couplings (LECs) is to match lattice results for the two-point
correlators with the predictions of the ChPT. Depending on the chiral regime,
the systematic uncertainty associated to the matching can vary considerably.
Exploring different regimes in a finite box can then be useful to give
reliable estimates of systematic errors associated to the LECs.
## 3 Lattice results
### 3.1 Setup and results for the Low Energy Couplings
We have carried out our simulations on CLS lattices of size $V=24^{3}\times
48$, with $\beta=5.3$ corresponding to a lattice spacing $a=0.0649(10)$ fm
[10]. We have considered three $p$-regime sea quark masses
$\kappa=0.13620,0.13625,0.13635$, labeling the sets by D4,D5,D6 respectively.
On these configurations we have built the Neuberger-Dirac operator, with
valence quark masses both in the $p$\- and in the $\epsilon$-regime. The
simulations parameters are collected in Table 1, where we report also the
effective sea pion masses in lattice units (see [8] for further details). For
valence quark masses in the $p$-regime, we have collected $\sim 80-160$
measurements. For the $\epsilon$-regime, we have in total $\sim 70-100$
measurements of the left correlator in the topological sectors
$|\nu|=0,1,2,3$.
$\beta=5.3$, $c_{SW}=1.90952$, $V=24^{3}\times 48$
---
lattice | $\kappa$ | $aM_{ss;{\rm eff}}$ | $am_{v}$ ($\epsilon$-reg.) | $am_{v}$ ($p$-reg.)
D4 | 0.13620 | 0.1695(14) | 0.002 | 0.012,0.02,0.028
D5 | 0.13625 | 0.1499(15) | 0.002 | 0.012,0.02
D6 | 0.13635 | 0.1183(37) | 0.002 | 0.006,0.015,0.025
Table 1: Simulation parameters
We have computed the two-point left current correlator defined in Eq. (1)
adopting the low-mode averaging technique of [11]. For the renormalization of
the local axial current we used the chiral Ward identity at fixed topological
charge saturated with the zero modes, following the strategy proposed in [12].
In particular we considered
$Z_{A}^{\rm mix}\lim_{m_{v}\rightarrow
0}(m_{v}V)\sum_{\vec{x}}\langle\partial_{0}P^{a}(x)J^{a}_{0}(0)\rangle_{\nu}=\lim_{m_{v}\rightarrow
0}(m_{v}^{2}V)\sum_{\vec{x}}\langle P^{a}(x)P^{a}(0)\rangle_{\nu},$ (10)
where $P^{a}=\bar{\psi}T^{a}\gamma_{5}\psi$ is the pseudoscalar density. The
right-hand side is completely saturated with zero modes and can be directly
computed without chiral extrapolation; on the left-hand side, the chiral limit
has to be taken numerically. The renormalization factor $Z_{A}^{\rm mix}$ can
then be extracted from ratios of correlations functions. For those correlators
we have collected a total of $\sim 70-140$ measurements for the topological
sectors $0<|\nu|<5$. On the left side of Fig. 1 we report the results for the
different sea quark masses corresponding to D4,D5,D6 and for $\nu=1,2,3,4$.
Apart from some fluctuations in the topological sector $|\nu|=1$, the data do
not depend on $\nu$ and on the sea quark masses. From this study we obtained
$Z_{A}^{\rm mix}=1.65(1).$ (11)
Figure 1: _Left_ : the renormalization constant of the local axial current
extracted through the chiral Ward identity at fixed topological charge
saturated with the zero modes. The blue,green and red points refer to the
lattices D4,D5,D6 respectively. _Right_ : $aF_{\epsilon;\rm{eff}}$ extracted
from the left correlator in the mixed $p$/$\epsilon$-regime as a function of
the topological charge. The gray bands represent the average over
$|\nu|=1,2,3$.
We then extracted $M_{vv;\rm{eff}}$, $F_{\rm eff}$, $F_{\epsilon;\rm{eff}}$
from the left correlators, according to Eqs. (2, 6); on the right side of Fig.
1 we show for instance the (bare) pseudoscalar decay constant computed from
the mixed $p$/$\epsilon$-regime correlators at $|\nu|=0,1,2,3$. As it should
be, results do not depend significantly on the topological charge;
$F_{\epsilon;\rm{eff}}$ has been determined by averaging over $|\nu|=1,2,3$
(gray band in the plot). We then performed a global fit of pseudoscalar masses
and decay constants using the NLO predictions of the PQChPT given in Eqs.
(3,4,7). From this fit we obtained the six parameters
$\displaystyle aF^{R}=0.0259(10)(8)$ , $\displaystyle aB^{R}=1.52(11)(9),$
$\displaystyle L_{4}^{R}=0.00054(10)(4)$ , $\displaystyle
L_{5}^{R}=0.00073(15)(2),$ (12)
$\displaystyle(L_{4}-2L_{6})^{R}=0.00033(5)(0)$ ,
$\displaystyle(L_{5}-2L_{8})^{R}=-0.00025(15)(5),$
where “R” means that the couplings are renormalized. The constants $L_{i}^{R}$
are given at the scale $\mu=770$ MeV, while $B^{R}$ is given in the
$\overline{\rm{MS}}$ scheme at $\mu=$ 2 GeV. For the tree-level pseudoscalar
masses we used $M_{ij}=(m_{i}^{R}+m_{j}^{R})B^{R}$; for the sea Wilson quark
masses we used the PCAC masses computed in [13, 10], with renormalization
constants $Z_{A}^{W}=0.78(1)$ [14], $Z_{P}^{W}(\overline{\rm
MS},\;2\;\rm{GeV})=0.543(8)$ [15]; for the overlap valence quarks we have
computed $Z_{S}^{\rm mix}(\overline{\rm MS},\;2\;\rm{GeV})=1.88(15)$ like in
[8]. The first error is statistical, while the second one reflects the
uncertainty on the PCAC quark masses.
Figure 2: Global chiral NLO fit for $M_{vv;\rm{eff}}$, $F_{\rm eff}$. The
squares in the right plot correspond to the $\epsilon$-regime results,
$F_{\epsilon;\rm{eff}}$.
The global fit is shown in Fig. 2. It is important to stress that having
valence quark masses in the $\epsilon$-regime allows us to extract the
pseudoscalar decay constant directly at $am_{v}=0$ and hence to have a good
control over the chiral extrapolation in $m_{v}$.
Appropriate combinations of the $L_{i}$ yield the conventional SU(2) LECs
$\bar{l}_{i}$; in particular we obtain
$\bar{l}_{3}=4.46(30)(14),\;\;\;\;\;\;\;\;\bar{l}_{4}=4.56(10)(4).$ (13)
The FLAG working group [16] gives an estimate $\bar{l}_{3}=3.2(8)$ and quotes
$\bar{l}_{4}=4.66(4)\binom{+4}{-33}$ [17] as best determination of
$\bar{l}_{4}$. Our determinations are in agreement with those results within
the uncertainties. By converting $aF^{R}$ and $aB^{R}$ in physical units we
obtain $F^{R}=79(4)$ MeV and $\Sigma^{R}(\overline{\rm
MS},\;2\;\rm{GeV})=B^{R}(F^{R})^{2}=(306(11)\;{\rm MeV})^{3}$, which also lie
in the right ballpark.
### 3.2 Unitary point
A natural definition of the unitary point corresponds to the matching of the
pseudoscalar masses
$M^{2}_{vv;\rm{eff}}|_{m_{v}=m_{v;\rm{un}}}=M^{2}_{ss;\rm{eff}}.$ (14)
We have extracted $m_{v;\rm{un}}$ through a linear interpolation of
$M^{2}_{vv;\rm{eff}}$. The pseudoscalar decay constants in the full Wilson
theory ($F^{R}_{s;\rm{eff}}$) and in the mixed action theory at
$m_{v;\rm{un}}$ ($F^{R}_{\rm eff}|_{m_{v}=m_{v;\rm{un}}}$) should then be
equal up to discretization effects. In Fig. 3 we show $F^{R}_{\rm
eff}|_{m_{v}=m_{v;\rm{un}}}$ and $F^{R}_{s;\rm{eff}}$ for the three lattices
D4,D5,D6. The Wilson on Wilson data have been taken from [13]. Within our
statistical accuracy, we do not observe large cutoff effects close to the
unitary point.
lat. | $am_{v;\rm{un}}$ | $F^{R}_{\rm eff}|_{m_{v}=m_{v;\rm{un}}}$ | $F^{R}_{s;\rm{eff}}$
---|---|---|---
D4 | 0.0241(5)(4) | 0.0413(6) | 0.0396(7)
D5 | 0.0196(13)(5) | 0.0391(11) | 0.0380(9)
D6 | 0.0088(6)(6) | 0.0320(9) | 0.0328(19)
Figure 3: Effective pseudoscalar decay constant computed in the full Wilson
theory (black points) and in the mixed action theory at the unitary point (red
points) as a function of the pseudoscalar mass. The empty square represents
the decay constant in the chiral limit extracted from the chiral fit, Eq.
(12).
Discretization effects in a mixed action setup can be investigated by means of
Mixed Action PQChPT [18]. If all mass scales are in the $p$-regime and one
assumes a power counting $a\sim m_{s,v}\sim p^{2}$ (_generically small
masses_), it is easy to see that “unitarity violations” of $O(a^{2})$
appearing in $M^{2}_{vs}$ and $(M^{2}_{ss}-M^{2}_{vv})$ affect our observables
only at NNLO, and hence are expected to be small. Our numerical data seem to
confirm this expectation, although a more detailed study is necessary in order
to have a precise estimate of those effects. 333A recent study with Neuberger
valence fermions and Wilson Twisted Mass sea quarks [6] indicates large cutoff
effects close to the unitary point.
## 4 $K\rightarrow\pi\pi$ amplitudes and $\Delta I=1/2$ rule
In [19], a strategy to study the rôle of the charm quark in the $\Delta I=1/2$
rule has been proposed. The basic idea is to start from the GIM limit
$m_{u}=m_{d}=m_{s}=m_{c}$, write down the $\Delta S=1$, CP-even effective weak
Hamiltonian and match it with the effective Hamiltonian in the SU(4) Chiral
Effective Theory. At leading order in ChPT, the ratio of amplitudes
corresponding to $\Delta I=1/2$ and $\Delta I=3/2$ $K\rightarrow\pi\pi$
transitions can be expressed in terms of given LECs $g_{1}^{\pm}$. As usual,
those couplings can be computed by matching suitable correlation functions
computed on the lattice with their analytic expressions obtained in ChPT (see
[19] for details). Quenched results obtained with Neuberger fermions have been
presented in [20]; this strategy, for which exact chiral symmetry at finite
lattice spacing is an essential ingredient, can be implemented also with our
mixed action setup. Preliminary results for the bare ratios $R^{\pm}$ of
three- and two-point functions defined in [20] have been obtained so far for
the lattice D4 (interpolated to the unitary point)
$R^{+}=0.64(6),\;\;\;R^{-}=1.5(2),$ (15)
of the same order as the quenched results for pseudoscalar masses in the same
range.
The second part of the strategy consists in studying the amplitudes ratio as a
function of $m_{c}>m_{u}=m_{d}=m_{s}$ and test if by increasing the charm
quark the $\Delta I=1/2$ enhancement observed experimentally is reproduced.
## 5 Conclusions and outlook
Mixed action simulations with exact chiral symmetry in the valence sector
provide a powerful tool to investigate QCD properties. Given the significant
errors associated to chiral extrapolations, it is important to explore several
chiral regimes where the systematic effects are different, so that such errors
can be properly quantified. In particular, we have considered valence quarks
both in the $p$\- and in the $\epsilon$-regime and matched the lattice results
with the PQChPT predictions at NLO. A global chiral fit of pseudoscalar masses
and decay constants allowed us to determine $F$, $B$, $\bar{l}_{3}$,
$\bar{l}_{4}$ with comparable accuracy with respect to other results present
in the literature. By matching pseudoscalar masses in the valence and in the
sea sector, we did not observed large cutoff effects close to the unitary
point. We stress that a precise determination of the LECs would require an
improvement of the statistical precision and a careful monitoring of
autocorrelations; furthermore, a larger set of volumes, lattice spacings and
quark masses would guarantee a solid estimate of the systematic uncertainties
associated to lattice artefacts, finite-volume effects and chiral
extrapolation.
As next step, we are implementing the strategy proposed in [19] to investigate
the rôle of the charm quark in the $\Delta I=1/2$ rule. This calculation is
more challenging and requires further computational efforts.
## References
* [1] LHP Collaboration, D.B. Renner et al., Nucl.Phys.Proc.Suppl. 140 (2005) 255.
* [2] C. Allton et al., PoS LAT2006 (2006) 202.
* [3] S. Dürr et al., PoS LAT2007 (2007) 115.
* [4] C. Aubin, J. Laiho and R.S. Van de Water, Phys.Rev. D77 (2008) 114501; J. Laiho and R.S. Van de Water, PoS LATTICE2010 (2010) 312.
* [5] xQCD Collaboration, A. Li et al., Phys.Rev. D82 (2010) 114501.
* [6] K. Cichy, G. Herdoiza and K. Jansen, Nucl.Phys. B847 (2011) 179.
* [7] S. Beane et al., (2011), 1108.1380 [hep-lat].
* [8] F. Bernardoni et al., Phys.Rev. D83 (2011) 054503.
* [9] F. Bernardoni and P. Hernández, JHEP 10 (2007) 033.
* [10] M. Marinkovic et al., PoS Lattice2011 (2011) 232.
* [11] L. Giusti et al., JHEP 0404 (2004) 013; T.A. DeGrand and S. Schaefer, Comput.Phys.Commun. 159 (2004) 185.
* [12] L. Giusti et al., JHEP 01 (2004) 003; P. Hernández et al., JHEP 0805 (2008) 043.
* [13] L. Del Debbio et al., JHEP 02 (2007) 082.
* [14] M. Della Morte, R. Sommer and S. Takeda, Phys.Lett. B672 (2009) 407.
* [15] ALPHA Collaboration, M. Della Morte et al., Nucl.Phys. B729 (2005) 117.
* [16] G. Colangelo et al., Eur.Phys.J. C71 (2011) 1695.
* [17] ETM Collaboration, R. Baron et al., JHEP 1008 (2010) 097.
* [18] O. Bär, G. Rupak and N. Shoresh, Phys.Rev. D67 (2003) 114505; O. Bär, G. Rupak and N. Shoresh, Phys.Rev. D70 (2004) 034508; M. Golterman, T. Izubuchi and Y. Shamir, Phys.Rev. D71 (2005) 114508; O. Bär, M. Golterman and Y. Shamir, Phys.Rev. D83 (2011) 054501.
* [19] L. Giusti et al., JHEP 11 (2004) 016.
* [20] L. Giusti et al., Phys.Rev.Lett. 98 (2007) 082003.
|
arxiv-papers
| 2011-10-05T08:45:13 |
2024-09-04T02:49:22.810502
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabio Bernardoni, Nicolas Garron, Pilar Hernandez, Silvia Necco,\n Carlos Pena",
"submitter": "Silvia Necco",
"url": "https://arxiv.org/abs/1110.0922"
}
|
1110.0989
|
# Neutrino Velocity Anomalies: A Resolution without a Revolution
Dmitry V. Naumov dnaumov@jinr.ru Dzhelepov Laboratory of Nuclear Problems,
Joint Institute for Nuclear Research, RU-141980 Dubna, Russia Vadim A. Naumov
vnaumov@theor.jinr.ru Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, RU-141980 Dubna, Russia
###### Abstract
We argue that the neutrino advance of time observed in MINOS and OPERA
experiments can be explained in the framework of the standard relativistic
quantum theory as a manifestation of the large effective transverse size of
the eigenmass neutrino wavepackets.
Neutrino; Charged currents; relativity; wavepacket;
###### pacs:
14.60.Lm, 14.60.Pq, 13.15.+g
## I Introduction
Recently, OPERA Collaboration reported that neutrinos from CERN arrive to the
Gran Sasso Underground Laboratory by $(60.7\pm 6.9_{\text{stat.}}\pm
7.4_{\text{sys.}})$ ns earlier than expected for almost massless particles
Autiero:2011hh . MINOS Collaboration also observed Adamson:2007zzb an earlier
arrival of neutrinos from FNAL to the Soudan Underground Laboratory by
$(126\pm 32_{\text{stat.}}\pm 64_{\text{sys.}})$ ns (68% C.L.). The
experiments have similar distances of about $730$ km between the neutrino
production and detection regions but different mean neutrino energies (17 and
3 GeV for, respectively, CNGS and NuMI beams) and different neutrino flavor
compositions of the beams. These remarkable results, being interpreted in
terms of neutrino velocity $v_{\nu}$, suggest a superluminal motion of
neutrinos with Footnote1
$v_{\nu}=1+\left\\{\begin{aligned} (5.1\pm 2.9)\times
10^{-5}&\enskip\text{(MINOS)},\\\ (2.48\pm 0.41)\times
10^{-5}&\enskip\text{(OPERA)}.\quad\end{aligned}\right.$
At the first blush this interpretation breaks the relativity principle – one
of the basis of modern physics. We will however argue that an earlier arrival
of neutrinos could be understood without any violation of relativity,
causality and other fundamental physical concepts and is just a manifestation
of quantum nature of neutrino. Namely, we will try to demonstrate that the
observed effect can be at least qualitatively explained by taking into account
that the quantum states of neutrinos with definite (small) masses are
described by the relativistic wavepackets having a finite and in fact very
large effective transversal size. Necessity of a wavepacket description of
neutrino propagation in vacuum and matter is now well understood (while not
yet commonly accepted) in the theory of neutrino flavor transitions
(“oscillations”) based on quantum mechanics or quantum field theory.
## II Neutrino wavepacket
Any wavepacket can be conventionally characterized by a most probable
4-momentum, 4-coordinate, and a set of parameters governing the shape of the
packet in the phase space. Apparently, a spherically symmetric wavepacket with
an effective spatial “size” $\sigma_{x}$ and momentum “width” $\sigma_{p}\sim
1/\sigma_{x}$ in its rest-frame becomes asymmetrical if it is boosted with a
Lorentz factor $\varGamma\gg 1$. The wavepacket spatial size in the boost
direction shrinks as $\sigma_{x}/\varGamma$ remaining unchanged in the
transverse plane. The momentum width increases in the boost direction as
$\sigma_{p}\varGamma$ remaining the same in the transverse plane. In our
previous paper Naumov:2010um we developed a covariant field-theoretical
approach to neutrino oscillations which operates with the relativistic
wavepackets describing initial and final states of particles involved into the
production and detection of neutrino. The neutrino in this approach is
described as a virtual mass eigenfield travelling between the macroscopically
separated production and detection vertices of Feynman graphs. Thus we make no
any assumption about its wavefunction. Instead, within our formalism we
compute the neutrino wavefunction which turns out to be a wavepacket with
spatial and momentum widths defined and functionally dependent on those of the
particles involved into the neutrino production and detection subprocesses.
Explicitly, up to a coordinate independent spinor factor, the effective
(outgoing) neutrino wavefunction reads Footnote2
$\psi_{\nu}^{*}=e^{iE_{\nu}\left(x_{0}-\mathbf{v}_{\nu}\mathbf{x}\right)-\sigma_{\nu}^{2}\varGamma_{\nu}^{2}\left(\mathbf{x}_{\parallel}-\mathbf{v}_{\nu}x_{0}\right)^{2}-\sigma_{\nu}^{2}\mathbf{x}_{\perp}^{2}}.$
(1)
Here $x_{0}$ is the time, $\mathbf{x}_{\parallel}$ and $\mathbf{x}_{\perp}$
are, respectively, the longitudinal and transverse (relative to the mean
velocity vector $\mathbf{v}_{\nu}=\mathbf{p}_{\nu}/E_{\nu}$) spatial
coordinates of the geometric center of the neutrino packet ($x_{0}$ and
$\mathbf{x}_{\parallel}+\mathbf{x}_{\perp}$ form a 4-vector);
$E_{\nu}=\sqrt{\mathbf{p}_{\nu}^{2}+m_{\nu}^{2}}$, and $m_{\nu}$ is the mass
of the neutrino mass eigenfield. In the most general case, the “spread”
$\sigma_{\nu}$ is a Lorentz invariant function of the most probable 3-momenta
$\mathbf{p}_{\varkappa}$, masses $m_{\varkappa}$, and momentum spreads
$\sigma_{\varkappa}=\text{const}$ of _all_ external in and out particles
$\varkappa$ involved into the neutrino production-detection process which are
described as asymptotically free relativistic wavepackets. It is shown in Ref.
Naumov:2010um that the center of any external wavepacket moves _in the mean_
along the classical trajectory
$\langle\mathbf{x}_{\varkappa}\rangle=\widetilde{\mathbf{x}}_{\varkappa}+\mathbf{v}_{\varkappa}x_{\varkappa}^{0}$
conserving energy, momentum and effective volume ($\propto
1/\sigma_{\varkappa}^{3}$); under certain conditions the packets remain stable
(nondiffluent) during the times much longer than their mean lifetimes (in case
of unstable particles) or the mean time between the two successive collisions
in the relevant ensemble (in case of stable particles).
Considering that the two-body decays of pions and kaons are the main processes
of neutrino production in the MINOS and OPERA experiments, we can neglect the
contributions into $\sigma_{\nu}$ coming from the particles, interacting with
neutrinos in the detector (reasonably assuming that their 4-momentum spreads
are much larger than $\sigma_{\pi}$, $\sigma_{K}$ and $\sigma_{\mu}$). With
this simplification, it can be proved that
$\sigma_{\nu}^{2}\approx
m_{\nu}^{2}\left(\frac{m_{a}^{2}}{\sigma_{a}^{2}}+\frac{m_{\mu}^{2}}{\sigma_{\mu}^{2}}\right)^{-1},\quad
a=\pi~{}\text{or}~{}K.$
Then from the above-mentioned conditions of stability for the meson and muon
wavepackets it follows that $\sigma_{\nu}$ must satisfy the following
conditions
$\sigma_{\nu}^{2}\ll
m^{2}_{\nu}\left(\frac{m_{\mu}}{\Gamma_{\mu}}+\frac{m_{a}}{\Gamma_{a}}\right)^{-1},$
where $\Gamma_{a}=1/\tau_{a}$ and $\Gamma_{\mu}=1/\tau_{\mu}$ are the full
decay widths of the meson $a$ and muon. Considering that for _any_ know meson
$m_{\mu}/\Gamma_{\mu}\gg m_{a}/\Gamma_{a}$, we conclude that
$\sigma_{\nu}^{2}/m^{2}_{\nu}\ll\Gamma_{\mu}/m_{\mu}\approx 2.8\times
10^{-18}.$ (2)
Therefore the neutrino momentum uncertainty is fantastically small Footnote3 .
From (2) one can immediately derive the lower bounds for the effective spatial
dimensions of the neutrino wavepacket:
$\displaystyle
d_{\perp}\gg\left(\frac{0.1~{}\text{eV}}{m_{\nu}}\right)~{}\text{km},$
$\displaystyle d_{\parallel}=\frac{d_{\perp}}{\varGamma_{\nu}}\gg
10^{-2}\left(\frac{10~{}\text{GeV}}{E_{\nu}}\right)\left(\frac{0.1~{}\text{eV}}{m_{\nu}}\right)~{}\mu\text{m},$
So the neutrino wavepacket appears as a huge but superfine disk of microscopic
(energy dependent) thickness in longitudinal direction, comparable with the
thickness of a soap-bubble skin, and macroscopically large (energy
independent) diameter in the transverse plane Footnote4 .
This is a key point in interpretation of the experiments with earlier arrival
of neutrino signal.
## III Qualitative estimations
In fact, neutrinos produced at accelerators arrive to the detector site widely
distributed across the plane transverse to the beam axis. Those neutrinos
which were misaligned with the neutrino detector nevertheless do have a chance
to interact within the detector due to macroscopically large transverse size
of its wavefunction. Moreover, its interaction probability very weakly depends
on the misalignment distance $r=\text{BC}$ (see Fig. 1) if it is small
compared to $d_{\perp}$.
Figure 1: Neutrinos are emitted from the “Source” and are registered in the
“Detector”. The centers of the neutrino wavepackets will arrive at the points
B and D simultaneously, while the signal from the neutrino wavepacket (shown
as an extremely oblate spheroid) which moves under the angle
$\theta=\angle\text{BAC}$ to the beam axis will arrive earlier since
$\text{DE}>0$. Neutrino velocity vector $\mathbf{v}_{\nu}$ lies in the plane
of the figure. Proportions do not conform to reality.
As is seen from Fig. 1, the misaligned neutrinos will interact systematically
earlier than those moving along the beam axis, due the huge transverse width
of their wavefunctions. The school-level planimetry suggests that the
advancing time is given by
$\delta t=L\left(1/\cos\theta-1\right)\approx r^{2}/(2L).$ (3)
Here we assume that (i) $1-v_{\nu}\lll 1$, (ii) the neutrino wavepacket
effective width is much larger than the detector dimensions, and (iii)
$\theta\ll 1$. Substituting numbers into (3) one obtains Fig. 2 from which
quantitative estimates for the time advance as a function of $r$ could be
drawn. For instance, a neutrino packet which moves 3 km away from the detector
will come earlier by about $20$ ns than that moving directly to the detector.
Figure 2: Advance $\delta t$ as a function of $r$.
What is the probability to find a neutrino at a distance $r$ from the beam
axis? This could be estimated taking into account that neutrino production is
dominated by two-particle decays of pions and kaons. The angular distribution
of massless neutrinos from these decays is well known:
$\frac{dI}{d\Omega}=\frac{1-v_{a}^{2}}{4\pi(1-v_{a}\cos\theta)^{2}}\approx\frac{1}{\pi(1+\varGamma_{a}^{2}\theta^{2})^{2}}.$
(4)
Here $\theta$ is the angle between the momenta of the meson $a$ and neutrino
($0\leq\theta\leq\pi$), $v_{a}$ is the meson velocity, and
$\varGamma_{a}=(1-v_{a}^{2})^{-1/2}=E_{a}/m_{a}$. The second approximate
equality in Eq. (4) holds for small angles and relativistic meson energies
($\theta\ll 1$, $4\varGamma_{a}^{2}\gg 1$). In the latter case, the main
contribution to the neutrino event rate comes from the narrow cone
$\theta\lesssim 1/\varGamma_{a}$. Considering that the mean neutrino energy,
$\overline{E}_{\nu}$, from the muonic decay of a meson with energy $E_{a}$ is
$\overline{E}_{\nu}=\varGamma_{a}E_{\nu}^{(a)}$, where
$E_{\nu}^{(a)}=(m_{a}^{2}-m_{\mu}^{2})/(2m_{a})$ is the neutrino energy in the
rest frame of the particle $a$, the characteristic angle can be defined as
$\theta_{(a)}=E_{\nu}^{(a)}/\overline{E}_{\nu}$.
In the case of OPERA, one can (very) roughly estimate the characteristic
angles for the “low-energy” (LE) range ($E_{\nu}<20$ GeV,
$\overline{E}_{\nu}\approx 13.9$ GeV) and “high-energy” (HE) range
($E_{\nu}>20$ GeV, $\overline{E}_{\nu}\approx 42.9$ GeV), assuming that the
main neutrino sources in these ranges are, respectively, $\pi_{\mu 2}$ and
$K_{\mu 2}$ decays:
$\displaystyle\theta_{\text{LE}}~{}\gtrsim~{}\theta_{(\pi)}=2.1\times
10^{-3},$ $\displaystyle\theta_{\text{HE}}~{}\lesssim~{}\theta_{(K)}=5.5\times
10^{-3}.$
This provides us with an order-of-magnitude estimate of the mean values of $r$
and advancing times ${\delta}t$:
$\displaystyle r_{\text{LE}}~{}\gtrsim~{}1.7~{}\text{km},\quad
r_{\text{HE}}~{}\lesssim~{}11~{}\text{km};$
$\displaystyle{\delta}t_{\text{LE}}~{}\gtrsim~{}5.6~{}\text{ns},\quad{\delta}t_{\text{HE}}~{}\lesssim~{}36.7~{}\text{ns}.$
Since the LE and HE ranges contribute almost equally to the CNGS $\nu_{\mu}$
beam, there must be a definite trend towards earlier neutrino arrival to OPERA
with approximately $21$ ns mean time-shift and a “tail” or, better to say,
“fore” of the same order coming from the “edges” of the CNGS beam.
Similar estimation for the low-energy NuMI beam at Fermilab producing
neutrinos for the MINOS experiment can be done with a better accuracy, since
the $\pi_{\mu 2}$ decay is here the dominant source of neutrinos and the
radial distribution of the beam is expected to be very flat. So, by using
$\overline{E}_{\nu}=3$ GeV we obtain
$r\approx 36.2~{}\text{km},\quad{\delta}t\approx 120.7~{}\text{ns}.$ (5)
The latter number is in surprisingly good agreement with the MINOS
observation. Obviously, MINOS should observe at the average a much earlier
arrival of neutrinos, in comparison with OPERA, because of the lower mean
neutrino energy which corresponds to a wider transverse beam distribution and
hence to a larger input from the misaligned neutrinos.
## IV Numerical results
Let us reevaluate the estimates given above with a somewhat detailed but still
simplified calculation. In particular, we could profit from the simulation of
expected radial distribution of $\nu_{\mu}$ charged current (CC) events
performed by the OPERA Collaboration NuFluxOPERA_Radial . This distribution
($P_{\text{\text{CC}}}(r)$) which we digitalized for our purposes is displayed
in Fig. 3. Being dominated by the $\pi_{\mu 2}$ and $K_{\mu 2}$ decays, the
transverse beam size at Gran Sasso is of the order of kilometres and the full
width at half maximum of the distribution is about 2.8 km.
Figure 3: Probability of neutrino charged current interactions expected in
OPERA as function of $r$.
The distribution $P_{\text{CC}}(r)$ transformed (with help of Eq. (3)) into
the $\delta t$ distribution as
$P_{\text{CC}}(\delta t)=\frac{rP_{\text{CC}}\left(r\left(\delta
t\right)\right)}{\displaystyle\int_{0}^{\infty}drrP_{\text{CC}}(r)}$
is shown in Fig. 4. Its average $\langle\delta t\rangle$ is about 20 ns with
similar variance and with the tail extending up to about 100 ns.
Figure 4: Advance $\delta t$ distribution expected in OPERA.
Figure 5 shows the integral distribution
$P_{\text{CC}}\left(<\delta t\right)=\int_{0}^{\delta
t}dt^{\prime}P_{\text{CC}}(t^{\prime}).$
Examination of Fig. 5 suggests that all CC events roughly equally populate the
following intervals in $\delta t$: $(0,20)$ ns, $(20,45)$ ns, and $(45,100)$
ns.
Figure 5: $P_{\text{CC}}(<\delta t)$ distribution expected in OPERA.
Finally, we compute the expected time distribution in OPERA, $g(t)$, as a
convolution of the probability density function of arrival time $f(t)$ shown
in top panel of Fig. 6 as solid line, taking into account an earlier arrival
of neutrino signal as follows:
$g(t)=\frac{\displaystyle\int_{0}^{\infty}f\left(t+\delta
t(r)\right)P_{\text{CC}}(r)rdr}{\displaystyle\int_{0}^{\infty}P_{\text{CC}}(r)rdr}.$
The resulting curve $g(t)$ is displayed superimposed in the top panel of Fig.
6 by dashed line. As is seen, on the average, it is shifted to the left by
about 20 ns. However, and this is even more important, the left front of the
signal is shifted by a larger amount as it accumulates the advance effect from
the total $f(t)$ distribution. One could verify that the solid and dashed
lines shown in the bottom panel of Fig. 6 for the left front of the OPERA time
distribution is shifted by about 50 ns (up to 60 ns on the tail) to the left,
while the right front is shifted only by 20–25 ns. In other words, the impact
of the misaligned neutrinos is predicted to be asymmetric in time.
Figure 6: Top panel: time probability density function for the first beam
extraction taken as an example, expected by the OPERA Collaboration
Autiero:2011hh and our calculation. In both cases, the systematic
“instrumental” shift is not shown since it does not change the shape of the
curves. Bottom left and right panels: zooms of the top panel for the left and
right fronts of the signal, respectively.
In the likelihood fit of the time distribution performed by the OPERA
Collaboration the fronts of the time distribution statistically play the major
role. Therefore, the time distortion evaluated in the present work seem to
explain the OPERA observations without any model parameter and without
introducing superluminal neutrinos or other exotics.
## V Conclusions
Large transverse size of the neutrino wavepacket and uncollimated beam of
neutrinos seem to explain the earlier arrival of the neutrino signal in OPERA
and MINOS. The neutrino signal is estimated to arrive in advance by about 20
ns in the mean (with a similar variance) for OPERA and by about 120 ns for
MINOS. In the case of the OPERA experiment only this effect essentially
reduces the statistical significance of its observation. Moreover, we have
evaluated the expected time distribution of the neutrino arrival in OPERA and
obtained that the left and right fronts are shifted to the left by about 50–60
ns and 20–25 ns, respectively. This probably explains the observed anomaly
all-in-all without any exotic hypothesis, like Lorentz violation and so on.
Let us underline that in our calculations we do not use any adjustable
parameter. In the case of the MINOS experiment there is also a surprisingly
good agreement between our expectation (5) and experimental result. Therefore,
we argue that observations of superluminal neutrinos by the OPERA and MINOS
experiments can be treated as a manifestation of the huge transverse size of
the neutrino wavefunction. This kind of effects could be investigated in the
future experiments (in particular, in the off-axis neutrino experiments) with
more details in order to prove or disprove our explanation.
Let us note that one should not expect an increase in the number of neutrino
induced events due to the misaligned neutrino interactions because this effect
will be compensated by the corresponding decrease of the number of aligned
neutrinos.
Let us briefly discuss the situation with the observed (anti)neutrino signal
from SN1987A. A proper treatment of these neutrinos should take care about the
dispersion of the neutrino wavepackets at astronomical distances. Deliberately
neglecting the dispersion, it appears that any terrestrial detector is
sensitive only to the aligned neutrinos, since the misaligned neutrinos will
have negligible impact due to the smallness of their wavepacket transverse
size relative to the astrophysical scale of about 50 kps. Therefore, no
advance signal should be expected. However this problem is not so simple and
needs in a more detailed theoretical analysis.
###### Acknowledgements.
This work was supported by the Federal Target Program “Scientific and
scientific-pedagogical personnel of the innovative Russia”, contract No.
02.740.11.5220.
## References
* (1) T. Adam et al. (OPERA Collaboration), arXiv:1109.4897 [hep-ex].
* (2) P. Adamson et al. (MINOS Collaboration), Phys. Rev. D 76, 072005 (2007) [arXiv:0706.0437 [hep-ex]].
* (3) The MINOS Collaboration providently concludes that their measurement is consistent with the speed of light to less than $1.8\sigma$ and the corresponding 99% C.L. bounds on $v_{\nu}$ are $-2.4\times 10^{-5}<v_{\nu}-1<12.6\times 10^{-5}$.
* (4) D. V. Naumov, V. A. Naumov, J. Phys. G G37, 105014 (2010) [arXiv:1008.0306 [hep-ph]].
* (5) In fact expression (1) has a limited range of applicability which is however deliberately comprises all terrestrial neutrino experiments.
* (6) By the way, Eq. (2) explains a success of the naive quantum-mechanical approach to neutrino oscillations which is explicitly based on the assumption that the massive neutrinos have definite 3-momenta and thus can be described by plain waves (infinitely large mathematical artifacts).
* (7) Note that the acceptable effective transverse dimensions of “normal” particle states (or, that is the same, the effective diameters of the packets in the intrinsic frame of reference) are not so huge as for neutrinos. For example, the limiting dimensions (minimal effective diameters of the wavepackets, necessary to keep them stable during the life of the particles) should be of the order of $10^{-5}$, $10^{-4}$, and $10^{-8}$ cm for $\pi^{\pm}$, $\mu^{\pm}$, and $\tau^{\pm}$, respectively. These dimensions are probably hard to measure in the current particle scattering experiments, since the wavepackets are reduced due to any measurement (interaction). Neutrinos provide the only (as yet) tool for such measurements, which are believed to be already realized by the MINOS and OPERA experiments.
* (8) http://proj-cngs.web.cern.ch/proj-cngs/Beam%20Performance/NeutrinoRadial.htm
|
arxiv-papers
| 2011-10-05T14:06:35 |
2024-09-04T02:49:22.828573
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dmitry V. Naumov, Vadim A. Naumov",
"submitter": "Vadim Naumov",
"url": "https://arxiv.org/abs/1110.0989"
}
|
1110.1091
|
# A simulation of the Neolithic transition in the Indus valley
###### Abstract
The Indus Valley Civilization (IVC) was one of the first great civilizations
in prehistory. This bronze age civilization flourished from the end of the
fourth millennium BC. It disintegrated during the second millennium BC;
despite much research effort, this decline is not well understood. Less
research has been devoted to the emergence of the IVC, which shows continuous
cultural precursors since at least the seventh millennium BC. To understand
the decline, we believe it is necessary to investigate the rise of the IVC,
i.e., the establishment of agriculture and livestock, dense populations and
technological developments 7000–3000 BC. Although much archaeological
information is available, our capability to investigate the system is hindered
by poorly resolved chronology, and by a lack of field work in the intermediate
areas between the Indus valley and Mesopotamia. We thus employ a complementary
numerical simulation to develop a consistent picture of technology,
agropastoralism and population developments in the IVC domain. Results from
this Global Land Use and technological Evolution Simulator show that there is
(1) fair agreement between the simulated timing of the agricultural transition
and radiocarbon dates from early agricultural sites, but the transition is
simulated first in India then Pakistan; (2) an independent agropastoralism
developing on the Indian subcontinent; and (3) a positive relationship between
archeological artifact richness and simulated population density which remains
to be quantified.
LEMMEN AND KHAN SIMULATION OF INDUS VALLEY NEOLITHIC C. Lemmen, Institute of
Coastal Research, Helmholtz-Zentrum Geesthacht, Max-Planck-Str. 1, 21205
Geesthacht, Germany, (carsten.lemmen@hzg.de) A. Khan, Institute of Coastal
Research, Helmholtz-Zentrum Geesthacht, Max-Planck-Str. 1, 21205 Geesthacht,
Germany, (aurangzeb.khan@hzg.de)
## 1 Introduction
The Indus Valley Civilization (IVC), often termed Harappan civilization after
its major type site, flourished along the banks of the river Indus and its
tributaries, including the adjacent coastal areas, between the fourth and
second millennium BC. The IVC is characterized by urban centers, bronze
technology and seals, trade networks with Mesopotamia and Arabia, and an as
yet undeciphered writing system (Shaffer, 1992; Possehl, 1998; Kenoyer, 2008).
Much research has been devoted to the disintegration of the IVC during the
second millennium BC, and much less to the emergence of this great
civilization which shows continuous cultural precursors at least since the
seventh millennium BC, such that some authors prefer to speak of the Indus
valley cultural tradition (e.g., Kenoyer, 2006).
The Indus valley is one of the two great river basins on the Indian
subcontinent, separated from the Ganges valley to the east by the Aravalli
mountain range and the Thar desert (Figure 1b). The region has been occupied
by anatomically modern humans for at least 34,000, but possibly $>$80,000
years (Petraglia et al., 2010). The late Paleolithic and the Mesolithic are
visible in stone and blade industries; microlith blades indicate composite
tools, differentiated occupation levels and a broad-spectrum diet of foraging
people from 26,000 years before present (a BP) until at least the second
millennium BC, well into the Bronze age (Petraglia et al., 2009).
Neolithic subsistence is based on a combination of barley and wheat dominated
agriculture and herding of sheep, goats, and cattle. Domesticates are of mixed
origin: barley and wheat have no local wild progenitor, but zebu cattle bears
a clear signature of local domestication (Chen et al., 2010; Fuller, 2011a).
The beginning of the Neolithic on the Indian subcontinent is regionally
diverse and ranges from 6500 BC in Baluchistan (Jarrige, 1995) to 3000 BC or
later on the central Indian plateau and in the Himalaya mountain range
(Samuel, 2000; Boivin et al., 2008). Our focus here is on the period which
spans from the Neolithic to the beginning of the Bronze age of the IVC
(7000–3200 BC), i.e., on the early food production and regionalization eras,
or the Mehrgarh phase and parts of the early Harappan phase according to the
different chronologies (Sharma et al., 1980; Fuller, 2006; Kenoyer, 2008).
Figure 1: Temporal and spatial domain for this study: a) Neolithic chronology
of Baluchistan, the Indus valley, and Gujarat based on Possehl (2002) with
updates from Fuller (2006). Relative resource shortage is based on palaeoproxy
evidence from Lemmen and Wirtz (2012) and a transient Holocene climate
simulation by Brovkin (2002). b-e) Geography, topography and site distribution
of artifacts (Law, 2008) typologically associated with the Neolithic.
Possehl (2002) separated these phases and eras of the pre-Harappan Neolithic
into two stages (Figure 1a): Early Neolithic and Developed Neolithic. The
Early Neolithic is divided in the Kili Ghul Mohammad (KGM) and the Burj
Basket-Marked (BBM) periods. Sites attributed to these periods are found in
the western borderlands of the greater Indus region and cluster in the hills
and piedmont of Baluchistan (Figure 1 b1–b2). During the BBM, soft pottery and
handmade pottery are introduced. Deposits of trash with burnt pebbles, ash,
animal bones, bone tools, hammer stones, polishers colored with red ochre, and
a large collection of blades, cores, and flint debris have been discovered,
and point to leather making, basket making, or weaving. Domesticated wheat,
barley and cotton were used, pulses may have been present; the use of copper
is visible in the form of arrowheads and beads (Moulherat et al., 2002;
Fuller, 2011a).
The Developed Neolithic is characterized by improved pottery, developed
agropastoral communities, growth, continuity, and geographical expansion.
During its first period (here defined as both Togau and Sheri Khan Tarakai,
SKT, Figure 1 b3), the settlement patterns show growth of village life;
technological innovations include the use of gold, the manufacture of
compartmented seals, glazed steatite, and beads. An eastward expansion is
visible in sites emerging along the ancient Ghaggar Hakra river. The second
period of the Developed Neolithic (Figure 1 b4) comprises several cultural
complexes. In the Kechi Beg complex in Baluchistan wheat replaces barley; an
irrigation canal near Mehrgarh may date to this period (Kenoyer, 2008). On the
Punjab plain, a change in the assemblage of pottery types characterizes the
Hakra Ware complex: Microlithic tools are abundant, and sites represented by a
light scatter of pottery without a buildup of a midden point to seasonal
occupations (Possehl, 2002). At the same time, the Anarta complex appears in
Gujarat with distinct pottery styles.
Our capability to investigate the system more closely and consistently in the
regional context is hindered by the poorly resolved chronology, and by a lack
of field work in the intermediate areas between the Indus valley and
Mesopotamia. We thus employ a complementary numerical simulation to the IVC
domain. Ackland et al. (2007) simulated the Neolithic transition in the IVC
with an advancing frontier spreading from Southwest Asia, and a competition
between migrating original farmers and resident converts. After 5000
simulation years (9000–4000 BC), their simulated spread of agriculture into
the Indian subcontinent exhibits a demarcation line along the Indus river,
separating original farmers to the west from converts to the east.
A local South Asian agricultural center was considered in the numerical model
by Patterson et al. (2010). They assumed a background hunting-gathering
population, and initial farmer populations near Mehrgarh and at the eastern
edge of their model domain. After 2000 simulation years, the population in the
Indus valley is dominated by expanding original farmers, a lesser share of
converts, and a tiny fraction of remaining hunter-gatherers.
We go beyond the biophysical approaches by Ackland et al. (2007) or Patterson
et al. (2010); in our Global Land Use and Technological Evolution Simulator
(GLUES, e.g., Lemmen et al., 2011), we include socio-technological innovation
in addition to migration, population growth and subsistence change to develop
a consistent picture of the dynamics of technology, agropastoralism and
population developments during the Mesolithic–Neolithic transition. This
simulated data is subsequently compared to a representative archaeological
data set (Law, 2008), where the chronology of Neolithic sites is used to
detect the regional transition from foraging to farming, and where the site
occurrence frequency is used to detect the Neolithic demographic transition.
## 2 Data and model description
### 2.1 Global Land Use and technological Evolution Simulator
GLUES is a socio-technological model which hindcasts technological evolution,
potential population density, and the timing of the transition to
agropastoralism for 685 world regions based on the geoenvironmental and
cultural contexts, and innovation and adaptation of regional populations. Each
local population in an approximately country-size region utilizes its natural
resources and interacts with its geographical neighbors via trade and
migration. The full model is described by Wirtz and Lemmen (2003), with
refinements of geography (Lemmen, 2009), migration and knowledge loss (Lemmen
and Wirtz, 2010), and climate events (Lemmen and Wirtz, 2012); it has been
validated against archaeological data by Lemmen et al. (2011). Key concepts
are summarized below:
Local societies are defined by their population density ($P$) and by three
characteristic traits: (1) technology ($T$), which describes efficiency of
food procurement; (2) share of agropastoral activities ($Q$) as the allocation
pattern of manpower to farming and herding as opposed to foraging; and (3)
economic diversity ($N$), a technology-related trait which represents the
different subsistence styles of a community. The temporal change of traits
follows the direction of increased benefit for success (i.e. growth) of its
associated population with a speed related to the trait’s variance (Fisher,
1930; Kisdi and Geritz, 2010). Population growth rate ($r$) depends on
resource availability ($E$), subsistence intensity, overexploitation of
resources, administrative overhead and health improvements; the latter four
terms are functions of $P,T,N$, and $Q$, while the resource availability $E$
is provided externally. $E$ is estimated from the net primary productivity,
which is in turn based on Climber-2 (Claussen et al., 1999) transient
temperature and precipitation anomalies from the IIASA climatological data
base (International Institute for Applied Systems Analysis, Leemans and
Cramer, 1991).
Abrupt climate deteriorations modulate $E$; they are incorporated by using the
extreme event anomalies found in 124 globally distributed palaeoclimate
proxies collected and analyzed by Wirtz et al. (2010); the estimation of
resource failure from proxy climate is described by Lemmen and Wirtz (2012).
Figure 1a shows the average relative decline of $E$ in the Indus valley
region; these anomalies are derived from regional abrupt changes found in
oxygen isotope anomalies from Kyrgyzstan (Ricketts et al., 2001), Tibet
(Fontes et al., 1996), the Arabian Sea (Doose-Rolinski et al., 2001; Gupta et
al., 2003), and Oman (Fleitmann et al., 2007).
We set up the eight global model parameters and initial values such that the
simulation is able to hindcast an accurate timing and location of the early
farming centers in Southwest Asia, northern China, and Mesoamerica (Smith,
1998), and a reasonable global pattern of the subsequent Neolithic transition.
The simulation is started at 9500 sim BC (simulation years BC; equivalent to
calendar years BC, but emphasizes the artificial model time scale, Lemmen et
al. 2011). All of the 685 biogeographically defined regions, including 32
around the Indus valley, are initially set with farming activity at $4\%$,
$1/4$ established agropastoral economies and unit technology; this setup
represents a low density Mesolithic population relying on a broad spectrum
foraging lifestyle with low unintentional farming activity. The regional
simulated transition to agropastoralism is recorded when $Q$ exceeds $0.5$.
### 2.2 Artifact database
We used the Indus Google Earth Gazetteer (version August 2008) compiled by Law
(2008) for the geolocation of artifacts relating to the Indus valley. From
this database of 21687 metal and stone artifacts, we use here the cultural
attribution and location of those 2028 artifacts and 368 sites that are in the
spatial and temporal domain of our study. For KGM, 18 sites contribute 90
artifacts to the data base; for BBM, there are 159 artifacts at 32 sites. We
combined the sites attributed to the Togau period with sites from the SKT
(both with starting date 4300 BC, Fuller 2006) and obtain 317 artifacts from
65 sites.
The subsequent period is represented by the Hakra Ware, Kechi Beg, and Anarta
complexes. Combined, there are 1250 artifacts (396, 670, and 184,
respectively) from 253 sites (99, 112, and 42, respectively). While the main
chronology is based on Possehl (2002), sites from the Anarta complex were
included in this period based on Fuller (2006). To our knowledge, this
gazetteer is the most representative data set of lithic and metal artifacts of
the Indus valley tradition. It has been used to investigate the trade and
distribution networks of the IVC by its author (Law, 2011).
In archaeological or palaeoenvironmental archives, the decrease of the number
of dates further back in time that is caused by gradual destruction of the
archive by post-depositional processes results in a taphonomic bias (Surovell
et al., 2009). We applied to the artifact density data a bias correction
derived from the comparison of volcanic eruption layers in the GISP2
(Greenland, no taphonomic bias) ice core with radiocarbon dated ash deposits;
the correction factor $c$ depends on time $t$ (in a BP): $c(t)=5.73\cdot
10^{6}(2175.4+t)^{-1.39}$ (Surovell et al., 2009). This method was
successfully validated with Canadian coin mintage data compared to a coin
collection and applied to North American radiocarbon dates by Peros et al.
(2010).
## 3 Results and Discussion
Figure 2: a) Timing of the Mesolithic–Neolithic transition from a GLUES
simulation (background color, simulation years BC) and inferred from the
presence of Neolithic sites (foreground triangles, calendar years BC) in the
greater Indus valley. b) Recovered artifact number in relation to simulated
population size. Histogram of artifact density in four pre-Harappan periods of
the Indus valley and Baluchistan based on Law’s (2008) Indus Google Earth
Gazetteer, and corrected for taphonomic bias (bars, annotated with $n$ x $c$,
where $n$ is the raw number of artifacts and $c$ the taphonomic correction
factor); GLUES-simulated population size in all areas within 200 km from where
Neolithic pre-Harappan artifacts were discovered (solid line).
The simulated transition to predominantly agropastoralism-based subsistence
occurs in the spatial domain of the Indus valley and surrounding areas between
6300 and 3800 sim BC (Figure 2a). There is a general east-west trend in the
simulated transition dates. Contemporary with the KGM, the earliest
transitions are simulated in India (before 5600 sim BC), including Rajasthan,
Gujarat, and the Ganges valley, and early transitions (before 5000 sim BC) in
southern Sindh and northern Pakistan, including parts of the Punjab and
Kashmir. During this first period, 15 out of 32 simulation regions undergo the
transition to agropastoralism. Most of the remaining regions (Pakistan,
eastern Afghanistan, eastern Iran and Makran coast) transition in the BBM
period before 4300 sim BC. Of these, agropastoralism is simulated first in the
Punjab and northern Baluchistan (before 4700 sim BC), then in northern Sindh
(before 4500 sim BC). The simulated transition occurs latest, during the Togau
period, in the eastern model domain, from central Afghanistan to the Pakistan-
Afghanistan-Iran border triangle (4300–3800 sim BC).
A detailed and quantitative comparison is limited foremost by the inherent
timing uncertainties in the typologically dated artifacts. This is closely
related to the secondary comparison difficulty, which is created by the lack
of temporal continuity of samples from the Mesolithic to the Neolithic. While
we have shown how these issues can be addressed in the context of plenty and
precise dates for Europe (Lemmen et al., 2011), the comparison with dates from
the Indian subcontinent can only be qualitative at this moment (see also
Patterson et al. 2010, who make a qualitative comparison not with single
sites, but with approximate timing contours).
In addition, the coexistence of Mesolithic and Neolithic subsistence in the
archaeological record complicates the comparison between model and data,
because the degree of both subsistence styles cannot be quantified with the
current data. This coexistence is evident in the long-term continued use of
(Mesolithic) microlithic tools, transhumance life style, i.e., the seasonal
back-and-forth movement of herders with their livestock, and the continuation
of foraging practice well into the Neolithic (Mughal, 1990; Fuller, 2011a), it
is not resolved in the model, where agropastoral life style is usually
dominant within less than 500 years after its emergence (Lemmen and Wirtz,
2012).
The model simulates first agropastoralism in northern Baluchistan and southern
Sindh in the KGM period consistent with earliest sites in these areas. For the
major part of Baluchistan, including the Mehrgarh site, the model places the
transition in the BBM period, which is later than the earliest site dates, but
agrees with dates from many other locations in Baluchistan. The most obvious
mismatch is the early simulated (KGM and BBM) transition along the Indus, the
Ghaggar-Hakra and Punjab rivers, where site dates are absent until the Kechi
Beg and Hakra periods.
While evidence for early archaeological sites may have been lost in the
geomorphologically dynamic Indus river floodplain, the lack of early sites in
Rajasthan is striking, considering that the model sees a biogeographically
favorable environment and cultural setting for an early Neolithic: one should
ask whether an agricultural center on the Indo-Gangetic divide is undetected
so far. First evidence for northern Indian agriculture could come from further
east, from Lahuredawa in the middle Ganges plains, where evidence for
domesticated rice appears as early as 6500 BC (Tewari et al., 2008) (but see
Fuller 2011a for a debate on the rice morphology). Less favored by the model
are the valleys along the Indo-Iranian plateau, where broad subsistence
possibilities are seen as one precondition for the rise of the IVC and where
agropastoralism arose before 6500 BC (Jarrige, 1995). The model might
underestimate the potential for agropastoralism in this area because of its
coarse spatial scale.
From a regional perspective, the simulated east-west direction of
agropastoralism contradicts the eastward trend seen in the site data. The
westward direction had been suggested in earlier literature on the Indus
valley tradition: Wheeler (1959) wrote of “movement of the Neolithic from
Burma if not behind.” In their simulation of the spread of agriculture into
the Indian continent, Patterson et al. (2010) assume such a Southeast Asian
center in addition the Mehrgarh location. While Wheeler’s (1959) earliest
dates (1000 BC) have been pushed further back by more recent datings, this
westward view cannot be upheld from either artifacts (Law, 2008) or the
description of sites in central India (3000 BC, Boivin et al. 2008), the Indus
plain (4000 BC, Mughal 1990), northern India (5000 BC, Sharma et al. 1980), or
Baluchistan (6500 BC, Jarrige 1995); all of these demonstrate the opposite
east- and then southward trend.
The simulated earliest agropastoralism in the Indus valley region is
independent of the two prescribed early centers of agropastoralism in Eurasia,
i.e., China and Southwest Asia; towards the end of KGM, however, a continuous
band of early agropastoralism ranges from the northern Indian subcontinent
through northern Southeast Asia to China (not shown). This South and East
Asian agricultural area connects to the Southwest Asian center during the BBM
period. The simulated independent emergence of South Asian agropastoralism
corresponds well to evidence for independent domestication of cattle and rice
(Chen et al., 2010; Fuller, 2011b). Though not completely detached from
Southwest Asian domesticates assemblage in the data, this separation is
visible in the delayed simulated transition to agropastoralism in the area
between the Indo-Gangetic and the Southwest Asian founder centers.
Building on the earlier idea of Sanders et al. (1979), we relate the number of
artifacts in the database (corrected for taphonomic bias) to population
estimates from our simulation (Figure 2b). We summed the total simulated
population in each half-degree model grid cell within 200 km distance to the
find sites and obtain a population trajectory which shows an increase from
$0.2$ million at 6000 sim BC to 2.5 million at 3500 sim BC; this corresponds
to a population density of $0.1$ and $1.3$ km-2, respectively. At 3500 sim BC
our estimate is higher than the one by Patterson et al. (2010, $0.2$ km-2), as
should be expected from our calculation from the surroundings (within 200 km)
of mostly settlement sites. To first approximation, the increases in simulated
population size and in reconstructed artifact density correspond: the model
correctly represents the emergence of dense settlements. Statistically,
however, no meaningful relationship should be derived from only four different
periods.
In summary, there is fair qualitative agreement between the reconstructed and
the simulated transition to agropastoralism, both in the spatio-temporal
distribution of agropastoral sites and simulation regions, as well as the
overall artifact and population density. Important lessons from the model are
that it corroborates an independent agropastoral center, and that it suggests
potential for earlier Indian agropastoralism. It will be left to further
studies to quantify the agreement and to elaborate on the reasons, both in
archaeology and in the simulation, of discrepancies.
## 4 Conclusion
We presented a novel numerical simulation of the pre-Harappan Neolithic of the
Indus valley tradition in the context of representative radiocarbon dates from
material culture. Simulated population size is qualitatively reflected in
artifact frequency in four pre-Harappan periods. Within the uncertainties of
the coarse chronology, the model predicts the spatio-temporal structure of the
Neolithic transition of this area fairly well. Our simulation points to a
possible earlier Neolithic in northern India than seen in the data, and it
corroborates an independent South Asian Neolithic. For quantitative model–data
comparisons and clarification of the connection to Southwest Asia, better
chronologic control of the pre-Harappan material is needed as well as more
evidence from the intervening Iranian plateau.
## Supplementary material
The simulated data have been permanently archived on and are freely accessible
from PANGAEA (Data Publisher for Earth & Environmental Science,
http://www.pangaea.de) as a netCDF dataset with reference “Lemmen, C. and
Khan, A. (2012): Simulated transition to agropastoralism in the Indus valley
7500–3000 BC. Dataset #779706, doi:XXXX”. GLUES is open source software and
can be downloaded from http://glues.sourceforge.net.
###### Acknowledgements.
We thank R.W. Williams for publicly providing his extensive data set for
further analysis; we thank three anonymous reviewers and the editor for their
critical and helpful comments on earlier versions of this manuscript. C.L.
acknowledges financial support from the DFG priority program Interdynamik (SPP
1266), A.K. acknowledges financial support from the Higher Education
Commission of Pakistan and the PACES program of the Helmholtz Gemeinschaft.
This paper benefitted greatly from ideas presented and discussed at the AGU
Chapman Conference on Climates, Past Landscapes, and Civilizations, held in
Santa Fe, USA, March 2011. Maps were prepared with Natural Earth free vector
and raster map data available from http://naturalearthdata.com.
## References
* Ackland et al. (2007) Ackland, G. J., M. Signitzer, K. Stratford, and M. H. Cohen, Cultural hitchhiking on the wave of advance of beneficial technologies., Proc. Natl. Acad. Sci. U.S.A., 104(21), 8714–9, 10.1073/pnas.0702469104, 2007.
* Boivin et al. (2008) Boivin, N., D. Q. Fuller, R. Korisettar, and M. D. Petraglia, First Farmers in south India: the role of internal processes and external influences in the emergence and transformation of south India’s earliest settled societies, Pragdhara, 18, 179–200, 2008.
* Brovkin (2002) Brovkin, V., Carbon cycle, vegetation, and climate dynamics in the Holocene: Experiments with the CLIMBER-2 model, Global Biogeochem. Cycles, 16(4), 10.1029/2001GB001662, 2002.
* Chen et al. (2010) Chen, S., et al., Zebu cattle are an exclusive legacy of the South Asia Neolithic, Molec. Biol. Evolut., 27(1), 1–6, 10.1093/molbev/msp213, 2010.
* Claussen et al. (1999) Claussen, M., C. Kubatzki, V. Brovkin, A. Ganopolski, P. Hoelzmann, and H.-J. Pachur, Simulation of an abrupt change in Saharan vegetation in the mid-Holocene, Geophys. Res. Lett., 26, 2037–2040, 1999.
* Doose-Rolinski et al. (2001) Doose-Rolinski, H., U. Rogalla, and G. Scheeder, High resolution temperature and evaporation changes during the late Holocene in the northeastern Arabian Sea, Paleoceanogr., 16(4), 358–367, 2001.
* Fisher (1930) Fisher, R. A., The Genetical Theory of Natural Selection, Dover, New York, 1930.
* Fleitmann et al. (2007) Fleitmann, D., et al., Holocene ITCZ and Indian monsoon dynamics recorded in stalagmites from Oman and Yemen (Socotra), Quatern. Sci. Rev., 26(1-2), 170–188, 10.1016/j.quascirev.2006.04.012, 2007.
* Fontes et al. (1996) Fontes, J.-C., F. Gasse, and E. Gibert, Holocene environmental changes in Lake Bangong basin (Western Tibet). Part 1: Chronology and stable isotopes of carbonates of a Holocene lacustrine core, Palaeogeogr. Palaeoclimatol. Palaeoecol., 120, 25–47, 1996.
* Fuller (2006) Fuller, D. Q., Agricultural Origins and Frontiers in South Asia: A Working Synthesis, J. World Prehist., 20(1), 1–86, 10.1007/s10963-006-9006-8, 2006.
* Fuller (2011a) Fuller, D. Q., Finding Plant Domestication in the Indian Subcontinent, Curr. Anthropol., 52(S4), S347–S362, 10.1086/658900, 2011a.
* Fuller (2011b) Fuller, D. Q., Pathways to Asian Civilizations: Tracing the Origins and Spread of Rice and Rice Cultures, Rice, 4(3-4), 78–92, 10.1007/s12284-011-9078-7, 2011b.
* Gupta et al. (2003) Gupta, A., D. Anderson, and J. Overpeck, Abrupt changes in the Asian southwest monsoon during the Holocene and their links to the North Atlantic ocean, Nature, 421, 354–356, 2003.
* Jarrige (1995) Jarrige, C., Mehrgarh: Field Reports 1974-1985, from Neolithic Times to the Indus Civilization, Dept. of Culture and Tourism, Govt. of Sindh; French Ministry of Foreign Affairs, 1995.
* Kenoyer (2006) Kenoyer, J. M., Cultures and Societies of the Indus Tradition, in Historical Roots in the Making of ‘The Aryan’, edited by R. Thapar, 1, pp. 10–19, National Book Trust, New Delhi, 2006.
* Kenoyer (2008) Kenoyer, J. M., The Indus Civilization, in Encyclopedia of Archaeology, edited by D. M. Pearsall, pp. 715–733, Academic Press, New York, 2008.
* Kisdi and Geritz (2010) Kisdi, E., and S. A. H. Geritz, Adaptive dynamics: a framework to model evolution in the ecological theatre, J. Math. Biol., 61(1), 165–9, 10.1007/s00285-009-0300-9, 2010.
* Law (2008) Law, R. W., Indus Google Earth Gazetteer, https://mywebspace.wisc.edu/rwlaw/, Presented at the Indus Archaeological Technical Training Workshop, 17–20 March 2007, Mahara Sayajirao University, Baroda, India, 2008.
* Law (2011) Law, R. W., Inter-regional interaction and urbanism in the ancient Indus valley, Linguistics, Archaeology, and the Human Past, vol. 11, Research Institute for Humanity and Nature, Kyoto, Japan, 2011.
* Leemans and Cramer (1991) Leemans, R., and W. P. Cramer, The IIASA database for mean monthly values of temperature, precipitation and cloudiness of a global terrestrial grid., Research report, International Institute of Applied Systems Analyses, Laxenburg, 1991.
* Lemmen (2009) Lemmen, C., World distribution of land cover changes during Pre- and Protohistoric Times and estimation of induced carbon releases, Géomorphol. : relief, processus, environ., (4), 303–312, 2009.
* Lemmen and Wirtz (2010) Lemmen, C., and K. W. Wirtz, Socio-technological revolutions and migration waves re-examining early world history with a mathematical model, in The Spread of the Neolithic to Central Europe, RGZM Tagungen, vol. 4, edited by D. Gronenborn and J. Petrasch, pp. 25–38, Römisch-Germanisches Zentralmuseum, Mainz, Germany, 2010.
* Lemmen and Wirtz (2012) Lemmen, C., and K. W. Wirtz, On the sensitivity of the simulated European Neolithic transition to climate extremes, J. Archaeol. Sci., submitted, 2012.
* Lemmen et al. (2011) Lemmen, C., D. Gronenborn, and K. W. Wirtz, A simulation of the Neolithic transition in Western Eurasia, J. Archaeol. Sci., 38(12), 3459–3470, 10.1016/j.jas.2011.08.008, 2011.
* Moulherat et al. (2002) Moulherat, C., M. Tengberg, J.-F. Haquet, and B. Mille, First Evidence of Cotton at Neolithic Mehrgarh, Pakistan: Analysis of Mineralized Fibres from a Copper Bead, J. Archaeol. Sci., 29(12), 1393–1401, 2002.
* Mughal (1990) Mughal, M., The decline of the Indus civilization and the Late Harappan period in the Indus valley, Lahore Museum Bulletin, 1990.
* Patterson et al. (2010) Patterson, M. A., G. R. Sarson, H. Sarson, and A. Shukurov, Modelling the Neolithic transition in a heterogeneous environment, J. Archaeol. Sci., 37(11), 2929–2937, 10.1016/j.jas.2010.07.003, 2010.
* Peros et al. (2010) Peros, M. C., S. E. Munoz, K. Gajewski, and A. E. Viau, Prehistoric demography of North America inferred from radiocarbon data, J. Archaeol. Sci., 37(3), 656–664, 10.1016/j.jas.2009.10.029, 2010.
* Petraglia et al. (2010) Petraglia, M. D., M. Haslam, D. Q. Fuller, N. Boivin, and C. Clarkson, Out of Africa: new hypotheses and evidence for the dispersal of Homo sapiens along the Indian Ocean rim, Ann. Human Biol., 37(3), 288–311, 10.3109/03014461003639249, 2010.
* Petraglia et al. (2009) Petraglia, M. D., et al., Population increase and environmental deterioration correspond with microlithic innovations in South Asia ca. 35,000 years ago, South Asia, pp. 1–6, 2009.
* Possehl (1998) Possehl, G. L., Sociocultural Complexity Without the State: The Indus Civilization, The Archaic States, pp. 261–292, 1998.
* Possehl (2002) Possehl, G. L., The Indus civilization: a contemporary perspective, 276 pp., AltaMira Press, Walnut Creek, CA, 2002.
* Ricketts et al. (2001) Ricketts, R. D., T. C. Johnson, E. T. Brown, K. A. Rasmussen, and V. V. Romanovsky, The Holocene paleolimnology of Lake Issyk-Kul, Kyrgyzstan: trace element and stable isotope composition of ostracodes, Palaeogeogr. Palaeoclimatol. Palaeoecol., 176, 207–227, 2001.
* Samuel (2000) Samuel, G., The Indus Valley civilization and early Tibet, Senri Ethnological Reports, vol. 15, 651–670 pp., National Museum of Ethnology, Osaka, Japan, 2000.
* Sanders et al. (1979) Sanders, W., J. Parsons, and R. Santley, The basin of Mexico, Academic Press, New York, USA, 1979.
* Shaffer (1992) Shaffer, J., The indus valley, baluchistan and helmand traditions: Neolithic through bronze age, in Chronologies in Old World archaeology, edited by R. Ehrich, 3 ed., pp. 441–464, University of Chicago Press, Chicago, USA, 1992\.
* Sharma et al. (1980) Sharma, G. R., J. D. Clark, B. K. Thapar, and D. Mandal (Eds.), History to Prehistory : Archaeology of the Ganga Valley and the Vindhyas, University of Allahabad, Department of Ancient History, Culture, and Archaeology, Allahabad, India, 1980.
* Smith (1998) Smith, B. D., The Emergence of Agriculture, 231 pp., Freeman and Company Publishers, New York, 1998.
* Surovell et al. (2009) Surovell, T. A., J. Byrd Finley, G. M. Smith, P. J. Brantingham, and R. Kelly, Correcting temporal frequency distributions for taphonomic bias, J. Archaeol. Sci., 36(8), 1715–1724, 10.1016/j.jas.2009.03.029, 2009.
* Tewari et al. (2008) Tewari, R., R. Srivastava, K. K. Sing, I. B. Sing, and K. Saraswat, Early farming at Lahuradewa, Pragdhara, 18, 347–373, 2008.
* Wheeler (1959) Wheeler, M., Early India and Pakistan: To Ashoka, 190 pp., Frederick A. Praeger, New York, USA, 1959.
* Wirtz and Lemmen (2003) Wirtz, K. W., and C. Lemmen, A global dynamic model for the Neolithic transition, Clim. Change, 59(3), 333–367, 2003.
* Wirtz et al. (2010) Wirtz, K. W., G. Lohmann, K. Bernhardt, and C. Lemmen, Mid-Holocene regional reorganization of climate variability: Analyses of proxy data in the frequency domain, Palaeogeogr. Palaeoclimatol. Palaeoecol., 298(3-4), 189–200, 10.1016/j.palaeo.2010.09.019, 2010.
|
arxiv-papers
| 2011-10-05T20:04:10 |
2024-09-04T02:49:22.837741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carsten Lemmen and Aurangzeb Khan",
"submitter": "Carsten Lemmen",
"url": "https://arxiv.org/abs/1110.1091"
}
|
1110.1116
|
# Characteristic Polynomial of Supersingular Abelian Varieties over Finite
Fields
Vijaykumar Singh, Gary McGuire, Alexey Zaytsev
###### Abstract.
In this article, we give a complete description of the characteristic
polynomials of supersingular abelian varieties over finite fields. We list
them for the dimensions upto $7$.
## 1\. Introduction
Supersingular abelian varieties have applications in cryptography and coding
theory and related areas. Identity based ecryption and computation of weights
of some Reed Muller codes are some of them. An important (isogeny) invariant
which carries most of the information about supersingular curves is the
characteristic polynomial of the Frobenius endomorphism. Here, we give a
complete description of the characteristic polynomials of supersingular
abelian varieties over finite fields. We list them for the dimensions upto
$7$.
## 2\. Background
###### Definition 2.1.
Let $A$ be an abelian variety of dimension $g$ over $\mathbb{F}_{q}$, where
$q=p^{n}$. For $l\neq p$, _the characteristic polynomial_ of Frobenius
endomorphism $\pi_{A}$ is defined as
$P_{A}(X):=\det(\pi_{A}-XId|_{V_{l}(A)}),$
The above definition is independent of choice of $l$. The coefficients of
$P_{A}(X)$ are in $\mathbb{Z}$. Moreover, $P_{A}(X)$ can be represented as
$P_{A}(X)=X^{2g}+a_{1}X^{2g-1}+\cdots+a_{i}X^{2g-i}+\cdots+a_{g}X^{g}+qa_{g-1}X^{g-1}+\cdots+q^{g-i}a_{i}X^{i}+\cdots+q^{g}.$
###### Definition 2.2.
A _Weil- $q$-number_ $\pi$ is defined to be an algebraic number such that, for
every embedding $\sigma:\mathbb{Q}[\pi]\hookrightarrow\mathbb{C}$,
$|\sigma\pi|=q^{\frac{1}{2}}$ holds.
For $q=p^{n}$, the real Weil polynomials are $X^{2}-q$ when $n$ is odd and
$X-\sqrt{q}$ when $n$ is even. Otherwise, the set of roots of $P_{A}(X)$ has
the form
$\\{\omega_{1},\overline{\omega_{1}},\dots,\omega_{g},\overline{\omega_{g}}\\}$,
where the $\omega_{i}$’s are Weil-$q$-numbers. A monic polynomial with integer
coefficients which satisfies this condition is called a _Weil polynomial_.
Thus, every Weil polynomial of degree $2g$ has the form above for certain
integers $a_{i}\in\mathbb{Z}$. The converse is false; indeed, since the
absolute value of the roots of $P(X)$ is prescribed (equal to $\sqrt{q}$), its
coefficients have to be bounded.
An abelian variety $A$ is _$k$ -simple_ if it is not isogenous to a product of
abelian varieties of lower dimensions over $k$. In that case, $P_{A}(X)$ is
either irreducible over $\mathbb{Z}$ or $P_{A}(X)=h(X)^{e}$, where
$h(X)\in\mathbb{Z}[X]$ is an irreducible, monic polynomial over $\mathbb{Z}$;
see Milne and Waterhouse in [1]. We have the following result from Tate [2].
###### Theorem 2.3.
Let $A$ and $B$ be abelian varieties defined over $\mathbb{F}_{q}$. Then $A$
is $\mathbb{F}_{q}$-isogenous to an abelian subvariety of $B$ if and only if
$P_{A}(X)$ divides $P_{B}(X)$ in $\mathbb{Q}[X]$. In particular,
$P_{A}(X)=P_{B}(X)$ if and only if $A$ and $B$ are $\mathbb{F}_{q}$-isogenous.
Let $W(q)$ be the set of Weil-$q$-number in $\mathbb{C}$. We say that two
elements $\pi$ and $\pi^{{}^{\prime}}$ are _conjugate_ , if $\pi$ and
$\pi^{{}^{\prime}}$ have the same minimum polynomial (called _Weil
polynomial_) over $\mathbb{Q}$.
The conjugacy class of $\pi_{A}$ depends only on the isogeny class of $A$,
more precisely, we have the following theorem from [3].
###### Theorem 2.4.
_(Honda-Tate theorem)_ The map $A\rightarrow\pi_{A}$ defines a bijection
$\\{simple~{}abelian~{}varieties/\mathbb{F}_{q}\\}/(isogeny)\mapsto
W(q)/(conjugacy).$
In otherwords, given an irreducible Weil polynomial $P(X)$, there exist an
unique abelian variety upto isogeny and $e$ a natural number such that
$P_{A}(X)=P(X)^{e}$.
A Weil-$q$ number $\pi$ is called a _supersingular Weil- $q$-number_ if
$\pi=\sqrt{q}\zeta$, where $\zeta$ is some root of unity.
###### Definition 2.5.
An abelian variety $A$ over $k$ is called _supersingular_ if any one of the
following (equivalent) conditions holds:(see [4, Theorem 4.2])
1. (1)
the eigenvalues of the Frobenius $\pi_{A}$ are supersingular Weil-$q$-numbers;
2. (2)
the Newton polygon of $A$ is a straight line of slope $1/2$;
3. (3)
$A$ is $\overline{k}$-isogenous to a power of a supersingular elliptic curve.
We have the following useful theorem from [3].
###### Theorem 2.6.
Let A be a simple abelian variety over $k=\mathbb{F}_{q}$, then
1. (1)
$End_{k}(A)\otimes\mathbb{Q}$ is a division algebra with center
$\mathbb{Q}(\pi_{A})$ and
$2dimA=[End_{k}(A)\otimes\mathbb{Q}:\mathbb{Q}(\pi_{A})]^{\frac{1}{2}}[\mathbb{Q}(\pi_{A}):\mathbb{Q}].$
2. (2)
The division algebra $End_{k}(A)\otimes\mathbb{Q}$ over $\mathbb{Q}(\pi_{A})$
has the following splitting behaviour
1. (a)
it splits at each divisor $\mathfrak{l}$ of $l$ in $\mathbb{Q}(\pi_{A})$, if
$l\neq p$,
2. (b)
the invariants at the divisors $\mathfrak{p}$ of $p$ in $\mathbb{Q}(\pi_{A})$
can be evaluated with
$inv_{\mathfrak{p}}(End_{k}(A)\otimes\mathbb{Q})\equiv\frac{v_{\mathfrak{p}}(\pi_{A})}{v_{\mathfrak{p}}(q)}[\mathbb{Q}(\pi_{A})_{\mathfrak{p}}:\mathbb{Q}_{p}]\mod\mathbb{Z},$
where $\mathbb{Q}(\pi_{A})_{\mathfrak{p}}$ denotes the completion of
$\mathbb{Q}(\pi_{A})$ at $\mathfrak{p}$ and $v_{\mathfrak{p}}$ denotes the
$\mathfrak{p}$-adic valuation.
3. (c)
it does not split at the real places of $\mathbb{Q}(\pi_{A})$.
If $A$ is a supersingular abelian variety, then $\pi_{A}=\sqrt{q}\zeta$, where
$\zeta$ is some root of unity. If $A$ is also simple, then it follows from
above theorem that
$inv_{\mathfrak{p}}(End_{k}(A)\otimes\mathbb{Q})\equiv\frac{1}{2}e_{i}f_{i}\mod\mathbb{Z}$,
where $e_{i}$ is the ramification index of $\mathfrak{p}$ over $p$ and $f_{i}$
is the residue class degree.
Also, $\pi\in\overline{\mathbb{Q}}$ be the supersingular Weil-$q$-number and
$P(X)$ be the corresponding minimal Weil polynomial. The invariants of
$End_{k}(A)\otimes\mathbb{Q}$ lie in $\mathbb{Q}/\mathbb{Z}$. They can be
evaluated from the minimal polynomial $P(X)$ of $\pi_{A}$ as follows. The only
real Weil numbers are $q^{1/2}$ and -$q^{1/2}$ , so there are hardly any real
places of $\mathbb{Q}(\pi_{A})$. We consider the polynomial $P(X)$ in
$\mathbb{Q}_{p}[X]$, i.e., over the $p$-adic numbers. Let
$P(X)=\displaystyle\prod_{i}f_{i}(X)$
be the decomposition in irreducible factors in $\mathbb{Q}_{p}[X]$. The
factors $f_{i}(X)$ correspond uniquely to the divisors $\mathfrak{p}_{i}$ of
$p$ in $\mathbb{Q}(\pi_{A})$. So to get the invariants, we have to factor
$P(X)$ over $\mathbb{Q}_{p}$. Indeed,
$\displaystyle
inv_{\mathfrak{p}_{i}}(End_{k}(A)\otimes\mathbb{Q})\equiv\frac{v_{p}(f_{i}(0))}{v_{p}(q)}\mod\mathbb{Z}.$
If $P(X)$ is a supersingular Weil polynomial, then
$f_{i}(0)=\displaystyle\prod_{j=1}^{\deg f_{i}}\sqrt{q}\zeta_{j}$, where
$\zeta_{j}$ is some root of unity. In that case,
$inv_{\mathfrak{p}_{i}}(End_{k}(A)\otimes\mathbb{Q})=\displaystyle\frac{n\frac{\deg
f_{i}}{2}}{n}=\frac{\deg{f_{i}}}{2}\mod\mathbb{Z}.$
Therefore the order of invariants is either $1$ or $2$ in
$\mathbb{Q}/\mathbb{Z}$ depending on whether $\deg f_{i}$ is even or odd
respectively. We use the invariants in order to evaluate the dimension of A as
follows.
Since $End_{k}(A)\otimes\mathbb{Q}$ is a division algebra and
$\mathbb{Q}(\pi_{A})$ is a number field, the number
$[End_{k}(A)\otimes\mathbb{Q}:\mathbb{Q}(\pi_{A})]^{\frac{1}{2}}$ is equal to
the order of $End_{k}(A)\otimes\mathbb{Q}$ in the Brauer group of
$\mathbb{Q}(\pi_{A})$ see theorem 18.6, [5], which turns to be equal to the
least common multiple of the orders of all the local invariants in
$\mathbb{Q}/\mathbb{Z}$; see theorem 18.5, [5]. This fact, along with the
Theorem 2.6, gives the dimension of $A$.
The rest of section is dedicated to background useful for computing
supersingular Weil polynomials.
Let $U(\mathbb{Z}/m\mathbb{Z})$ denote the multiplicative group of integers
modulo $m$ and $\zeta_{m}$ be a primitive $m$-th root of unity. Then there
exist a natural isomorphism of groups
$U(\mathbb{Z}/m\mathbb{Z})\rightarrow
Gal(\mathbb{Q}(\zeta_{m})/\mathbb{Q}),~{}~{}~{}a\rightarrow\sigma_{a}$
where $\sigma_{a}(\zeta_{m})=\zeta_{m}^{a}$. There is a unique nontrivial
homomorphism $U(\mathbb{Z}/p\mathbb{Z})\rightarrow\\{\pm 1\\}$ ($p$ an odd
prime), the Legendre symbol $a\rightarrow(\frac{a}{p})$. The group
$U(\mathbb{Z}/8\mathbb{Z})=\\{\pm 1,\pm 3\\}$ admits three nontrivial
quadratic characters (homomorphisms to $\\{\pm 1\\}$):
1. (1)
$\varepsilon$, defined by $\varepsilon(a)\equiv a\mod 4$.
2. (2)
$\chi_{2}$, given by $\chi_{2}(\pm 1)=1,~{}\chi_{2}(\pm 3)=-1$.
3. (3)
$\chi_{-2}$, given by
$\chi_{-2}(1)=\chi_{-2}(3)=1,\chi_{2}(-1)=~{}\chi_{2}(-3)=-1$.
For any odd prime $p$, $\sqrt{\pm p}\in\mathbb{Q}(\zeta_{4p})$ and
$\displaystyle\sigma_{a}(\sqrt{\pm
p})=\left\\{\begin{array}[]{ll}(\frac{a}{p})\sqrt{\pm p}&\mbox{if $\pm p\equiv
1\mod 4$ };\\\ \varepsilon(a)(\frac{a}{p})\sqrt{\pm p}&\mbox{if $\pm p\equiv
3\mod 4$ }.\end{array}\right.$
$\sqrt{\pm 2}\in\mathbb{Q}(\zeta_{8})$ and for $a\in
U(\mathbb{Z}/8\mathbb{Z})$,
$\sigma_{a}(\sqrt{2})=\chi_{2}(a)\sqrt{2},~{}\sigma_{a}(\sqrt{-2})=\chi_{-2}(a)\sqrt{-2}$
For an odd prime $p$ we let
$p^{*}=(\frac{-1}{p})p=\left\\{\begin{array}[]{ll}p&\mbox{if $p\equiv 1\mod 4$
};\\\ -p&\mbox{if $p\equiv 3\mod 4$ }.\end{array}\right.$
For $p$ an odd prime and $t$ any odd number let
$\displaystyle\Psi_{p,t}(X):=\prod_{a\in
U(\mathbb{Z}/pt\mathbb{Z})}(X-(\frac{a}{p})\zeta_{pt}^{a})$
$\displaystyle\Psi_{2,t}(X):=\prod_{a\in
U(\mathbb{Z}/t\mathbb{Z})}(X-\zeta_{8}\zeta_{t}^{a})(X-\zeta_{8}^{-1}\zeta_{t}^{a})$
$\displaystyle\Psi_{-2,t}(X):=\prod_{a\in
U(\mathbb{Z}/t\mathbb{Z})}(X-\zeta_{8}\zeta_{t}^{a})(X-\zeta_{8}^{3}\zeta_{t}^{a}).$
The polynomials below
$\Psi_{p,t}(X)\in\mathbb{Q}(\sqrt{p^{*}})[X]$
$\Psi_{2,t}(X)\in\mathbb{Q}(\sqrt{2})[X]$
$\Psi_{-2,t}(X)\in\mathbb{Q}(\sqrt{-2})[X]$
factor as a product of 2 irreducibles over
$\mathbb{Q}(\sqrt{p^{*}})[X],~{}\mathbb{Q}(\sqrt{2})[X],~{}\mathbb{Q}(\sqrt{-2})[X]$
respectively. For example $\Psi_{p,t}(X)=\Psi_{p,t}^{1}(X)\Psi_{p,t}^{-1}(X)$
where
$\Psi_{p,t}^{1}(X)=\prod_{\frac{a}{p}=1}(X-(\frac{a}{p}=1)\zeta_{pt}^{a})$ and
$\Psi_{p,t}^{-1}(X)=\prod_{\frac{a}{p}=-1}(X+(\frac{a}{p})\zeta_{pt}^{a}).$
If $K$ is a field, $f(X)\in K[X]$ and $a\in K^{\times}$, then let
$f^{[a]}(X):=a^{\deg f}f(\frac{X}{a})$. Then $\Psi_{p,t}^{[\sqrt{p^{*}}]}(X)$,
$\Psi_{2,t}^{[\sqrt{2}]}(X)$, $\Psi_{-2,t}^{[\sqrt{-2}]}(X)\in\mathbb{Q}[X]$
and are all irreducible over $\mathbb{Q}$.
## 3\. Odd case
###### Theorem 3.1.
Let $A$ be a supersingular simple abelian variety over $\mathbb{F}_{q}$ with
$q=p^{r}$ with $r$ odd. If the characteristic polynomial of $A$, $P_{A}(X)$,
has no real root, then $P_{A}(X)$ is irreducible.
###### Proof.
If $A$ is a supersingular abelian variety then it follows from the Theorem 2.6
that,
$inv_{\mathfrak{p}}(End_{k}(A)\otimes\mathbb{Q})\equiv\frac{1}{2}e_{i}f_{i}\mod\mathbb{Z}$,
where $e_{i}$ is the ramification index of $\mathfrak{p}$ over $p$ and $f_{i}$
is the residue class degree.
But $e_{i}=e=v_{\mathfrak{p}}(p)=2v_{\mathfrak{p}}(\sqrt{p})$. This implies
$inv_{\mathfrak{p}}(End_{k}(A)\otimes\mathbb{Q})=0\mod\mathbb{Z}$. In
otherwords, order of invariants is zero and $2\dim A$ is equal to the degree
of the Weil polynomial. This along with the definition of the characteristic
polynomial implies $P_{A}(X)$ is irreducible. ∎
###### Remark 3.2.
If $P(X)$ is a supersingular Weil polynomial with all real roots, over
$\mathbb{F}_{q}$ with $q=p^{n}$, $n$ is odd, then $P(X)=X^{2}-q$, then we have
least common multiple of local invariants as $2$, hence $P(X)^{2}$ is a
characteristic polynomial of simple supersingular abelian variety of dimension
$2$, see [6].
Let $p$ be a prime and $q=(\pm p)^{r}$ with $r$ odd. We wish to calculate the
minimal polynomial over $\mathbb{Q}$ of $\sqrt{q}\zeta_{m}$. Without lost of
generality we assume that $m=4t$ for some $t$ since
$\sqrt{q}\zeta_{m}=\sqrt{-q}\zeta_{4}\zeta_{m}=\sqrt{-q}\zeta_{m^{{}^{\prime}}}$.
###### Theorem 3.3.
Let $\theta=\sqrt{q}\zeta_{4t}$. We distinguish two cases.
1. (1)
Normal case If either
1. (a)
$q$ is odd and
1. (i)
$t$ is even or
2. (ii)
$p\nmid t$ or
3. (iii)
$q\equiv 1\mod 4$
or
2. (b)
$q$ is even and $t\not\equiv 2\mod 4$.
then the minimal polynomial of $\theta$ over $\mathbb{Q}$ is
$\Phi_{4t}^{[\sqrt{q}]}(X)$.
2. (2)
exceptional case
1. (a)
If $q\equiv 3\mod 4$ and $p|t$ and $t$ is odd, then the minimal polynomial of
$\theta$ is $\Psi_{p,\frac{t}{p}}^{[\sqrt{q^{*}}]}(X)$.
2. (b)
If $q>0$ is even and $t=2s\equiv 2\mod 4$, then the minimal polynomial of
$\theta$ is $\Psi_{2,s}^{[\sqrt{q}]}(X)$.
3. (c)
If $q<0$ is even and $t=2s\equiv 2\mod 4$, then the minimal polynomial of
$\theta$ is $\Psi_{-2,s}^{[\sqrt{q}]}(X)$.
###### Proof.
Case (A) First suppose $p\nmid t$. Then $\sqrt{q}\notin\mathbb{Q}(\zeta_{4t})$
and
$\Gamma:=Gal(\mathbb{Q}(\sqrt{q},\zeta_{4t})/\mathbb{Q})=<\sigma>\times
Gal(\mathbb{Q}(\zeta_{4t})/\mathbb{Q})$ where $\sigma(\sqrt{q})=-\sqrt{q}$,
$\sigma(\zeta_{4t})=\zeta_{4t}$.
Let $\tau\in\Gamma$ and suppose $\tau(\theta)=\theta$.
´Case 1: $\tau=\sigma_{a},~{}a\in U(\mathbb{Z}/4t\mathbb{Z})$ implies
$\sqrt{q}\zeta_{4t}=\sqrt{q}\zeta_{4t}^{a}$ which implies $a=1$ which implies
$\tau=1$.
Case 2: $\tau=\sigma\sigma_{a}$ implies
$\sqrt{q}\zeta_{4t}=-\sqrt{q}\zeta_{4t}^{a}$ implies
$\zeta_{4t}^{a}=-\zeta_{4t}=\zeta_{4t}^{1+2t}$ which implies $a=2t+1$. So the
subgroup of $\Gamma$ fixing $\theta$ is $\\{1,\sigma\sigma_{1+2t}\\}$ and
$Gal(\mathbb{Q}(\theta)/\mathbb{Q})\cong
Gal(\mathbb{Q}(\zeta_{4t})/\mathbb{Q})$. In particular, the conjugates of
$\theta$ over $\mathbb{Q}$ are $\sigma_{a}(\theta)=\sqrt{q}\zeta_{4t}^{a}$,
and the minimal polynomial of $\theta$ over $\mathbb{Q}$ is
$\displaystyle\prod_{a\in
U(\mathbb{Z}/4t\mathbb{Z})}(X-\sqrt{q}\zeta_{4t}^{a})=\Phi_{4t}^{[\sqrt{q}]}(X)$.
Case B.
Now suppose $p$ is odd and $p|t$. Then $\sqrt{q}\in\mathbb{Q}(\zeta_{4t})$ and
if $a\in U(\mathbb{Z}/4t\mathbb{Z})$ we have
$\displaystyle\sigma_{a}(\theta)=\sigma_{a}(\sqrt{q}\zeta_{4t})=\left\\{\begin{array}[]{ll}(\frac{a}{p})\sqrt{q}\zeta_{4t}^{a}&\mbox{if
$q\equiv 1\mod 4$ };\\\
\varepsilon(a)(\frac{a}{p})\sqrt{q}\zeta_{4t}^{a}&\mbox{if $q\equiv 3\mod 4$
}.\end{array}\right.$
Suppose $q\equiv\mod 4$ and $\sigma_{a}(\theta)=\theta$. Then
$(\frac{a}{p})\zeta_{4t}^{a}=\zeta_{4t}$. If $(\frac{a}{p})=1$ then $a=1$ in
$U(\mathbb{Z}/4t\mathbb{Z})$ implies $\sigma_{a}=1$. However
$(\frac{a}{p})=-1$ implies $\zeta_{4t}^{a}=\zeta_{4t}^{1+2t}$ which implies
$a=1+2t$ or $a\equiv 1\mod p$. This implies $(\frac{a}{p})=1$ which is a
contradiction.
So $\sigma_{a}=1$ and
$Gal(\mathbb{Q}(\theta)/\mathbb{Q})=(\mathbb{Q}(\zeta_{4t})/\mathbb{Q})$
whence the minimal polynomial of $\theta$ is $\Phi_{4t}^{[\sqrt{q}]}(X)$.
Now suppose $q\equiv 3\mod 4$ and $\sigma_{a}(\theta)=\theta$. Then
$\zeta_{4t}=\varepsilon(a)(\frac{a}{p})\sqrt{q}\zeta_{4t}^{a}$. Since
$\varepsilon(a)(\frac{a}{p})\in\\{1,-1\\}$ we have $a\in\\{1,1+2t\\}$. If
$a=1+2t$ then $(\frac{a}{p})=1$ and $\varepsilon(a)=1+2t\mod 4=-1$ if and only
if $t$ is odd. Thus if $t$ is even, the minimal polynomial of $\theta$ is
$\Phi_{4t}^{[\sqrt{q}]}(X)$.
However, if $t$ is odd, then $\theta$ is fixed by $\sigma_{1+2t}$ and hence
the degree of its minimal polynomial over $\mathbb{Q}$ is
$\frac{1}{2}\phi(4t)=\phi(t).$ However, in this case
$\sqrt{q}\zeta_{4t}=\pm\sqrt{-q}\zeta_{t}=\pm\sqrt{q^{*}}\zeta_{t}\in\mathbb{Q}(\zeta_{t})$
and thus its minimal polynomial is $\prod_{a\in
U(\mathbb{Z}/t\mathbb{Z})}(X-(\frac{a}{p})\zeta_{t}^{a})=\Psi_{p,\frac{t}{p}}^{[\sqrt{q^{*}}]}(X)$.
Case C: $q$ is even and $2|t$. So $4t=8s$. Let $\zeta=\zeta_{4t}=\zeta_{8s}$.
Then $\theta\in\mathbb{Q}{\zeta}$ and
$\displaystyle\sigma_{a}(\theta)=\sigma_{a}(\sqrt{q}\zeta)=\left\\{\begin{array}[]{ll}(\chi_{2}(a)\sqrt{q}\zeta^{a}&\mbox{if
$q>0$ };\\\ \chi_{-2}(a)\sqrt{q}\zeta^{a}&\mbox{if $q<0$ }.\end{array}\right.$
Suppose $\sigma_{a}(\theta)=\theta$. Then $\chi_{\pm 2}(a)=-1$ Then $a=1+4s\in
U(\mathbb{Z}/8s\mathbb{Z})$. If $s$ is even then $a\equiv 1\mod 8$ which
implies $\chi_{\pm 2}(a)=1$ which a contradiction. Thus if $s$ is even, the
minimal polynomial of $\theta$ is $\Phi_{4t}^{[\sqrt{q}]}(X)$.
We can suppose that $4t=8s$ with $s$ odd. Then $1+4s=-3\in
U(\mathbb{Z}/8s\mathbb{Z})$. It follows that $\theta$ is fixed by the
$\\{1,\sigma_{1+4s}\\}$ and so its minimal polynomial has degree
$\frac{1}{2}\phi(4t)$. We can write $\zeta$ as a product $\zeta_{8}\zeta_{s}$.
So $\theta=\sqrt{q}\zeta_{8}\zeta_{s}$.
If $q>0$, the conjugates of $\theta$ are $\sqrt{q}\zeta_{8}\zeta_{s}^{a}$ and
$\sqrt{q}\zeta_{8}^{-1}\zeta_{s}^{a}$ as $a$ ranges over
$U(\mathbb{Z}/s\mathbb{Z})$. Thus the minimal polynomial is
$\displaystyle\prod_{a\in
U(\mathbb{Z}/s\mathbb{Z})}(X-\sqrt{q}\zeta_{8}\zeta_{s}^{a})(X-\sqrt{q}\zeta_{8}^{-1}\zeta_{s}^{a})=\Psi_{2,s}^{[\sqrt{q}]}(X)$.
Similarly, if $q<0$, the minimal polynomial of $\theta$ is
$\Psi_{-2,s}^{[\sqrt{q}]}(X)$.
This proves the theorem. ∎
###### Remark 3.4.
1. (1)
In the normal case, the degree is $\phi(4t)$ and in the exception case the
degree is $\frac{1}{2}\phi(4t)$.
2. (2)
Since $\Psi_{p,t}^{[\sqrt{p^{*}}]}(X)\in\mathbb{Q}[X]$, it follows
$\Psi_{p,t}(X)\in\mathbb{Q}(\sqrt{p^{*}})[X]$ and the coefficients of even
powers are rational (even integers) and the coefficients of odd powers are
integer multiples of $\sqrt{p^{*}}$. Similiar is true for $\Psi_{\pm 2,t}(X).$
3. (3)
If $p$ is odd, then $\Psi_{p,t}$ is a reciprocal polynomial when $p\equiv
1\mod 4$ and anti-reciprocal (i.e;
$X^{\deg\Psi_{p,t}}\Psi_{p,t}(-\frac{1}{X}))=\Psi_{p,t}(X))$ when $p\equiv
3\mod 4$.
4. (4)
$\Psi_{p,t}(X)$ acutally depends on the choice of $\zeta_{pt}$. If we replace
it with $\zeta_{pt}^{a}$ where $(\frac{a}{p})=-1$ then we get
$\Psi_{p,t}(-X)$. Similiar is true for $\Psi_{\pm 2,t}(X).$ We have
$\Psi_{p,t}(X)\Psi_{p,t}(-X)=\Phi_{pt}(X^{2}).$
5. (5)
It easily follows that polynomial $\Psi_{p,t}$ can be calculated recursively
in the same way as cyclotomic polynomials.
$\Psi_{p,tp^{k}}(X)=\Psi_{p,t}(X^{p^{k-1}}).$
While if $l$ is a prime not dividing $t$ then
$\Psi_{p,l^{k}}(X)=\frac{\Psi_{p,t}(X^{l^{k}})}{\Psi_{p,t}(X^{l^{k-1}})}.$
## 4\. Dimension 1
### 4.1. Normal Case
In this case $\phi(4t)=2g=2$ which implies $t=1$. Therefore $X^{2}\pm q$ is a
Weil polynomial.
### 4.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=2$. That implies $4t\in\\{8,12\\}$ or
$t\in\\{2,3\\}$. In case $t=2$ then $q=(\pm 2)^{r}$ with $r$ odd. Then,
1. (1)
If $q>0$ then the minimal polynomial of $\theta$ is
$\Psi_{2,1}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{2,1}(X)=(X-\zeta_{8})(X-\zeta_{8}^{-1})=X^{2}\pm(\zeta_{8}+\zeta_{8}^{-1})X+1=X^{2}\pm\sqrt{2}X+1$.
Therefore $\Psi_{2,1}^{[\sqrt{q}]}(X)=X^{2}\pm\sqrt{2q}X+q$.
2. (2)
If $q<0$, then the minimal polynomial of $\theta$ is
$\Psi_{-2,1}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{2,1}(X)=(X-\zeta_{8})(X-\zeta_{8}^{3})=X^{2}\pm(\zeta_{8}+\zeta_{8}^{3})X-1=X^{2}\pm\sqrt{-2}X-1$.
Therefore $\Psi_{-2,1}^{[\sqrt{q}]}(X)=X^{2}\pm\sqrt{-2q}X-q$.
In case $t=3$, then $p=3$ and $q=3^{r}$ with $r$ odd. In that case, the
minimal polynomial of $\theta$ is $\Psi_{3,1}^{[\sqrt{q}]}(X).$
But, $\displaystyle\Psi_{3,1}(X)=\prod_{a\in
U(\mathbb{Z}/3\mathbb{Z})}(X-(\frac{a}{3})\zeta_{3}^{a})=X^{2}-\sum_{a=1}^{2}(\frac{a}{p})X+1=X^{2}\pm\sqrt{-3}X-1$.
Therefore $\Psi_{3,1}^{[\sqrt{q^{*}}]}(X)=X^{2}\pm\sqrt{3q}X+q$.
## 5\. Dimension 2
### 5.1. Normal Case
In this case $\phi(4t)=2g=4$. That implies $4t\in\\{8,12\\}$ or
$t\in\\{2,3\\}$. Therefore either $t=2$, $p$ odd and minimal polynomial is
$\Phi_{8}^{[\sqrt{q}]}(X)=X^{4}+q^{2},$
or $t=3$, $q\neq 3^{r}$, $r~{}odd$ and minimal polynomial is
$\Phi_{12}^{[\sqrt{q}]}(X)=X^{4}-qX^{2}+q^{2}.$
### 5.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=4$. That implies $4t\in\\{16,20,24\\}$ or
$t\in\\{4,5,6\\}$.
1. (1)
Either $p$ is odd, $t$ is odd, $p$ divides $t$, $q\equiv 3\mod 4$.
In that case, $t=p=5$, $q=(-5)^{r}$ with $r$ odd and
$\Psi_{5,1}^{[\sqrt{q^{*}}]}(X)$ is the minimal polynomial of $\theta$. We
have $\Psi_{5,1}(X)\Psi_{5,1}(-X)=\Phi_{5}(X^{2})$. Therefore,
$\Psi_{5,1}(X)=X^{4}\pm\sqrt{5}X^{3}+3X^{2}\pm\sqrt{5}X+1$ and
$\Psi_{5,1}^{[\sqrt{q^{*}}]}(X)=X^{4}\pm\sqrt{-5q}X^{3}-3qX^{2}\mp
q\sqrt{-5q}X+q^{2}.$
2. (2)
$t\equiv 2\mod 4,~{}q=(\pm 2)^{r}$ with $r$ odd. In that case, $t=6=2\cdot
3\mod 4$. Then
1. (a)
If $q>0$ then the minimal polynomial of $\theta$ is
$\Psi_{2,3}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{2,3}(X)=\frac{\Psi_{2,1}(X^{3})}{\Psi_{2,1}(X)}=X^{4}\pm\sqrt{2}X^{3}+X^{2}\pm\sqrt{2}X+1$
and hence
$\Psi_{2,3}^{[\sqrt{q}]}(X)=X^{4}\pm\sqrt{2q}X^{3}+qX^{2}\pm
q\sqrt{2q}X+q^{2}.$
2. (b)
If $q<0$, then the minimal polynomial of $\theta$ is
$\Psi_{-2,3}^{[\sqrt{q}]}(X)$.
But, $\Psi_{-2,3}(X)=X^{4}\pm\sqrt{-2}X^{3}-X^{2}\mp\sqrt{-2}X+1$ and hence
$\Psi_{-2,3}^{[\sqrt{q}]}(X)=X^{4}\pm\sqrt{-2q}X^{3}-qX^{2}\mp
q\sqrt{-2q}X+q^{2}$.
## 6\. Dimension 3
### 6.1. Normal Case
In this case $\phi(4t)=2g=6$ and there is no such $t$.
### 6.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=6$. That implies $4t\in\\{28,36\\}$ or
$t\in\\{7,9\\}$. In that case, $p$ is odd, $t$ is odd, $p$ divides $t$,
$q\equiv 3\mod 4$.
1. (1)
$t=7,~{}p=7,~{}q=7^{r}$ with $r$ odd, then $\Psi_{7,1}^{[\sqrt{q^{*}}]}(X)$ is
the minimal polynomial of $\theta$.
But $\Psi_{7,1}(X)\Psi_{7,1}(-X)=\Phi_{7}(X^{2})$ and therefore,
$\Psi_{7,1}(X)=X^{6}\pm\sqrt{-7}X^{5}-3X^{4}\mp\sqrt{-7}X^{3}+3X^{2}\pm\sqrt{-7}X-1.$
This gives,
$\Psi_{7,1}^{[\sqrt{q^{*}}]}(X)=X^{6}\pm\sqrt{7q}X^{5}+3qX^{4}\pm
q\sqrt{7q}X^{3}+3q^{2}X^{2}\pm q^{2}\sqrt{7q}X+q^{3}$
2. (2)
$t=9,~{}p=3,~{}q=3^{r}$ with $r$ odd, then $\Psi_{3,3}^{[\sqrt{q^{*}}]}(X)$ is
the minimal polynomial of $\theta$.
But,
$\Psi_{3,3}(X)=\Psi_{3,1}(X^{3})=X^{6}\pm\sqrt{-3}X^{3}-1$
and hence,
$\Psi_{3,3}^{[\sqrt{q^{*}}]}(X)=X^{6}\pm q\sqrt{3q}X^{3}+q^{3}.$
## 7\. Dimension 4
### 7.1. Normal Case
In this case $\phi(4t)=2g=8$. That implies $4t\in\\{16,20,24\\}$ or
$t\in\\{4,5,6\\}$. We have the following possibilities.
1. (1)
If $t=4$ then,
1. (a)
$q$ is odd and $t$ is even,
2. (b)
$q$ is even and $t\not\equiv 2\mod 4$.
Therefore the minimal polynomial is
$\Phi_{16}^{[\sqrt{q}]}(X)=X^{8}+q^{4}$
for all primes $p$.
2. (2)
If $t=5$ then,
1. (a)
$q$ is odd and $p\nmid t$ but $p\equiv 1\mod 4$, this implies
$q\neq(-5)^{r},~{}r$ odd.
2. (b)
$q$ is even and $t\not\equiv 2\mod 4$.
Therefore the minimal polynomial is
$\Phi_{20}^{[\sqrt{q}]}(X)=X^{8}-qX^{6}+q^{2}X^{4}-q^{3}X^{2}+q^{4}$
for all $q\neq(-5)^{r},~{}r$ odd.
3. (3)
If $t=6$ then,
1. (a)
$q$ is odd and $t$ is even,
Therefore the minimal polynomial is
$\Phi_{24}^{[\sqrt{q}]}(X)=X^{8}-q^{2}X^{4}+q^{4}$
for all primes $p\neq 2$.
### 7.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=8$. That implies $4t\in\\{32,40,48,60\\}$
or $t\in\\{8,10,12,15\\}$. We have the following possibilities.
1. (1)
Either $p$ is odd, $t$ is odd, $p$ divides $t$, $q\equiv 3\mod 4$. In that
case $t=15$ then
1. (a)
either $p=3,~{}q=3^{r}$ with $r$ odd and $\Psi_{3,5}^{[\sqrt{q^{*}}]}(X)$ is
the minimal polynomial of $\theta$.
Since $\Psi_{3,5}(X)=\frac{\Psi_{3,1}(X^{5})}{\Psi_{3,1}(X)}$, we have
$\Psi_{3,5}^{[\sqrt{q^{*}}]}(X)=X^{8}\mp\sqrt{3q}X^{7}+2qX^{6}\mp
q\sqrt{3q}X^{5}+q^{2}X^{4}\mp\sqrt{3q}X^{3}+2X^{2}\pm q^{3}\sqrt{3q}X+q^{4}$
or
2. (b)
$p=5,~{}q=(-5)^{r}$ with $r$ odd and $\Psi_{5,3}^{[\sqrt{q^{*}}]}(X)$ is the
minimal polynomial of $\theta$.
We have,
$\Psi_{5,3}(X)=\frac{\Psi_{5,1}(X^{3})}{\Psi_{5,1}(X)}=X^{8}\pm\sqrt{5}X^{7}+2X^{6}\pm\sqrt{5}X^{5}+3X^{4}\pm\sqrt{5}X^{3}+2X^{2}\pm\sqrt{5}X+1$
and hence $\Psi_{5,3}^{[\sqrt{q^{*}}]}(X)=X^{8}\pm\sqrt{-5q}X^{7}-2qX^{6}\mp
q\sqrt{-5q}X^{5}+3q^{2}X^{4}\pm q^{2}\sqrt{-5q}X^{3}-2q^{3}X^{2}\mp
q^{3}\sqrt{-5q}X+q^{4}.$
2. (2)
$t\equiv 2\mod 4,~{}q=(\pm 2)^{r}$ with $r$ odd. In that case $t=10=2\cdot
5\mod 4$.
1. (a)
If $q>0$ then the minimal polynomial of $\theta$ is
$\Psi_{2,5}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{2,5}(X)=\frac{\Psi_{2,1}(X^{5})}{\Psi_{2,1}(X)}=X^{8}\pm\sqrt{2}X^{7}+X^{6}-X^{4}+X^{2}\pm\sqrt{2}X+1$
and hence
$\Psi_{2,5}^{[\sqrt{q}]}(X)=X^{8}\pm\sqrt{2q}X^{7}+qX^{6}-q^{2}X^{4}+q^{3}X^{2}\pm
q^{3}\sqrt{2q}X+q^{4}.$
2. (b)
If $q<0$, then the minimal polynomial of $\theta$ is
$\Psi_{-2,5}^{[\sqrt{q}]}(X)$.
But, $\Psi_{-2,5}(X)=X^{8}\pm\sqrt{-2}X^{7}-X^{6}-X^{4}-X^{2}\mp\sqrt{-2}X+1$.
Therefore
$\Psi_{-2,5}^{[\sqrt{q}]}(X)=X^{8}\pm\sqrt{-2q}X^{7}-qX^{6}-q^{2}X^{4}-q^{3}X^{2}\mp
q^{3}\sqrt{-2q}X+q^{4}$.
## 8\. Dimension 5
### 8.1. Normal Case
In this case $\phi(4t)=2g=10$ and there is no such $t$.
### 8.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=10$. That implies $4t=44$ or $t=11$. In
that case, $p=11,~{}q=11^{r}$, then $\Psi_{11,1}^{[\sqrt{q^{*}}]}(X)$ is the
minimal polynomial of $\theta$. But
$\Psi_{11,1}(X)\Psi_{11,1}(-X)=\Phi_{11}(X^{2})$ which implies
$\Psi_{11,1}(X)=X^{10}\mp\sqrt{-11}X^{9}-5X^{8}\pm\sqrt{-11}X^{7}-X^{6}\pm\sqrt{-11}X^{5}+X^{4}\pm\sqrt{-11}X^{3}+5X^{2}\mp\sqrt{-11}X-1$.
Hence, $\Psi_{11,1}^{[\sqrt{q^{*}}]}(X)=X^{10}\mp\sqrt{11q}X^{9}+5qX^{8}\mp
q\sqrt{11q}X^{7}-q^{2}X^{6}\pm q^{2}\sqrt{11q}X^{5}-q^{3}X^{4}\mp
q^{3}\sqrt{11q}X^{3}+5q^{4}X^{2}\mp q^{4}\sqrt{11q}X+q^{5}$.
## 9\. Dimension 6
### 9.1. Normal Case
In this case $\phi(4t)=2g=12$. That implies $4t\in\\{28,36\\}$ or
$t\in\\{7,9\\}$. We have the following possibilities
1. (1)
If $t=7$ then,
1. (a)
$q$ is odd and $p\nmid t$ but $-7\equiv 1\mod 4$ i.e; $q\neq 7^{r},~{}r$ odd.
2. (b)
$q$ is even and $t\not\equiv 2\mod 4$.
Therefore the minimal polynomial is
$\Phi_{28}^{[\sqrt{q}]}(X)=X^{12}-qX^{10}+q^{2}X^{8}-q^{3}X^{6}+q^{4}X^{4}-q^{5}X^{2}+q^{6}$
for $q\neq 7^{r},~{}r$ odd.
2. (2)
If $t=9$ then,
1. (a)
$q$ is odd and $p\nmid t$ but $-3\equiv 1\mod 4$ i.e; $q\neq 3^{r},~{}r$ odd.
2. (b)
$q$ is even and $t\not\equiv 2\mod 4$.
Therefore the minimal polynomial is
$\Phi_{36}^{[\sqrt{q}]}(X)=X^{12}-q^{3}X^{6}+q^{6}$
for $q\neq 3^{r},~{}r$ odd.
### 9.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=12$. That implies
$4t\in\\{52,56,72,84\\}$ or $t\in\\{13,14,18,21\\}$.
1. (1)
Either $p$ is odd, $t$ is odd, $p$ divides $t$, $q\equiv 3\mod 4$
1. (a)
$t=p=13$, $q=(-13)^{r}$ and $\Psi_{13,1}^{[\sqrt{q^{*}}]}(X)$ is the minimal
polynomial of $\theta$. But $\Psi_{13,1}(X)\Psi_{13,1}(-X)=\Phi_{13}(X^{2})$
which implies $\Psi_{13,1}(X)=X^{12}\pm\sqrt{13}X^{11}+7X^{10}\pm
3\sqrt{13}X^{9}+15X^{8}\pm 5\sqrt{13}X^{7}+19X^{6}\pm
5\sqrt{13}X^{5}+15X^{4}\pm 3\sqrt{13}X^{3}+7X^{2}\pm\sqrt{13}X+1$ and
therefore
$\Psi_{13,1}^{[\sqrt{q^{*}}]}(X)=X^{12}\pm\sqrt{-13q}X^{11}-7qX^{10}\mp
3q\sqrt{-13q}X^{9}+15q^{2}X^{8}\pm 5q^{2}\sqrt{-13q}X^{7}-19q^{3}X^{6}\mp
5q^{3}\sqrt{-13q}X^{5}+15q^{4}X^{4}\pm 3q^{4}\sqrt{-13q}X^{3}-7q^{5}X^{2}\mp
q^{5}\sqrt{-13q}X+q^{6}$.
2. (b)
$t=21$, then
1. (i)
$p=3,~{}q=3^{r}$ with $r$ odd and $\Psi_{3,7}^{[\sqrt{q^{*}}]}(X)$ is the
minimal polynomial of $\theta$.
But
$\Psi_{3,7}(X)=\frac{\Psi_{3,1}(X^{7})}{\Psi_{3,1}(X)}=X^{12}\pm\sqrt{-3}X^{11}-2X^{10}\mp\sqrt{-3}X^{9}+X^{8}+X^{6}+X^{4}\pm\sqrt{-3}X^{3}-2X^{2}\mp\sqrt{-3}X+1$
and hence $\Psi_{3,7}^{[\sqrt{q^{*}}]}(X)=X^{12}\pm\sqrt{3q}X^{11}+2qX^{10}\pm
q\sqrt{3q}X^{9}+q^{2}X^{8}-q^{3}X^{6}+q^{4}X^{4}\pm
q^{4}\sqrt{3q}X^{3}+2q^{5}X^{2}\pm q^{5}\sqrt{3q}X+q^{6}$ or
2. (ii)
$p=7,~{}q=7^{r}$ with $r$ odd and $\Psi_{7,3}^{[\sqrt{q^{*}}]}(X)$ is the
minimal polynomial of $\theta$.
$\Psi_{7,3}(X)=\frac{\Psi_{7,1}(X^{3})}{\Psi_{7,1}(X)}=X^{12}\pm\sqrt{-7}X^{11}-4X^{10}\mp\sqrt{-7}X^{9}-X^{8}\mp
2\sqrt{-7}X^{7}+7X^{6}\pm
2\sqrt{-7}X^{5}-X^{4}\pm\sqrt{-7}X^{3}-4X^{2}\mp\sqrt{-7}X+1.$ Hence,
$\Psi_{7,3}^{[\sqrt{q^{*}}]}(X)=X^{12}\pm\sqrt{7q}X^{11}+4qX^{10}\pm
q\sqrt{7q}X^{9}-q^{2}X^{8}\mp 2q^{2}\sqrt{7q}X^{7}-7q^{3}X^{6}\mp
2q^{3}\sqrt{7q}X^{5}-q^{4}X^{4}\pm q^{4}\sqrt{7q}X^{3}+4q^{5}X^{2}\pm
q^{5}\sqrt{7q}X+q^{6}$.
2. (2)
$t\equiv 2\mod 4,~{}q$ even.
1. (a)
$t=14=2\cdot 7\mod 4$.
1. (i)
If $q>0$ then the minimal polynomial of $\theta$ is
$\Psi_{2,7}^{[\sqrt{q}]}(X)$.
But
$\Psi_{2,7}(X)=\frac{\Psi_{2,1}(X^{7})}{\Psi_{2,1}(X)}=X^{12}\mp\sqrt{2}X^{11}+X^{10}-X^{8}\pm\sqrt{2}X^{7}-X^{6}\pm\sqrt{2}X^{5}-X^{4}+X^{2}\mp\sqrt{2}X+1$
and hence,
$\Psi_{2,7}^{[\sqrt{q}]}(X)=X^{12}\mp\sqrt{2q}X^{11}+qX^{10}-q^{2}X^{8}\pm
q^{2}\sqrt{2q}X^{7}-q^{3}X^{6}\pm q^{3}\sqrt{2q}X^{5}-q^{4}X^{4}+q^{5}X^{2}\mp
q^{5}\sqrt{2q}X+q^{6}$.
2. (ii)
If $q<0$, then the minimal polynomial of $\theta$ is
$\Psi_{-2,7}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{-2,7}(X)=X^{12}\pm\sqrt{-2}X^{11}-X^{10}-X^{8}\mp\sqrt{-2}X^{7}+X^{6}\pm\sqrt{-2}X^{5}-X^{4}-X^{2}\mp\sqrt{-2}X+1$
$\Psi_{-2,7}^{[\sqrt{q}]}(X)=X^{12}\pm\sqrt{-2q}X^{11}-qX^{10}-q^{2}X^{8}\mp
q^{2}\sqrt{-2q}X^{7}+q^{3}X^{6}\pm
q^{3}\sqrt{-2q}X^{5}-q^{4}X^{4}-q^{5}X^{2}\mp q^{5}\sqrt{-2q}X+q^{6}$
2. (b)
$t=18=2\cdot 9\mod 4$ with $q$ even. In that case,
1. (i)
If $q>0$ then the minimal polynomial of $\theta$ is
$\Psi_{2,9}^{[\sqrt{q}]}(X)$.
But,
$\Psi_{2,9}(X)=\frac{\Psi_{2,1}(X^{3^{2}})}{\Psi_{2,1}(X^{3})}=X^{12}\pm\sqrt{2}X^{9}+X^{6}\pm\sqrt{2}X^{3}+1.$
and hence,
$\Psi_{2,9}^{[\sqrt{q}]}(X)=X^{12}\pm q\sqrt{2q}X^{9}+q^{3}X^{6}\pm
q^{4}\sqrt{2q}X^{3}+q^{6}.$
2. (ii)
If $q<0$, then the minimal polynomial of $\theta$ is
$\Psi_{-2,9}^{[\sqrt{q}]}(X)$.
But, $\Psi_{-2,9}(X)=X^{12}\pm\sqrt{-2}X^{9}-X^{6}\mp\sqrt{-2}X^{3}+1$.
Therefore $\Psi_{-2,9}^{[\sqrt{q}]}(X)=X^{12}\pm
q\sqrt{-2q}X^{9}-q^{3}X^{6}\pm q^{4}\sqrt{-2q}X^{3}+q^{6}.$
## 10\. Dimension 7
### 10.1. Normal Case
In this case $\phi(4t)=2g=14$ and there is no such $t$.
### 10.2. Exceptional Case
In this case $\frac{1}{2}\phi(4t)=2g=14$ and there is no such $t$.
## 11\. Even Case
If $P(X)$ is a supersingular Weil polynomial with all real roots, over
$\mathbb{F}_{q}$ with $q=p^{n}$, $n$ is even, then $P(X)=X\pm\sqrt{q}$, then
we have least common multiple of local invariants as $2$, hence $P(X)^{2}$ is
a characteristic polynomial of simple supersingular abelian variety of
dimension $1$. We have the following Theorem which states all the possible
characteristic polynomials of abelian varieties over $\mathbb{F}_{q}$.
###### Theorem 11.1.
Let $\phi(m)=2g$. Then $(\Phi_{m}^{[\sqrt{q}]}(X))^{e}$ is a characteristic
polynomial of Frobenius endomorphism of a simple supersingular abelian variety
of dimension $ge$ over $\mathbb{F}_{q},~{}q=p^{n},~{}n$ even, where
$r=\left\\{\begin{array}[]{ll}\mbox{order of $p$ in the multiplicative group
$U(\mathbb{Z}/m\mathbb{Z})$ },&\mbox{if $(p,m)=1$};\\\
\mbox{$f(p^{k}-p^{k-1})$, $f$ is order of $p$ in the multiplicative group
$U(\mathbb{Z}/s\mathbb{Z})$}&\mbox{if $m=p^{k}s$}.\end{array}\right.$
and
$e=\left\\{\begin{array}[]{ll}\mbox{$1$},&\mbox{if $r$ is even};\\\
\mbox{$2$},&\mbox{if $r$ is odd}.\end{array}\right.$
###### Proof.
Let $P(X)$ be an irreducible supersingular Weil-$q$-polynomial of degree $2g$.
Since $q$ is an even power of prime, $\sqrt{q}\in\mathbb{Z}$. Using Honda-Tate
theorem 2.4, we get $\frac{1}{q^{g}}(P(\sqrt{q}X))$ is an irreducible
cyclotomic polynomial $\Phi_{m}$ over $\mathbb{Z}$ of degree $2g$, i.e.
$\phi(m)=2g$. Therefore $P(X)=\Phi_{m}^{[\sqrt{q}]}(X)$ is an irreducible
supersingular Weil polynomial. To determine the dimension of the corresponding
abelian variety to $P(X)$, we will need to factorise $P(X)$ over
$\mathbb{Q}_{p}$. If $P(X)=\displaystyle\prod_{i}f_{i}(X)$ is the
factorisation over $\mathbb{Q}_{p}[X]$, then
$inv_{\mathfrak{p}_{i}}(End_{k}(A)\otimes\mathbb{Q})\displaystyle\equiv\frac{v_{p}(f_{i}(0))}{v_{p}(q)}\mod\mathbb{Z}\equiv\frac{\deg
f_{i}}{2}\mod\mathbb{Z}$.
Since $P(X)=\Phi_{m}^{[\sqrt{q}]}(X)$,
$inv_{\mathfrak{p}_{i}}(End_{k}(A)\otimes\mathbb{Q})\displaystyle=\frac{\deg
r_{i}}{2}\mod\mathbb{Z}$, where $\Phi_{m}(X)=\displaystyle\prod_{i}r_{i}(X)$
over $\mathbb{Q}_{p}$. But it follows from the chapter IV.4 in [7] that $\deg
r_{i}=r$, where
$r=\left\\{\begin{array}[]{ll}\mbox{order of $p$ in the multiplicative group
$U(\mathbb{Z}/m\mathbb{Z})$ },&\mbox{if $(p,m)=1$};\\\
\mbox{$f(p^{k}-p^{k-1})$, where $f$ is order of $p$ in the multiplicative
group $U(\mathbb{Z}/s\mathbb{Z})$,}&\mbox{if $m=p^{k}s$}.\end{array}\right.$
Hence
$inv_{\mathfrak{p}_{i}}(End_{k}(A)\otimes\mathbb{Q})=\left\\{\begin{array}[]{ll}1&\mbox{if
$r$ is even};\\\ 2&\mbox{if $r$ is odd}.\end{array}\right.$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=~{}e$
.
From Theorem 2.6, $2\dim A=e\deg P(X)$ or $\dim A=eg$. ∎
## 12\. Summary of results for dimensions $1$ to $7$
We gather the list of characteristic polynomials, from dimensions $1$ to $7$.
###### Theorem 12.1.
Let $A$ be an supersingular simple abelian variety over $\mathbb{F}_{q}$,
where $q=p^{n}$, $n$ odd. Then characteristic polynomial of $A$ is given by
1. (1)
Dimension 1 _(Deuring and Waterhouse[8, 1])_
1. (a)
$p=2:X^{2}\pm\sqrt{2q}X+q$,
2. (b)
$p=3:X^{2}\pm\sqrt{3q}X+q$,
3. (c)
$X^{2}+q$.
2. (2)
Dimension 2 _(C.Xing, D.Maisner and E.Nart[9, 10])_
1. (a)
$p\neq 3:X^{4}-qX^{2}+q^{2}$,
2. (b)
$X^{4}+qX^{2}+q^{2}$,
3. (c)
$p=2:X^{4}\pm\sqrt{pq}X^{3}+qX^{2}\pm q\sqrt{pq}X+q^{2}$,
4. (d)
$p=5:X^{4}\pm\sqrt{pq}X^{3}+3qX^{2}\pm q\sqrt{pq}X+q^{2}$,
5. (e)
$(X^{2}-q)^{2}$,
6. (f)
$p\neq 2:X^{4}+q^{2}$.
3. (3)
Dimension 3 _(E.Nart, C.Ritzenthaler and S.Haloui[11, 12])_
1. (a)
$p=3:X^{6}\pm q\sqrt{pq}X^{3}+q^{3}$, or
2. (b)
$p=7:X^{6}\pm\sqrt{pq}X^{5}+3qX^{4}\pm q\sqrt{pq}X^{3}+3q^{2}X^{2}\pm
q^{2}\sqrt{pq}X+q^{3}$.
4. (4)
Dimension 4 _( S.Haloui, V.Singh[13])_
1. (a)
$X^{8}+q^{4}$,
2. (b)
$X^{8}-qX^{6}+q^{2}X^{4}-q^{3}X^{2}+q^{4}$,
3. (c)
$p\neq 5:X^{8}+qX^{6}+q^{2}X^{4}+q^{3}X^{2}+q^{4}$,
4. (d)
$p\neq 2:X^{8}-q^{2}X^{4}+q^{4}$,
5. (e)
$p=3:X^{8}\pm\sqrt{3q}X^{7}+2qX^{6}\pm q\sqrt{3q}X^{5}+q^{2}X^{4}\pm
q^{2}\sqrt{3q}X^{3}+2q^{3}X^{2}\pm q^{3}\sqrt{3q}X+q^{4}$,
6. (f)
$p=5:X^{8}\pm\sqrt{5q}X^{7}+2qX^{6}\pm q\sqrt{5q}X^{5}+3q^{2}X^{4}\pm
q^{2}\sqrt{5q}X^{3}+2q^{3}X^{2}\pm q^{3}\sqrt{5q}X+q^{4}$,
7. (g)
$p=2:X^{8}\pm\sqrt{2q}X^{7}+qX^{6}-q^{2}X^{4}+q^{3}X^{2}\pm
q^{3}\sqrt{2q}X+q^{4}$.
5. (5)
Dimension 5
1. (a)
$X^{10}\mp\sqrt{11q}X^{9}+5qX^{8}\mp q\sqrt{11q}X^{7}-q^{2}X^{6}\pm
q^{2}\sqrt{11q}X^{5}-q^{3}X^{4}\mp q^{3}\sqrt{11q}X^{3}+5q^{4}X^{2}\mp
q^{4}\sqrt{11q}X+q^{5}$.
6. (6)
Dimension 6
1. (a)
$X^{12}+qX^{10}+q^{2}X^{8}+q^{3}X^{6}+q^{4}X^{4}+q^{5}X^{2}+q^{6}$,
2. (b)
$p\neq 7:X^{12}-qX^{10}+q^{2}X^{8}-q^{3}X^{6}+q^{4}X^{4}-q^{5}X^{2}+q^{6}$,
3. (c)
$X^{12}+q^{3}X^{6}+q^{6}$,
4. (d)
$p\neq 3:X^{12}-q^{3}X^{6}+q^{6}$,
5. (e)
$X^{12}\pm\sqrt{13q}X^{11}+7qX^{10}\pm 3q\sqrt{13q}X^{9}+15q^{2}X^{8}\pm
5q^{2}\sqrt{13q}X^{7}+19q^{3}X^{6}\pm 5q^{3}\sqrt{13q}X^{5}+15q^{4}X^{4}\pm
3q^{4}\sqrt{13q}X^{3}+7q^{5}X^{2}\pm q^{5}\sqrt{13q}X+q^{6}$
6. (f)
$X^{12}\pm\sqrt{3q}X^{11}+2qX^{10}\pm
q\sqrt{3q}X^{9}+q^{2}X^{8}-q^{3}X^{6}+q^{4}X^{4}\pm
q^{4}\sqrt{3q}X^{3}+2q^{5}X^{2}\pm q^{5}\sqrt{3q}X+q^{6}.$
7. (g)
$X^{12}\pm\sqrt{7q}X^{11}+4qX^{10}\pm q\sqrt{7q}X^{9}-q^{2}X^{8}\mp
2q^{2}\sqrt{7q}X^{7}-7q^{3}X^{6}\mp 2q^{3}\sqrt{7q}X^{5}-q^{4}X^{4}\pm
q^{4}\sqrt{7q}X^{3}+4q^{5}X^{2}\pm q^{5}\sqrt{7q}X+q^{6}$,
8. (h)
$X^{12}\mp\sqrt{2q}X^{11}+qX^{10}-q^{2}X^{8}\pm
q^{2}\sqrt{2q}X^{7}-q^{3}X^{6}\pm q^{3}\sqrt{2q}X^{5}-q^{4}X^{4}+q^{5}X^{2}\mp
q^{5}\sqrt{2q}X+q^{6}$,
9. (i)
$X^{12}\pm q\sqrt{2q}X^{9}+q^{3}X^{6}\pm q^{4}\sqrt{2q}X^{3}+q^{6}$.
7. (7)
Dimension 7
There is no supersingular simple abelian variety of dimension $7$.
###### Theorem 12.2.
Let $A$ be an supersingular simple abelian variety over $\mathbb{F}_{q}$,
where $q=p^{n}$, $n$ even. Then characteristic polynomial of $A$ is given by
1. (1)
Dimension 1 _(Deuring and Waterhouse[8, 1])_
1. (a)
$X^{2}+X\sqrt{q}+q$, $~{}~{}p\not\equiv 1\mod 3$,
2. (b)
$X^{2}+q$, $~{}~{}p\not\equiv 1\mod 4$,
3. (c)
$X^{2}-X\sqrt{q}+q$ , $~{}~{}p\not\equiv 1\mod 6$,
4. (d)
$(X\pm\sqrt{q})^{2}$.
2. (2)
Dimension 2 _(C.Xing, D.Maisner and E.Nart[9, 10])_
1. (a)
$(X^{2}+X\sqrt{q}+q)^{2}$, $~{}~{}p\equiv 1\mod 3$,
2. (b)
$(X^{2}+q)^{2}$, $~{}~{}p\equiv 1\mod 4$,
3. (c)
$(X^{2}-X\sqrt{q}+q)^{2}$ , $~{}~{}p\equiv 1\mod 6$,
4. (d)
$X^{4}+\sqrt{q}X^{3}+qX^{2}+q^{3/2}X+q^{2}$, $~{}~{}p\not\equiv 1\mod 5$,
5. (e)
$X^{4}+q^{2}$, $~{}~{}p\not\equiv 1\mod 8$,
6. (f)
$X^{4}-\sqrt{q}X^{3}+qX^{2}-q^{3/2}X+q^{2}$, $~{}~{}p\not\equiv 1\mod 10$,
7. (g)
$X^{4}-qX^{2}+q^{2}$, $~{}~{}p\not\equiv 1\mod 12$.
3. (3)
Dimension 3 _(E.Nart, C.Ritzenthaler and S.Haloui[11, 12])_
1. (a)
$X^{6}+\sqrt{q}X^{5}+qX^{4}+q^{3/2}X^{3}+q^{2}X^{2}+q^{5/2}X+q^{3}$,
$~{}~{}p\not\equiv 1,~{}2,~{}4\mod 7$,
2. (b)
$X^{6}+q^{3/2}X^{3}+q^{3}$, $~{}~{}p\not\equiv 1~{},~{}4,~{}7\mod 9$,
3. (c)
$X^{6}-\sqrt{q}X^{5}+qX^{4}-q^{3/2}X^{3}+q^{2}X^{2}-q^{5/2}X+q^{3}$,
$~{}~{}p\not\equiv 1,~{}9,~{}11\mod 14$,
4. (d)
$X^{6}-q^{3/2}X^{3}+q^{3}$, $~{}~{}p\not\equiv 1,7,13\mod 18$.
4. (4)
Dimension 4 _( S.Haloui, V.Singh[13])_
1. (a)
$(X^{4}+\sqrt{q}X^{3}+qX^{2}+q^{3/2}X+q^{2})^{2}$, $~{}~{}p\equiv 1\mod 5$,
2. (b)
$(X^{4}+q^{2})^{2}$, $~{}~{}p\equiv 1\mod 8$,
3. (c)
$(X^{4}-\sqrt{q}X^{3}+qX^{2}-q^{3/2}X+q^{2})^{2}$, $~{}~{}p\equiv 1\mod 10$,
4. (d)
$(X^{4}-qX^{2}+q^{2})^{2}$, $~{}~{}p\equiv 1\mod 12$,
5. (e)
$X^{8}-\sqrt{q}X^{7}+q^{3/2}X^{5}-q^{2}X^{4}+q^{5/2}X^{3}-q^{7/2}X+q^{4}$,
$p\not\equiv 1\mod 15$,
6. (f)
$X^{8}+q^{4}$, $p\not\equiv 1\mod 16$,
7. (g)
$X^{8}-qX^{6}+q^{2}X^{4}-q^{3}X^{2}+q^{4}$, $p\not\equiv 1\mod 20$,
8. (h)
$X^{8}-q^{2}X^{4}+q^{4}$, $p\not\equiv 1\mod 24$,
9. (i)
$X^{8}+\sqrt{q}X^{7}-q^{3/2}X^{5}-q^{2}X^{4}-q^{5/2}X^{3}+q^{7/2}X+q^{4}$,
$p\not\equiv 1\mod 30$.
5. (5)
Dimension 5
1. (a)
$X^{10}+\sqrt{q}X^{9}+qX^{8}+q^{3/2}X^{7}+q^{2}X^{6}+q^{5/2}X^{5}+q^{3}X^{4}+q^{7/2}X^{3}+q^{4}X^{2}+q^{9/2}X+q^{5}$,
$~{}~{}p\not\equiv 1,3,4,5,9\mod 11$,
2. (b)
$X^{10}-\sqrt{q}X^{9}+qX^{8}-q^{3/2}X^{7}+q^{2}X^{6}-q^{5/2}X^{5}+q^{3}X^{4}-q^{7/2}X^{3}+q^{4}X^{2}-q^{9/2}X+q^{5}$,
$~{}~{}p\not\equiv 1,3,5,9,15\mod 22$.
6. (6)
Dimension 6
1. (a)
$(X^{6}+\sqrt{q}X^{5}+qX^{4}+q^{3/2}X^{3}+q^{2}X^{2}+q^{5/2}X+q^{3})^{2}$,
$~{}~{}p\equiv 1~{},~{}2,~{}4\mod 7$,
2. (b)
$(X^{6}+q^{3/2}X^{3}+q^{3})^{2}$, $~{}~{}p\equiv 1~{},~{}4,~{}7\mod 9$,
3. (c)
$(X^{6}-\sqrt{q}X^{5}+qX^{4}-q^{3/2}X^{3}+q^{2}X^{2}-q^{5/2}X+q^{3})^{2}$,
$~{}~{}p\equiv 1~{},~{}9,~{}11\mod 14$,
4. (d)
$(X^{6}-q^{3/2}X^{3}+q^{3})^{2}$, $~{}~{}p\equiv 1,7,13\mod 18$,
5. (e)
$X^{12}+\sqrt{q}X^{11}+qX^{10}+q^{3/2}X^{9}+q^{2}X^{8}+q^{5/2}X^{7}+q^{3}X^{6}+q^{7/2}X^{5}+q^{4}X^{4}+q^{9/2}X^{3}+q^{5}X^{2}+q^{11/2}X+q^{6}$,
$~{}~{}p\not\equiv 1,3,9\mod 13$,
6. (f)
$X^{12}-\sqrt{q}X^{11}+q^{3/2}X^{9}-q^{2}X^{8}+q^{3}X^{6}-q^{4}X^{4}+q^{9/2}X^{3}-q^{11/2}X+q^{6}$,
$~{}~{}p\not\equiv 1,4,16\mod 21$,
7. (g)
$X^{12}-\sqrt{q}X^{11}+qX^{10}-q^{3/2}X^{9}+q^{2}X^{8}-q^{5/2}X^{7}+q^{3}X^{6}-q^{7/2}X^{5}+q^{4}X^{4}-q^{9/2}X^{3}+q^{5}X^{2}-q^{11/2}X+q^{6}$,
$~{}~{}p\not\equiv 1,3,9\mod 26$,
8. (h)
$X^{12}-qX^{10}+q^{2}X^{8}-q^{3}X^{6}+q^{4}X^{4}-q^{5}X^{2}+q^{6}$,
$~{}~{}p\not\equiv 1,9,25\mod 28$,
9. (i)
$X^{12}-q^{3}X^{6}+q^{6}$,$~{}~{}p\not\equiv 1,13,25\mod 36$,
10. (j)
$X^{12}+\sqrt{q}X^{11}-q^{3/2}X^{9}-q^{2}X^{8}+q^{3}X^{6}-q^{4}X^{4}-q^{9/2}X^{3}+q^{11/2}X+q^{6}$,
$~{}~{}p\not\equiv 1,25,37\mod 42$.
7. (7)
Dimension 7
There is no supersingular simple abelian variety of dimension $7$.
###### Proof.
The proof of this theorem, follows from the straightforward calculations using
Theorem 11.1. ∎
## 13\. Existence of supersingular abelian varieties
There are two important existential questions.
1. (1)
Given a positive integer $d$ and $q=p^{n}$, does there exist a simple
supersingular abelian variety of dimension $d$ over $\mathbb{F}_{q}$?
2. (2)
How many isogeny classes of simple supersingular abelian varieties are there
over $\mathbb{F}_{q}$ for a given dimension?
In this section, we will give partial answers to these questions.
###### Theorem 13.1.
Let $g>2$ be an odd positive integer and $q=p^{n},~{}n$ even. Then the
characteristic polynomial of a supersingular simple abelian variety over
$\mathbb{F}_{q}$ dimension $g$ is irreducible.
###### Proof.
The characteristic polynomials of supersingular simple abelian variety of
dimension $g$ are $P(X)^{e}$, where $e=1$ or $e=2$. When $e=1$, $P(X)$ is a
supersingular Weil polynomial of degree $2g$ and when $e=2$, $P(X)$ is a
supersingular Weil polynomial of degree $g$. Also all non linear Weil
polynomials have even degree. Since $g>2$ and $g$ is odd, $e=2$ is not
possible and hence the theorem follows. ∎
###### Theorem 13.2.
Let $g>2$. Then there is no simple supersingular abelian variety over
$\mathbb{F}_{q}$, with $q=p^{n}$, $n$ even, of dimension $g$ if and only if
$\phi^{-1}(g)$ and $\phi^{-1}(2g)$ are empty sets.
###### Proof.
A supersingular irreducible Weil polynomial of degree $2g$ is given by
$\Phi_{m}^{[\sqrt{q}]}(X)$, where $\phi(m)=2g$. The characteristic polynomials
of supersingular simple abelian variety of dimension $g$ are either
irreducible supersingular Weil polynomials of degree $2g$ or the square of an
irreducible supersingular Weil polynomials of degree $g$. There are no
supersingular irreducible Weil polynomials of degree $g$ and $2g$ if
$\phi^{-1}(g)$ and $\phi^{-1}(2g)$ are empty sets, respectively. Conversely
let $\phi^{-1}(g)$ or $\phi^{-1}(2g)$ be non empty set. If $\phi^{-1}(g)$ is
not empty and $g>2$ this implies $g$ is even. If $\phi(m)=g$, then then by
Theorem 11.1 $(\Phi_{m}^{[\sqrt{q}]}(X))^{e}$ is a characteristic polynomial
of simple supersingular abelian variety over $\mathbb{F}_{q}$ of dimension
$\frac{ge}{2}$, where $e$ the order of $p\mod m+1$ i.e; where $e=1$ or $e=2$.
For $p$ such that order of $p$ modulo $m$ is odd,
$(\Phi_{m}^{[\sqrt{q}]}(X))^{2}$ is a characteristic polynomial of simple
supersingular abelian variety over $\mathbb{F}_{q}$ of dimension $g$.
Similarly if $\phi^{-1}(2g)$ is non empty, for all primes $p$ such that order
of $p$ modulo $m$ is even, and repeating the argument above, we get
$\Phi_{m}^{[\sqrt{q}]}(X)$ is characteristic polynomial of a simple
supersingular abelian variety of dimension $g$. Hence the theorem follows. ∎
###### Remark 13.3.
Given an integer $m>2$, there are infinitely many primes which have an even
order modulo $m$ and there are infinitely many primes which have an odd order
modulo $m$.
###### Theorem 13.4.
Let $g>2$. Then there is no simple supersingular abelian variety over
$\mathbb{F}_{q}$, with $q=p^{n}$, $n$ odd, of dimension $g$ if $\phi^{-1}(g)$
and $\phi^{-1}(2g)$ are all empty sets.
###### Proof.
Let $q=p^{n}$, $n$ odd. By Theorem 3.3 and Remark 3.4, there are no
irreducible supersingular Weil polynomial of degree $2g$ if $\phi(4t)=2g$ with
and $\frac{1}{2}\phi(4t)=2g$ has no solution for $t$. If $2|m$ then we have
$\phi(2m)=2\phi(m)$. Therefore, $\phi(4t)=2g$ and $\phi(4t)=4g$ has solutions
for $t$ if and only if $\phi(2t)=g$ $\phi(2t)=2g$ has solutions for $t$.
Therefore there is no irreducible supersingular Weil polynomial of degree $2g$
if $\phi^{-1}(2g)$ and $\phi^{-1}(g)$ has no solution. Since the
characteristic polynomial of a simple supersingular abelian variety of
dimension $g>2$ is irreducible, the result follows.
∎
Though the complete characterisation of values of inverse of Euler function is
a difficult problem, the above two theorems provide the following partial
result.
###### Corollary 13.5.
If $p$ is prime greater than $2$ such that $2p+1$ is not prime(i.e. $p$ is not
a Sophie Germain prime) then is no simple supersingular abelian variety of
dimension $p$ over finite fields.
Therefore it follows from Theorems 13.2 and 13.4, that for the following
dimensions $g\leq 100$ there are no simple supersingular abelian varieties.
prime : $7,13,17,19,31,37,43,47,61,67,71,73,79,97.$
composite: $25,27,34,38,45,57,62,63,76,77,85,87,91,93,94,95.$
###### Theorem 13.6.
There are infinitely many positive integers $g$ such that there is no simple
supersingular abelian variety of dimension $g$ over finite fields.
###### Proof.
We will show the Theorem using the fact that by there are infinitely many
primes satisfying Corollary 13.5. It is well known that there are infinitely
many primes congruent to $5\mod 6$ which implies $2p+1\equiv 4\mod 6$. Hence
$2p+1$ in this family is not prime and the Theorem follows. ∎
The following result is in contrast to Theorem 13.6.
###### Theorem 13.7.
There are infinitely many positive integers $g$ such that there are simple
supersingular abelian varieties of dimension $g$.
###### Proof.
There are infinitely many positive integers $g$ such that $\phi^{-1}(g)$ or
$\phi^{-1}(2g)$ are not all empty sets. From Theorem 13.2 and Remark 13.3,
there is a supersingular simple abelian variety of dimension $g$ over
$\mathbb{F}_{q}$, where $q$ is a square for infinitely many primes $p$ and
hence this result follows. ∎
Let $G_{q,g}$ denote the number of isogeny classes of supersingular simple
abelian varieties of dimension $g$ over $\mathbb{F}_{q}$ and
$A(m):=\\#\\{x|\phi(x)=m\\}$.
###### Theorem 13.8.
If $g>1$ and $q=p^{n},~{}n$ even, then
$G_{q,g}=A(2g)(o(p,2g)+1)+A(g)(o(p,g)),$
where $o(p,k)=$ the order of $p\mod k$, taken$\mod 2$.
###### Proof.
It follows easily from Theorem 11.1. ∎
###### Theorem 13.9.
If$g>2$ $q=p^{n},~{}n$ odd, then
$G_{q,g}\leq((-1)^{g+1}+1)A(2g)+2\sum_{n\in A(2g)}~{}\omega(n),$
where $\omega(n)$ is the number of distinct prime factors of $n$.
###### Proof.
The proof follows from counting the supersingular irreducible Weil polynomial
of degree $2g$ both in exceptional and normal case from the Theorem 3.3.
∎
## Acknowledgements
We would like to thank Kevin Hutchison, H.G. Rück and Christophe Ritzenthaler
for the useful suggestions.
## References
* [1] W. Waterhouse and J. Milne, “Abelian varieties over finite fields,” _Ann. Sci. Ecole Norm. Sup.(4)_ , vol. 2, pp. 521–560, 1969.
* [2] J. Tate, “Endomorphisms of abelian varieties over finite fields,” _Inventiones mathematicae_ , vol. 2, no. 2, pp. 134–144, 1966.
* [3] ——, “Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda),” _Séminaire Bourbaki vol. 1968/69 Exposés 347-363_ , pp. 95–110, 1971.
* [4] F. Oort, “Subvarieties of moduli spaces,” _Inventiones mathematicae_ , vol. 24, no. 2, pp. 95–119, 1974.
* [5] R. Pierce, _Associative algebras_. Springer Verlag, 1982.
* [6] K. Eisenträger, “The theorem of Honda and Tate,” _http://www.math.psu.edu/eisentra/_.
* [7] J. Serre, _Local fields_. Springer, 1979.
* [8] M. Deuring, “Die typen der multiplikatorenringe elliptischer funktionenkörper,” in _Abhandlungen aus dem mathematischen Seminar der Universität Hamburg_ , vol. 14, no. 1. Springer, 1941, pp. 197–272.
* [9] C. Xing, “On supersingular abelian varieties of dimension two over finite fields,” _Finite Fields and Their Applications_ , vol. 2, no. 4, pp. 407–421, 1996.
* [10] D. Maisner and E. Nart, “Abelian surfaces over finite fields as Jacobians,” _Experimental mathematics_ , vol. 11, no. 3, pp. 321–337, 2002.
* [11] S. Haloui, “The characteristic polynomials of abelian varieties of dimensions 3 over finite fields,” _Journal of Number Theory_ , vol. 130, no. 12, pp. 2745–2752, 2010.
* [12] E. Nart and C. Ritzenthaler, “Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2,” _Finite Fields and their applications_ , vol. 14, no. 3, pp. 676–702, 2008.
* [13] S. Haloui and V. Singh, “The characteristic polynomials of abelian varieties of dimensions 4 over finite fields,” _Arxiv preprint arXiv:1101.5070v1_ , 2011.
|
arxiv-papers
| 2011-10-05T23:15:32 |
2024-09-04T02:49:22.847771
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vijaykumar Singh, Gary McGuire, Alexey Zaytsev",
"submitter": "Vijaykumar Singh Mr.",
"url": "https://arxiv.org/abs/1110.1116"
}
|
1110.1146
|
# Convergence of quantum electrodynamics on the Poincaré group
V. V. Varlamov Siberian State Industrial University, Kirova 42, Novokuznetsk
654007, Russia
###### Abstract
Extended particles are considered in terms of the fields on the Poincaré
group. Dirac like wave equations for extended particles of any spin are
defined on the various homogeneous spaces of the Poincaré group. Free fields
of the spin 1/2 and 1 (Dirac and Maxwell fields) are considered in detail on
the eight-dimensional homogeneous space, which is equivalent to a direct
product of Minkowski spacetime and two-dimensional complex sphere. It is shown
that a massless spin-1 field, corresponding to a photon field, should be
defined within principal series representations of the Lorentz group.
Interaction between spin-1/2 and spin-1 fields is studied in terms of a
trilinear form. An analogue of the Dyson formula for $S$-matrix is introduced
on the eight-dimensional homogeneous space. It is shown that in this case
elements of the $S$-matrix are defined by convergent integrals.
Keywords: extended particles, fields on the Poincaré group, homogeneous
spaces, wave equations, quantum electrodynamics
PACS numbers: 02.30.Gp, 02.60.Lj, 03.65.Pm, 12.20.-m
## 1 Introduction
It is well-known that the representation of a particle as an idealized point
has been used in physics for a long time. Historically, such a description
follows from classical (celestial) mechanics, where the distances between
explored objects are much greater than their sizes. However, in quantum field
theory a point-like particle description meets with some conceptual
difficulties. For example, ultra-violet divergences follow directly from the
point-like description. Refusing from the point-like idealization, we come to
_extended objects_.
Our consideration based on the concept of generalized wave functions
introduced by Ginzburg and Tamm in 1947 [24], where the wave function depends
both coordinates $x_{\mu}$ and additional internal variables $u_{\mu}$ which
describe spin of the particle, $\mu=0,1,2,3$. In 1955, Finkelstein showed [18]
that elementary particles models with internal degrees of freedom can be
described on manifolds larger then Minkowski spacetime (homogeneous spaces of
the Poincaré group). The quantum field theories on the Poincaré group were
discussed in the papers [38, 35, 12, 5, 36, 61, 39, 16, 25, 29]. A
consideration of the field models on the homogeneous spaces leads naturally to
a generalization of the concept of wave function (fields on the Poincaré
group). The general form of these fields relates closely with the structure of
the Lorentz and Poincaré group representations [20, 43, 11, 25] and admits the
following factorization $f(x,{\bf z})=\phi^{n}({\bf z})\psi_{n}(x)$, where
$x\in T_{4}$ and $\phi^{n}({\bf z})$ form a basis in the representation space
of the Lorentz group. At this point, four parameters $x^{\mu}$ correspond to
position of the point-like object, whereas remaining six parameters ${\bf
z}\in\operatorname{{\bf Spin}}_{+}(1,3)$ define orientation in quantum
description of orientable (extended) object [26, 27] (see also [33]). It is
obvious that the point-like object has no orientation, therefore, orientation
is an intrinsic property of the extended object. Taking it into account, we
come to consideration of physical quantity as an extended object, the
generalized wave function of which is described by the field
$\boldsymbol{\psi}(\boldsymbol{\alpha})\equiv\langle
x,g\,|\boldsymbol{\psi}\rangle$
on the homogeneous space of some orthogonal group $\operatorname{SO}(p,q)$,
where $x\in T_{n}$ (position) and $g\in\operatorname{{\bf Spin}}_{+}(p,q)$
(orientation), $n=p+q$. So, in [54, 55] Segal and Zhou proved convergence of
quantum field theory, in particular, quantum electrodynamics, on the
homogeneous space $R^{1}\times S^{3}$ of the conformal group
$\operatorname{SO}(2,4)$, where $S^{3}$ is the three-dimensional real sphere.
On the other hand, measurements in quantum field theory lead to extended
objects. As is known, loop divergences emerging in the Green functions in
quantum field theory originate from correspondence of the Green functions to
_unmeasurable_ (and hence unphysical) point-like quantities. This is because
no physical quantity can be measured in a point, but in a region, the size of
which (or ‘diameter’111For example, this diameter can be defined by a
characteristic rest-frame kernel dimension $\sim\;\hbar/mc$ (Compton
wavelength). The magnitude of the ‘kernel’ radius of extended particles was
first proposed by de Broglie [15], see also [60, 52]. of the extended object)
is constrained by the resolution of measuring equipment [2]. Ordinary quantum
field theory defines the field function $\phi(x)$ as a scalar product of the
state vector of the system and the state vector corresponding to the
localization at the point:
$\phi(x)\equiv\langle x|\phi(x)\rangle.$
This field describes the point-like object. On the other hand, _resolution-
dependent fields_
$\phi_{a}(x)\equiv\langle x,a;g|\phi(x)\rangle,$
introduced in the work [2], describe in essence extended objects, here
$\langle x,a;g|$ is the bra-vector corresponding to localization of the
measuring device around the point with the spatial resolution $a$, $g$ labels
the apparatus function of the equipment (an aperture). It is easy to see that
the resolution-dependent fields have the same mathematical structure as the
fields on the Poincaré group (more generally, _fields on the groups_).
The present paper is organized as follows. Basic facts concerning fields on
the Poincaré group are considered in the section 2. Dirac-like wave equations
for extended objects are presented in the section 3. Solutions for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of spin 1/2 and 1 (Dirac and
Maxwell fields) are given in terms of associated hyperspherical functions
defined on the two-dimensional complex sphere, where
$\mathfrak{g}\in\operatorname{{\bf Spin}}_{+}(1,3)$. Interaction between these
fields is introduced in the section 4 in terms of a trilinear form.
## 2 Fields on the Poincaré group
As it mentioned above, Ginzburg and Tamm proposed to consider new internal
continuous variables $u_{\mu}$. It allows one to generalize the Klein-Gordon
wave equation. The Ginzburg-Tamm equation has the form
$\left[\Box-m^{2}+\frac{\beta}{2}M_{\mu\nu}M_{\mu\nu}\right]\Psi(x_{\rho},u_{\gamma})=0,$
(1)
where
$M_{\mu\nu}=-i\left(u_{\mu}\frac{\partial}{\partial
u_{\nu}}-u_{\nu}\frac{\partial}{\partial
u_{\mu}}\right)\quad(\mu,\nu=0,1,2,3)$
are rotation generators of the four-dimensional space, the constant $\beta$ is
analogous to the momentum $J$ in the rotator hamiltonian ${\cal
H}=(2J)^{-1}L_{ik}L_{ik}$, where $L_{ik}$ are generators of the Lorentz group,
$m^{2}$ is a constant. The wave function $\Psi$ depends on coordinates
$x_{\mu}$ of mass centre of the particle and internal variables $u_{\mu}$. The
4-vector $u_{\mu}$ is a 4-vector of relative motion of the structural
component of the particle around its mass centre. The wave function $\Psi$ can
be factored:
$\Psi(x_{\mu},u_{\mu})=\Psi(x_{\mu})\Phi(u_{\mu}).$
Then the Ginzburg-Tamm equation is decomposed in the following two equations:
$(\Box-m^{2}+\lambda_{1}\beta)\Psi(x_{\mu})=0,\quad
L_{1}\Phi(u_{\mu})=\lambda_{1}\Phi(u_{\mu}).$ (2)
At this point, the function $\Phi(u_{\mu})$ is an eigenfunction of the
operator $L_{1}$,
$L_{1}=M_{\mu\nu}M_{\mu\nu}/2,\quad L_{1}\Phi=\lambda_{1}\Phi.$
The first equation from (2) has the following solution:
$\Psi(x_{\mu})=C\exp(-im_{0}t),$
where
$m^{2}_{0}=m^{2}-\lambda_{1}\beta.$
Thus, a mass spectrum, described by the equation (1), is defined by an
eigenvalue of the operator $L_{1}$. The mass operator $L_{1}$ in spherical
coordinates on the one-sheeted hyperboloid $H^{3}$,
$\begin{array}[]{ccl}u_{0}&=&r\sinh\chi;\\\
u_{1}&=&r\cosh\chi\sin\theta\cos\phi;\end{array}\quad\begin{array}[]{ccl}u_{2}&=&r\cosh\chi\sin\theta\sin\phi;\\\
u_{3}&=&r\cosh\chi\cos\theta,\end{array}$
is defined by an expression
$L_{1}=\frac{1}{r}\frac{\partial}{\partial
r}\left(r^{3}\frac{\partial}{\partial
r}\right)-\frac{1}{\cosh^{2}\chi}\frac{\partial}{\partial\chi}\left(\cosh^{2}\chi\frac{\partial}{\partial\chi}\right)-\\\
-\frac{1}{\cosh^{2}\chi}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]-\left(r\frac{\partial}{\partial
r}\right)^{2}-2r\frac{\partial}{\partial r}.$ (3)
_The hyperboloid $H^{3}$ is a homogeneous space of the Lorentz group_.
Solutions of the Ginzburg-Tamm equation have the form of the fields on the
Poincaré group:
$\Psi(x_{\mu},u_{\mu})=C\exp(-im_{0}t)Y_{l,m}(\theta,\phi)P^{j}_{l}(\tanh\chi),$
where $Y_{l,m}(\theta,\phi)$ is a spherical function, and
$P^{j}_{l}(\tanh\chi)$ is a Legendre polynomial.
Further, fields on the Poincaré group present itself a natural generalization
of the concept of wave function. These fields (generalized wave functions)
were introduced independently by several authors [24, 8, 70, 56] mainly in
connection with constructing relativistic wave equations. In essence, this
generalization consists in replacing the Minkowski space by a larger space on
which the Poincaré group acts. If this action is to be transitive, one is lead
to consider the homogeneous spaces of the Poincaré group. All the homogeneous
spaces of this type were listed by Finkelstein [18] and by Bacry and Kihlberg
[6] and the fields on these spaces were considered in the works [38, 7, 45,
34, 35, 61, 25].
A homogeneous space ${\cal M}$ of a group $G$ has the following properties:
a) It is a topological space on which the group $G$ acts continuously, that
is, let $y$ be a point in ${\cal M}$, then $gy$ is defined and is again a
point in ${\cal M}$ ($g\in G$).
b) This action is transitive, that is, for any two points $y_{1}$ and $y_{2}$
in ${\cal M}$ it is always possible to find a group element $g\in G$ such that
$y_{2}=gy_{1}$.
There is a one-to-one correspondence between the homogeneous spaces of $G$ and
the coset spaces of $G$. Let $H_{0}$ be a maximal subgroup of $G$ which leaves
the point $y_{0}$ invariant, $gy_{0}=y_{0}$, $g\in H_{0}$, then $H_{0}$ is
called the stabilizer of $y_{0}$. Representing now any group element of $G$ in
the form $g=g_{c}g_{0}$, where $g_{0}\in H_{0}$ and $g_{c}\in G/H_{0}$, we see
that, by virtie of the transitivity property, any point $y\in{\cal M}$ can be
given by $y=g_{c}g_{0}y_{0}=g_{c}y$. Hence it follows that the elements
$g_{c}$ of the coset space give a parametrization of ${\cal M}$. The mapping
${\cal M}\leftrightarrow G/H_{0}$ is continuous since the group multiplication
is continuous and the action on ${\cal M}$ is continuous by definition. The
stabilizers $H$ and $H_{0}$ of two different points $y$ and $y_{0}$ are
conjugate, since from $H_{0}g_{0}=g_{0}$, $y_{0}=g^{-1}y$, it follows that
$gH_{0}g^{-1}y=y$, that is, $H=gH_{0}g^{-1}$.
Returning to the Poincaré group ${\cal P}$, we see that the enumeration of the
different homogeneous spaces ${\cal M}$ of ${\cal P}$ amounts to an
enumeration of the subgroups of ${\cal P}$ up to a conjugation. Following to
Finkelstein, we require that ${\cal M}$ always contains the Minkowski space
$\bb R^{1,3}$ which means that four parameters of ${\cal M}$ can be denoted by
$x\,(x^{\mu})$. This means that the stabilizer $H$ of a given point in ${\cal
M}$ can never contain an element of the translation subgroup of ${\cal P}$.
Thus, the stabilizer must be a subgroup of the homogeneous Lorentz group
$\mathfrak{G}_{+}$.
In such a way, studying different subgroups of $\mathfrak{G}_{+}$, we obtain a
full list of homogeneous spaces ${\cal M}={\cal P}/H$ of the Poincaré group.
In the present paper we restrict ourselves by a consideration of the following
four homogeneous spaces:
$\displaystyle{\cal M}_{10}$ $\displaystyle=$ $\displaystyle\bb
R^{1,3}\times\mathfrak{L}_{6},\quad H=0;$ $\displaystyle{\cal M}_{8}$
$\displaystyle=$ $\displaystyle\bb R^{1,3}\times S^{2},\quad H=\Omega_{\psi};$
$\displaystyle{\cal M}_{7}$ $\displaystyle=$ $\displaystyle\bb R^{1,3}\times
H^{3},\quad H=\operatorname{SU}(2);$ $\displaystyle{\cal M}_{6}$
$\displaystyle=$ $\displaystyle\bb R^{1,3}\times S^{2},\quad
H=\\{\Omega_{\psi},\Omega_{\tau},\Omega_{\epsilon}\\}.$
Hence it follows that a group manifold of the Poincaré group, ${\cal
M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$, is a maximal homogeneous space of
${\cal P}$, $\mathfrak{L}_{6}$ is a group manifold of the Lorentz group. The
fields on the manifold ${\cal M}_{10}$ were considered by Lurçat [38]. These
fields depend on all the ten parameters of ${\cal P}$:
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\mathfrak{g}_{1},\mathfrak{g}_{2},\mathfrak{g}_{3},\mathfrak{g}_{4},\mathfrak{g}_{5},\mathfrak{g}_{6}),$
where an explicit form of $\psi(x)$ is given by the exponentials, and the
functions $\psi(\mathfrak{g})$ are expressed via the generalized
hyperspherical functions $\mathfrak{M}^{l}_{mn}(\mathfrak{g})$. As is known,
the universal covering of the proper Poincaré group is isomorphic to a
semidirect product $\operatorname{SL}(2;\bb C)\odot T_{4}$ or
$\operatorname{{\bf Spin}}_{+}(1,3)\odot T_{4}$. Since the group $T_{4}$ is
Abelian, then all its representations are one-dimensional. Thus, all the
finite-dimensional representations of the proper Poincaré group in essence are
equivalent to the representations $\mathfrak{C}$ of the group
$\operatorname{{\bf Spin}}_{+}(1,3)$. In the case of finite-dimensional
representations of $\operatorname{{\bf Spin}}_{+}(1,3)$,
$T_{\mathfrak{g}}q(\xi,\overline{\xi})=(\gamma\xi+\delta)^{l_{0}+l_{1}-1}\overline{(\gamma\xi+\delta)}^{l_{0}-l_{1}+1}q\left(\frac{\alpha\xi+\beta}{\gamma\xi+\delta};\frac{\overline{\alpha\xi+\beta}}{\overline{\gamma\xi+\delta}}\right),$
(4)
basic functions (matrix elements) of the finite-dimensional representation of
${\cal P}$ have the form
$t^{l}_{mn}(\boldsymbol{\alpha})=e^{-imx}\mathfrak{M}^{l}_{mn}(\mathfrak{g}),$
where
$\mathfrak{M}^{l}_{mn}(\mathfrak{g})=e^{-im\varphi^{c}}\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})e^{-in\psi^{c}}.$
_Hyperspherical functions_ $\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})$ can be
expressed via the hypergeometric functions [65]. So, at $m\geq n$ and $m\geq
k,\;k\geq n$ we have
$\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})=i^{m-n}\sqrt{\frac{\Gamma(l+m+1)\Gamma(l-n+1)}{\Gamma(l-m+1)\Gamma(l+n+1)}}\cos^{2l}\frac{\theta}{2}\cosh^{2l}\frac{\tau}{2}\times\\\
\times\sum^{l}_{k=-l}\tan^{m-k}\frac{\theta}{2}\tanh^{k-n}\frac{\tau}{2}\times\\\
\times\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}m-l,-k-l\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{-\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}k-l,-n-l\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}k-n+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}.$
(5)
There also exist three expressions of the hypergeometric type for the
functions $\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})$ with the index values $m\geq
n,\;m\geq k,\;n\geq k$, $n\geq m,\;k\geq m,\;n\geq k$ and $n\geq m,\;k\geq
m,\;k\geq n$. For the unitary representations, that is, in the case of
principal series representations of the group $\operatorname{SO}_{0}(1,3)$,
there exists an analogue of the formula (4),
$V_{a}f(z)=(a_{12}z+a_{22})^{\frac{\lambda}{2}+i\frac{\rho}{2}-1}\overline{(a_{12}z+a_{22})}^{-\frac{\lambda}{2}+i\frac{\rho}{2}-1}f\left(\frac{a_{11}z+a_{21}}{a_{12}z+a_{22}}\right),$
(6)
where $f(z)$ is a measurable function of the Hilbert space $L_{2}(Z)$,
satisfying the condition $\int|f(z)|^{2}dz<\infty$, $z=x+iy$. A totality of
all representations $a\rightarrow T^{\alpha}$, corresponding to all possible
pairs $\lambda$, $\rho$, is called a principal series of representations of
the group $\operatorname{SO}_{0}(1,3)$ and denoted as
$\mathfrak{S}_{\lambda,\rho}$. At this point, a comparison of (6) with the
formula (4) for the spinor representation $\mathfrak{S}_{l}$ shows that the
both formulas have the same structure; only the exponents at the factors
$(a_{12}z+a_{22})$, $\overline{(a_{12}z+a_{22})}$ and the functions $f(z)$ are
different. In the case of spinor representations the functions $f(z)$ are
polynomials $p(z,\bar{z})$ in the spaces $\operatorname{Sym}_{(k,r)}$, and in
the case of a representation $\mathfrak{S}_{\lambda,\rho}$ of the principal
series $f(z)$ are functions from the Hilbert space $L_{2}(Z)$. Further, we
know that a representation $S_{l}$ of the group $\operatorname{SU}(2)$ is
realized in terms of the functions
$t^{l}_{mn}(u)=e^{-im\varphi}P^{l}_{mn}(\cos\theta)e^{-in\psi}$ [68]. It is
well known that the representation $S_{k}$ of $\operatorname{SU}(2)$ is
contained in $\mathfrak{S}_{\lambda,\rho}$ not more then one time [43]. At
this point, $S_{k}$ is contained in $\mathfrak{S}_{\lambda,\rho}$, when
$\frac{\lambda}{2}$ is one from the numbers $-k,-k+1,\ldots,k$. Matrix
elements of the principal series representations of the group
$\operatorname{SO}_{0}(1,3)$ have the form [65]
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}=e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\times\\\\[5.69046pt]
\sum^{l_{0}}_{t=-l_{0}}i^{m-t}\sqrt{\Gamma(l_{0}-m+1)\Gamma(l_{0}+m+1)\Gamma(l_{0}-t+1)\Gamma(l_{0}+t+1)}\times\\\
\cos^{2l_{0}}\frac{\theta}{2}\tan^{m-t}\frac{\theta}{2}\times\\\\[5.69046pt]
\sum^{\min(l_{0}-m,l_{0}+t)}_{j=\max(0,t-m)}\frac{i^{2j}\tan^{2j}\frac{\theta}{2}}{\Gamma(j+1)\Gamma(l_{0}-m-j+1)\Gamma(l_{0}+t-j+1)\Gamma(m-t+j+1)}\times\\\\[5.69046pt]
\sqrt{\Gamma(\tfrac{1}{2}+i\rho-n)\Gamma(\tfrac{1}{2}+i\rho+n)\Gamma(\tfrac{1}{2}+i\rho-t)\Gamma(\tfrac{1}{2}+i\rho+t)}\cosh^{-1+2i\rho}\frac{\tau}{2}\tanh^{n-t}\frac{\tau}{2}\times\\\\[5.69046pt]
\sum^{\infty}_{s=\max(0,t-n)}\frac{\tanh^{2s}\frac{\tau}{2}}{\Gamma(s+1)\Gamma(\frac{1}{2}+i\rho-
n-s)\Gamma(\frac{1}{2}+i\rho+t-s)\Gamma(n-t+s+1)},$ (7)
where $l_{0}=\left|\frac{\lambda}{2}\right|$ and $\frac{\lambda}{2}$ is one
from the numbers $-k,-k+1,\ldots,k$. It is obvious that
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ cannot be
attributed as matrix elements to single irreducible representation. From the
latter expression it follows that relativistic spherical functions
$f(\mathfrak{g})$ of the principal series can be defined by means of the
function
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=e^{-m(\epsilon+i\varphi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\cos\theta^{c})e^{-n(\varepsilon+i\psi)},$
(8)
where
$\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\cos\theta^{c})=\sum^{l_{0}}_{t=-l_{0}}P^{l_{0}}_{mt}(\cos\theta)\mathfrak{P}^{-\frac{1}{2}+i\rho}_{tn}(\cosh\tau).$
In turn, the functions
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ can be expressed
via the hypergeometric functions. So, at $m\geq t,\;t\geq n$ we have [65]
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=i^{m-n}e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\sqrt{\frac{\Gamma(l_{0}+m+1)\Gamma(i\rho-n+\frac{1}{2})}{\Gamma(l_{0}-m+1)\Gamma(i\rho+n+\frac{1}{2})}}\times\\\
\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1+2i\rho}\frac{\tau}{2}\sum^{l_{0}}_{t=-l_{0}}\tan^{m-t}\frac{\theta}{2}\tanh^{t-n}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}m-l_{0},-t-l_{0}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m-t+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{-\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}t-i\rho+\frac{1}{2},-n-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}t-n+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}.$
(9)
There also exist three expressions of the hypergeometric type for the
functions $\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ with
the index values $m\geq t,\;n\geq t$, $t\geq m,\;n\geq t$ and $t\geq m,\;t\geq
n$.
The following eight-dimensional homogeneous space ${\cal M}_{8}=\bb
R^{1,3}\times S^{2}$ is a direct product of the Minkowski space $\bb R^{1,3}$
and the complex two-sphere $S^{2}_{c}$. In this case the stabilizer $H$
consists of the subgroup $\Omega^{c}_{\psi}$ of the diagonal matrices
$\begin{pmatrix}e^{\frac{i\psi^{c}}{2}}&0\\\
0&e^{-\frac{i\psi^{c}}{2}}\end{pmatrix}$. Bacry and Kihlberg [6] claimed that
the space ${\cal M}_{8}$ is the most suitable for a description of both half-
integer and integer spins. The fields, defined in ${\cal M}_{8}$, depend on
the eight parameters of ${\cal P}$:
$\langle
x,\varphi^{c},\theta^{c}|\boldsymbol{\psi}\rangle=\psi(x)\psi(\varphi^{c},\theta^{c})=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\varphi,\epsilon,\theta,\tau),$
(10)
where the functions $\psi(\varphi^{c},\theta^{c})$ are expressed via the
associated hyperspherical functions defined on the surface of the complex two-
sphere $S^{2}_{c}$. The sphere $S^{c}_{2}$ can be constructed from the
quantities $z_{k}=x_{k}+iy_{k}$, $\overset{\ast}{z}_{k}=x_{k}-iy_{k}$
$(k=1,2,3)$ as follows:
$S^{c}_{2}:\;z^{2}_{1}+z^{2}_{2}+z^{2}_{3}={\bf x}^{2}-{\bf y}^{2}+2i{\bf
x}{\bf y}=r^{2}.$ (11)
The complex conjugate (dual) sphere $\dot{S}^{c}_{2}$ is
$\dot{S}^{c}_{2}:\;\overset{\ast}{z}_{1}{}^{2}+\overset{\ast}{z}_{2}{}^{2}+\overset{\ast}{z}_{3}{}^{2}={\bf
x}^{2}-{\bf y}^{2}-2i{\bf x}{\bf y}=\overset{\ast}{r}{}^{2}.$ (12)
For more details about the two-dimensional complex sphere see [30, 31, 59].
The surface of $S^{c}_{2}$ is homeomorphic to the space of the pairs
$(z_{1},z_{2})$, where $z_{1}$ and $z_{2}$ are the points of a _complex
projective line_ , $z_{1}\neq z_{2}$. This space is a homogeneous space of the
Lorentz group [21]. In Euler parametrization we have
$(z_{1},z_{2})\sim(\varphi^{c},\theta^{c})=(\varphi-i\epsilon,\theta-i\tau)$.
It is well-known that both quantities ${\bf x}^{2}-{\bf y}^{2}$, ${\bf x}{\bf
y}$ are invariant with respect to the Lorentz transformations, since a surface
of the complex sphere is invariant (Casimir operators of the Lorentz group are
constructed from such quantities). Moreover, since the real and imaginary
parts of the complex two-sphere transform like the electric and magnetic
fields, respectively, the invariance of ${\bf z}^{2}\sim({\bf E}+i{\bf
B})^{2}$ under proper Lorentz transformations is evident. At this point, the
quantities ${\bf x}^{2}-{\bf y}^{2}$, ${\bf x}{\bf y}$ are similar to the well
known electromagnetic invariants $E^{2}-B^{2}$, ${\bf E}{\bf B}$. This
intriguing relationship between the Laplace-Beltrami operators, Casimir
operators of the Lorentz group and electromagnetic invariants
$E^{2}-B^{2}\sim{\bf x}^{2}-{\bf y}^{2}$, ${\bf E}{\bf B}\sim{\bf x}{\bf y}$
leads naturally to a Riemann-Silberstein representation of the electromagnetic
field (see, for example, [69, 57, 10]). Further, associated hyperspherical
functions
$\mathfrak{M}^{m}_{l}(\varphi^{c},\theta^{c},0)=e^{-im\varphi^{c}}\mathfrak{Z}^{m}_{l}(\cos\theta^{c}),$
(13)
where
$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})=\sum^{l}_{k=-l}P^{l}_{mk}(\cos\theta)\mathfrak{P}^{k}_{l}(\cosh\tau),$
are defined on the surface of the two-dimensional complex sphere (11). In
turn, the functions
$\mathfrak{M}^{\dot{m}}_{\dot{l}}(\dot{\varphi}^{c},\theta^{c},0)=e^{i\dot{m}\dot{\varphi}^{c}}\mathfrak{Z}^{\dot{m}}_{\dot{l}}(\cos\dot{\theta}^{c})$
are defined on the surface of the dual sphere (12). Explicit expressions of
the hypergeometric type for the functions
$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})$ and
$\mathfrak{Z}^{\dot{m}}_{\dot{l}}(\cos\dot{\theta}^{c})$ follow directly from
the previous expressions (5) and (9) at $n=0$. In the case of the principal
series representations of the group $\operatorname{SO}_{0}(1,3)$ we have
$\mathfrak{Z}^{m}_{-\frac{1}{2}+i\rho,l_{0}}(\cos\theta^{c})=\sum^{l_{0}}_{t=-l_{0}}P^{l_{0}}_{mt}(\cos\theta)\mathfrak{P}^{t}_{-\frac{1}{2}+i\rho}(\cosh\tau),$
(14)
where $\mathfrak{P}^{t}_{-\frac{1}{2}+i\rho}(\cosh\tau)$ are conical functions
(see [9]).
In turn, a seven-dimensional homogeneous space ${\cal M}_{7}=\bb R^{1,3}\times
H^{3}$ is a direct product of $\bb R^{1,3}$ and a three-dimensional timelike
(two-sheeted) hyperboloid $H^{3}$. The stabilizer $H$ consists of the subgroup
of three-dimensional rotations, $\operatorname{SU}(2)$. Quantum field theory
on the space ${\cal M}_{7}$ was studied by Boyer and Fleming [12]. They showed
that the fields built over ${\cal M}_{7}$ are in general nonlocal and become
local only when the finite dimensional representations of the Lorentz group
are used. It is easy to see that the fields $\boldsymbol{\psi}\in{\cal M}_{7}$
depend on the seven parameters of the Poincaré group:
$\langle
x,\tau,\epsilon,\varepsilon|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\tau,\epsilon,\varepsilon),$
where the functions $\psi(\tau,\epsilon,\varepsilon)$ are expressed via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{l}_{mn}(\cosh\tau)$ in the case of
finite dimensional representations, and also via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{-\frac{1}{2}+i\rho}_{mn}(\cosh\tau)$
in the case of principal series of unitary representations, and via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{l,\pm}_{mn}(\cosh\tau)$ in the
case of the discrete representation series of the subgroup
$\operatorname{SU}(1,1)$. The spherical function
$\mathfrak{P}^{l}_{mn}(\cosh\tau)$ on the group $\operatorname{SU}(1,1)$ has
the form
$\mathfrak{P}^{l}_{mn}(\cosh\tau)=\sqrt{\Gamma(l-n+1)\Gamma(l+n+1)\Gamma(l-m+1)\Gamma(l+m+1)}\times\\\
\cosh^{2l}\frac{\tau}{2}\tanh^{n-m}\frac{\tau}{2}\times\\\
\sum^{\min(l-n,l+m)}_{s=\max(0,m-n)}\frac{\tanh^{2s}\frac{\tau}{2}}{\Gamma(s+1)\Gamma(l-n-s+1)\Gamma(n-m+s+1)\Gamma(l+m-s+1)}.$
(15)
Further, a six-dimensional space ${\cal M}_{6}=\bb R^{1,3}\times S^{2}$ is a
minimal homogeneous space of the Poincaré group, since the real two-sphere
$S^{2}$ has a minimal possible dimension among the homogeneous spaces of the
Lorentz group. In this case, the stabilizer $H$ consists of the subgroup
$\Omega^{c}_{\psi}$ and the subgroups $\Omega_{\tau}$ and $\Omega_{\epsilon}$
formed by the matrices
$\begin{pmatrix}\cosh\frac{\tau}{2}&\sinh\frac{\tau}{2}\\\
\sinh\frac{\tau}{2}&\cosh\frac{\tau}{2}\end{pmatrix}$ and
$\begin{pmatrix}e^{\frac{\epsilon}{2}}&0\\\
0&e^{-\frac{\epsilon}{2}}\end{pmatrix}$, respectively. Field models on the
homogeneous space ${\cal M}_{6}$ have been considered in the works [36, 39,
16]. In the paper [16] Drechsler considered the real two-sphere as a ‘spin
shell’ $S^{2}_{r=2s}$ of radius $r=2s$, where
$s=0,\frac{1}{2},1,\frac{3}{2},\ldots$. The fields, defined on ${\cal M}_{6}$,
depend on the six parameters of ${\cal P}$:
$\langle
x,\varphi,\theta|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\varphi,\theta),$
where the functions $\psi(\varphi,\theta)$ are expressed via the generalized
spherical functions of the type $e^{-im\varphi}P^{l}_{m0}(\cos\theta)$ or via
the Wigner $D$-functions, here
$P^{l}_{mn}(\cos\theta)=e^{-i(m\varphi+n\psi)}i^{m-n}\sqrt{\Gamma(l-m+1)\Gamma(l+m+1)\Gamma(l-n+1)\Gamma(l+n+1)}\times\\\
\cos^{2l}\frac{\theta}{2}\tan^{m-n}\frac{\theta}{2}\times\\\
\sum^{\min(l-n,l+n)}_{j=\max(0,n-m)}\frac{i^{2j}\tan^{2j}\frac{\theta}{2}}{\Gamma(j+1)\Gamma(l-m-j+1)\Gamma(l+n-j+1)\Gamma(m-n+j+1)}$
(16)
is a spherical function on the group $\operatorname{SU}(2)$.
Further, let ${\cal H}$ be a Hilbert space and let $U(g)$ be a square
integrable representation of a locally compact Lie group $G$ acting
transitively on ${\cal H}$, $\forall\phi\in{\cal H}$, $g\in
G:\,U(g)\phi\in{\cal H}$. Let there exist such a vector
$\mid\psi\rangle\in{\cal H}$ that satisfies the admissibility condition:
$C_{\psi}=||\psi||^{-2}\int_{g\in G}|\langle\mid
U(g)\psi\rangle|^{2}d\mu(g)\;<\;\infty,$
where $d\mu(g)$ is the left-invariant Haar measure on $G$. Then, any vector
$|\phi\rangle\in{\cal H}$ can be represented in the form
$|\phi\rangle=C^{-1}_{\psi}\int_{G}\mid U(g)\psi\rangle d\mu(g)\langle\psi
U^{\ast}(g)\mid\phi\rangle.$
At the restriction of $G$ on the affine subgroup $x^{\prime}=ax+b$ (the group
of translations and dilatations of the real axis) a harmonic analysis on the
group $G$ is reduced to the wavelet-transformation:
$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\psi}}\int\frac{1}{a^{d}}\psi\left(\frac{x-b}{a}\right)\phi_{a}(b)\frac{dad^{d}b}{a},$
$\displaystyle\phi_{a}(b)$ $\displaystyle=$
$\displaystyle\int\frac{1}{a^{d}}\overline{\psi\left(\frac{x-b}{a}\right)}\phi(x)d^{d}x,$
$\displaystyle C_{\psi}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}|\widetilde{\psi}(ak)|^{2}\frac{da}{a}\;<\;\infty.$
(17)
At this restriction the field $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,g\,|\boldsymbol{\psi}\rangle$ on the group $G$ is reduced to the field
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,a\,|\boldsymbol{\psi}\rangle\sim\langle x,a;g|\phi(x)\rangle$ on the affine
group, where $a$ is a scale factor which can be associated with the resolution
of measuring equipment [2]. It should be noted that further restriction of $G$
on the translation subgroup leads to the restriction of
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,g\,|\boldsymbol{\psi}\rangle$ to the field $\boldsymbol{\psi}(x)=\langle
x\,|\boldsymbol{\psi}\rangle$ describing the point-like object, and the
wavelet-transformation in this case is reduced to the Fourier-transformation.
Thus, the affine group is a minimal group which can be used for description of
the extended object. The restriction of $G$ to the affine subgroup is
equivalent formally to the Faddeev-Popov method considering translation and
dilatation invariance in quantum field calculations related with the
renormalization group, at this point, $a=1/\Lambda$, where $\Lambda$ is a cut-
off parameter. Moreover, in the framework of wavelet-analysis, loop Green
functions emerging in $\phi^{4}$-model, are free from ultra-violet divergences
[1, 2].
## 3 Free fields on the two-dimensional complex sphere
We will start with a more general homogeneous space of the group ${\cal P}$,
${\cal M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$ (group manifold of the
Poincaré group). Let $\mathcal{L}(\boldsymbol{\alpha})$ be a Lagrangian on the
group manifold ${\cal M}_{10}$ (in other words,
$\mathcal{L}(\boldsymbol{\alpha})$ is a 10-dimensional point function), where
$\boldsymbol{\alpha}$ is the parameter set of this group. Then an integral for
$\mathcal{L}(\boldsymbol{\alpha})$ on some 10-dimensional volume $\Omega$ of
the group manifold we will call an action on the Poincaré group:
$A=\int\limits_{\Omega}d\boldsymbol{\alpha}\mathcal{L}(\boldsymbol{\alpha}),$
where $d\boldsymbol{\alpha}$ is a Haar measure on the group ${\cal P}$.
Let $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ be a function on the group manifold
${\cal M}_{10}$ (now it is sufficient to assume that
$\boldsymbol{\psi}(\boldsymbol{\alpha})$ is a square integrable function on
the Poincaré group) and let
$\frac{\partial\mathcal{L}}{\partial\boldsymbol{\psi}}-\frac{\partial}{\partial\boldsymbol{\alpha}}\frac{\partial\mathcal{L}}{\partial\frac{\partial\boldsymbol{\psi}}{\partial\boldsymbol{\alpha}}}=0$
(18)
be Euler-Lagrange equations on ${\cal M}_{10}$ (more precisely speaking, the
equations (18) act on the tangent bundle $T{\cal
M}_{10}=\underset{\boldsymbol{\alpha}\in{\cal
M}_{10}}{\cup}T_{\boldsymbol{\alpha}}{\cal M}_{10}$ of the manifold ${\cal
M}_{10}$, see [4]). Let us introduce a Lagrangian
$\mathcal{L}(\boldsymbol{\alpha})$ depending on the field function
$\boldsymbol{\psi}(\boldsymbol{\alpha})$ as follows
$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}B_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{11}\boldsymbol{\psi}(\boldsymbol{\alpha}),$
where $B_{\nu}$ ($\nu=1,2,\ldots,10$) are square matrices. The number of rows
and columns in these matrices is equal to the number of components of
$\boldsymbol{\psi}(\boldsymbol{\alpha})$, $\kappa$ is a non-null real
constant.
Further, if $B_{11}$ is non-singular, then we can introduce the matrices
$\Gamma_{\mu}=B^{-1}_{11}B_{\mu},\quad\mu=1,2,\ldots,10,$
and represent the Lagrangian $\mathcal{L}(\boldsymbol{\alpha})$ in the form
$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Gamma_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha}),$
(19)
where
$\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})=\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{11}.$
Varying independently $\psi(x)$ and $\overline{\psi}(x)$, we obtain from (19)
in accordance with (18) the following equations:
$\begin{array}[]{ccc}\Gamma_{i}\dfrac{\partial\psi(x)}{\partial
x_{i}}+\kappa\psi(x)&=&0,\\\
\Gamma^{T}_{i}\dfrac{\partial\overline{\psi}(x)}{\partial
x_{i}}-\kappa\overline{\psi}(x)&=&0.\end{array}\quad(i=0,\ldots,3)$ (20)
The matrix $\Gamma_{0}$ in equations (20) can be written in the form (see [20,
3, 48])
$\Gamma_{0}=\text{diag}\left(C^{0}\otimes I_{1},C^{1}\otimes
I_{3},\ldots,C^{s}\otimes I_{2s+1},\ldots\right)$ (21)
for integer spin and
$\Gamma_{0}=\text{diag}\left(C^{\frac{1}{2}}\otimes
I_{2},C^{\frac{3}{2}}\otimes I_{4},\ldots,C^{s}\otimes I_{2s+1},\ldots\right)$
(22)
for half-integer spin, where $C^{s}$ is a spin block. If the spin block
$C^{s}$ has non-null roots, then the particle possesses the spin $s$. The spin
block $C^{s}$ in (21)–(22) consists of the elements
$c^{s}_{\boldsymbol{\tau}\boldsymbol{\tau}^{\prime}}$, where
$\boldsymbol{\tau}_{l_{1},l_{2}}$ and
$\boldsymbol{\tau}_{l^{\prime}_{1},l^{\prime}_{2}}$ are interlocking
irreducible representations of the Lorentz group, that is, such
representations, for which $l^{\prime}_{1}=l_{1}\pm\frac{1}{2}$,
$l^{\prime}_{2}=l_{2}\pm\frac{1}{2}$. At this point, the block $C^{s}$
contains only the elements
$c^{s}_{\boldsymbol{\tau}\boldsymbol{\tau}^{\prime}}$ corresponding to such
interlocking representations $\boldsymbol{\tau}_{l_{1},l_{2}}$,
$\boldsymbol{\tau}_{l^{\prime}_{1},l^{\prime}_{2}}$ which satisfy the
conditions
$|l_{1}-l_{2}|\leq s\leq l_{1}+l_{2},\quad|l^{\prime}_{1}-l^{\prime}_{2}|\leq
s\leq l^{\prime}_{1}+l^{\prime}_{2}.$
The two most full schemes of the interlocking irreducible representations of
the Lorentz group for integer and half-integer spins are shown on the Fig. 1
and Fig. 2.
$\dgARROWLENGTH=0.5em\dgHORIZPAD=1.7em\dgVERTPAD=2.2ex\begin{diagram}$
Fig. 1: Interlocking representation scheme for the fields of integer spin
(Bose-scheme).
$\dgARROWLENGTH=0.5em\dgHORIZPAD=1.7em\dgVERTPAD=2.2ex\begin{diagram}$
Fig. 2: Interlocking representation scheme for the fields of half-integer spin
(Fermi-scheme).
As follows from Fig. 1 the simplest field is the scalar field
$(0,0).$
This field is described by the Fock-Klein-Gordon equation. In its turn, the
simplest field from the Fermi-scheme (Fig. 2) is the electron-positron
(spinor) field corresponding to the following interlocking scheme:
$\dgARROWLENGTH=2.5em\begin{diagram}.$
This field is described by the Dirac equation. Further, the next field from
the Bose-scheme (Fig. 1) is a photon field (Maxwell field) defined within the
interlocking scheme
$\dgARROWLENGTH=2.5em\begin{diagram}.$
This interlocking scheme leads to the Maxwell equations. The fields
$(1/2,0)\oplus(0,1/2)$ and $(1,0)\oplus(0,1)$ (Dirac and Maxwell fields) are
particular cases of fields $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of the type $(l,0)\oplus(0,l)$,
where $\mathfrak{g}\in\operatorname{{\bf Spin}}_{+}(1,3)$. Wave equations for
such fields and their general solutions were found in the works [62, 63, 64].
It is easy to see that the interlocking scheme, corresponding to the Maxwell
field, contains the field of tensor type:
$\left(\frac{1}{2},\frac{1}{2}\right).$
Further, the next interlocking scheme (see Fig. 2)
$\dgARROWLENGTH=2.5em\begin{diagram},$
corresponding to the Pauli-Fierz equations [19], contains a chain of the type
$\dgARROWLENGTH=2.5em\begin{diagram}.$
In such a way we come to wave equations for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of tensor type
$(l_{1},l_{2})\oplus(l_{2},l_{1})$. Wave equations for such fields and their
general solutions were found in the work [67].
Further, varying independently $\psi(\mathfrak{g})$ and
$\overline{\psi}(\mathfrak{g})$ one gets from (19) the following equations:
$\begin{array}[]{ccc}\Gamma_{k}\dfrac{\partial\psi(\mathfrak{g})}{\partial\mathfrak{g}_{k}}+\kappa\psi(\mathfrak{g})&=&0,\\\
\Gamma^{T}_{k}\dfrac{\overline{\psi}(\mathfrak{g})}{\partial\mathfrak{g}_{k}}-\kappa\overline{\psi}(\mathfrak{g})&=&0,\end{array}\quad(k=1,\ldots,6)$
(23)
where
$\psi(\mathfrak{q})=\begin{pmatrix}\psi(\mathfrak{q})\\\
\dot{\psi}(\mathfrak{q})\end{pmatrix},\quad\Gamma_{k}=\begin{pmatrix}0&\Lambda^{\ast}_{k}\\\
\Lambda_{k}&0\end{pmatrix}.$
Non-zero elements of the matrices $\Lambda_{k}$ and $\Lambda^{\ast}_{k}$ have
the form (for more details see [62])
${\Lambda_{1}:\quad\left\\{\begin{array}[]{ccc}a^{k^{\prime}k}_{l-1,l,m-1,m}&=&-\frac{c_{l-1,l}}{2}\sqrt{(l+m)(l+m-1)},\\\
a^{k^{\prime}k}_{l,l,m-1,m}&=&\frac{c_{ll}}{2}\sqrt{(l+m)(l-m+1)},\\\
a^{k^{\prime}k}_{l+1,l,m-1,m}&=&\frac{c_{l+1,l}}{2}\sqrt{(l-m+1)(l-m+2)},\\\
a^{k^{\prime}k}_{l-1,l,m+1,m}&=&\frac{c_{l-1,l}}{2}\sqrt{(l-m)(l-m-1)},\\\
a^{k^{\prime}k}_{l,l,m+1,m}&=&\frac{c_{ll}}{2}\sqrt{(l+m+1)(l-m)},\\\
a^{k^{\prime}k}_{l+1,l,m+1,m}&=&-\frac{c_{l+1,l}}{2}\sqrt{(l+m+1)(l+m+2)}.\end{array}\right.}$
(24)
${\Lambda_{2}:\quad\left\\{\begin{array}[]{ccc}b^{k^{\prime}k}_{l-1,l,m-1,m}&=&-\frac{ic_{l-1,l}}{2}\sqrt{(l+m)(l+m-1)},\\\
b^{k^{\prime}k}_{l,l,m-1,m}&=&\frac{ic_{ll}}{2}\sqrt{(l+m)(l-m+1)},\\\
b^{k^{\prime}k}_{l+1,l,m-1,m}&=&\frac{ic_{l+1,l}}{2}\sqrt{(l-m+1)(l-m+2)},\\\
b^{k^{\prime}k}_{l-1,l,m+1,m}&=&-\frac{ic_{l-1,l}}{2}\sqrt{(l-m)(l-m-1)},\\\
b^{k^{\prime}k}_{l,l,m+1,m}&=&-\frac{ic_{ll}}{2}\sqrt{(l+m+1)(l-m)},\\\
b^{k^{\prime}k}_{l+1,l,m+1,m}&=&\frac{ic_{l+1,l}}{2}\sqrt{(l+m+1)(l+m+2)}.\end{array}\right.}$
(25)
${\Lambda_{3}:\quad\left\\{\begin{array}[]{ccc}c^{k^{\prime}k}_{l-1,l,m}&=&c^{k^{\prime}k}_{l-1,l}\sqrt{l^{2}-m^{2}},\\\
c^{k^{\prime}k}_{l,l,m}&=&c^{k^{\prime}k}_{ll}m,\\\
c^{k^{\prime}k}_{l+1,l,m}&=&c^{k^{\prime}k}_{l+1,l}\sqrt{(l+1)^{2}-m^{2}}.\end{array}\right.}$
(26)
${\Lambda^{\ast}_{1}:\quad\left\\{\begin{array}[]{ccc}d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}-1,\dot{m}}&=&-\frac{c_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}-1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}-1,\dot{m}}&=&\frac{c_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}+1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}-1,\dot{m}}&=&\frac{c_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m}+1)(\dot{l}-\dot{m}+2)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}+1,\dot{m}}&=&\frac{c_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m})(\dot{l}-\dot{m}-1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}+1,\dot{m}}&=&\frac{c_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}-\dot{m})},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}+1,\dot{m}}&=&-\frac{c_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}+\dot{m}+2)}.\end{array}\right.}$
(27)
${\Lambda^{\ast}_{2}:\quad\left\\{\begin{array}[]{ccc}e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}-1,\dot{m}}&=&-\frac{ic_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}-1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}-1,\dot{m}}&=&\frac{ic_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}+1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}-1,\dot{m}}&=&\frac{ic_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m}+1)(\dot{l}-\dot{m}+2)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}+1,\dot{m}}&=&\frac{-ic_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m})(\dot{l}-\dot{m}-1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}+1,\dot{m}}&=&\frac{-ic_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}-\dot{m})},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}+1,\dot{m}}&=&-\frac{ic_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}+\dot{m}+2)}.\end{array}\right.}$
(28)
${\Lambda^{\ast}_{3}:\quad\left\\{\begin{array}[]{ccc}f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l}}\sqrt{\dot{l}^{2}-\dot{m}^{2}},\\\
f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}\dot{l}}\dot{m},\\\
f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l}}\sqrt{(\dot{l}+1)^{2}-\dot{m}^{2}}.\end{array}\right.}$
(29)
In general, the matrix $\Lambda_{3}$ must be a reducible representation of the
proper Lorentz group $\mathfrak{G}_{+}$, and can always be written in the form
$\Lambda_{3}=\begin{bmatrix}\Lambda^{l_{1}}_{3}&&&&\\\
&\Lambda^{l_{2}}_{3}&&\text{\huge 0}&\\\ &&\Lambda^{l_{3}}_{3}&&\\\
&\text{\huge 0}&&\ddots&\\\ &&&&\Lambda^{l_{n}}_{3}\end{bmatrix},$ (30)
where $\Lambda^{l_{i}}_{3}$ is a spin block (the matrix
$\overset{\ast}{\Lambda}_{3}$ has the same decompositions). It is obvious that
the matrices $\Lambda_{1}$, $\Lambda_{2}$ and $\overset{\ast}{\Lambda}_{1}$,
$\overset{\ast}{\Lambda}_{2}$ admit also the decompositions of the type (30)
by definition. If the spin block $\Lambda^{l_{i}}_{3}$ has non-null roots,
then the particle possesses the spin $l_{i}$. Corresponding to the
decomposition (30), the wave function also decomposes into a direct sum of
component wave functions which we write
$\boldsymbol{\psi}=\psi_{l_{1}m_{1}}+\psi_{l_{2}m_{2}}+\psi_{l_{3}m_{3}}+\ldots.$
According to a de Broglie theory of fusion [14], interlocking representations
give rise to indecomposable relativistic wave equations. Otherwise, we have
decomposable equations. As is known, the indecomposable wave equations
correspond to composite particles. A relation between indecomposable wave
equations and composite particles will be studied in a separate work.
Returning to the equations (23), we see that the first equation can be written
in the form
$\displaystyle\sum^{3}_{j=1}\Lambda^{\ast}_{j}\frac{\partial\dot{\psi}}{\partial\widetilde{a}_{j}}+i\sum^{3}_{j=1}\Lambda^{\ast}_{j}\frac{\partial\dot{\psi}}{\partial\widetilde{a}^{\ast}_{j}}+\kappa^{c}\psi$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum^{3}_{j=1}\Lambda_{j}\frac{\partial\psi}{\partial
a_{j}}-i\sum^{3}_{j=1}\Lambda_{j}\frac{\partial\psi}{\partial
a^{\ast}_{j}}+\dot{\kappa}^{c}\dot{\psi}$ $\displaystyle=$ $\displaystyle 0,$
(31)
where $\mathfrak{g}_{1}=a_{1}$, $\mathfrak{g}_{2}=a_{2}$,
$\mathfrak{g}_{3}=a_{3}$, $\mathfrak{g}_{4}=ia_{1}$,
$\mathfrak{g}_{5}=ia_{2}$, $\mathfrak{g}_{6}=ia_{3}$,
$a^{\ast}_{1}=-i\mathfrak{g}_{4}$, $a^{\ast}_{2}=-i\mathfrak{g}_{5}$,
$a^{\ast}_{3}=-i\mathfrak{g}_{6}$, and $\widetilde{a}_{j}$,
$\widetilde{a}^{\ast}_{j}$ are the parameters corresponding to the dual basis.
In essence, the equations (31) are defined in a three-dimensional complex
space $\bb C^{3}$. In turn, the space $\bb C^{3}$ is isometric to a
6-dimensional bivector space $\bb R^{6}$ (a parameter space of the Lorentz
group [47]). The bivector space $\bb R^{6}$ is a tangent space of the group
manifold $\mathfrak{L}_{6}$ of the Lorentz group, that is, the manifold
$\mathfrak{L}_{6}$ in the each its point is equivalent locally to the space
$\bb R^{6}$. Thus, for all $\mathfrak{g}\in\mathfrak{L}_{6}$ we have
$T_{\mathfrak{g}}\mathfrak{L}_{6}\simeq\bb R^{6}$. There exists a close
relationship between the metrics of the Minkowski spacetime $\bb R^{1,3}$ and
the metrics of $\bb R^{6}$ (see Appendix). In the case of $\bb R^{1,3}$ with
the metric tensor
$g_{\alpha\beta}=\begin{pmatrix}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}$
in virtue of (86) for the metric tensor of $\bb R^{6}$ we obtain
$g_{ab}=\begin{bmatrix}-1&0&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&-1&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\end{bmatrix},$ (32)
where the order of collective indices in $\bb R^{6}$ is $23\rightarrow 0$,
$10\rightarrow 1$, $20\rightarrow 2$, $30\rightarrow 3$, $31\rightarrow 4$,
$12\rightarrow 5$. The system (31) in the bivector space is written as
follows:
$\sum_{i}\left[g^{-}_{i1}\Lambda_{1}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}+g^{-}_{i2}\Lambda_{2}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}+g^{-}_{i3}\Lambda_{3}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}-\right.\\\
\left.-ig^{-}_{i1}\Lambda_{1}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}-ig^{-}_{i2}\Lambda_{2}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}-ig^{-}_{i3}\Lambda_{3}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}\right]+\kappa^{c}T^{-1}_{\mathfrak{g}}\psi^{\prime}=0,\\\
\sum_{i}\left[g^{+}_{i1}\Lambda^{\ast}_{1}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+g^{+}_{i2}\Lambda^{\ast}_{2}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+g^{+}_{i3}\Lambda^{\ast}_{3}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+\right.\\\
\left.+ig^{+}_{i1}\Lambda^{\ast}_{1}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}+ig^{+}_{i2}\Lambda^{\ast}_{2}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}+ig^{+}_{i3}\Lambda^{\ast}_{3}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}\right]+\dot{\kappa}^{c}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\dot{\psi}^{\prime}=0,$
(33)
where
$\displaystyle\frac{\partial}{\partial a_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\sin\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\varphi}+\frac{\cos\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\theta}+\cos\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (34) $\displaystyle\frac{\partial}{\partial a_{2}}$ $\displaystyle=$
$\displaystyle\frac{\cos\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\varphi}+\frac{\sin\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\theta}+\sin\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (35) $\displaystyle\frac{\partial}{\partial a_{3}}$ $\displaystyle=$
$\displaystyle-\frac{\sin\theta^{c}}{r}\frac{\partial}{\partial\theta}+\cos\theta^{c}\frac{\partial}{\partial
r}.$ (36) $\displaystyle\frac{\partial}{\partial a^{\ast}_{1}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{1}}=-\frac{\sin\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\epsilon}+\frac{\cos\varphi^{c}\sin\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\cos\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (37) $\displaystyle\frac{\partial}{\partial a^{\ast}_{2}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{2}}=\frac{\cos\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\epsilon}+\frac{\sin\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\sin\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (38) $\displaystyle\frac{\partial}{\partial a^{\ast}_{3}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{3}}=-\frac{\sin\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\cos\theta^{c}\frac{\partial}{\partial
r}.$ (39) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{1}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\varphi}+\frac{\cos\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\cos\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (40) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{2}}$
$\displaystyle=$
$\displaystyle\frac{\cos\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\varphi}+\frac{\sin\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\sin\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (41) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{3}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\cos\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}}.$ (42)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{1}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{1}}=\frac{\sin\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\epsilon}-\frac{\cos\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\cos\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (43)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{2}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{2}}=-\frac{\cos\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\epsilon}-\frac{\sin\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\sin\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (44)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{3}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{3}}=\frac{\sin\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\cos\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}}$ (45)
are derivatives defined on the spheres (11) and (12). Solutions of wave
equations for extended object are found in the form of series in associated
hyperspherical functions which defined on the spheres $S^{2}_{c}$ and
$\dot{S}^{2}_{c}$:
$\displaystyle\psi^{k}_{lm;\dot{l}\dot{m}}$ $\displaystyle=$
$\displaystyle\boldsymbol{f}^{l_{0}}_{lmk}(r)\mathfrak{M}^{l_{0}}_{mn}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\psi^{\dot{k}}_{\dot{l}\dot{m};lm}$ $\displaystyle=$
$\displaystyle\boldsymbol{f}^{\dot{l}_{0}}_{\dot{l}\dot{m}\dot{k}}(r^{\ast})\mathfrak{M}^{\dot{l}_{0}}_{\dot{m}\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$
where $l_{0}\geq l$, $-l_{0}\leq m$, $n\leq l_{0}$ and
$\dot{l}_{0}\geq\dot{l}$, $-\dot{l}_{0}\leq\dot{m}$, $\dot{n}\leq\dot{l}_{0}$.
_We claim that a system of Dirac like wave equations_
${\left\\{\begin{array}[]{cll}\Gamma_{i}\dfrac{\partial\psi(x)}{\partial
x_{i}}+\kappa\psi(x)&=&0,\\\
\Gamma^{T}_{i}\dfrac{\partial\overline{\psi}(x)}{\partial
x_{i}}-\kappa\overline{\psi}(x)&=&0,\quad(i=0,\ldots,3);\\\
\Gamma_{k}\dfrac{\partial\psi(\mathfrak{g})}{\partial\mathfrak{g}_{k}}+\kappa\psi(\mathfrak{g})&=&0,\\\
\Gamma^{T}_{k}\dfrac{\overline{\psi}(\mathfrak{g})}{\partial\mathfrak{g}_{k}}-\kappa\overline{\psi}(\mathfrak{g})&=&0\quad(k=1,\ldots,6)\end{array}\right.}$
(46)
_describes extended particles of any spin on the group manifold ${\cal
M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$ of the Poincaré group ${\cal P}$_
(in particular, on the homogeneous space ${\cal M}_{8}=\bb R^{1,3}\times
S^{2}$). In its turn, the Ginzburg-Tamm system of Klein-Gordon like wave
equations (2) describes extended particles on the homogeneous space ${\cal
M}=\bb R^{1,3}\times H^{3}$ of ${\cal P}$, where $H^{3}$ is a 3-dimensional
one-sheeted hyperboloid. A relation between the systems (46) and (2) is
similar to a relation between usual Dirac and Klein-Gordon wave equations for
the point-like particles. Namely, equations (46) (differential equations of
the first order) can be understood as ‘square roots’ of the equations (2)
(differential equations of the second order).
### 3.1 The Dirac field
In accordance with the general Fermi-scheme (Fig. 1) of the interlocking
representations of $\mathfrak{G}_{+}$ the field $(1/2,0)\oplus(0,1/2)$ is
defined within the following chain:
$\dgARROWLENGTH=2.5em\begin{diagram}.$
We start with the Lagrangian (19) on the group manifold ${\cal M}_{10}$:
$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial
x_{\mu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial
x_{\mu}}\Gamma_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Upsilon_{\nu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}\Upsilon_{\nu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha}),$
(47)
where $\boldsymbol{\psi}(\boldsymbol{\alpha})=\psi(x)\psi(\mathfrak{g})$
($\mu=0,1,2,3,\;\nu=1,\ldots,6$), and
$\gamma_{0}=\begin{pmatrix}\sigma_{0}&0\\\
0&-\sigma_{0}\end{pmatrix},\;\;\gamma_{1}=\begin{pmatrix}0&\sigma_{1}\\\
-\sigma_{1}&0\end{pmatrix},\;\;\gamma_{2}=\begin{pmatrix}0&\sigma_{2}\\\
-\sigma_{2}&0\end{pmatrix},\;\;\gamma_{3}=\begin{pmatrix}0&\sigma_{3}\\\
-\sigma_{3}&0\end{pmatrix},$ (48)
$\Upsilon_{1}=\begin{pmatrix}0&\Lambda^{\ast}_{1}\\\
\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{2}=\begin{pmatrix}0&\Lambda^{\ast}_{2}\\\
\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{3}=\begin{pmatrix}0&\Lambda^{\ast}_{3}\\\
\Lambda_{3}&0\end{pmatrix},$ (49)
$\Upsilon_{4}=\begin{pmatrix}0&i\Lambda^{\ast}_{1}\\\
i\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{5}=\begin{pmatrix}0&i\Lambda^{\ast}_{2}\\\
i\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{6}=\begin{pmatrix}0&i\Lambda^{\ast}_{3}\\\
i\Lambda_{3}&0\end{pmatrix},$ (50)
where $\sigma_{i}$ are the Pauli matrices, and the matrices $\Lambda_{j}$ and
$\Lambda^{\ast}_{j}$ are derived from (24)–(26) and (27)–(29) at $l=1/2$:
$\displaystyle\Lambda_{1}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\Lambda_{2}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\Lambda_{3}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$
$\displaystyle\Lambda^{\ast}_{1}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\Lambda^{\ast}_{2}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\Lambda^{\ast}_{3}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (51)
It is easy to see that these matrices coincide with the Pauli matrices
$\sigma_{i}$ when $c_{\frac{1}{2}\frac{1}{2}}=2$. The Dirac bispinor
$\psi=(\psi_{1},\psi_{2},\dot{\psi}_{1},\dot{\psi}_{2})^{T}$ is defined on
${\cal M}_{8}=\bb R^{1,3}\times S^{2}$ by the following components [63]:
$\displaystyle\psi^{l}_{1n}(\boldsymbol{\alpha})=\psi^{+}_{1}(x)\psi^{l}_{1n}(\mathfrak{g})=u_{1}({\bf
p})e^{-ipx}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\mathfrak{M}^{l}_{\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\psi^{l}_{2n}(\boldsymbol{\alpha})=\psi^{+}_{2}(x)\psi^{l}_{2n}(\mathfrak{g})=\pm
u_{2}({\bf p})e^{-ipx}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\mathfrak{M}^{l}_{-\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\dot{\psi}^{\dot{l}}_{1\dot{n}}(\boldsymbol{\alpha})=\psi^{-}_{1}(x)\dot{\psi}^{\dot{l}}_{1\dot{n}}(\mathfrak{g})=\mp
v_{1}({\bf
p})e^{ipx}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\mathfrak{M}^{\dot{l}}_{\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\dot{\psi}^{\dot{l}}_{2\dot{n}}(\boldsymbol{\alpha})=\psi^{-}_{2}(x)\dot{\psi}^{\dot{l}}_{2\dot{n}}(\mathfrak{g})=v_{2}({\bf
p})e^{ipx}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\mathfrak{M}^{\dot{l}}_{-\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$
where
$\displaystyle u_{1}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}1\\\ 0\\\
\frac{p_{z}}{E+m}\\\ \frac{p_{+}}{E+m}\end{pmatrix},\quad u_{2}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}0\\\ 1\\\
\frac{p_{-}}{E+m}\\\ \frac{-p_{z}}{E+m}\end{pmatrix},$ $\displaystyle
v_{1}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}\frac{p_{z}}{E+m}\\\
\frac{p_{+}}{E+m}\\\ 1\\\ 0\end{pmatrix},\quad v_{2}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}\frac{p_{-}}{E+m}\\\
\frac{-p_{z}}{E+m}\\\ 0\\\ 1\end{pmatrix},$
here $p_{\pm}=p_{x}\pm ip_{y}$. Radial functions have the form
$\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)=C_{1}\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,rJ_{l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r\right)+C_{2}\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,rJ_{-l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm Re}\,r\right),$
$\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})=\frac{C_{1}}{2}\sqrt{\frac{\dot{\kappa}^{c}}{\kappa^{c}}}\mbox{\rm
Re}\,r^{\ast}J_{l+1}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r^{\ast}\right)-\frac{C_{2}}{2}\sqrt{\frac{\dot{\kappa}^{c}}{\kappa^{c}}}\mbox{\rm
Re}\,r^{\ast}J_{-l-1}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r^{\ast}\right),$
where $J_{l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm Re}\,r\right)$
are the Bessel functions of half-integer order, and
$\displaystyle l=\frac{1}{2},\;\frac{3}{2},\;\frac{5}{2},\ldots;$
$\displaystyle\dot{l}=\frac{1}{2},\;\frac{3}{2},\;\frac{5}{2},\ldots;$
$\mathfrak{M}_{l}^{\pm\frac{1}{2}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp\frac{1}{2}(\epsilon+i\varphi)}\mathfrak{Z}_{l}^{\pm\frac{1}{2}}(\theta,\tau),$
$\mathfrak{Z}_{l}^{\pm\frac{1}{2}}(\theta,\tau)=\cos^{2l}\frac{\theta}{2}\cosh^{2l}\frac{\tau}{2}\sum^{l}_{k=-l}i^{\pm\frac{1}{2}-k}\tan^{\pm\frac{1}{2}-k}\frac{\theta}{2}\tanh^{-k}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-l+1,1-l-k\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}-l+1,1-l-k\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$},$
(52)
$\mathfrak{M}_{\dot{l}}^{\pm\frac{1}{2}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp\frac{1}{2}(\epsilon-i\varphi)}\mathfrak{Z}_{\dot{l}}^{\pm\frac{1}{2}}(\theta,\tau),$
$\mathfrak{Z}_{\dot{l}}^{\pm\frac{1}{2}}(\theta,\tau)=\cos^{2\dot{l}}\frac{\theta}{2}\cosh^{2\dot{l}}\frac{\tau}{2}\sum^{\dot{l}}_{\dot{k}=-\dot{l}}i^{\pm\frac{1}{2}-\dot{k}}\tan^{\pm\frac{1}{2}-\dot{k}}\frac{\theta}{2}\tanh^{-\dot{k}}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-\dot{l}+1,1-\dot{l}-\dot{k}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}-\dot{l}+1,1-\dot{l}-\dot{k}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}$
(53)
are the associated hyperspherical functions defined on the spheres $S^{c}_{2}$
and $\dot{S}^{c}_{2}$. General solutions are found in the form of generalized
Fourier integrals
$\displaystyle\psi_{1}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{l=\frac{1}{2}}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\sum^{l}_{n=-l}\int\limits_{T_{4}}u_{1}({\bf
p})e^{ipx}\alpha^{\frac{1}{2}}_{l,n}\mathfrak{M}^{l}_{\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\psi_{2}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\pm\sum^{\infty}_{l=\frac{1}{2}}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\sum^{l}_{n=-l}\int\limits_{T_{4}}u_{2}({\bf
p})e^{ipx}\alpha^{-\frac{1}{2}}_{l,n}\mathfrak{M}^{l}_{-\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\dot{\psi}_{1}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\mp\sum^{\infty}_{\dot{l}=\frac{1}{2}}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\sum^{\dot{l}}_{\dot{n}=-\dot{l}}\int\limits_{T_{4}}v_{1}({\bf
p})e^{-ipx}\alpha^{\frac{1}{2}}_{\dot{l},\dot{n}}\mathfrak{M}^{\dot{l}}_{\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\dot{\psi}_{2}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{\dot{l}=\frac{1}{2}}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\sum^{\dot{l}}_{\dot{n}=-\dot{l}}\int\limits_{T_{4}}v_{2}({\bf
p})e^{-ipx}\alpha^{-\frac{1}{2}}_{\dot{l},\dot{n}}\mathfrak{M}^{\dot{l}}_{-\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
where
$\displaystyle\alpha^{\pm\frac{1}{2}}_{l,n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}(2l+1)(2\dot{l}+1)}{32\pi^{4}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,a)}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}F_{\pm\frac{1}{2}}(\boldsymbol{\alpha})e^{-ipx}\mathfrak{M}^{l}_{\pm\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g},$
$\displaystyle\alpha^{\pm\frac{1}{2}}_{\dot{l},\dot{n}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\dot{n}}(2l+1)(2\dot{l}+1)}{32\pi^{4}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,a^{\ast})}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}\dot{F}_{\pm\frac{1}{2}}(\boldsymbol{\alpha})e^{ipx}\mathfrak{M}^{\dot{l}}_{\pm\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g}.$
### 3.2 The Maxwell field
In accordance with the general Bose-scheme of the interlocking representations
of $\mathfrak{G}_{+}$ (see Fig. 1), the field $(1,0)\oplus(0,1)$ is defined
within the following interlocking scheme:
$\dgARROWLENGTH=2.5em\begin{diagram}.$
We start with the Lagrangian (19) on the group manifold ${\cal M}_{10}$. Let
us rewrite (19) in the form
$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial
x_{\mu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial
x_{\mu}}\Gamma_{\mu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Upsilon_{\nu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}\Upsilon_{\nu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right),$
(54)
where $\boldsymbol{\phi}(\boldsymbol{\alpha})=\phi(x)\phi(\mathfrak{g})$
($\mu=0,1,2,3,\;\nu=1,\ldots,6$), and
$\Gamma_{0}=\begin{pmatrix}0&I\\\
I&0\end{pmatrix},\;\;\Gamma_{1}=\begin{pmatrix}0&-\alpha_{1}\\\
\alpha_{1}&0\end{pmatrix},\;\;\Gamma_{2}=\begin{pmatrix}0&-\alpha_{2}\\\
\alpha_{2}&0\end{pmatrix},\;\;\Gamma_{3}=\begin{pmatrix}0&-\alpha_{3}\\\
\alpha_{3}&0\end{pmatrix},$ (55)
$\Upsilon_{1}=\begin{pmatrix}0&\Lambda^{\ast}_{1}\\\
\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{2}=\begin{pmatrix}0&\Lambda^{\ast}_{2}\\\
\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{3}=\begin{pmatrix}0&\Lambda^{\ast}_{3}\\\
\Lambda_{3}&0\end{pmatrix},$ (56)
$\Upsilon_{4}=\begin{pmatrix}0&i\Lambda^{\ast}_{1}\\\
i\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{5}=\begin{pmatrix}0&i\Lambda^{\ast}_{2}\\\
i\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{6}=\begin{pmatrix}0&i\Lambda^{\ast}_{3}\\\
i\Lambda_{3}&0\end{pmatrix},$ (57)
where
$\alpha_{1}=\begin{pmatrix}0&0&0\\\ 0&0&i\\\
0&-i&0\end{pmatrix},\quad\alpha_{2}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},\quad\alpha_{3}=\begin{pmatrix}0&i&0\\\ -i&0&0\\\
0&0&0\end{pmatrix},$ (58)
and the matrices $\Lambda_{j}$ and $\Lambda^{\ast}_{j}$ are derived from
(24)–(26) and (27)–(29) at $l=1$:
$\Lambda_{1}=\frac{c_{11}}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&1&0\end{pmatrix},\quad\Lambda_{2}=\frac{c_{11}}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{pmatrix},\quad\Lambda_{3}=c_{11}\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&-1\end{pmatrix}.$ (59)
$\Lambda^{\ast}_{1}=\frac{\sqrt{2}}{2}\dot{c}_{11}\begin{pmatrix}0&1&0\\\
1&0&1\\\
0&1&0\end{pmatrix},\quad\Lambda^{\ast}_{2}=\frac{\sqrt{2}}{2}\dot{c}_{11}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\
0&i&0\end{pmatrix},\quad\Lambda^{\ast}_{3}=\dot{c}_{11}\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&-1\end{pmatrix}.$ (60)
At this point, electromagnetic field should be defined in the Riemann-
Silberstein representation [69, 57, 10]. The Riemann-Silberstein (Majorana-
Oppenheimer) representation considered during long time by many authors [40,
46, 28, 42, 51, 41, 13, 23]. The interest to this formulation of
electrodynamics has grown in recent years [32, 58, 22, 17]. One of the main
advantages of this approach lies in the fact that Dirac and Maxwell fields are
derived similarly from the Dirac-like Lagrangians. These fields have the
analogous mathematical structure, namely, they are the functions on the
Poincaré group. This circumstance allows us to consider the fields
$(1/2,0)\oplus(0,1/2)$ and $(1,0)\oplus(0,1)$ on an equal footing, from the
one group theoretical viewpoint.
In 1949, Newton and Wigner [44] showed that for the photon there exist no
localized states. Therefore, a massless field $(1,0)\oplus(0,1)$ should be
considered within the unitary infinite-dimensional representation of the
Lorentz group. In accordance with the Naimark theorem [43], the representation
$S_{k}$ of the subgroup $\operatorname{SU}(2)$ is contained in
$\mathfrak{S}_{\lambda,\rho}$ not more then one time. Matrix elements of the
representation $\mathfrak{S}_{\lambda,\rho}$ are defined by the functions (7).
_We suppose that the photon field is described within the infinite-dimensional
representation $\mathfrak{S}_{\lambda,\rho}$_. This representation contains a
finite-dimensional representation associated with the field
$(1,0)\oplus(0,1)$.
Since longitudinal solutions
$\phi^{-\frac{1}{2}+i\rho,l_{0}}_{0,n}(\boldsymbol{\alpha})$ and
$\dot{\phi}^{-\frac{1}{2}-i\rho,l_{0}}_{0,\dot{n}}(\boldsymbol{\alpha})$ do
not contribute to a real photon due to their transversality conditions, then
particular solutions for the Majorana-Oppenheimer bispinor
$\boldsymbol{\phi}=(\phi_{1},\phi_{2},\phi_{3},\dot{\phi}_{1},\dot{\phi}_{2},\dot{\phi}_{3})^{T}$
on ${\cal M}_{8}$ are defined by the following expressions [64]:
$\phi^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\boldsymbol{\alpha})=\phi_{\pm}({\bf k};{\bf
x},t)\phi^{-\frac{1}{2}+i\rho,l_{0}}_{\pm 1,n}(\mathfrak{g})=\\\
\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\begin{pmatrix}\varepsilon_{\pm}({\bf
k})\\\ \varepsilon_{\pm}({\bf k})\end{pmatrix}\exp[i({\bf k}\cdot{\bf
x}-\omega t)]\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(r)\mathfrak{M}_{\pm
1,n}^{-\frac{1}{2}+i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0),$ (61)
$\dot{\phi}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{\pm
1,\dot{n}}(\boldsymbol{\alpha})=\phi^{\ast}_{\pm}({\bf k};{\bf
x},t)\dot{\phi}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{\pm
1,\dot{n}}(\mathfrak{g})=\\\
\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\begin{pmatrix}\varepsilon^{\ast}_{\pm}({\bf
k})\\\ \varepsilon^{\ast}_{\pm}({\bf k})\end{pmatrix}\exp[-i({\bf k}\cdot{\bf
x}-\omega t)]\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(r^{\ast})\mathfrak{M}_{\pm
1,\dot{n}}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}(\varphi,\epsilon,\theta,\tau,0,0),$
(62)
where
$\displaystyle\varepsilon_{\pm}({\bf k})$ $\displaystyle=$
$\displaystyle\left\\{2|{\bf
k}|^{2}(k^{2}_{1}+k^{2}_{2})\right\\}^{-\frac{1}{2}}\begin{bmatrix}-k_{1}k_{3}\pm
ik_{2}|{\bf k}|\\\ -k_{2}k_{3}\mp ik_{1}|{\bf k}|\\\
k^{2}_{1}+k^{2}_{2}\end{bmatrix}$
are the polarization vectors of a photon. Radial functions have the form
$\displaystyle\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,1}(r)$
$\displaystyle=$ $\displaystyle
C\sqrt{r}+\sqrt{\left(l_{0}+i\rho+\frac{1}{2}\right)\left(\frac{l_{0}}{2}+i\frac{\rho}{2}+\frac{5}{4}\right)}r,$
$\displaystyle\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,1}(r^{\ast})$
$\displaystyle=$
$\displaystyle\dot{C}\sqrt{r^{\ast}}+\sqrt{\left(\dot{l}_{0}-i\rho+\frac{3}{2}\right)\left(\frac{\dot{l}_{0}}{2}-i\frac{\rho}{2}+\frac{7}{4}\right)}r^{\ast},$
and
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp(\epsilon+i\varphi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\theta,\tau),$ $\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\theta,\tau)=\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1+2i\rho,l_{0}}\frac{\tau}{2}\sum^{l_{0}}_{k=-l_{0}}i^{\pm
1-k}\tan^{\pm 1-k}\frac{\theta}{2}\tanh^{n-k}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm
1-l_{0}+1,1-l_{0}-k\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\pm
1-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}n-i\rho+\frac{1}{2},\frac{1}{2}-i\rho-k\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}n-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$},$
(63) $\mathfrak{M}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp(\epsilon-i\varphi)}\mathfrak{Z}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\theta,\tau),$ $\mathfrak{Z}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\theta,\tau)=\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1-2i\rho,l_{0}}\frac{\tau}{2}\sum^{l_{0}}_{\dot{k}=-l_{0}}i^{\pm
1-\dot{k}}\tan^{\pm
1-\dot{k}}\frac{\theta}{2}\tanh^{\dot{n}-\dot{k}}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm
1-l_{0}+1,1-l_{0}-\dot{k}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\pm
1-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\dot{n}+i\rho+\frac{1}{2},\frac{1}{2}+i\rho-\dot{k}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\dot{n}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}$
(64)
are associated hyperspherical functions defined on the spheres $S^{2}_{c}$ and
$\dot{S}^{2}_{c}$. General solutions are found via the following generalized
Fourier integrals:
$\boldsymbol{\phi}_{\pm
1}(\boldsymbol{\alpha})=\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\sum^{\infty}_{l_{0}=1}\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(r)\sum^{l_{0}}_{n=-l_{0}}\int\limits_{T_{4}}\begin{pmatrix}\varepsilon_{\pm}({\bf
k})\\\ \varepsilon_{\pm}({\bf k})\end{pmatrix}e^{ikx}\alpha^{\pm
1}_{l_{0},n}\mathfrak{M}_{\pm
1,n}^{-\frac{1}{2}+i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\dot{\boldsymbol{\phi}}_{\pm
1}(\boldsymbol{\alpha})=\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\sum^{\infty}_{l_{0}=1}\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(r^{\ast})\sum^{l_{0}}_{\dot{n}=-l_{0}}\int\limits_{T_{4}}\begin{pmatrix}\varepsilon^{\ast}_{\pm}({\bf
k})\\\ \varepsilon^{\ast}_{\pm}({\bf k})\end{pmatrix}e^{-ikx}\alpha^{\pm
1}_{l_{0},\dot{n}}\mathfrak{M}_{\pm
1,\dot{n}}^{-\frac{1}{2}-i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
where
$\displaystyle\alpha^{\pm 1}_{l_{0},n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}(2l_{0}+1)^{2}}{32\pi^{4}\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(a)}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}F_{\pm
1}(\boldsymbol{\alpha})e^{-ikx}\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g},$
$\displaystyle\alpha^{\pm 1}_{l_{0},\dot{n}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\dot{n}}(2l_{0}+1)^{2}}{32\pi^{4}\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(a^{\ast})}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}\dot{F}_{\pm
1}(\boldsymbol{\alpha})e^{ikx}\mathfrak{M}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g}.$
## 4 Interaction
Up to now we analyze free Dirac and Maxwell fields. Let us consider an
interaction between these fields. As usual, interactions between the fields
are described by an interaction Lagrangian $\mathcal{L}_{I}$. In our case we
take the following Lagrangian [66]:
$\mathcal{L}_{I}(\boldsymbol{\alpha})=\mu(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\sigma^{D}_{(\mu\nu)^{k}}\boldsymbol{\psi}(\boldsymbol{\alpha}))(\Xi^{M}_{(\rho)k}\boldsymbol{\phi}(\boldsymbol{\alpha})),$
(65)
where
$\sigma^{D}_{\mu\nu}=\frac{1}{2}(\Xi^{D}_{\mu}\Xi^{D}_{\nu}-\Xi^{D}_{\nu}\Xi^{D}_{\mu})$
and
$\Xi^{D}=(\Gamma^{D}_{0},\Gamma^{D}_{1},\Gamma^{D}_{2},\Gamma^{D}_{3},\Upsilon^{D}_{1},\Upsilon^{D}_{2},\Upsilon^{D}_{3},\Upsilon^{D}_{4},\Upsilon^{D}_{5},\Upsilon^{D}_{6})$,
$\Xi^{M}=(\Gamma^{M}_{0},\Gamma^{M}_{1},\Gamma^{M}_{2},\Gamma^{M}_{3},\Upsilon^{M}_{1},\Upsilon^{M}_{2},\Upsilon^{M}_{3},\Upsilon^{M}_{4},\Upsilon^{M}_{5},\Upsilon^{M}_{6})$,
here $\Gamma^{D}$ and $\Upsilon^{D}$ are the matrices (48) and (49)–(50), and
$\Gamma^{M}$ and $\Upsilon^{M}$ are the matrices (55) and (56)–(57).
The full Lagrangian of interacting Dirac and Maxwell fields equals to a sum of
the free field Lagrangians and the interaction Lagrangian:
$\mathcal{L}(\boldsymbol{\alpha})=\mathcal{L}_{D}(\boldsymbol{\alpha})+\mathcal{L}_{M}(\boldsymbol{\alpha})+\mathcal{L}_{I}(\boldsymbol{\alpha}),$
where $\mathcal{L}_{D}(\boldsymbol{\alpha})$ and
$\mathcal{L}_{M}(\boldsymbol{\alpha})$ are of the type (47) and (54),
respectively. Or,
$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Xi^{D}_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Xi^{D}_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Xi^{M}_{\mu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Xi^{M}_{\mu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right)-\\\
-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha})+\mu(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\sigma^{D}_{(\mu\nu)^{k}}\boldsymbol{\psi}(\boldsymbol{\alpha}))(\Xi^{M}_{(\rho)k}\boldsymbol{\phi}(\boldsymbol{\alpha})).$
(66)
Since the Lagrangian (65) does not contain derivatives on the field functions,
then for a Hamiltonian density we have ${\cal
H}_{I}(\boldsymbol{\alpha})=-\mathcal{L}_{I}(\boldsymbol{\alpha})$.
As is known, in the standard quantum field theory the $S$-matrix is expressed
via the Dyson formula [53]
$S=T\left[\exp\left(-\frac{i}{\hbar c}\int\limits^{+\infty}_{-\infty}{\cal
H}_{I}(x)d^{4}x\right)\right],$ (67)
where $T$ is the time ordering operator.
In our case, the electron-positron and photon fields are defined on the space
${\cal M}_{8}=\bb R^{1,3}\times S^{2}$ which larger then the Minkowski space
$\bb R^{1,3}$. With a view to define a formula similar to the equation (67) it
is necessarily to replace $d^{4}x$ by the following invariant measure on
${\cal M}_{8}$:
$d^{8}\mu=d^{4}xd^{4}\mathfrak{g},$
where
$d^{4}\mathfrak{g}=\sin\theta^{c}d\theta d\tau d\varphi d\epsilon.$
Therefore, an analogue of the Dyson formula (67) on the manifold ${\cal
M}_{8}$ can be written as follows
$S=T\left[\exp\left(-\frac{i}{\hbar
c}\int\limits_{T_{4}}\int\limits_{S^{2}_{c}}{\cal
H}_{I}(\boldsymbol{\alpha})d^{4}xd^{4}\mathfrak{g}\right)\right].$
Let us consider an integral on trilinear form, defining the interaction, over
the two-dimensional complex sphere:
$\int\limits_{S^{2}_{c}}{\cal
H}_{I}(\boldsymbol{\alpha})dS^{2}_{c}\sim\int\limits_{S^{2}_{c}}\mathfrak{M}^{\dot{m}_{e}}_{\dot{l}_{e}}(\varphi,\epsilon,\theta,\tau,0,0)\mathfrak{M}^{m_{e}}_{l_{e}}(\varphi,\epsilon,\theta,\tau,0,0)\mathfrak{M}^{m_{f}}_{-\frac{1}{2}+i\rho}(\varphi,\epsilon,\theta,\tau,0,0)\sin\theta^{c}d\theta^{c}d\varphi^{c},$
that is, to investigate convergence of the integral
$I_{1}=\int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty}e^{-i(m_{e}\varphi^{c}-\dot{m}_{e}\dot{\varphi}^{c}-m_{f}\varphi^{c})}\mathfrak{Z}^{\dot{m}_{e}}_{\dot{l}_{e}}(\cos\dot{\theta}^{c})\mathfrak{Z}^{m_{e}}_{l_{e}}(\cos\theta^{c})\mathfrak{Z}^{m_{f}}_{-\frac{1}{2}+i\rho}(\cos\theta^{c})\sin\theta^{c}d\theta^{c}d\varphi^{c}.$
Here the symbols $l_{e}$, $m_{e}$ correspond to the electron field
$\boldsymbol{\psi}(\mathfrak{g})$, $\dot{l}_{e}$, $\dot{m}_{e}$ correspond to
the positron field $\overline{\boldsymbol{\psi}(\mathfrak{g})}$, and $m_{f}$
corresponds to the photon field $\boldsymbol{\phi}(\mathfrak{g})$. It is
obvious that convergence of the integral
$I_{2}=\int\limits^{+\infty}_{-\infty}e^{-i(m_{e}\varphi^{c}-\dot{m}_{e}\dot{\varphi}^{c}-m_{f}\varphi^{c})}d\varphi^{c}$
is not difficult to investigate. Let us consider the following integral:
$I_{3}=\int\limits^{+\infty}_{-\infty}\mathfrak{Z}^{\dot{m}_{e}}_{\dot{l}_{e}}(\cos\dot{\theta}^{c})\mathfrak{Z}^{m_{e}}_{l_{e}}(\cos\theta^{c})\mathfrak{Z}^{m_{f}}_{-\frac{1}{2}+i\rho}(\cos\theta^{c})\sin\theta^{c}d\theta^{c}.$
Rewriting hyperspherical functions via the hypergeometric functions, we obtain
$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{e}+1)\Gamma(\dot{m}_{e}+1)\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\int\limits^{+\infty}_{-\infty}\sin^{\dot{m}_{e}+m_{e}+m_{f}}\frac{\theta^{c}}{2}\cos^{\dot{m}_{e}+m_{e}+m_{f}}\frac{\theta^{c}}{2}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\dot{l}_{e}+\dot{m}_{e}+1,\dot{m}_{e}-\dot{l}_{e}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}\dot{m}_{e}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}l_{e}+m_{e}+1,m_{e}-l_{e}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{e}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c}.$
(68)
Further, using an explicit expression for the associated hyperspherical
function
$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})=i^{m}\sqrt{\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}}\cos^{m}\frac{\theta^{c}}{2}\sin^{m}\frac{\theta^{c}}{2}\times\\\
\sum^{l-m}_{j=0}\frac{(-1)^{j}\Gamma(l+m+j+1)}{\Gamma(j+1)\Gamma(m+j+1)\Gamma(l-m-j+1)}\sin^{2j}\frac{\theta^{c}}{2},$
(69)
we rewrite the first two hypergeometric functions in the integral (68). Then
$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\sum^{l_{e}-m_{e}}_{j_{1}=0}\sum^{\dot{l}_{e}-\dot{m}_{e}}_{j_{2}=0}\frac{(-1)^{j_{1}+j_{2}}\Gamma(l_{e}+m_{e}+j_{1}+1)}{\Gamma(j_{1}+1)\Gamma(j_{2}+1)\Gamma(m_{e}+j_{1}+1)\Gamma(\dot{m}_{e}+j_{2}+1)}\times\\\
\int\limits^{+\infty}_{-\infty}\cos^{m_{e}+\dot{m}_{e}+m_{f}}\frac{\theta^{c}}{2}\sin^{m_{e}+\dot{m}_{e}+m_{f}+2j_{1}+2j_{2}}\frac{\theta^{c}}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c}.$
(70)
Making the substitution $z=\cos\theta^{c}$ in the integral
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}}}\int\limits^{+\infty}_{-\infty}\sin^{m_{e}+\dot{m}_{e}+m_{f}}\theta^{c}\sin^{2j_{1}+2j_{2}}\frac{\theta^{c}}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c},$
(71)
we obtain
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\int\limits^{+\infty}_{-\infty}(1-z^{2})^{\frac{m_{e}+\dot{m}_{e}+m_{f}}{2}}\left(\frac{1-z}{2}\right)^{j_{1}+j_{2}}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1-z}{2}}\right)$}dz.$
(72)
Or,
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}(-1)^{q}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{q!(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\int\limits^{+\infty}_{-\infty}z^{k}\left(\frac{1-z}{2}\right)^{j_{1}+j_{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1-z}{2}}\right)$}dz.$
(73)
Introducing a new variable $t=(1-z)/2$, we find
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}(-1)^{q+1}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{q!(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\int\limits^{+\infty}_{-\infty}(1-2t)^{q}t^{j_{1}+j_{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(74)
Decomposing $(1-2t)^{q}$ via the Newton binomial, we obtain
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(-1)^{q+p+1}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\frac{2^{p}}{p!(k-p)!}\int\limits^{+\infty}_{-\infty}t^{j_{1}+j_{2}+p}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}+\frac{1}{2},m_{f}-i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(75)
With the aim to calculate the latter integral we use the following formula
[49]:
$I_{5}=\int
t^{n}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}a,b\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}c\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt=\\\
=n!\sum^{n+1}_{k=1}(-1)^{k+1}\frac{(c-k)_{k}t^{n-k+1}}{(a-k+1)!(a-k)_{k}(b-k)_{k}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}a-k,b-k\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}c-k\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}.$
(76)
Then
$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(j_{1}+j_{2}+p)!\times\\\
\sum^{j_{1}+j_{2}+p}_{k=1}\frac{(-1)^{q+p+k}2^{p}(m_{e}+\dot{m}_{e}+m_{f})!}{p!(k-p)!(m_{e}+\dot{m}_{e}+m_{f}-q)!(m_{f}+i\rho+\frac{3}{2}-k)!}\times\\\
\frac{(m_{f}-k+1)_{k}t^{j_{1}+j_{2}+p-k+1}}{(m_{f}+i\rho-k+\frac{1}{2})_{k}(m_{f}-k+1)_{k}}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}i\rho+m_{f}-k+\frac{1}{2},m_{f}-i\rho-k+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}m_{f}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(77)
With the aim to investigate the convergence of (68) let us apply the following
asymptotic expansion for the hypergeometric function [9]:
$\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}a,b\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}c\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}(-t)^{-a}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}a,1-c+a\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}1-b+a\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}+\\\
+\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}(-t)^{-b}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}b,1-c+b\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}1-a+b\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}.$
(78)
Thus,
$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\sum^{l_{e}-m_{e}}_{j_{1}=0}\sum^{\dot{l}_{e}-\dot{m}_{e}}_{j_{2}=0}\frac{(-1)^{j_{1}+j_{2}}\Gamma(l_{e}+m_{e}+j_{1}+1)}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}\Gamma(j_{1}+1)\Gamma(j_{2}+1)\Gamma(m_{e}+j_{1}+1)\Gamma(\dot{m}_{e}+j_{2}+1)}\times$
(79)
$\frac{\Gamma(\dot{l}_{e}+\dot{m}_{e}+j_{2}+1)}{\Gamma(l_{e}-m_{e}-j_{1}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}-j_{2}+1)}\times\\\
\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(j_{1}+j_{2}+p)!\sum^{j_{1}+j_{2}+p}_{k=1}\frac{(-1)^{q+p+k}2^{p}(m_{e}+\dot{m}_{e}+m_{f})!}{p!(k-p)!(m_{e}+\dot{m}_{e}+m_{f}-q)!(m_{f}+i\rho+\frac{3}{2}-k)!}\times$
(80)
$\frac{(m_{f}-k+1)_{k}}{(m_{f}+i\rho-k+\frac{1}{2})_{k}(m_{f}-k+1)_{k}}\left[(-1)^{k-m_{f}-i\rho-\frac{1}{2}}\frac{\Gamma(m_{f}-k+1)\Gamma(-2i\rho)}{\Gamma(m_{f}-i\rho-k+\frac{1}{2})\Gamma(\frac{1}{2}-i\rho)}\times\right.\\\
t^{j_{1}+j_{2}+p-m_{f}-i\rho+\frac{1}{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}m_{f}+i\rho-k+\frac{1}{2},i\rho+\frac{1}{2}\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}2i\rho+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}+$
(81)
$\left.(-1)^{k-m_{f}+i\rho-\frac{1}{2}}\frac{\Gamma(m_{f}-k+1)\Gamma(2i\rho)}{\Gamma(m_{f}+i\rho-k+\frac{1}{2})\Gamma(\frac{1}{2}-i\rho)}\times\right.\\\
\left.t^{j_{1}+j_{2}+p-m_{f}+i\rho+\frac{1}{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}m_{f}+i\rho-k+\frac{1}{2},\frac{1}{2}-i\rho\end{array}\hfil\lx@intercol\\\\[2.84526pt]
\lx@intercol\hfil\begin{array}[]{c}1-2i\rho\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}\right],$
(82)
where $t=\sin^{2}\frac{\theta^{c}}{2}$. Since $I_{3}\sim 1/t^{M}$ (the
hypergeometric function ${}_{2}F_{1}$ can be written as a power series in
$1/t$) and $M\rightarrow\infty$, then $I_{3}$ converges at
$M>j_{1}+j_{2}+p-m_{f}+i\rho+\frac{1}{2}$ and, therefore, in accordance with
(67) the elements of $S$-matrix are defined by convergent expressions.
## 5 Summary
We have proved convergence of quantum electrodynamics on the Poincaré group in
the case of homogeneous space ${\cal M}_{8}=\bb R^{1,3}\times S^{2}$. We
considered Dirac like equations for extended objects on ${\cal M}_{8}$ and
their particular solutions corresponding to the fields of spin 1/2 and 1. At
this point, the spin-1 field (Maxwell field) is defined within an infinite-
dimensional representation of the Lorentz group. We showed that an analogue of
the Dyson formula for $S$-matrix in the case of ${\cal M}_{8}$ is defined by
convergent integrals. It would be interesting to consider quantum field models
and their convergence on other homogeneous spaces of the Poincaré group, such
as ${\cal M}_{6}=\bb R^{1,3}\times S^{2}$, ${\cal M}_{7}=\bb R^{1,3}\times
H^{3}$ and ${\cal M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$. It would be
interesting also to consider wave equations and field models for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{q}\,|\boldsymbol{\psi}\rangle$ on the de Sitter group, where $x\in
T_{5}$ and $\mathfrak{q}\in\operatorname{{\bf
Spin}}_{+}(1,4)\simeq\operatorname{Sp}(1,1)$, and for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{c}\,|\boldsymbol{\psi}\rangle$ on the conformal group, where $x\in
T_{6}$ and $\mathfrak{c}\in\operatorname{{\bf
Spin}}_{+}(2,6)\simeq\operatorname{SU}(2,2)$. Our next paper will be devoted
to these questions.
## Appendix: Spinor groups and bivector spaces
As is known, rotations of pseudo-Euclidean spaces $\bb R^{p,q}$ are defined by
spinor groups [37]
$\operatorname{{\bf
Spin}}(p,q)=\left\\{s\in\boldsymbol{\Gamma}^{+}_{p,q}\;|\;N(s)=\pm
1\right\\},$
where $\boldsymbol{\Gamma}^{+}_{p,q}=\boldsymbol{\Gamma}_{p,q}\cap
C\kern-1.99997pt\ell^{+}_{p,q}$ is a special Lipschitz group, $s\in
C\kern-1.99997pt\ell^{+}_{p,q}$ is an even invertible element of the real
Clifford algebra $C\kern-1.99997pt\ell_{p,q}$. In more detail, the element $s$
is a linear combination of even basic elements, that is,
$s=\sum_{k}a^{i_{1}\ldots i_{2k}}\mbox{\bf e}_{i_{1}\ldots i_{2k}}.$
The condition $N(s)=\pm 1$ means that
$\sum_{k}\sigma(i_{1})\cdots\sigma(i_{2k})\left(a^{i_{1}\ldots
i_{2k}}\right)^{2}=\pm 1.$ (83)
On the other hand, from the fact that the elements
$s\in\boldsymbol{\Gamma}^{+}_{p,q}$ are even products of
$\nu=\sum\nu^{i}\mbox{\bf e}_{i}$, it is easy to derive that coordinates of
$s\in\operatorname{{\bf Spin}}(p,q)$ are related by the following conditions
[50]:
$a^{i_{1}i_{2}\ldots
i_{2k}}(a)^{k-1}=(2k-1)!!a^{[i_{1}i_{2}}a^{i_{2}i_{3}}\cdots
a^{i_{2k-1}i_{2k}]}.$ (84)
Conditions (84) can be rewritten in the form
$\left.\begin{array}[]{rcl}aa^{i_{1}i_{2}i_{3}i_{4}}&=&3!!a^{[i_{1}i_{2}}a^{i_{3}i_{4}]},\\\
aa^{i_{1}i_{2}i_{3}i_{4}i_{5}i_{6}}&=&5!!a^{[i_{1}i_{2}}a^{i_{3}i_{4}i_{5}i_{6}]},\\\
[2]{3}\\\ aa^{i_{1}i_{2}\ldots
i_{2k}}&=&(2k-1)!!a^{[i_{1}i_{2}\ldots}a^{i_{3}i_{4}\ldots
i_{2k-1}i_{2k}]}.\end{array}\right\\}$ (85)
Conditions (83) and (84) express all coordinates of the elements $s$,
belonging to the spinor group $\operatorname{{\bf Spin}}(p,q)$, via $n(n-1)/2$
coordinates $a^{ij}$, where $n=p+q$. The number of the coordinates $a^{ij}$
coincides with the number of parameters of the rotation group of the space
$\bb R^{p,q}$. This fact shows that expressions (83) and (84) form a full
system of the conditions separating the spinor group $\operatorname{{\bf
Spin}}(p,q)$ from the algebra $C\kern-1.99997pt\ell_{p,q}$.
Further, parameters of the rotation group of the space $\bb R^{p,q}$ form a
bivector space $\bb R^{N}$, where $N=\frac{n(n-1)}{2}$. Indeed, Let $\bb
R^{p,q}$ be the $n$-dimensional pseudo-Euclidean space, $p+q=n$. Let us evolve
in $\bb R^{p,q}$ all the tensors satisfying the following two conditions: 1) a
rank of the tensors is even; 2) covariant and contravariant indexes are
divided into separate skewsymmetric pairs. Such tensors can be exemplified by
bivectors (skewsymmetric tensors of the second rank). The set of all bivector
tensor fields in $\bb R^{p,q}$ is called a bivector set, and its
representation in a given point of $\bb R^{p,q}$ is called a local bivector
set. In any tensor from the bivector set we take the each skewsymmetric pair
$\alpha\beta$ as one collective index. At this point, from the two possible
pairs $\alpha\beta$ and $\beta\alpha$ we fix only one, for example,
$\alpha\beta$. The number of all collective indexes is equal to
$N=\frac{n(n-1)}{2}$. In a given point, the bivector set of the space $\bb
R^{p,q}$ with the contravariant components defines in the collective indexes a
vector set, and the each vector of this set has $N$ components. Identifying
these vectors with the points of the $N$-dimensional manifold, we come to an
affine manifold $E^{N}$ if and only if this manifold admits a Klein geometry
with the group
$\displaystyle\eta^{a^{\prime}}=A^{a^{\prime}}_{a}\eta^{a},\quad\eta^{a}=A^{a}_{a^{\prime}}\eta^{a^{\prime}},$
$\displaystyle\det A^{a^{\prime}}_{a}\neq 0,\quad
A^{a}_{b^{\prime}}A^{b^{\prime}}_{c}=\delta^{a}_{c},$
where
$A^{a^{\prime}}_{a}\longrightarrow
A^{\alpha^{\prime}}_{[\alpha}A^{\beta^{\prime}}_{\beta]}.$
Thus, any local bivector set of the space $\bb R^{p,q}$ ($p+q=n$) can be
mapped onto the affine space $E^{N}$ . Therefore, $E^{N}$ is related with the
each point of the space $\bb R^{p,q}$. The space $E^{N}$ is called a bivector
space. It should be noted that the bivector space is a particular case of the
most general mathematical construction called a Grassmannian manifold (a
manifold of $m$-dimensional planes of the affine space). In the case $m=2$ the
manifold of two-dimensional planes is isometric to the bivector space, and the
Grassmann coordinates in this case are called Pluecker coordinates.
The metrization of the bivector space $E^{N}$ is given by the formula (see
[47])
$g_{ab}\longrightarrow g_{\alpha\beta\gamma\delta}\equiv
g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma},$ (86)
where $g_{\alpha\beta}$ is a metric tensor of the space $\bb R^{p,q}$, and the
collective indexes are skewsymmetric pairs $\alpha\beta\rightarrow a$,
$\gamma\delta\rightarrow b$. After introduction of $g_{ab}$, the bivector
affine space $E^{N}$ is transformed to a metric space $\bb R^{N}$.
As is known, any transformation from the rotation group of the space $\bb
R^{p,q}$ can be represented via $\frac{n(n-1)}{2}$ transformations in the
planes $(x_{1},x_{2})$, $(x_{1},x_{3})$, $\ldots$, $(x_{p+q},x_{1})$. The full
number of the planes $(x_{i},x_{j})$ is equal to $N=\frac{n(n-1)}{2}$, and the
each plane $(x_{i},x_{j})$ corresponds to the coordinate $a^{ij}$ of the
spinor group $\operatorname{{\bf Spin}}(p,q)$.
## References
* [1] Altaisky, M. V.: _Wavelet-Based Quantum Field Theory_. SIGMA 3, 105–118 (2007); arXiv:0711.1671 [hep-th] (2007).
* [2] Altaisky, M. V.: _Quantum field theory without divergences_. Phys. Rev. D. 81, 125003 (2010); arXiv:1002.2566 [hep-th] (2010).
* [3] Amar, V., Dozzio, U.: Gel’fand-Yaglom Equations with Charge or Energy Density of Definite Sign. Nuovo Cimento A11, 87–99 (1972).
* [4] Arnold, V. I.: Mathematical Methods of Classical Mechanics. Nauka, Moscow (1989) [in Russian].
* [5] Arodź, H.: Metric tensors, Lagrangian formalism and Abelian gauge field on the Poincaré group. Acta Phys. Pol. B. 7, 177–190 (1976).
* [6] Bacry, H., Kihlberg, A.: Wavefunctions on homogeneous spaces. J. Math. Phys. 10, 2132–2141 (1969).
* [7] Bacry, H., Nuyts, J.: Mass-Spin Relation in a Lagrangian Model. Phys. Rev. 157, 1471–1472 (1967).
* [8] Bargmann, V., Wigner, E. P.: Group theoretical discussion of relativistic wave equations. Proc. Nat. Acad. USA 34, 211–223 (1948).
* [9] Bateman, H., Erdélyi, A.: Higher Transcendental Functions. vol. I. Mc Grow-Hill Book Company, New York (1953).
* [10] Bialynicki-Birula, I.: Photon wave function. Progress in Optics, Vol. XXXVI, Ed. E. Wolf, Elsevier, Amsterdam (1996); arXiv:quant-th/0508202 (2005).
* [11] Biedenharn, L. C., Braden, H. W, Truini, P., van Dam, H.: Relativistic wavefunctions on spinor spaces. J. Phys. A: Math. Gen. 21, 3593–3610 (1988).
* [12] Boyer, C. P., Fleming, G. N.: Quantum field theory on a seven-dimensional homogeneous space of the Poincaré group. J. Math. Phys. 15, 1007–1024 (1974).
* [13] Da Silveira.: Dirac-like equations for the photon. Z. Naturforsh A34, 646–647 (1979).
* [14] de Broglie L. Theorie Generale des Particules a Spin (Methode de Fusion). Gauthier-Villars, Paris (1943).
* [15] de Broglie, L.: La théorie de la mesure en mécanique ondulatorie. Gauthier-Villars, Paris (1957).
* [16] Drechsler, W.: Geometro-stohastically quantized fields with internal spin variables. J. Math. Phys. 38, 5531–5558 (1997); arXiv:gr-qc/9610046 (1996).
* [17] Esposito, S.: Covariant Majorana Formulation of Electrodynamics. Found. Phys. 28, 231–244 (1998); arXiv:hep-th/9704144 (1997).
* [18] Finkelstein, D.: Internal Structure of Spinning Particles. Phys. Rev. 100, 924–931 (1955).
* [19] Fierz, M., Pauli, W.: On Relativistic Wave Equations of Particles of Arbitrary Spin in an Electromagnetic Field. Proc. Roy. Soc. (London) A. 173, 211–232 (1939).
* [20] Gel’fand, I. M., Minlos, R. A., Shapiro, Z. Ya.: Representations of the Rotation and Lorentz Groups and their Applications. Pergamon Press, Oxford (1963).
* [21] Gel’fand, I. M., Graev, M. I., Vilenkin, N. Ya.: Generalized Functions Vol. 5. Integral Geometry and Representation Theory. Academic Press, New York, London (1986).
* [22] Gersten, A.: Maxwell equations as one-photon quantum equation. Found. Phys. Lett. 12, 291–298 (1998); arXiv:quant-ph/9911049 (1999).
* [23] Giannetto, E.: A Majorana-Oppenheimer Formulation of Quantum Electrodynamics. Lettere al Nuovo Cimento 44, 140–144 (1985).
* [24] Ginzburg, V. L., Tamm, I. E.: On the theory of spin. Zh. Ehksp. Teor. Fiz. 17, 227–237 (1947).
* [25] Gitman, D. M., Shelepin, A. L.: Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations. Int. J. Theor. Phys. 40, 3, 603–684 (2001); arXiv:hep-th/0003146 (2000).
* [26] Gitman, D. M., Shelepin, A. L.: _Field on the Poincaré group and quantum description of orientable objects_. Eur. Phys. J. C. 61, 111-139 (2009); arXiv:0901.2537 [hep-th] (2009).
* [27] Gitman, D. M., Shelepin, A. L.: _Classification of quantum relativistic orientable objects_. Phys. Scr. 83, 015103 (2011);arXiv: 1001.5290 [hep-th] (2010).
* [28] Good, R. H.: Particle aspect of the electromagnetic field equations. Phys. Rev. 105, 1914 (1957).
* [29] Grandpeix, J.-Y., Lurçat, F.: Particle description of zero energy vacuum. Found. Phys. 32, 109–158 (2002); arXiv:hep-th/0106229 (2001).
* [30] Huszar, M.: Angular Momentum and Unitary Spinor Bases of the Lorentz Group. Preprint JINR No. E2-5429, Dubna (1970).
* [31] Huszar, M., Smorodinsky, J.: Representations of the Lorentz Group on the Two-Dimensional Complex Sphere and Two-Particle States. Preprint JINR No. E2-5020, Dubna (1970).
* [32] Inagaki, T.: Quantum-mechanical approach to a free photon. Phys. Rev. A49, 2839–2843 (1994).
* [33] Kaiser, G.: _Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis_. arXiv: 0910.0352 [math-ph] (2009).
* [34] Kihlberg, A.: Internal Co-ordinates and Explicit Representations of the Poincaré Group. Nuovo Cimento A53, 592–609 (1968).
* [35] Kihlberg, A.: Fields on a homogeneous space of the Poincaré group. Ann. Inst. Henri Poincaré. 13, 57–76 (1970).
* [36] Kuzenko, S. M., Lyakhovich, S. L., Segal, A. Yu.: A geometric model of the arbitrary spin massive particle. Int. J. Mod. Phys. A. 10, 1529–1552 (1995); arXiv:hep-th/9403196 (1994).
* [37] Lipschitz R.: Untersuchungen über die Summen von Quadraten. Max Cohen und Sohn, Bonn (1886).
* [38] Lurçat, F.: Quantum field theory and the dynamical role of spin. Physics. 1, 95 (1964).
* [39] Lyakhovich, S. L., Segal, A. Yu., Sharapov, A. A.:Universal model of a $D=4$ spinning particles. Phys. Rev. D. 54, 5223–5238 (1996); arXiv:hep-th/9603174 (1996).
* [40] Majorana, E.: Scientific Papers, unpublished, deposited at the “Domus Galileana”, Pisa, quaderno 2, p.101/1; 3, p.11, 160; 15, p.16;17, p.83, 159.
* [41] Mignani, R., Recami, E., Baldo, M.: About a Dirac-Like Equation for the Photon according to Ettore Majorana. Lettere al Nuovo Cimento 11, 568–572 (1974).
* [42] Moses, H. E.: Solution of Maxwell’s Equations in Terms of a Spinor Notation: the Direct and Inverse Problem. Phys. Rev. 113, 1670–1679 (1959).
* [43] Naimark, M. A.: Linear Representations of the Lorentz Group. Pergamon, London (1964).
* [44] Newton, T. D., Wigner, E. P.: Localized states for elementary systems. Rev. Mod. Phys. 21, 400 (1949).
* [45] Nilsson, J., Beskow, A.: The concept of wave function and irreducible representations of the Poincaré group. Arkiv för Fysik 34, 307–324 (1967).
* [46] Oppenheimer, J. R.: Note on light quanta and the electromagnetic field. Phys. Rev. 38, 725 (1931).
* [47] Petrov, A. Z.: Einstein Spaces. Pergamon Press, Oxford (1969).
* [48] Pletyukhov, V. A., Strazhev, V. I.: On Dirac-like relativistic wave equations. Russian J. Phys. n.12, 38-41 (1983).
* [49] Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I.: Integrals and Series: Supplementary Chapters [in Russian]. Nauka, Moscow (1981).
* [50] Rozenfeld, B. A.: Geometry of Lie groups. Dordrecht-Boston-London (1997).
* [51] Sachs, M., Schwebel, S. L.: On covariant formulation of the Maxwell-Lorentz theory of electromagnetism. J. Math. Phys. 3, 843–848 (1962).
* [52] Sastry, R. R.: _Quantum Mechanics of Extended Objects_. arXiv:quant-ph/9903025 (1999).
* [53] Schweber, S. S.: An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York (1961).
* [54] Segal, I. E., Zhou, Z.: _Convergence of nonlinear massive quantum field theory in the Einstein universe_. Ann. Phys. 218, 2, 279–292 (1992).
* [55] Segal, I. E., Zhou, Z.: _Convergence of Quantum Electrodynamics in a Curved Deformation of Minkowski Space_. Ann. Phys. 232, 1, 61–87 (1994).
* [56] Shirokov, Yu. M.: Relativistskaia teoria spina. Zh. Ehksp. Teor. Fiz. 21, 748–760 (1951).
* [57] Silberstein, L.: Elektromagnetische Grundgleichungen in bivectorieller Behandlung. Ann. d. Phys. 22, 579 (1907).
* [58] Sipe, J. E.: Photon wave functions. Phys. Rev. A52, 1875–1883 (1995).
* [59] Smorodinsky, Ya. A., Huszar, M.: Representations of the Lorentz group and the generalization of helicity states. Teor. Mat. Fiz. 4, 3, 328–340 (1970).
* [60] Todorov, N. S.: _Extended Particles Part I: Reformulation and Reinterpretation of the Dirac and Klein-Gordon Theories_. Annales de la Fondation Louis de Broglie 25, 41–66 (2000).
* [61] Toller, M.: Free quantum fields on the Poincaré group. J. Math. Phys. 37, 2694–2730 (1996); arXiv:gr-qc/9602031 (1996).
* [62] Varlamov, V. V.: _General Solutions of Relativistic Wave Equations_. Int. J. Theor. Phys. 42, 3, 583–633 (2003); arXiv:math-ph/0209036 (2002).
* [63] Varlamov, V. V.: _Relativistic wavefunctions on the Poincaré group_. J. Phys. A: Math. Gen. 37, 5467–5476 (2004); arXiv:math-ph/0308038 (2003).
* [64] Varlamov, V. V.: _Maxwell field on the Poincaré group_. Int. J. Mod. Phys. A. 20, 17, 4095–4112 (2005); arXiv:math-ph/0310051 (2003).
* [65] Varlamov V. V.: _Relativistic spherical functions on the Lorentz group_. J. Phys. A: Math. Gen. 39, 805–822 (2006); arXiv:math-ph/0507056 (2005).
* [66] Varlamov V. V.: _Towards the Quantum Electrodynamics on the Poincaré Group_. New Topics in Mathematical Physics Research (Ed. C. V. Benton) New York. – Nova Science Publishers, 2006. – P. 109–179; arXiv:hep-th/0403070 (2004).
* [67] Varlamov, V. V.: _General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains_. Int. J. Theor. Phys. 46, 4, 741–805 (2007); arXiv:math-ph/0503058 (2005).
* [68] Vilenkin, N. Ya.: Special Functions and the Theory of Group Representations. AMS, Providence (1968).
* [69] Weber, H.: Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann’s Vorlesungen. Friedrich Vieweg und Sohn, Braunschweig (1901).
* [70] Yukawa, H.: Quantum theory of non-local fields. I. Free fields. Phys. Rev. 77, 219–226 (1950).
|
arxiv-papers
| 2011-10-06T03:37:47 |
2024-09-04T02:49:22.859058
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Varlamov",
"submitter": "Vadim Varlamov",
"url": "https://arxiv.org/abs/1110.1146"
}
|
1110.1347
|
$Author:diego$ $Date$ $Rev:398$
# A Dual-based Method for Resource Allocation in OFDMA-SDMA Systems with
Minimum Rate Constraints
Diego Perea-Vega, André Girard and Jean-François Frigon (Corresponding author)
Department of Electrical Engineering
École Polytechnique de Montréal
C.P. 6079, succ. centre-ville, Montréal, QC, Canada, H3C 3A7
Email: enrique.perea@polymtl.ca, andre.girard@gerad.ca, j-f.frigon@polymtl.ca
Tel: 1-514-340-4711 ext. 3642
Fax: 1-514-340-5892
(Rev. on — )
###### Abstract
We consider multi-antenna base stations using orthogonal frequency-division
multiple access (OFDMA) and space division multiple access (SDMA) techniques
to serve single antenna users, where some of those users have minimum rate
requirements and must be served in the current time slot (real time users),
while others do not have strict timing constraints (non real time users) and
are served on a best effort basis. The resource allocation problem is to find
the user assignment to subcarriers and the transmit beamforming vectors that
maximize a linear utility function of the user rates subject to power and
minimum rate constraints. The exact optimal solution to this problem can not
be reasonably obtained for practical parameters values of the communication
system. We thus derive a dual problem formulation whose optimal solution
provides an upper bound to all feasible solutions and can be used to benchmark
the performance of any heuristic method used to solve this problem. We also
derive from this dual optimal solution a primal-feasible dual-based method to
solve the problem and we compare its performance and computation time against
a standard weight adjustment method. We find that our method follows the dual
optimal bound more closely than the weight adjustment method. This off-line
algorithm can serve as the basis to develop more efficient heuristic methods.
## I Introduction
Multi-antenna base stations using orthogonal frequency-division multiple
access (OFDMA) and space division multiple access (SDMA) can simultaneously
transmit to different sets of users on multiple subcarriers. In OFDMA-SDMA
systems, multi-user diversity allows an increase in the system throughput by
assigning transmitting resources to users with good channel conditions. High
data rates are thus made possible by exploiting the degrees of freedom of the
system in time, frequency and space dimensions. OFDMA-SDMA is also supported
by WiMAX and LTE-Advanced systems which are the technologies that will most
likely be used to implement fourth generation (4G) cellular networks [1, 2].
Due to the increased degrees of freedom it is critical to use a dynamic and
efficient resource allocation (RA) mechanism that takes full advantage of all
OFDMA-SDMA transmitting resources [3]. The role of an RA and scheduling
algorithm is to allocate the resources for transmission required to meet the
quality of service (QoS) requested from upper layers. In this paper, we focus
on resource allocation policy for an OFDMA-SDMA system supporting real time
traffic with minimum rate requirements.
### I-A State of the Art
The combinatorial nature of the RA problem makes it NP-complete [4]. For an
OFDMA-SDMA system with a practical number of subcarriers, users and transmit
antennas, it is thus almost impossible to solve the RA problem directly.
Therefore, most research work focuses on developing heuristic and near-optimal
RA algorithms.
Traffic in the system can be divided into two main groups: delay sensitive
real time (RT) services and delay-insensitive non real time (nRT) services.
Early work on OFDMA-SDMA systems focused on solving the RA problem for only
nRT services,where the objective was to maximize the total throughput with
only power constraints and possibly minimum BER constraints. In [4] the
complete optimization problem was divided into a per-subcarrier user selection
problem and a power allocation problem, which are both solved heuristically. A
similar approach using Zero Forcing (ZF) beamforming was reported in [5]. The
work of [4, 5] does not solve the complete optimization problem because of its
computational complexity. Instead, it is separated into uncoupled subproblems
that provide non-optimal solutions. For this reason, optimal or good
approximations to near-optimal solutions are important to benchmark the
heuristic algorithm performance. Several methods to obtain near optimum
solutions have been proposed for the RA problem with nRT traffic only. For
example, in [6] genetic algorithms are proposed, while [7, 8, 9, 10] provide
methods to compute a near-optimal solution based on dual decomposition
methods. In addition to providing a benchmark, near optimal algorithms can
also lead to the design of efficient RA methods as shown in [10] where
heuristic algorithms derived from the dual decomposition methods are proposed.
Several approaches have been proposed to solve the OFDMA-SDMA RA algorithm
with RT traffic services. In some works, the design of the RA algorithm
supporting real time QoS is intertwined with the scheduler design. The
scheduler decides at each time slot which users must be served and sets their
priorities in the RA utility function [11]. Priority can thus be given
according to the current deadlines for RT services by increasing their weight
in the utility function while achieving some degree of fairness for nRT
services [12]. The utility function is used as the objective to maximize by
the RA problem without any explicit minimum rate constraints for the RT users.
Similarly, RA heuristic algorithms for RT services have also been proposed
where the users are served in priority according to the packet waiting time
and urgency to be served [13, 14, 15]. Note that with these _reactive_
approaches, the minimum rate requirements are not modelled as hard constraints
in the per-slot RA problem. Instead, RT users with poor channel conditions are
backlogged until their delay is close to the deadline and then the RA reacts
and allocates resources to them. This causes an increase in the average delay
and delay jitter, which is not suitable for RT services.
Another approach is to use constraints on the average rate delivered to a user
[16]. However, unlike the work presented in [17] where a near optimal solution
is provided for the single antenna OFDMA RA problem with average rate
constraint, the algorithm presented in [16] is a heuristic approximation. Note
that with average rate constraints, RT users tend to be served when they have
good channel conditions which can create unwanted delay violations and jitter.
In [18], an heuristic algorithm for the RA problem with proportional rate
constraints is proposed. Although rate constraints are used on a per slot
basis, the actual rate is relative to the total rate and does not offer a
guaranteed minimum rate to RT users.
Heuristic algorithms have also been proposed for the RA problem with minimum
rate constraints
[19, 20]. For example, in [19] a subcarrier exchange heuristic is proposed to
satisfy the minimum rates.
### I-B Paper Contribution and Organization
To guarantee the required QoS for RT users it is preferable to integrate their
minimum rate requirements into the RA optimization problem rather than using
reactive methods without explicit rate constraints. Some heuristics have been
proposed for the OFDMA SDMA RA problem with minimum rate requirements but no
optimal or near-optimal solution has been derived, as was done for the OFDMA-
SDMA RA problem for nRT traffic. Having such a solution is important to
benchmark heuristics and can also lead to the design of more efficient
heuristics. The main contribution of this paper is therefore an efficient
method that provides a near-optimal solution to the following OFDMA-SDMA RA
problem for mixed RT and nRT traffic: for a given time slot, find the user
selection and beamforming vectors that maximize a linear utility function of
the users rates, given a power constraint and minimum rate constraints for RT
users. The user weights in the linear utility function are arbitrary and can
be the result of a prioritization or fairness policy by the scheduler. We
focus on the solution of the RA optimization problem by using a Lagrange dual
decomposition method. The solution to the dual function provides an upper
bound to the primal RA problem. We show, for small cases where it is possible
to find the optimal solution to the primal problem, that the duality gap is
small. We also propose a simple off-line near optimal algorithm which, based
on the solution obtained from the dual decomposition method, provides a
feasible solution to the RA problem. We study several cases where we compare
the performance of the dual upper bound, the feasible solution provided by the
proposed method and a solution obtained using weight adjustments in the
utility function. The results indicate that the proposed method is close to
the upper bound while methods adjusting the user weights in the utility
function in order to prioritize RT users lead to significantly sub-optimal
solutions for the OFDMA-SDMA RA problem.
The paper is organized as follows. We describe the system and formulate the
optimization problem we seek to solve in Section II. We present in Section III
the dual-based method. In Section IV, we present an algorithm that finds a
feasible solution based on the dual optimal solution and we present two other
alternative methods: one that solves the problem by finding the exact
solution, the other by using the weight adjustment technique. In Section V, we
compare the performance and computation time of our method against the two
alterative methods and in Section VI we present our conclusions.
## II System description and Problem Formulation
We consider the resource allocation problem for the downlink transmission in a
multi-carrier multi-user multiple input single output (MISO) system with a
single base station (BS) serving $K$ users, where some of those users have RT
traffic with minimum rate requirements while the others have nRT traffic. The
base station is equipped with $M$ transmit antennas and each user has one
receive antenna. In this configuration, the BS can transmit in the downlink
data to different users on each subcarrier by performing linear beamforming
precoding at the BS. At each OFDM symbol, the base station can change the
beamforming vector for each user on each subcarrier to maximize some
performance function. In this paper, we assume that capacity achieving channel
coding is employed and the data rate units are in terms of bits per OFDM
symbol or equivalently bits per second per Hertz (bps/Hz).
### II-A Signal Model
First we describe the model used to compute the bit rate received by each
user. Define
$s_{k,n}$
the symbol transmitted by user $k$ on subcarrier $n$. We assume that the
$s_{k,n}$ are independently identically distributed random variables with
$s_{k,n}\sim\mathcal{CN}(0,1)$.
${\mathbf{w}}_{k,n}$
a $M$-component column vector representing the beamforming for user $k$ on
subcarrier $n$. Unless otherwise noted, we denote ${\mathbf{w}}$ the vector
made up by the column stacking of the vectors ${\mathbf{w}}_{k,n}$.
$\mathbf{x}_{n}$
a $M$-component column vector representing the signal sent by the array of $M$
antennas at the BS for each subcarrier $n$.
$\mathbf{h}_{k,n}$
a $M$-component row vector representing the channel between the $M$ antennas
at the BS and the receive antenna at user $k$ for each subcarrier $n$.
$z_{k,n}$
the additive white gaussian noise at the receiver for user $k$ on subcarrier
$n$. The $z_{k,n}$ are independently identically distributed (i.i.d.) and
without loss of generality we assume that $z_{k,n}\sim\mathcal{CN}(0,1)$.
$y_{k,n}$
the signal received by user $k$ on subcarrier $n$.
$r_{k,n}^{0}$
the rate of user $k$ on subcarrier $n$.
The signal vector ${\mathbf{x}}_{n}$ is built by a linear precoding scheme
which is a linear transformation of the information symbols ${s}_{k,n}$:
${\mathbf{x}}_{n}=\sum_{k}{\mathbf{w}}_{k,n}s_{k,n}.$ (1)
The signal received by user $k$ on subcarrier $n$ is then given by
$\displaystyle y_{k,n}$
$\displaystyle={\mathbf{h}}_{k,n}{\mathbf{x}}_{n}+z_{k,n}$
$\displaystyle={\mathbf{h}}_{k,n}{\mathbf{w}}_{k,n}s_{k,n}+\sum_{j\neq
k}{\mathbf{h}}_{k,n}{\mathbf{w}}_{j,n}s_{j,n}+z_{k,n}.$ (2)
The second and third terms in the right side of (2) correspond to the
interference and noise terms, respectively. Since the signals and noise are
Gaussian, the interference plus noise term is also Gaussian and the data rate
of user $k$ for subcarrier $n$ is given by the Shannon channel capacity for an
additive white Gaussian noise channel:
$r_{k,n}^{0}\left({\mathbf{w}}\right)=\log_{2}\left(1+\frac{\|{\mathbf{h}}_{k,n}{\mathbf{w}}_{k,n}\|^{2}}{1+\sum_{j\neq
k}\|{\mathbf{h}}_{k,n}{\mathbf{w}}_{j,n}\|^{2}}\right).$ (3)
### II-B Rate Maximization Problem
The general rate maximization problem corresponding to the OFDMA-SDMA RA
problem with mixed RT and nRT traffic is to find a set of beamforming vectors
${\mathbf{w}}_{k,n}$ that will maximize the users weighted sum rate. This is
limited by the total power available for the transmission at the base station
and some users with real time QoS requirements must receive a minimum rate.
More precisely, we assume that we know
$K$
Number of users in the cell.
$\mathcal{K}$
Set of users in the cell: $\\{1,\ldots,K\\}$.
$\mathcal{D}$
A subset of $\mathcal{K}$ containing the users that have minimum rate
requirements. We define $D=|\mathcal{D}|$.
$M$
Number of antennas at the BS.
$\check{d}_{k}$
Minimum rate requirement for user $k$.
$N$
Number of subcarriers available.
$\check{P}$
Total power available at the base station for transmitting over all channels.
$c_{k}$
Weight factors that are used by the scheduler to implement prioritization or
fairness.
We then want to solve the following optimization problem to obtain the
resource allocation
$\displaystyle\max_{{\mathbf{w}}}U^{0}$
$\displaystyle=\sum_{n=1,k=1}^{N,K}c_{k}r_{k,n}^{0}({\mathbf{w}})$ (4)
$\displaystyle\sum_{n=1,k=1}^{N,K}\|{{\mathbf{w}}}_{k,n}\|^{2}$
$\displaystyle\leq\check{P}$ (5)
$\displaystyle\sum_{n=1}^{N}r_{k,n}^{0}({\mathbf{w}})$
$\displaystyle\geq\check{d}_{k},\quad k\in{\mathcal{D}}.$ (6)
The power used by the transmitter is represented by the sum of the squared
norms of the beamforming vectors in constraint (5). The achievable rate over
all subcarriers should be higher or equal than the required minimum rate per
user as in (6).
Problem (4-6) is a non-convex, nonlinear optimization problem. Using an exact
algorithm to find a global optimal solution is very hard considering the size
of a typical problem where there can be up to a hundred users and hundreds of
sub-channels. Another option is to use a standard non-linear program (NLP)
solver to compute a local optimal solution and use different starting points
in the hope of finding a good global solution. The problem with this approach
is that 1) we don’t know how close we are to the true optimum and 2) the
technique is quite time-consuming. Albeit those problems, we explored this
approach and observe that most users end up with a zero beamforming vector and
only a small subset of users ($\leq M$) actually get some rate. Furthermore,
in accordance to what was reported for the SDMA problem in [21], we observed
that at high SNR the ZF solution is very close to the local optimum. This ZF
solution is easily computed by channel diagonalization and water-filling power
allocation. For these reasons, we now turn to the so-called _Zero-Forcing_
beamforming strategy.
### II-C Zero-Forcing Beamforming
In general, user $k$ is subject to the interference from other users which
reduces its bit rate, as indicated by the denominator in (3). Zero-forcing
beamforming is a strategy that completely eliminates interference from other
users. For each subcarrier $n$, we choose a set $\phi$ of $g\leq M$ users
which are allowed to transmit. This is called a _SDMA_ set. We then impose the
condition that for each user $k$ in this set, the beamforming vector of user
$k$ must be orthogonal to the channel vectors of all the other users of the
set. This amounts to adding the orthogonality constraints
${\mathbf{h}}_{k,n}{\mathbf{w}}_{j,n}=0\quad j\not=k,\,\,j,k\in\phi$ (7)
and the user $k$ data rate for subcarrier $n$ simplifies to:
$r_{k,n}^{0}\left({\mathbf{w}}_{k,n}\right)=\log_{2}\left(1+\|{\mathbf{h}}_{k,n}{\mathbf{w}}_{k,n}\|^{2}\right).$
(8)
With zero-forcing, the beamforming problem is now made up of two parts. We
need to select a SDMA set for each subcarrier and for each selected SDMA set,
we must compute the beamforming vectors in such a way that the total rate
received by all users is maximized. Because of this, we need to add another
set of decision variables
$\alpha_{k,n}$
a binary variable that is 1 if we allow user $k$ to transmit on subcarrier $n$
and zero otherwise. We denote the collection of $\alpha_{k,n}$ by the vector
$\boldsymbol{\alpha}$ .
This results in the ZF problem
$\displaystyle\max_{{\mathbf{w}},\boldsymbol{\alpha}}U^{1}$
$\displaystyle=\sum_{n=1,k=1}^{N,K}c_{k}r_{k,n}^{0}({\mathbf{w}}_{k,n})$ (9)
$\displaystyle\sum_{n=1,k=1}^{N,K}\|{\mathbf{w}}_{k,n}\|^{2}$
$\displaystyle\leq\check{P}$ (10)
$\displaystyle\sum_{n=1}^{N}r_{k,n}^{0}({\mathbf{w}}_{k,n})$
$\displaystyle\geq\check{d}_{k},\quad k\in{\mathcal{D}}$ (11)
$\displaystyle\sum_{k}\alpha_{k,n}$ $\displaystyle\leq M,\quad\forall n$ (12)
$\displaystyle\left({\mathbf{h}}_{k,n}{\mathbf{w}}_{j,n}\right)^{2}$
$\displaystyle\leq
B\left[(1-\alpha_{k,n})+(1-\alpha_{j,n})\right],\quad\forall n,\,\forall
k,\,\forall j,\,k\not=j$ (13) $\displaystyle\|{\mathbf{w}}_{k,n}\|$
$\displaystyle\leq A\alpha_{k,n}$ (14)
$\displaystyle\alpha_{k,n}\in\left\\{0,1\right\\}$ (15)
where $A$ and $B$ are some large positive constants. Constraint (12)
guarantees that we do not choose more than $M$ users for each subcarrier and
constraint (13) guarantees that if we have chosen two users $k$ and $j$, they
meet the zero-forcing constraints and that the constraints are redundant for
other choices of users. Constraint (14) guarantees that the beamforming vector
is zero for users that are not chosen. It would seem that the zero-forcing
model is not improving things much: We have gone from a non-convex nonlinear
program to a non-convex mixed nonlinear program. However, as we will explain
in the next section, this allows us to design an efficient and accurate
algorithm.
## III Dual-Based Solution Method
We cannot solve problem (9–15) fast enough to use it for a real time
algorithm. Nevertheless, we need to compute solutions so that we can use them
as benchmarks to evaluate the quality of real time heuristic approximations.
We now present an off-line solution technique that is tractable for problems
of moderate size and that can give either near-optimal solutions or at least a
bound when the optimum cannot be reached.
Solving the zero-forcing problem will require some form of search over the
$\alpha$ variables. Note that this ranges over all subsets of users smaller
than $M\times N$ so that the search space is going to be fairly large. Our
first transformation is thus to separate the problem into single-subcarrier
subproblems. For this, we dualize the constraints (10) and (11) since they are
the ones that couple the subcarriers. Define the dual variables
$\lambda$
Lagrange multiplier for power constraint (10).
$\mu_{k}$
Lagrange multipliers for minimum rate constraint (11) of user $k$. The
collection of $\mu_{k}$ is denoted $\boldsymbol{\mu}$.
In order to simplify the derivation, we define the dual variables $\mu_{k}$
for all users $k\in\mathcal{K}$. For users with no minimum rate requirements
($k\notin\mathcal{D}$), we have $\mu_{k}=0$. In what follows, we use the
standard form of Lagrangian duality which is expressed in terms of
minimization with inequality constraints of the form $\leq$. Under these
conditions, the multipliers $\lambda,\boldsymbol{\mu}\geq 0$. We get the
partial Lagrangian
$\displaystyle\mathcal{L}$
$\displaystyle=-\sum_{n=1,k=1}^{N,K}c_{k}r_{k,n}^{0}({\mathbf{w}}_{k,n})+\lambda\left[\sum_{n=1,k=1}^{N,K}\|{\mathbf{w}}_{k,n}\|^{2}-\check{P}\right]+\sum_{k\in\mathcal{D}}\mu_{k}\left[\sum_{n=1}^{N}-r_{k,n}^{0}({\mathbf{w}}_{k,n})+\check{d}_{k}\right]$
$\displaystyle=-\lambda\check{P}+\sum_{k}\mu_{k}\check{d}_{k}+\sum_{n}\left\\{-\sum_{k}(c_{k}+\mu_{k})r_{k,n}^{0}({\mathbf{w}}_{k,n})+\lambda\sum_{k}\|{\mathbf{w}}_{k,n}\|^{2}\right\\}.$
(16)
The value of the dual function $\Theta$ at some point
$(\lambda,\boldsymbol{\mu})$ is obtained by minimizing the Lagrange function
over the primal variables
$\Theta(\lambda,\boldsymbol{\mu})=\min_{{\mathbf{w}},\boldsymbol{\alpha}}\mathcal{L}(\lambda,\boldsymbol{\mu},{\mathbf{w}},\boldsymbol{\alpha})$
(17)
and the dual problem is
$\displaystyle\max_{\lambda,\boldsymbol{\mu}}\ $
$\displaystyle\Theta(\lambda,\boldsymbol{\mu})$ (18)
$\displaystyle\lambda,\boldsymbol{\mu}$ $\displaystyle\geq 0$ (19)
which we can solve by the well known subgradient algorithm [22]. From now on,
we concentrate on the calculation of the subproblem (17).
### III-A Subchannel Subproblem
Because of the relaxation of the carriers coupling constraints, the
subproblems in (17) decouple by subcarrier since the objective (16) is
separable in $n$ and so are constraints (12–14). Computing the dual function
then requires the solution of $N$ independent subproblems. For subcarrier $n$,
after dropping the $n$ subscript, this has the form
$\displaystyle\min_{{\mathbf{w}},\boldsymbol{\alpha}}f_{n}$
$\displaystyle=-\sum_{k}(c_{k}+\mu_{k})r_{k}^{0}({\mathbf{w}}_{k})+\lambda\sum_{k}\|{\mathbf{w}}_{k}\|^{2}$
(20) $\displaystyle\sum_{k}\alpha_{k}$ $\displaystyle\leq M,$ (21)
$\displaystyle\left({\mathbf{h}}_{k}{\mathbf{w}}_{j}\right)^{2}$
$\displaystyle\leq B\left[(1-\alpha_{k})+(1-\alpha_{j})\right],\quad\forall
k,\,\forall j,\,k\not=j$ (22) $\displaystyle\|{\mathbf{w}}_{k}\|$
$\displaystyle\leq A\alpha_{k}$ (23)
$\displaystyle\alpha_{k}\in\left\\{0,1\right\\}$
Problem (20–23) is still a mixed NLP, albeit of a smaller size.
### III-B SDMA Subproblem
A simple solution procedure is to enumerate all possible choices for
$\alpha_{k,n}$ that meet constraint (12). This is called the _extensive_
formulation of the problem. Each such choice defines a SDMA set which we will
denote by $s$ and $\kappa=|s|$. For each $s$, we solve the optimal beamforming
problem
$\displaystyle\max_{{\mathbf{w}}}f_{n,s}$ $\displaystyle=\sum_{k\in
s}c_{k}^{\prime}\log_{2}\left(1+({\mathbf{h}}_{k}{\mathbf{w}}_{k})^{2}\right)-\lambda\|{\mathbf{w}}_{k}\|^{2}$
(24) $\displaystyle{\mathbf{h}}_{j}{\mathbf{w}}_{k}$ $\displaystyle=0\quad
j\not=k\,\ j,k\in s$ (25)
where $c^{\prime}_{k}=c_{k}+\mu_{k}$. Note that constraint (21) is
automatically satisfied by the construction of $s$, constraint (23) simply
drops out since ${\mathbf{w}}_{k}=0$ for $k\not\in s$ and constraint (22)
remains only for $k\in s$, but we write it as (25) because we are considering
only users that belong to SDMA set $s$.
This is certainly not a feasible real time algorithm, but for realistic values
of $K$, say around 100, and $M=4$, the number of cases is still manageable.
This can give near-optimal solutions against which to compare heuristics. This
is possible only if the SDMA beamforming sub-problem (24–25) can be solved
efficiently.
### III-C Beamforming Subproblem
This is in fact the case because it separates into $\kappa$ independent
problems, one for each user in the SDMA set. Here again we drop the $k$ index
to simplify the discussion. We have to compute the beamforming vector
${\mathbf{w}}$ for a given user. For this user, we know the set of channel
vectors for the other members of $s$. We denote these vectors by the
$(\kappa-1)\times M$ matrix ${\mathbf{H}}$. We also denote the channel vector
for the user under consideration by ${\mathbf{h}}$. The problem is then
$\displaystyle\max_{{\mathbf{w}}}f_{n,s,k}$
$\displaystyle=c^{\prime}_{k}\log_{2}\left(1+\left({\mathbf{h}}{\mathbf{w}}\right)^{2}\right)-\lambda\|{\mathbf{w}}\|^{2}$
(26) $\displaystyle{\mathbf{H}}{\mathbf{w}}$ $\displaystyle=0$ (27)
so that for realistic values, this is a small nonlinear program. There are $M$
variables and $\kappa-1$ linear constraints. It can be solved quickly by a
number of techniques. Still, the overall computation load can be quite large.
There will be $\kappa$ such problems to solve for each SDMA set, and there are
$S=\sum_{i=1}^{\kappa}\binom{K}{i}$ such sets for each of the $N$ subcarriers
so that we have to solve the problem $\kappa\times S\times N$ times and this
for each iteration of the subgradient algorithm. Clearly, any simplification
of the beamforming subproblem can reduce the overall computation time
significantly.
### III-D Approximate Solution of the Beamforming Problem
This can be done by the following construction. Instead of searching in the
whole orthogonal subspace of ${\mathbf{H}}$ as defined by (27), we pick a
direction vector in that subspace and search only on its support. This will
give a good approximation to the extent that the direction vector is close to
the optimal vector. The choice of direction is motivated by the fact that the
objective function depends only on the product ${\mathbf{h}}{\mathbf{w}}$. We
then introduce a new independent variable
$q={\mathbf{h}}{\mathbf{w}}$ (28)
and because this variable is not independent of ${\mathbf{w}}$, we add Eq.
(28) as a constraint. We then get the equivalent problem
$\displaystyle\max_{{\mathbf{w}},q}f$
$\displaystyle=c^{\prime}\log_{2}\left(1+q^{2}\right)-\lambda\|{\mathbf{w}}\|^{2}$
(29) $\displaystyle\left({\mathbf{h}}{\mathbf{w}}\right)$ $\displaystyle=q$
(30) $\displaystyle{\mathbf{H}}{\mathbf{w}}$ $\displaystyle=0$ (31)
which we can rewrite in the standard form
${\mathbf{G}}{\mathbf{w}}={\mathbf{b}}$ where the ${\mathbf{G}}$ matrix is the
concatenation of ${\mathbf{h}}$ and ${\mathbf{H}}$ and
${\mathbf{b}}^{T}=[q,0,0\ldots 0]^{T}$.
Since we are proposing to transform the constrained optimization over the
$\kappa$ variables into an unconstrained optimization over $q$ only, we must
be able to express ${\mathbf{w}}$ as a function of $q$. Since the linear
system is under-determined, this is obviously not unique. We use
${\mathbf{G}}^{+}$, the pseudo-inverse of ${\mathbf{G}}$, for this back
transformation. We then have ${\mathbf{w}}={\mathbf{G}}^{+}{\mathbf{b}}$. A
well known property of the pseudo-inverse is that it picks the vector of
minimum norm compatible with the linear system. In other words, choosing this
transformation will _minimize_ $|{\mathbf{w}}|$ so that it is minimizing the
power term in the objective function. Because $\lambda\geq 0$, this has the
effect of contributing to the maximization of $f$.
Expanding the matrix equation for ${\mathbf{w}}$, we find that
$w_{i}={\mathbf{G}}^{+}_{i,1}q\quad i=1\ldots\kappa$ (32)
which can then be replaced in the objective function. We get the unconstrained
problem
$\max_{q}c^{\prime}\log_{2}\left(1+q^{2}\right)-\lambda q^{2}\gamma^{2}$ (33)
where $\gamma=\|{\mathbf{G}}^{+}_{1}\|$ and ${\mathbf{G}}^{+}_{1}$ denotes the
first column of ${\mathbf{G}}^{+}$. Changing the variables $p=q^{2}$ and
adding the constraint $p\geq 0$, we get the equivalent problem
$\displaystyle\max_{p}\ $ $\displaystyle
c^{\prime}\log(1+p)-\lambda\gamma^{2}p$ (34) $\displaystyle p$
$\displaystyle\geq 0$ (35)
which has the solution
$p=\max\left\\{0,\frac{c^{\prime}}{\lambda\gamma^{2}}-1\right\\}$ (36)
so that the computation time is basically the evaluation of
${\mathbf{G}}^{+}$.
### III-E Computing the Dual Function
To summarize, after reinstating all indices, we can write
$\displaystyle\Theta(\lambda,\boldsymbol{\mu})$
$\displaystyle=-\left(\lambda\check{P}-\sum_{k}\mu_{k}\check{d}_{k}+\sum_{n}f_{n}\right)$
(37) $\displaystyle f_{n}$ $\displaystyle=\max_{s}\left\\{f_{n,s}\right\\}$
(38) $\displaystyle f_{n,s}$ $\displaystyle=\sum_{k\in s}f_{n,s,k}$ (39)
$\displaystyle f_{n,s,k}$
$\displaystyle=c^{\prime}_{k}\log\left(1+p_{n,s,k}\right)-\lambda\|{\mathbf{w}}_{n,s,k}\|^{2}$
(40) $\displaystyle p_{n,s,k}$
$\displaystyle=\max\left\\{0,\frac{c^{\prime}_{k}}{\lambda\gamma_{n,s,k}^{2}}-1\right\\}$
(41) $\displaystyle\gamma_{n,s,k}$
$\displaystyle=\|{\mathbf{G}}_{1}^{+}\|_{n,s,k}$ (42)
$\displaystyle{\mathbf{w}}_{n,s,k}$
$\displaystyle={\mathbf{G}}^{+}_{n,s,k}{\mathbf{b}}_{n,s,k}$ (43)
$\displaystyle{\mathbf{b}}_{n,s,k}$ $\displaystyle=[p_{n,s,k},0,\ldots 0]^{T}$
(44)
and ${\mathbf{G}}^{+}_{n,s,k}$ is the pseudo-inverse of the concatenation of
${\mathbf{h}}_{n,k}$ and ${\mathbf{H}}_{n,s,k}$. We denote $s^{*}(n)$,
$n=1,\dots,N$, the solution of the maximization operation over $s$ in (38).
This is the optimal SDMA set for subchannel $n$ for the current values of the
multipliers. We also denote ${\mathbf{w}}^{*}_{n,k}$ the optimal beamforming
vectors for the users $k\in s^{*}(n)$. The largest part of the computation to
evaluate the dual function is the calculation of ${\mathbf{G}}^{+}_{n,s,k}$
which has to be done for each subchannel, each SDMA set and each user in these
SDMA sets. The number of evaluations can become quite large but the size of
each matrix is relatively small so that the calculation remains feasible for
medium-size networks. Another advantage is that while solving the dual problem
requires multiple subgradient iterations, the calculation of the pseudo-
inverses is _independent_ of the value of the multipliers. This means that can
be done only once in the initialization step of the subgradient procedure.
For convenience we define $\Phi(\lambda,\boldsymbol{\mu})$ as the negative of
the dual function,
$\displaystyle\Phi\doteq-\Theta=\lambda\check{P}-\sum_{k}\mu_{k}\check{d}_{k}+\sum_{n}\max_{s}\left\\{f_{n,s}\right\\}$
(45)
and minimize this function when solving the dual problem (18). Algorithm 1
finds the optimal dual variables $(\lambda^{*},\boldsymbol{\mu}^{*})$ that
solve the dual problem (18) using the subgradient method [22] with a fixed
step size $\delta$ . The optimum value $\Phi^{*}$ is a bound for the primal
objective and thus for any feasible point of the primal problem (9). If
$U^{1}$ is the objective achieved by any feasible point in the primal problem
(9) and $U^{*}$ its optimum, the following inequalities hold [23]
$\Phi(\lambda,\boldsymbol{\mu})\geq\Phi^{*}\geq U^{*}\geq U^{1}$ (46)
The dual optimum found $\Phi^{*}$, or any approximation to it, is thus an
_upper bound_ to the optimum value of the primal problem which can be used to
benchmark other solution methods.
Construct the set $\mathcal{S}$ of all subsets of users of size
$1\leq\kappa\leq M$
for all $n=1\ldots N$ do
for all $s\in\mathcal{S}$ do
for all $k\in s$ do
Compute the pseudo-inverse $G^{+}_{n,s,k}$ and $\gamma_{n,s,k}$
end for
end for
end for
Choose an initial value $\lambda^{0}$ and $\boldsymbol{\mu}^{0}$
Subgradient iterations. We set a limit of $I_{m}$ on the number of iterations
for all $i=1\ldots I_{m}$ do
Solve the $N$ subproblems (38)
Compute the subgradients:
$g_{\mu}^{(k)}=\check{d}_{k}-\sum_{n}r_{n,k}$
$g_{\lambda}=\sum_{n}\sum_{k\in
s^{*}(n)}\|{\mathbf{w}}^{*}_{n,k}\|^{2}-\check{P}$
if $\|\boldsymbol{g}_{\mu}\|\leq\epsilon$ and $\|g_{\lambda}\|\leq\epsilon$
then
Break
else
Update the multipliers
$\lambda^{i+1}=[\lambda^{i}+\delta g_{\lambda}]^{+}$
$\boldsymbol{\mu}^{i+1}=[\boldsymbol{\mu}^{i}+\delta\boldsymbol{g}_{\mu}]^{+}$
end if
end for
Algorithm 1 Calculation of the dual solution
### III-F Performance of the Dual Method
We study in this section the convergence speed and computation time of the
dual algorithm for single random channel realizations (the average performance
is studied in Section V). We present in Figure 1 the value of the dual
function and Lagrange multipliers as a function of the number of iterations
while the total transmit power and the rate received by user 1 (single user
with minimum rate requirements) are shown in Figure 2. We can see that the
method converges very quickly to a solution that is both close to the minimum
value and feasible. We observed a similar behavior for several other
configurations.
Figure 1: Dual function and multipliers for $M=3$, $K=16$, $N=16$,
$\check{P}=80$, $D=1$ and $\check{d}_{1}=$20 bps/Hz. Figure 2: Power and rate
constraints for $M=3$, $K=16$, $N=16$, $\check{P}=80$, $D=1$ and
$\check{d}_{1}=$20 bps/Hz.
We can see the computation time (i.e., the time used by the CPU) required to
solve a problem as a function of the number of users in Figure 3 and as a
function of the number of channels in Figure 4. In Figure 4 we further
separated the CPU time between the CPU time required to compute the pseudo-
inverse at the initialization and the CPU time required to find the dual
solution. There is a striking difference since the CPU time growth is much
faster than linear as a function of $K$ while it is very linear as a function
of $N$. This is obviously because the Lagrangian decomposition separates the
overall problem into $N$ independent subproblems and for fixed $K$, the CPU
time will grow as the number of subproblems. As a side note, the last value
plotted in Figure 3 is much lower because for this random channel realization
the problem had no active rate constraints, thus no extra iterations were
required. Another important point however is that calculating the pseudo-
inverses is much more time-consuming that solving the dual itself. Notice that
the two curves in Figure 4 almost overlap but they are plotted on a different
scale.
Figure 3: Computation time as a function of the number of users $K$ for $M=3$,
$N=4$, $\check{P}=80$, $D=1$ and $\check{d}_{1}=$20 bps/Hz. Figure 4:
Computation time as a function of the number of subcarriers $N$ for $M=3$,
$K=16$, $\check{P}=80$, $D=1$ and $\check{d}_{1}=$20 bps/Hz.
## IV Other Solution Algorithms
The SDMA set selection and beamforming vectors found by algorithm 1 do not
always provide a primal feasible solution. The rate or power constraints might
be violated whenever the algorithm stops because the number of iterations has
been reached before the convergence rule is met. In this section we present
three different approaches to solve the primal problem. In Section IV-A, a
direct method where we enumerate all the variables $\boldsymbol{\alpha}$ and
solve the problem for the beamforming vectors ${\mathbf{w}}$ in each case is
explained. In Section IV-B, we propose a simple procedure to obtain a feasible
solution from the dual solution found with algorithm 1. Finally, in Section
IV-C we propose a method based on weight adjustments of the utility function
to meet the minimum rate requirements. In Section IV-D we study the difference
between the dual-based and the weight adjustment methods.
### IV-A Exact Solution
One way to evaluate the accuracy of the dual algorithm is to compare it with
an exact solution. Problem (9–14) is a nonlinear MIP for which we can do a
complete enumeration of the binary variables $\boldsymbol{\alpha}$, the set of
${\mathbf{w}}$ variables is determined by the choice of a particular
$\boldsymbol{\alpha}$ and constraints (14) are automatically satisfied. The
same remark goes for Eq. (13) where the only remaining constraints are the
ones for which $\alpha_{k,n}=1$. The ZF constraints are now written as (50)
where $s(n)$ is the SDMA set for subchannel $n$, given by the value of
${\mathbf{\alpha}}$. For each given value of $\boldsymbol{\alpha}$, we then
need to solve the optimal beamforming problem
$\displaystyle\max_{{\mathbf{w}}}\sum_{n=1,k=1}^{N,K}c_{k}r_{k,n}^{0}({\mathbf{w}}_{k,n})$
(47) $\displaystyle\sum_{n=1,k=1}^{N,K}\|{\mathbf{w}}_{k,n}\|^{2}$
$\displaystyle\leq\check{P}$ (48)
$\displaystyle\sum_{n=1}^{N}r_{k,n}^{0}({\mathbf{w}}_{k,n})$
$\displaystyle\geq\check{d}_{k}\quad k\in{\mathcal{D}}$ (49)
$\displaystyle\left({\mathbf{h}}_{k,n}{\mathbf{w}}_{j,n}\right)^{2}$
$\displaystyle=0\quad\,j,k\in s(n),\,\,k\not=j\,\,\forall n$ (50)
where the optimization variables are the beamforming vectors of users in
$s(n)$ for each subchannel $n$. The main difference with the dual method is
that here, we have to enumerate the set of all possible choices of subsets of
$K$ users, $M$ antennas and $N$ channels. This is a much larger set than for
the dual method where the enumeration is done separately for each channel and
for this reason, we cannot expect to solve very large problems with the exact
solution approach.
Each beamforming sub-problem (47) is relatively small but it is not convex. We
use the same technique as in Section III-D to simplify it. First, we group the
user vectors belonging to SDMA set $s(n)$ in a $|s|\times M$ matrix
${\mathbf{H}}_{n}$ and we assume a given $1\times|s|$ user power vector
${\mathbf{p}}_{n}$. The ZF constraints (50) are written in matrix form as
$\displaystyle{{\mathbf{H}}_{n}}{\mathbf{W}}_{n}=\text{diag}(\sqrt{{\mathbf{p}}_{n}}),\quad\forall
n$ (51)
and we restrict the beamforming vectors to be in the direction given by the
pseudo-inverse matrix
${\mathbf{W}}_{n}={{\mathbf{H}}_{n}}^{{\dagger}}\text{diag}({\sqrt{{\mathbf{p}}_{n}}}),\quad\forall
n.$ (52)
The problem then reduces to the optimization over the vector
${\mathbf{p}}_{n}$
$\displaystyle\max_{{\mathbf{p}}_{n}}U^{2}$ $\displaystyle=\sum_{k\in
s(n)}c_{k}\sum_{n=1}^{N}\log_{2}(1+p_{k,n})$ (53)
$\displaystyle\sum_{n=1}^{N}\sum_{k\in s(n)}\beta_{k,n}p_{k,n}$
$\displaystyle\leq\check{P}$ (54)
$\displaystyle\sum_{n=1}^{N}\log_{2}(1+p_{k,n})$
$\displaystyle\leq\check{d}_{k},\quad k\in{\mathcal{D}}$ (55)
$\displaystyle\beta_{k,n}$
$\displaystyle={\left[{({\mathbf{H}}_{n}^{{\dagger}})}^{H}{\mathbf{H}}_{n}^{{\dagger}}\right]}_{k,k}$
$\displaystyle{\mathbf{p}}_{n}$ $\displaystyle\geq 0.$ (56)
Problem (53–56) is convex since we are maximizing a concave function over a
convex set and can be solved by standard techniques. The overall procedure to
find an exact solution by enumeration is summarized in algorithm (2).
MAX $\leftarrow 0$
for $i=1$ to $S^{N}$ do
Solve problem (53–56) for given SDMA set assignment $s_{i}(n)$ with objective
function $U^{2}$
if $U^{2}\geq$ MAX then
MAX $\leftarrow U^{2}$
$s^{o}\leftarrow s_{i}$
end if
end for
Algorithm 2 Enumeration Algorithm
### IV-B Dual-Based Feasible Solution
Algorithm 1 can provide an optimal solution
$(\lambda^{*},\boldsymbol{\mu}^{*})$ to the dual problem (18). However, the
subgradient algorithm is known to converge slowly and, in some cases, we need
to stop the iterations before the algorithm has reached an optimal dual
solution. In these cases, the optimal solution may not be feasible. In this
section we are proposing a refinement of the dual method to construct a
feasible solution starting from the final solution found for the dual problem.
Algorithm 3 summarizes this method. The algorithm begins by solving the dual
problem (18) using algorithm 1. If the solution is not feasible either
directly or by recomputing the power allocation using (53–56) for the SDMA set
assignment found in the dual problem, the algorithm performs a search by
increasing the dual variables associated to the users whose QoS constraints
are not met until a new SDMA set assignment is found. It then solves the power
allocation problem (53–56) for this new SDMA set assignment and checks the
solution feasibility with regards to the minimum rate constraints. The search
for new SDMA sets continues using this method until a feasible SDMA set
assignment is found or a maximum number of iteration is reached.
Solve the dual problem (18) using algorithm 1. This yields the optimal dual
variables $\lambda^{*},\boldsymbol{\mu}_{k}^{*}$ and a SDMA set assignment
vector ${s}^{*}(n)$ for each subchannel $n$.
Set $s^{o}_{0}(n)=s^{*}(n)$
Evaluate total power and user rate constraints (10–11)
if All constraints are met then
Exit. A feasible solution has been found.
end if
Compute power allocation problem (53–56) for $s^{o}_{0}(n)$ and evaluate total
power and user rate constraints (10–11)
if All constraints are met then
Exit. A feasible solution has been found.
end if
Compute the multipliers $\mu_{k}$ for users $k$ such that
$r_{k}<\check{d}_{k}$
for $j=1$ to $\overline{J}$ do
$\mu_{k}=\mu_{k}+\delta$
Find $s^{o}_{j}=\hbox{arg\ }\max_{s}\\{f_{n,s}\\}$ where $f_{n,s}$ is given by
(39) for the current dual variables $\lambda,\boldsymbol{\mu}$
Let $s^{o}_{j}(n)$ be the SDMA assignment found
if $s^{o}_{j}(n)\not=s^{o}_{j-1}(n)$ then
We have found a new SDMA assignment
Compute power allocation problem (53–56) for $s^{o}_{j}(n)$ and evaluate total
power and user rate constraints (10–11)
if All constraints are met then
Exit. A feasible solution has been found.
end if
end if
end for
Exit. A feasible solution was not found.
Algorithm 3 Calculating a feasible point from the dual solution
In contrast to the enumeration method described in Section IV-A which performs
an enumeration of all possible SDMA set assignments, the dual-based algorithm
3 is a method that finds a SDMA set assignment close to the dual optimal and
then uses it to solve one sub-problem (53–56). This makes the search for a
near-optimal feasible solution much faster than finding the exact solution.
### IV-C Weight Adjustment Method
In Section I-A we discussed several RA algorithms that support RT traffic
which increase the user weights in the utility function until such users
receive transmission resources. In this section we thus propose a weight
adjustment method to evaluate the efficiency of algorithms that use this
approach. The objective of the proposed method is to find a set of weights in
the utility function (9) for which, when we solve problem (9–14) without the
rate constraints (11), the rate requirements of the RT users are met. We also
want the set of weights to have the least deviation between users in order to
maximize the multi-user diversity gain. Algorithm 4 implements a generic
method for weight adjustment to achieve this objective. The algorithm
increases the user weights for RT users until enough resources are allocated
to meet the minimum rate requirements. The parameter $\epsilon$ controls how
much the weights are increased with respect to the rate bounds.
Solve RA problem (9) without minimum rate constraints constraints (11)
${\mathbf{c}}^{\prime}\leftarrow{\mathbf{c}}$
Let $r_{k}$ be the achieved rate for user $k$ at every iteration
iteration $\leftarrow$ 1
while ($r_{k}<\check{d}_{k}$ for one or more users $k\in{\mathcal{D}}$) AND
(iteration $\leq$ MAX_iterations) do
Increase user weight using
$c_{k}^{\prime}=c_{k}^{\prime}+\epsilon\left(\check{d}_{k}-r_{k}\right)$ for
users in need, where $0<\epsilon\leq 1$
Solve RA problem (9) without minimum rate constraints (11) using user weights
${\mathbf{c}}^{\prime}$
iteration $\leftarrow$ iteration $+1$
end while
Algorithm 4 Weight adjustment algorithm
### IV-D Comparison of the Weight Adjustment and the Dual-Based Methods
The rates achieved by weight adjustment algorithm 4 and the dual-based
algorithm 3 are different since they solve different problems. That is,
algorithm 4 can be seen as solving problem (53–56) by a linear penalty method
for constraints (55) of the form
$P_{k}=\min\left\\{0,r_{k}-\check{d}_{k}\right\\}$
The modified objective function is then
$\displaystyle U_{P}$ $\displaystyle=\sum_{k}c_{k}r_{k}+P_{k}$
$\displaystyle=\sum_{k}c_{k}r_{k}+\epsilon\sum_{k|r_{k}<\check{d}}(r_{k}-\check{d})$
(57)
At each iteration of the penalty method, whenever rate constraints are active,
the solution of (57) cannot be smaller than that of (53–56) since it is a
relaxation. Notice that problem (57) is quite simple since it has a single
constraint (54) but it has to be solved many times to adjust the weights of
the real time users. In weight adjustment algorithms such as [13, 14], the
user weights are increased at each time slot using an increasing function of
the packets delay, so the computation task is distributed over time. However,
this distributed approach does not guarantee that the rate requirements are
met in a given time slot and leads to delay violations and jitter.
To illustrate numerically the difference between solving the problem with
explicit rate constraints versus modifying the user weights, we simulated the
performance of both approaches for a single random channel realizations. We
set a minimum rate constraint of $\check{d}_{1}=8$ bps/Hz for one RT user. We
first solved the problem using the dual-based algorithm 3 using equal weights
for all users. The sum rate is shown with the dash line in the top plot of
Figure 5 while the user 1 rate is shown with the dash line in the bottom plot
of Figure 5. To specifically study the impact of the utility function weights
on the performance, we then solved problem (9–14) without the rate constraints
(11) for different weight assignments as follows: All user weights were set to
1 initially and we varied the user 1 weight from 1 to 10. The corresponding
user 1 and sum rates are indicated in Figure 5 with the star line. We can
observe three regions as a function of the user 1 weight $c_{1}$. For low
values of $c_{1}$, user 1 does not get the required minimum rate so that the
total rate is much larger that the optimal value, which is to be expected for
an unfeasible solution. For middle values, the user 1 rate is much better but
is not feasible and the total rate is correspondingly smaller but still larger
than the optimum. Finally, if we increase $c_{1}$ such that the required rate
must be satisfied to a high accuracy, the total rate is lower than the value
obtained with the dual-based method. Note that the total rate and user 1 rate
discontinuities observed for the weight adjustment method are due to the fact
that the selected SDMA set changes as we increase the weight and the solution
moves from one region to another.
Figure 5: Comparison between weight adjustment and dual-based methods for
$M=3$, $K=6$, $N=4$, and $\check{P}=20$.
Another example is shown in Figures 6 and 7 where we plot the total objective
and the user rate, respectively, as a function of $c_{1}$. We consider two
cases for the user 1 rate requirement, one where $\check{d}_{1}=48$ bps/Hz and
the other where $\check{d}_{1}=66$ bps/Hz.
Figure 6: Total rate for weight adjustment and dual-based methods for $M=3$,
$K=8$, $N=8$, and $\check{P}=20$. Figure 7: User 1 rate for weight adjustment
and dual-based methods for $M=3$, $K=8$, $N=8$, and $\check{P}=20$.
The results show that for a value of $c_{1}\approx 1.5$ the weight adjustment
approach provides the same total rate as the dual-based method and the user 1
rate requirement is met. However, it is interesting to note that the range of
$c_{1}$ over which this is possible is quite narrow: A slightly lower value
produces an unfeasible solution and a slightly higher value, a much larger
rate allocation for the user 1 with a lower total rate. In other words, one
would need many iterations of the weight adjustment algorithm to get the
required accuracy to find the optimum. For the other case, where the rate
constraint is higher, we see that the weight adjustment method is unable to
find a solution that is both feasible and reasonably close to the optimal
value. This is a clear indication that the rate adjustment algorithm should be
used with care, especially in those cases where the rate constraints are
important.
These examples show that even if we were to perform an exhaustive search to
determine the weights ${\mathbf{c}}^{\prime}$, the performance would not be as
good as the dual-based method. For the cases where the performance is
comparable, we also observed that the weight adjustment method performance is
very sensitive to the weight value, which is not a desirable property.
However, if an efficient method to adjust the weights is used, the
computational load of the weight adjustment method would be lower than that of
the proposed dual-based method because it solves a problem with a single
constraint.
## V Numerical Results
### V-A Parameter Setup and Methodology
In this section, we compare the performance and computation time of the
different methods to solve problem (9). We used a Rayleigh fading model to
generate the user channels and assumed independent fading between users,
antennas and subcarriers. Unless noted otherwise, we used a configuration with
$M=3$ transmit antennas, $K=16$ users, $N=16$ subcarriers, and one RT user
(i.e., $D=1$). We also fixed the power constraint to $\check{P}=20$ and used a
large scale attenuation of 0 dB for all users. The user weights in (9) were
set to $c_{k}=1$ for all users.
We compared the performance of the different methods for various scenarios
where we increased the resource requirements for the RT users until the
solution was unfeasible (i.e., the minimum rate requirements can no longer be
respected for all RT users). In the first scenario, we increased the minimum
rate $\check{d}_{k}$ for a single user with RT QoS requirements. In the second
scenario we fixed the minimum rate but increased the large scale attenuation
of the single RT user to study the case where RT users travel away from the BS
while still having minimum rate requirements. In the last scenario, we
increased the number of RT users with the same minimum rate requirements.
For each scenario and channel realization, the dual solution upper bound was
computed using algorithm 1 described in Section III-E. For small system
configurations we also found the exact solution using the primal enumeration
algorithm 2 given in Section IV-A. We could only find this exact solution for
small configurations because the computation time is prohibitive for large
system configurations. We also computed the solutions given by dual-based
primal feasible algorithm 3 and the weight adjustment algorithm 4 described in
Section IV-B and IV-C, respectively. The provided results are the average
total rate and computation time over 100 independent channel realizations.
Furthermore, whenever a result is provided for a given configuration and
algorithm, the solution was feasible with respect to the minimum rate
requirements. We also use the bound given by the dual optimal solution as a
reference given that the exact solution is generally not available except for
very small size cases.
Method | Minimum rate (bps/Hz)
---|---
| $13.33$ | $16.66$ | $20$
Dual-based upper bound (bps/Hz) | 49.13 | 47.12 | 40.8
Primal enum. gap (%) | 0.57 | 0.55 | 0.10
Dual-based feas. gap (%) | 0.57 | 0.59 | 0.04
Weight mod. gap (%) | 0.68 | 0.71 | 0.15
TABLE I: Average performance gap against the dual optimal upper bound for
small system configuration.
### V-B Total Rate Performance
First we present in Table I the average gap in percent between the three
different methods to find feasible solutions against the dual optimal upper
bound for a small system configuration with $K=4$ users and $N=2$ subcarriers.
We increased the minimum rate requirement for one RT user from 13.33 to 20
bps/Hz. As the minimum rate increases, the upper bound decreases as more
resources need to be assigned to the RT user until the problem is no longer
feasible. For this small configuration, we see that all methods give excellent
results and the duality gap is very small. Also note that due to the solver
numerical accuracy and the fact that all solutions were close to each other,
we even had cases where the dual-based feasible solution was better than the
solution given by the primal enumeration method.
Method | Minimum rate (bps/Hz)
---|---
| 80 | 100 | 120
Total rate gap against the upper bound (%)
Dual-based feas. | 0.24 | 0.23 | 0.21
Weight mod. | 9.49 | 7.30 | 3.36
Computation time (sec)
Dual-based feas. | 5.30 | 5.09 | 4.22
Weight mod. | 1.34 | 1.34 | 1.34
TABLE II: Average total rate gap and computation time as a function minimum
rate requirement. Figure 8: Average total rate as a function of the minimum
rate requirements
In the remaining results we use the larger system configuration with $K=16$
users and $N=16$ subcarriers, where it is no longer feasible to compute the
solution using the primal enumeration method. We present in Table II the
difference in percentage between the solutions of the dual-based feasible
algorithm and the weight adjustment algorithm and the dual upper bound. The
dual-based feasible algorithm provides a solution within $0.25$% of the dual
upper bound. The weight adjustment method, on the other hand, is much worse
and the difference can be almost 10%. This is due to the fact that the
solution found by the weight adjustment algorithm does not change much when
the minimum rate is increased, as can be seen from Figure 8 which shows the
sum rate achieved by the dual-based feasible algorithm and the weight
modification method against the minimum rate requirement.
Figure 9: Average total rate as a function of RT user large scale channel attenuation. Method | RT user attenuation (dB)
---|---
| 0 | 5 | 10
Total rate gap against the upper bound (%)
Dual-based feas. | 0.16 | 0.70 | 0.82
Weight mod. | 9.53 | 32.95 | 52.35
Computation time (sec)
Dual-based feas. | 4.73 | 5.38 | 4.58
Weight mod. | 1.48 | 1.35 | 1.55
TABLE III: Average total rate gap and computation time as a function of RT
user large scale channel attenuation.
Figure 9 shows the average total rate when the large scale channel attenuation
of the RT user varies from 0 to 15 dB. As the user moves away from the BS, the
RA algorithm dedicates more resources to this RT user until the problem is
unfeasible. The results show that for all SNR, the dual based method provides
a solution much closer to the upper bound than the weight adjustment method.
Table III shows the error in percentage between the objective and the upper
bound. For an attenuation of 15 dB, neither method is able to find a feasible
solution.
Method | Number of RT users
---|---
| 1 | 2 | 3 | 4 | 5 | 6 | 7
| Total rate gap against the upper bound (%)
Dual Feas. | 0.16 | 0.61 | 2.09 | 2.41 | 3.52 | 3.20 | 3.43
Weight mod. | 3.5 | 3.5 | 6.52 | 13.86 | 22.71 | - | -
| Computation time (sec)
Dual Feas. | 4.73 | 7.13 | 9.68 | 10.80 | 13.29 | 15.25 | 18.45
Weight mod. | 1.48 | 1.50 | 1.60 | 1.69 | 2.06 | - | -
TABLE IV: Average total rate gap and computation time as a function of the
number of RT users Figure 10: Average total rate as a function of the number
of RT users.
Finally, Figure 10 shows the optimal dual bound and the solution given by the
dual-based feasible and weight adjustment methods as a function of the number
of RT users. Table IV lists the performance gap against the dual bound in
percentage. The dual feasible method exhibits a much lower performance gap
than the weight adjustment method for all values of the number of RT users.
Moreover, we can see that the weight adjustment method performance quickly
degrades when the number of RT users increases. It is not even able to find
feasible points when the number of RT users is $6$ or $7$ while the dual-based
feasible algorithm provides solutions within $3.52\%$ of the upper bound.
We can observe from all the results presented in this section (similar trends
were also observed for other configurations that we studied) that the
difference between the dual-based upper bound and the dual-based feasible
solution is very small. This is an indication that the duality gap is very
small and that it is possible to find feasible solutions close to the optimal,
albeit with an off-line algorithm. Furthermore, the dual-based feasible
solution is always better than the weight modification method and this
difference becomes more significant as the resource requirements to meet the
RT users needs increase. This shows that the weight modification method should
be used carefully for RA in OFDMA-SDMA systems with RT users and that more
efficient heuristics should be developed to approach the performance of the
dual-based feasible solution.
### V-C Computation Time
We compare the computation time of the algorithms based on the time required
by an Intel dual-core 660 CPU to solve the problem. The primal enumeration
method is not included in the comparison due to its computation inefficiency.
The average computation time to find a solution for the three scenarios (i.e.
variation of minimum rate requirement, large scale channel attenuation and
number of RT users) is presented in Tables II to IV.
In Table II, where the minimum rate constraints are varied, the computation
time remains constant for the weight adjustment method, requiring an average
of two main iterations of algorithm 4. The dual-based method takes on average
$3.63$ times more time than the weight adjustment method. However, the total
rate that it produces is much closer to the upper bound. We have similar
results in Table III, where we varied the SNR of the RT user. The dual-based
method produces a higher total sum rate than that of the weight adjustment
method at the expense of a larger CPU time.
Table IV shows the computation time as a function of the number of RT users.
For the weight adjustment method the computation time is almost constant and
slightly increases until the number of RT users is five, after which it cannot
find feasible solutions. In contrast, the dual-based method computation time
grows steadily with the number of RT users and it is approximately five times
higher.
This comparison highlights the main difference between these two methods:
While the weight adjustment method performs several iterations solving a
problem with a single constraint, the dual method takes into account the rate
constraints so that the computation time increases with the number of RT
users. However, the performance gap against the dual upper bound of the dual
method is very low and approximately constant. In contrast, the performance
gap of the weight adjustment method is larger and increases with the number of
RT users. Moreover, the dual method can find feasible solutions for cases
where the weight adjustment method cannot.
## VI Conclusion
In this work, we proposed a method to compute the beamforming vectors and the
user selection in an OFDMA-SDMA MISO system with minimum rates for some RT
users. We used a Lagrangian relaxation of the power and rate constraints and
solved the dual problem using a subgradient algorithm. The Lagrange
decomposition yields sub-problems separated per subcarrier, SDMA sets and
users which substantially reduces the computational complexity. We also used a
closed-form expression of the beamforming subproblem based on a pseudo-inverse
condition on the beamforming vectors. The dual function is expressed in terms
of the dual variables and the dual optimum is found using a subgradient
algorithm. The dual optimum can then be used as a benchmark to compare against
other solution methods and heuristics.
We then proposed, in addition to the complete enumeration approach which is
not computationally practical for normal size problems, two algorithms to find
feasible solution to the RA problem with minimum rate requirements. The first
algorithm starts from the dual-based optimum solution and finds a feasible
solution by searching along the rate requirement dual variables, while the
second uses weight adjustments in the objective function to achieve the
required rates. Our results show that the dual-based method provides solutions
much closer to the upper bound than the weight adjustment method. The
difference is more significant when the SNR of the RT users is low or the
number of RT users is high. However, the computational load of the dual-based
method increases when the number of users with minimum rate requirements
increases, whereas it is almost constant for the weight adjustment method.
This reflects the trade-off between the two methods. The weight adjustment
method is computationally more efficient but the solutions provided by the
proposed dual-based method are much closer to the optimal. This indicates that
there is an advantage when including the minimum rate constraints in the
resource allocation problem. In addition, the weight adjustment method
requires many time slots to adjust the weights and schedule real time users.
Our method explicitly includes the minimum rate constraints which allows RT
users to be scheduled in the current slot, which decreases the average packet
delay.
To implement the RA algorithm in real time, we still need to design fast
methods to reduce the number of SDMA sets to be searched. The design of these
heuristic algorithms is outside the scope of this paper but it is part of our
current efforts. Finally, the upper bound given by the dual function
minimization provides a very useful benchmark to compare the performance of
these heuristics and the dual-based algorithm can also guide the design of
efficient novel heuristics.
## Acknowledgments
This research project was partially supported by NSERC grant CRDPJ 335934-06.
## References
* [1] 3GPP, “Further advancements for EUTRA: Physical layer aspects Rel. 9, 2010,” 3GPP TR 36.814 V1.2.1, Tech. Spec.n Group Radio Access Network.
* [2] IEEE, “Draft amendment to IEEE standard for local and metropolitan area networks part 16: Air interface for fixed and mobile broadband wireless access systems: Multi-hop relay specification, 2009,” Standard IEEE P802.16j/D9-2009, Institute of Electrical and Electronic Engineers.
* [3] K. Letaief and Y. Zhang, “Dynamic multiuser resource allocation and adaptation for wireless systems,” _IEEE Wireless Communications Magazine_ , vol. 13, no. 4, pp. 38–47, Aug. 2006.
* [4] D. Bartolomé and A. Pérez-Neira, “Practical implementation of bit loading schemes for multiantenna multiuser wireless OFDM systems,” _IEEE Transactions on Communications_ , vol. 55, no. 8, pp. 1577–1587, Aug. 2007.
* [5] T. F. Maciel and A. Klein, “A resource allocation strategy for SDMA/OFDMA systems,” in _Proc. of IST Mobile and Wireless Communications Summit_ , Jul. 2007, pp. 1–5.
* [6] B. Ozbek and D. L. Ruyet, “Adaptive resource allocation for SDMA-OFDMA systems with genetic algorithm,” in _6th International Symposium on Wireless Communication Systems, ISWCS_ , Sep. 2009, pp. 483–442.
* [7] Y. Tsang and R. Cheng, “Optimal resource allocation in SDMA/multi-input-single-output/OFDM systems under QoS and power constraints,” in _Proc. of WCNC_ , Mar. 2004, pp. 1595–1600.
* [8] P. Chan and R. Cheng, “Capacity maximization for zero-forcing MIMO-OFDMA downlink systems with multiuser diversity,” _IEEE Transactions on Wireless Communications_ , vol. 6, no. 5, pp. 1880 – 1889, 2007.
* [9] L. Xingmin, T. Hui, S. Qiaoyun, and L. Lihua, “Utility based scheduling for downlink OFDMA/SDMA systems with multimedia traffic,” in _Proc. IEEE Wireless Communications and Networking Conference, WCNC_ , Mar. 2010, pp. 130–134.
* [10] D. Perea-Vega, J. Frigon, and A. Girard, “Near-optimal and efficient heuristic algorithms for resource allocation in MISO-OFDM systems,” in _IEEE International Conference on Communications ICC_ , May 2010, pp. 1–6.
* [11] L. Lee, C. Chang, Y. Chen, and S. Shen, “A utility-approached radio resource allocation algorithm for downlink in OFDMA cellular systems,” in _Proc. IEEE 61st Vehicular Technology Conference_ , vol. 3, 2005, pp. 1798–1802.
* [12] G. Song and Y. Li, “Cross-layer optimization for OFDM wireless networks part II: Algorithm development,” _IEEE Transactions in Wireless Communications_ , vol. 4, no. 2, pp. 625–634, Mar. 2005.
* [13] C. Tsai, C. Chang, F. Ren, and C. Yen, “Adaptive radio resource allocation for downlink OFDMA/SDMA systems with multimedia traffic,” _IEEE Transactions on Wireless Communications_ , vol. 7, no. 5, pp. 1734–1743, 2008\.
* [14] W. Chung, L. Wang, and C. Chang, “A low-complexity beamforming-based scheduling for downlink OFDMA/SDMA systems with multimedia traffic,” in _Proc. of IEEE GLOBECOM_ , Nov. 2009, pp. 1–5.
* [15] W. Huang, K. Sun, and T. Bo, “A new weighted proportional fair scheduling algorithm for SDMA/OFDMA systems,” in _Proc. 3rd Int. Conf. on Communications and Networking in China, ChinaCom_ , Aug. 2008, pp. 538–541.
* [16] V. Tralli, P. Henarejos, and A. Perez-Neira, “A low complexity scheduler for multiuser MIMO-OFDMA systems with heterogeneous traffic,” in _Proc. International Conference on Information Networking (ICOIN)_ , Jan. 2011, pp. 251–256.
* [17] X. Wang and G. Giannakis, “Ergodic capacity and average rate-guaranteed scheduling for wireless multiuser OFDM systems,” in _IEEE International Symposium on Information Theory, ISIT 2008._ , Jul. 2008, pp. 1691–1695.
* [18] L. Zhu and K. Yeung, “Optimization of resource allocation for the downlink of multiuser MISO-OFDM Systems,” in _Proc. IEEE 17th International Conference on Telecommunications (ICT)_ , vol. 1, Apr. 2010, pp. 266–271.
* [19] I. Koutsopoulos and L. Tassiulas, “Adaptive resource allocation in SDMA-based wireless broadband networks with OFDM signaling,” in _Proc. INFOCOM 2002. Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies_ , vol. 3, 2002, pp. 1376–1385.
* [20] S. K. V. Papoutsis, I. Fraimis, “User selection and resource allocation algorithm with fairness in MISO-OFDMA,” _IEEE Communications Letters_ , vol. 14, no. 5, pp. 411–413, 2010.
* [21] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” _IEEE Journal on Selected Areas in Communications_ , vol. 24, no. 3, pp. 528–541, Mar. 2006.
* [22] D. Bertsekas, _Convex Analysis and Optimization_. Athena Scientific – Belmont, MA, 2003.
* [23] L. V. S. Boyd, _Convex Optimization_. Cambridge University Press, 2004.
|
arxiv-papers
| 2011-10-06T18:47:05 |
2024-09-04T02:49:22.887135
|
{
"license": "Public Domain",
"authors": "Diego Perea-Vega and Andr\\'e Girard and Jean-Fran\\c{c}ois Frigon",
"submitter": "Diego Perea Mr.",
"url": "https://arxiv.org/abs/1110.1347"
}
|
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